SlideShare una empresa de Scribd logo
1 de 14
Descargar para leer sin conexión
IEEE/ACM TRANSACTIONS ON NETWORKING                                                                                                                  1




  Random Access Game and Medium Access Control
                    Design
                          Lijun Chen, Member, IEEE, Steven H. Low, Fellow, IEEE, and John C. Doyle



   Abstract—Motivated partially by a control-theoretic viewpoint,                   channel. In the backoff mechanism, channel access probability
we propose a game-theoretic model, called random access game,                       is implemented through a backoff algorithm, and each wireless
for contention control. We characterize Nash equilibria of random                   node maintains a contention window and waits for a random
access games, study their dynamics, and propose distributed algo-
                                                                                    amount of time bounded by the contention window before a
rithms (strategy evolutions) to achieve Nash equilibria. This pro-
vides a general analytical framework that is capable of modeling a                  transmission. When simultaneous accesses to the channel by




                            rs f
large class of system-wide quality-of-service (QoS) models via the                  different nodes cause contention, the persistence probability or
specification of per-node utility functions, in which system-wide                    contention window is adjusted appropriately so that contention




                         Ve o
fairness or service differentiation can be achieved in a distributed                is reduced. Different medium access control methods differ
manner as long as each node executes a contention resolution al-                    in terms of how they adjust these parameters in response to




                              ion
gorithm that is designed to achieve the Nash equilibrium. We thus                   contention and what contention measure they use. For example,
propose a novel medium access method derived from carrier sense
                                                                                    the standard IEEE 802.11 DCF uses a backoff mechanism and a
multiple access/collision avoidance (CSMA/CA) according to dis-




                              ro
tributed strategy update mechanism achieving the Nash equilib-                      binary contention signal—packet collision or successful trans-
rium of random access game. We present a concrete medium access                     mission—in which each wireless node doubles its contention
method that adapts to a continuous contention measure called con-                   window upon a collision (binary exponential backoff) and sets
ditional collision probability, stabilizes the network into a steady                it to the base value upon a successful transmission [2].
state that achieves optimal throughput with targeted fairness (or                       The choice of contention measure and contention resolution
service differentiation), and can decouple contention control from                  algorithm is key to the performance of medium access methods.
handling failed transmissions. In addition to guiding medium ac-
                     int P
cess control design, the random access game model also provides
                                                                                    “Inappropriate” choice of these two components will result in
an analytical framework to understand equilibrium and dynamic                       poor performance. For example, in high-load scenarios, 802.11
properties of different medium access protocols.                                    DCF results in excessive collisions and, hence, low throughput
                                                                                    because setting to the base contention window upon successful
  Index Terms—Contention-based medium access, control-the-
oretic analysis, game theory, Nash equilibrium, strategy update                     transmission is too drastic and each new transmission starts with
mechanism.                                                                          the base contention window independent of the contention level
                                                                                    in the network. It also has short-term unfairness problem due
                         EE

                                                                                    to oscillation in the contention window. The binary exponential
                                                                                    backoff directly causes short-term unfairness. However, this os-
                                                                                    cillation in the contention window is unavoidable because DCF
                            I. INTRODUCTION                                         uses a binary contention signal. In order to achieve high effi-
                                                                                    ciency (high throughput and low collision) and better fairness,
         IRELESS channel is a shared medium that is interfer-                       we need to stabilize the network into a steady state that sustains
W        ence-limited. Contention-based medium access control
(contention control) is a distributed strategy to access and share
                                                                                    an appropriate contention window size (or equivalently, channel
                                                                                    access probability) for each node. Furthermore, how we can es-
wireless channel among contending wireless nodes. From a                            timate and implement the contention measure is important. Al-
                      IE


control-theoretic point of view, it consists of two components:                     most all medium access methods, including 802.11 DCF, adapt
a contention resolution algorithm that dynamically adjusts                          to packet collisions. However, they cannot distinguish collisions
channel access probability in response to contention in the                         from corrupted frames that are common in wireless networks.
network and a feedback mechanism that updates a contention                          This leads to increased unfairness and lower throughput. To en-
measure and sends it back to wireless nodes. Contention                             sure good performance, we need to use a contention measure
resolution is usually achieved through two mechanisms: per-                         whose estimation is not based on packet collisions and decouple
sistence and backoff [1]. In the persistence mechanism, each                        contention control from handling failed transmissions.
wireless node maintains a persistence probability and accesses                          The main motivation of this work is to provide an analytical
                                                                                    framework to systematically study contention control and de-
                   Pr



the channel with this probability when it perceives an idle
                                                                                    sign medium access methods that could stabilize the network
   Manuscript received August 20, 2008; revised May 21, 2009; approved by
                                                                                    around a steady state with a target fairness (or service differ-
IEEE/ACM TRANSACTIONS ON NETWORKING Editor R. Mazumdar.                             entiation) and high efficiency. To this end, we define a gen-
   The authors are with the Engineering and Applied Science Division, Cali-         eral game-theoretic model, called random access game, to cap-
fornia Institute of Technology, Pasadena, CA 91125 USA (e-mail: chen@cds.           ture the interaction among wireless nodes in wireless networks
caltech.edu; slow@caltech.edu; doyle@cds.caltech.edu).
   Color versions of one or more of the figures in this paper are available online   with contention-based medium access. The basic idea is to re-
at http://ieeexplore.ieee.org.                                                      gard the process of contention control as carrying out a dis-
   Digital Object Identifier 10.1109/TNET.2010.2041066                               tributed strategy update algorithm to achieve the equilibrium of
                                                                 1063-6692/$26.00 © 2010 IEEE
2                                                                                                   IEEE/ACM TRANSACTIONS ON NETWORKING



random access game. Hence the equilibrium (or steady state)         tributed strategy update algorithms derived from the random ac-
and dynamic properties of a MAC can be understood or de-            cess game as prescriptive control approach for channel access.1
signed through the specification of the underlying random ac-
cess game.                                                                                    II. RELATED WORK
   Specifically, in random access games, a wireless node’s
                                                                       Game-theoretic approach has been applied extensively to
strategy is its channel access probability, and its payoff func-
                                                                    study random access (see, e.g., [4]–[10]). The work closest to
tion includes both utility gain from channel access and cost
                                                                    ours is Jin et al. [4], which studies noncooperative equilibrium
from packet collision. Through the specification of per-node
                                                                    of Aloha networks and their local convergence, and Altman
utility function, we can model a large class of system-wide
                                                                    et al. [6] and Borkar et al. [7], which study distributed scheme
quality-of-service (QoS) models, similar to that in utility
                                                                    for adapting random access. Our motivation, model, and results
framework for network flows [3]. We characterize the Nash
                                                                    are different from those works. Our random access game model
equilibrium of random access games, study their dynamics, and
                                                                    is intended as an analytical framework to reverse-engineer con-
propose algorithms (strategy evolutions) to achieve the Nash
                                                                    tention control as well as to guide the design of new medium
equilibrium. We show that system-wide fairness or service
                                                                    access methods to achieve system-wide performance objec-




                        rs f
differentiation can be achieved in a distributed manner as long
                                                                    tives. The specific structure of random access game is derived
as each node executes a contention resolution algorithm that
                                                                    from a control-theoretic viewpoint of contention control, and




                     Ve o
is designed to achieve the Nash equilibrium. We thus propose
                                                                    the utility functions are derived from the steady operating
a novel medium access method derived from carrier sense




                          ion
                                                                    points of existing protocols or the desired operating points we
multiple access/collision avoidance (CSMA/CA) in which
                                                                    want medium access control to achieve. Inaltekin et al. [10]
each node estimates its conditional collision probability and
                                                                    analyze Nash equilibria in multiple access with selfish nodes.




                          ro
adjusts its channel access probability accordingly. Our method       ˇ
                                                                    Cagalj et al. [8] study selfish behavior in CSMA/CA networks
adapts to continuous feedback signal (conditional collision
                                                                    and propose a distributed protocol to guide multiple selfish
probability) rather than binary contention signal, and each node
                                                                    nodes to a Pareto-optimal Nash equilibrium. We do not con-
tries to keep a fixed channel access probability (persistence
                                                                    sider such selfish behaviors of wireless users that tamper with
probability or equivalently contention window size) specified
                                                                    wireless interfaces to increase their share of channel access
by the Nash equilibrium of random access game. In addition
                                                                    as in [8]. In contrast, we use game-theoretic model to capture
                 int P
to controllable performance objectives via the specification of
                                                                    the information and implementation constraints encountered in
per-node utility functions, as wireless nodes can estimate con-
                                                                    real networks and design games to guide distributed users to
ditional collision probabilities by observing consecutive idle
                                                                    achieve system-wide performance objectives.
slots between transmissions, our access method can decouple
                                                                       There are lots of works on various enhancements and im-
contention control from handling failed transmissions. As a
                                                                    provements to 802.11 DCF. We will only briefly discuss some
case study of medium access control design in the proposed
                                                                    designs that propose better contention resolution algorithms and
game-theoretic framework, we present a concrete medium
                                                                    that improve throughput by tuning the contention window ac-
                     EE

access method and show that it achieves optimal throughput,
                                                                    cording to the number of contending nodes. Aad et al. [11] in-
low collision, and good short-term fairness and can provide
                                                                    troduce a slow decrease method to improve efficiency and fair-
flexible service differentiations among wireless nodes.
                                                                    ness. Kwon et al. [12] propose a fast collision resolution al-
   Game-theoretic approach has recently been applied to net-
                                                                    gorithm for throughput improvement. Our design is different
work design and control. There are two complementary per-
                                                                    in terms of both contention measure and contention resolution
spectives to the game-theoretic approach: economic perspec-
                                                                    algorithm. Bianchi et al. [13] and Cali et al. [14] propose to
tive and engineering perspective. The economic perspective as-
                                                                    choose and compute an optimal contention window to maxi-
sumes network components or users are selfish and try to thwart
                                                                    mize the throughput. They need sophisticated methods to es-
selfish behaviors or induce cooperation using external mecha-
                                                                    timate the number of contending nodes in the system, while our
                  IE


nisms such as pricing. The engineering perspective envisions a
                                                                    method does not require that information. There also exists ex-
scenario where network components or users are willing to co-
                                                                    tensive work on 802.11 QoS provisioning (see, e.g., [15]). Our
operate but only have limited information about network states
                                                                    access method can provide more flexible service differentiations
due to various practical constraints in real networks. In such a
                                                                    through the specification of per-node utility functions, except
situation, the best an agent can do is to optimize some local or
                                                                    for manipulating the length of interframe space.
private objective and adjust its action based on limited informa-
                                                                       Related work also includes [16] and [17]. Heusse et al. [16]
tion about the network state. We use noncooperative game to
                                                                    propose a novel idle sense access method for a single-cell wire-
“model” such a situation (especially to capture the information
                                                                    less LAN, which compares the mean number of idle slots be-
structure of the system) and let network agents behave “self-
               Pr



                                                                    tween transmission attempts to the optimal value and adopts an
ishly” according to the game that is designed to guide individual
                                                                    additive increase and multiplicative decrease algorithm to dy-
agents to seek an equilibrium achieving some system-wide per-
                                                                    namically control the contention window in order to improve
formance objective. Thus, from the engineering perspective, the
                                                                    throughput and short-term fairness. In our access method, wire-
focus is not on incentive issues, but on the implementation in
                                                                    less nodes estimate conditional collision probabilities by ob-
practical networks. There are some controversies around the
                                                                    serving consecutive idle slots between transmissions. Therefore,
engineering perspective. Nonetheless, game-theoretic technique
                                                                    like idle sense access method, our access method can decouple
and language provide a powerful framework to reason about the
engineering systems and guide their design and control. In this       1However, our random access game model can also serve as a descriptive

paper, we take the engineering perspective and implement dis-       model of existing contention control protocols with selfish wireless nodes. This
                                                                    part is consistent with the economic perspective.
CHEN et al.: RANDOM ACCESS GAME AND MEDIUM ACCESS CONTROL DESIGN                                                                                       3



contention control from handling failed transmissions. Hu et al.                   Define the utility function of each node as
[17] use a similar idea to propose a novel contention control
method to maximize the bandwidth utilization and achieve pro-                                                                                       (5)
portional bandwidth allocation. This method observes two ref-
erence variables, the number of consecutive idle slots and the
number of collisions between two consecutive successful trans-                     Being an integral, and since                       ,       is a con-
missions, based on which to control the dequeueing rate of wire-                   tinuous and nondecreasing function. It is reasonable to assume
less nodes instead of dynamically tuning their contention win-                     that        is a decreasing function: The larger the contention, the
dows. Our access method dynamically adjusts the contention                         smaller the channel access probability. This implies that
window and achieves higher throughput (almost no throughput                        is strictly concave.
degradation compared to the optimal throughput) and provides                          With the above utility function, we define a random access
a better guarantee of service differentiation.                                     game as follows.
   A comparison with TCP congestion control is also in place.                         Definition 1: A random access game                 is defined as
Contention control has a striking similarity with congestion con-                  a triple                                        , where          is a
                                                                                   set of players (wireless nodes), player                      strategy




                            rs f
trol. They need to handle almost the same issues such as conges-
tion/contention measure, load control (e.g., window update) al-                                                    with                            , and
                                                                                   payoff function                                with utility function




                         Ve o
gorithm, and decoupling load control from handling failed trans-
missions, etc. However, the interaction among wireless nodes is                             and contention measure         .




                              ion
different from that among TCP flows, which means a different                           We constrain the strategy              to be strictly less than 1
framework is needed to study contention control. Actually, one                     in order to prevent a node from exclusively occupying wire-
                                                                                   less channel. Note that the throughput of node is proportional




                              ro
of the motivations of this work is to try to develop a parallel
story for contention control to what has been done for TCP con-                    to if there is no collision, and can be seen as contention
gestion control in the utility maximization framework (see, e.g.,                  price for node . Thus, the payoff function           has a nice eco-
[3]).                                                                              nomic interpretation: the net gain of utility from channel access,
                                                                                   discounted by contention cost. The key to understanding con-
                                                                                   tention control is to regard (1)–(2) as the strategy update algo-
                     III. RANDOM ACCESS GAME                                       rithm to achieve the equilibrium of the random access game.
                     int P
   Consider a set     of wireless nodes in a wireless LAN with                     Hence, the equilibrium (or steady-state) properties of a con-
contention-based medium access. Associated with each wire-                         tention control protocol can be understood and designed through
less node         is its channel access probability         2 and a                the specification of the utility function         and the contention
certain contention measure               it can observe at time .                  measure (e.g., collision probability). Their specification de-
Node can observe its own access probability               and con-                 fines the underlying random access game whose equilibrium
tention measure       , but not those of other nodes. It adjusts its               determines the steady-state properties such as throughput, fair-
                         EE

channel access probability         based only on        and                        ness, and collision of the contention control protocol. The adap-
                                                                                   tation of channel access probability can be specified through
                                                                            (1)             and corresponds to different strategies to approach the
                                                                                   equilibrium of the game.
The contention measure                 depends on the channel access                  Random access game             is defined in a rather general
probabilities                                 chosen by the wireless               manner. As we saw above, it can be reverse engineered from
nodes                                                                              given medium access control methods. Regarding the design
                                                                                   of medium access control (i.e., forward-engineering), we can
                                                                            (2)    choose to implement any contention measures and specify any
                                                                                   utility functions we think appropriate. To make our presentation
                      IE


Here,     models the contention resolution algorithms, and                         and theoretical development concrete, in this paper we choose
models the contention measure update mechanisms. For conve-                        conditional collision probability as contention measure, i.e.,
nience, we will also denote       by        .
   We assume that (1) and (2) have an equilibrium         . The                                                                                     (6)
fixed point of (1) defines an implicit relation between the equi-
librium channel access probability and contention measure
                                                                                   where denotes the set of nodes that interfere with the trans-
                                                                            (3)    mission of node . As will become clear later, such a choice of
                   Pr



                                                                                   contention measure has two nice properties, among others. First,
Assume      is continuously differentiable and                in                   conditional collision probability is an accurate measure of con-
     . Then, by the implicit function theorem [18], there exists                   tention in the network. Each measurement of conditional colli-
a unique continuously differentiable function such that                            sion probability provides multibit information of the contention.
                                                                                   This makes it easier for an equation-based contention resolution
                                                                            (4)    algorithm to stabilize the network into a steady state with a target
                                                                                   fairness (or service differentiation) and high efficiency. Second,
   2In our description of contention control, each wireless node’s control vari-
able is its channel access probability. For the same variable, we may have dif-
                                                                                   wireless nodes can estimate conditional collision probabilities
ferent ways to implement it in practical networks. Window-based control is seen    without using explicit feedback, which enables the decoupling
as a way of “implementing” channel access probability.                             of contention control from handling failed transmissions.
4                                                                                                            IEEE/ACM TRANSACTIONS ON NETWORKING



   In the rest of this section and Section IV, we focus on single-               and
cell wireless LANs,3 and will consider multicell wireless LANs
in Section V. Before proceeding, we summarize the assump-
tions that will be used in this paper as follows.
                                                                                 Thus, under assumption A1
      A0: The utility function             is continuously differen-
      tiable, strictly concave, and with finite curvatures that are
      bounded away from zero, i.e., there exist some constants
      and such that                                       .                      By second-order conditions [20],          is a strictly concave
      A1: Let                                and denote the smallest             function over the strategy space. Therefore, the optimization
      eigenvalue of            over by          . Then,                          problem (8) has a unique optimal, and hence the random access
        .                                                                        game has a unique Nash equilibrium.
      A2: Functions                                              are                The equilibrium condition (7) implies that, at the Nash equi-
      all strictly increasing or all strictly decreasing.                        librium,    either takes value at the boundaries of the strategy
Assumption A0 is a standard assumption in economics and can                      space    or satisfies




                           rs f
also be seen as derived from those assumptions on function
      . A1 guarantees the uniqueness of the Nash equilibrium of                                                                                  (10)




                        Ve o
the random access game. A2 guarantees the uniqueness of non-                     We call a Nash equilibrium a nontrivial equilibrium if, for all
trivial Nash equilibrium, which we will define later.




                             ion
                                                                                 nodes ,     satisfies (10), and trivial equilibrium otherwise. In
A. Nash Equilibrium                                                              the remainder of this section, we will mainly focus on nontrivial
                                                                                 Nash equilibria. It follows from (10) that at nontrivial Nash equi-




                             ro
    We now analyze the equilibrium of random ac-                                 librium
cess game. The solution concept we use is the
Nash equilibrium [19]. Denote the strategy (channel                                                                                              (11)
access probability) selection for all nodes but                  by
                                               and write                         Note that the right-hand side of the above equation is indepen-
for the strategy profile                                          .A              dent of . Thus,                    for any        .
                                                                                    Theorem 4: Suppose assumption A2 holds. If the random
                    int P
vector of access probability       is a Nash equilibrium if, for all
nodes          ,                               for all         . We              access game has a nontrivial Nash equilibrium, it must be
see that the Nash equilibrium is a set of strategies for which no                unique.
player has an incentive to change unilaterally.                                        Proof: Suppose that there are two nontrivial Nash equi-
    Theorem 2: Under assumption A0, there exists a Nash equi-                    libria and . From (11), we require that there exist
librium for random access game .                                                 such that, for all
       Proof: Since the strategy spaces       are compact convex
                        EE

sets, and the payoff functions are continuous and concave in
   , there exists a Nash equilibrium [19].
    Since payoff function       is concave in , at the Nash equi-
                                                                                 Since          is one-to-one,          . Without loss of generality,
librium,      satisfies
                                                                                 assume that          is increasing and           . Thus,          for
                                                                                 all . By (10),                                       , which contra-
                                                                          (7)
                                                                                 dicts the fact that        is a decreasing function. Thus, random
                                                                                 access game has a unique nontrivial Nash equilibrium.
Define function                                                 .                    Theorem 4 complements Theorem 3. Although A1 guaran-
It is easy to verify that (7) is an optimality condition for the                 tees the uniqueness of nontrivial Nash equilibrium when there
                     IE


following optimization problem [20]:                                             is one, it only gives a sufficient condition for the uniqueness.
                                                                                 When there exist nontrivial Nash equilibria while A1 does not
                                                                          (8)    hold, A2 guarantees the uniqueness of nontrivial equilibrium.
                                                                                    Each node can choose any utility function            it thinks ap-
i.e., the Nash equilibria of the random access game are optimal                  propriate. If all nodes have the same utility functions, the system
points of the problem (8).                                                       is said to have homogeneous users. If the nodes have different
   Theorem 3: Suppose additionally assumption A1 holds.                          utility functions, the system is said to have heterogeneous users.
Then, random access game has a unique Nash equilibrium.                          The motivation for studying systems of heterogeneous users is
                  Pr



       Proof: The Hessian of function       is written as                        to provide differentiated services to different wireless nodes. To
                                                                                 this end, we further differentiate among symmetric and asym-
                                                                          (9)    metric equilibria as follows.
                                                                                    Definition 5: A Nash equilibrium is said to be a symmetric
Note that                                                                        equilibrium if              for all           , and an asymmetric
                                                                                 equilibrium otherwise.
                                                                                    For a system of homogeneous users, both symmetric and
                                                                                 asymmetric Nash equilibria are possible. For example, take
  3Single-cell means that every wireless node can hear every other node in the                                 with           and                 . In
network—for example, a single-cell wireless access network.                      addition to nontrivial symmetric Nash equilibrium                that
CHEN et al.: RANDOM ACCESS GAME AND MEDIUM ACCESS CONTROL DESIGN                                                                                      5



satisfies                                ,      , when the number of                  Clearly, if the above dynamics reach a steady state, then this
wireless nodes             there exists a family of nontrivial asym-                 state is a Nash equilibrium. Nonetheless, there are no conver-
metric Nash equilibria         that satisfy                          ,               gence results for general games using these dynamics.
                                                                                        We thus consider an alternative strategy update mechanism
                 for all         and        .                                        called gradient play [22]. Compared to “best response” strategy,
   Since by symmetry there must be multiple asymmetric Nash                          gradient play can be viewed as a “better response.” In gradient
equilibria if there exists any, the following result follows directly                play, every player adjusts a current channel access probability
from Theorems 3 and 4.                                                               gradually in a gradient direction suggested by observations of
   Corollary 6: For a system of homogeneous users, suppose                           other player actions. Mathematically, each node         updates
assumption A1 or A2 holds. If the random access game has a                           its strategy according to
nontrivial Nash equilibrium, it must be unique and symmetric.
More generally, for a system with several classes of homoge-                                                                                      (13)
neous users, under the same assumption, if the random access
game has a nontrivial Nash equilibrium, it must be unique and                        where the stepsize              can be a function of the strategy
symmetric.4 5                                                                        of player and “ ” denotes the projection onto the player




                            rs f
   Corollary 6 guarantees the uniqueness of nontrivial Nash                          strategy space. The gradient play admits a nice economic inter-
equilibrium, and moreover, it guarantees fair sharing of wireless                    pretation if we interpret the conditional collision probability




                         Ve o
channel among the same class of wireless nodes and provides                          as contention price for node : If the marginal utility          is




                              ion
service differentiation among different classes of wireless                          greater than contention price, we increase the access probability,
nodes. This will facilitate the analysis of dynamic property of                      and if the marginal utility is less than contention price, we de-
random access games and the design of medium access control.




                              ro
                                                                                     crease the access probability. The following result is immediate.
   Remark: Since some player takes a strategy at the boundary                           Lemma 7: By the equilibrium condition (7), the Nash equi-
of the strategy space at trivial Nash equilibrium, a trivial Nash                    libria of random access game are fixed points of the gradient
equilibrium usually has great unfairness or low payoff. There-                       play (13), and vice versa.
fore, nontrivial Nash equilibrium is desired. If there does not                         Theorem 8: Suppose assumptions A0 and A1 hold. The gra-
exist any nontrivial Nash equilibrium for a random access game,                      dient play (13) converges to the unique Nash equilibrium of
we may need to look for an alternative solution other than the                       random access game if, for any                , the stepsize
                     int P
Nash equilibrium. For example, we may use Nash bargaining                                      .
framework in cooperative game theory to derive a desired equi-                             Proof: Consider function
librium solution.                                                                                   . Define a matrix                         . We have


B. Dynamics of Random Access Game
                         EE

                                                                                     Since       is symmetric,                          , and hence
   The dynamics of game studies how interacting players could
converge to a Nash equilibrium. It is a difficult problem in gen-
eral, as pointed out in [19] that “game theory lacks a general
and convincing argument that a Nash outcome will occur.” In
the setting of random access, players (wireless nodes) can ob-
serve the outcome (e.g., packet collision or successful transmis-
sion) of the actions of others, but do not have direct knowledge
of other player actions and payoffs. We consider repeated play
of random access game and look for an update mechanism in
                      IE


which players repeatedly adjust strategies in response to obser-                     By Taylor expansion, we have
vations of other player actions so as to achieve the Nash equi-
librium.
   The simplest strategy update mechanism is of best response
sort: At each stage, every node chooses the best response to the
actions of all the other nodes in the previous round. Mathemat-
ically, at stage       , node          chooses a channel access
probability
                   Pr



                                                                            (12)     where                                                      . Now,
                                                                                     by the projection theorem [20]
   4For a system with several classes of users, a Nash equilibrium is symmetric
if the users of the same class choose the same strategy at equilibrium.
   5There are many ways to give conditions on the existence of nontrivial Nash
equilibrium ( see [21]) for one example condition. Here, we do not do that since     from which we obtain
those conditions intended for the game with arbitrary utility functions are re-
strictive. It is more sensible to give the existence condition for a concrete game
with concrete utility functions; see Section IV-A for an example.
6                                                                                                                 IEEE/ACM TRANSACTIONS ON NETWORKING



Thus, we have                                                                        If           is nonsingular, function

                                                                                                                                              satisfies the
                                                                                     conditions of the implicit function theorem [18]. The result fol-
                                                                                     lows directly from the implicit function theorem.
                                                                                        Theorem 10: Assume the same conditions as in Theorem 8.
                                                                                     For any        , there exists a      such that if           , then
                                                                                     the gradient play (15) with estimation errors converges into an
Thus, if                    ,                              . We
                                                                                      -neighborhood of nontrivial Nash equilibrium.
see that       will keep increasing until the system reaches a
                                                                                          Proof: By Theorem 9, for any          , there exists a such
fixed point of (13). By Theorem 3 and Lemma 7, under assump-
                                                                                     that            implies that                            .
tions A0 and A1, (13) has a unique fixed point in strategy space.
                                                                                        Now, under the gradient play (15), we have
Hence, the gradient play (13) converges to the unique Nash equi-
librium of random access game .




                            rs f
   Theorem 8 guarantees the convergence of distributed gradient
play to the Nash equilibrium. If a backoff mechanism is imple-




                         Ve o
mented, each node          updates its contention window      as




                              ion
follows:




                              ro
                                                                            (14)

Equation (14) follows from the relation                  that re-
lates channel access probability       to a constant contention
window       . This relation can be derived under the decou-
pling approximation for a set of wireless nodes with constant
                     int P
contention windows (see, e.g., [23] and [24]). The decoupling                        If                                                             ,
approximation is an extremely accurate approximation, as                             will keep increasing. Therefore, the gradient play (15) will
validated by extensive simulations reported in, e.g., [23] and                       enter at least once into a region defined by
[24]                                                                                                        . Take any
   In practice, there will be estimation error in the contention                              , and we have                   , i.e., there exits a such
measure. Then, the gradient play can be written as                                   that            and                                       . It follows
                         EE

                                                                                     that the gradient play (15) will enter at least once into a neigh-
                                                                                     borhood of defined by                                                  .
                                                                                     Once it enters into this neighborhood, the gradient play will
                                                                            (15)
                                                                                     stay inside a larger neighborhood defined by                         .
                                                                                        Theorem 10 is the robust verification of the gradient play to
where      denotes the estimation error in node ’s conditional                       the estimation error. It holds even under weaker condition that
collision probability.6 We assume that the estimation errors are                     only requires the estimation errors to eventually remain within a
bounded, i.e., there exists a constant       such that                                -neighborhood of the origin. Being a dynamic feedback-based
  . We will show that under certain conditions, the gradient play
                                                                                     protocol, contention control usually operates at a region where
(15) with estimation error converges to a neighborhood of non-
                      IE


                                                                                     the equilibrium condition is approximately satisfied due to var-
trivial Nash equilibrium, where the size of the neighborhood de-
                                                                                     ious practical factors or constraints. Theorem 10 guarantees that
pends on the accuracy of the contention measure estimation.
                                                                                     this region is in a small neighborhood of the equilibrium, and
    Theorem 9: Let be a nontrivial Nash equilibrium. Suppose
                                                                                     thus the performance of the medium access method derived
that           is nonsingular. Then, there exist a and a unique
                                                                                     from the random access game is determined by the equilibrium.
continuously differentiable function                      defined
on a -neighborhood of the origin such that
                                                                                     C. Medium Access Control Design

                                                                            (16)        Our ultimate purpose for studying random access games is to
                   Pr



                                                                                     design medium access method with better performance. Corol-
      Proof: At nontrivial Nash equilibrium                 , we have                lary 6 and Theorem 8 suggest that random access games provide
                                                                                     a general analytical framework to model a large class of system-
                                                                                     wide QoS models (mainly in terms of throughput) via the spec-
                                                                            (17)
                                                                                     ification of per-node utility functions, and system-wide fairness
   6For the simplicity of presentation, we choose the same stepsize for all wire-    or service differentiation can be achieved in a distributed manner
less nodes and neglect the projection operation. However, similar results as those   as long as each node executes a contention resolution algorithm
of Theorems 9 and 10 can be established for the general case with heteroge-
neous stepsizes and the projection operation. The key is to note that function
                                                                                     that is designed to achieve the nontrivial Nash equilibrium.
[p + f 1 (rV (p) 0 e)]                  0 p is one-to-one for fixed e, and then
                                                                                        Based on this understanding of the equilibrium and dy-
apply the implicit function theorem for nonsmooth functions.                         namics of random access games, we propose a novel medium
CHEN et al.: RANDOM ACCESS GAME AND MEDIUM ACCESS CONTROL DESIGN                                                                                      7



                             TABLE I                                          With this estimation of the contention measure, our access
              MEDIUM ACCESS METHOD VIA GRADIENT PLAY                          method can decouple contention control from handling packet
                                                                              losses and is immune to those problems incurred in methods
                                                                              that infer channel contention from packet collisions.
                                                                                 In Section IV, we will study a concrete random access game
                                                                              and the corresponding medium access control design as a case
                                                                              study for the proposed design methodology in game-theotic
                                                                              framework. We will discuss more on the design of our medium
                                                                              access method.

                                                                                                       IV. A CASE STUDY
                                                                                 We have mostly ignored the design of utility functions in
                                                                              the previous section. In the following, we will derive utility
                                                                              functions and random access games from the desired operating




                           rs f
                                                                              points and design medium access methods accordingly.
                                                                                 Consider a single-cell wireless network with    classes of




                        Ve o
                                                                              users. Each class is associated with a weight , and without
                                                                              loss of generality, we assume                            . We




                             ion
                                                                              want to achieve maximal throughput under the weighted fair-
                                                                              ness constraint




                             ro
access method derived from CSMA/CA7: instead of executing
exponential backoff upon collisions, each node estimates its
conditional collision probability and adjusts its channel access
probability and contention window according to the gradient
                                                                              where is the throughput of a class- node. Let               .
play (13)–(14); see Table I for a formal description (see
                                                                              Under the assumption of the Poisson arrival process [25], the
Section IV-C for more explanation on notations). Our method
                                                                              channel idle probability is approximately
                    int P
adapts to continuous feedback signal (conditional collision
probability) rather than binary feedback (packet collision)                                                                                        (19)
and stabilizes the network around a steady state specified by
the Nash equilibrium of random access game. Therefore, it
can achieve controllable performance objectives by choosing                   and the aggregate successful packet transmission probability is
appropriate per-node utility functions. Our access method                          . Hence, the aggregate throughout can be written as
is an equation-based control, and its performance (such as
                        EE

throughput, collision and fairness) is determined by the Nash
equilibrium. Note that                               specifies how
far the current state is from the equilibrium. The contention                 which achieves maximum at             that solves
window adjustment is small when the current state is close to
the equilibrium, and large otherwise, independent of where                                                                                         (20)
the equilibrium is. This is in sharp contrast to the approach
taken by 802.11 DCF, where window adjustment depends on                       where is the packet payload, is the duration of an idle slot,
just the current window size and is independent of where the                  and      and     are the durations the channel is sensed busy be-
current state is with respect to the target equilibrium. Thus, our            cause of a successful transmission and during a collision, re-
                     IE


access method can achieve better contention control and better                spectively.
short-term fairness.                                                             Now, consider how to achieve the weighted fairness. When
   Now, consider how wireless nodes can estimate conditional                  there is a large number of nodes accessing the channel, each user
collision probabilities. Let denote the number of consecu-                    should sense approximately the same environment on average,
tive idle slots between two transmissions. Here, “a transmis-                 and we can assume that each user has the same conditional col-
sion” corresponds to a busy period in the channel when only a                 lision probability.8 Thus, with equal packet payload sizes the
node transmits (i.e., a successful transmission) or multiple nodes            throughput ratio between users of different classes is approxi-
transmit simultaneously (i.e., a collision). Since has the geo-               mately the ratio between their channel access probabilities, i.e.,
metric distribution with parameter         , its mean is given by             we require
                  Pr



                         . Thus, each node can estimate its condi-
tional collision probability by observing the average number of                                                                                    (21)
consecutive idle slots according to
                                                                              The users of each class are associated with the same utility
                                                                      (18)          and thus the same function                              .
                                                                              Note that at the nontrivial Nash equilibrium,
   7We can also design medium access methods based on the ordinary Aloha-     for all             . Thus, condition (21) can be achieved at a
type access method [25]. One of the main differences would be that, without
CSMA, wireless nodes have to estimate the contention measure based on ex-       8This assumption is similar to the decoupling approximation made in [23]
plicit feedback (i.e., failed transmissions).                                 and other works in performance analysis of 802.11 DCF.
8                                                                                               IEEE/ACM TRANSACTIONS ON NETWORKING



nontrivial Nash equilibrium if we design utility functions such         Theorem 12: Suppose additionally                  . Then,
that                                                                 random access game    has a unique and nontrivial Nash equi-
                                                                     librium.
                                                             (22)         Proof: We have

By (19)–(20), a nontrivial equilibrium    that achieves maximal
throughput should satisfy                                                                                                           (26)
                                                                     and
                                                             (23)
                                                                                                                                    (27)
Equations (22)–(23) plus assumption A2 are the constraints on
the desired utility functions that achieve the maximal throughput    where                             . Note that
under the weighted fairness constraint at the nontrivial equilib-               .      is a positive semidefinite matrix of rank 1, and




                        rs f
rium. Note that when the number of wireless nodes is large,
should be very small. A convenient choice that approximately         thus                        . Therefore, if                 ,
                                                                                    . It follows from Theorem 3 that random access




                     Ve o
satisfies the above constraints is
                                                                     game     has a unique Nash equilibrium. Since, by Lemma 11,




                          ion
                                                                        has a nontrivial Nash equilibrium, the unique equilibrium
                                                             (24)
                                                                     must be nontrivial.




                          ro
from which we derive a utility function                              B. Dynamics
                                                                        Assume that each node            adjusts its strategy according
                                                             (25)    to gradient play

With this utility function, we define a random access game
                 int P
where all players have the same strategy space         with
and different user classes are identified with different choices of                                                                  (28)
the utility functions (25) with different weights.
                                                                                                                                    (29)
   Remark: The way we derive utility function (25) follows
the same idea as in reverse-engineering (4)-(5): to derive the
utility function and game from the equilibrium. To see this,         where the stepsize             . Note that the Nash equilibria of
note that the equilibrium condition (4) or equivalently (10) is      random access game         are the fixed points of (28), and vice
                     EE

encapsulated in (11). In the above utility function design, we       versa. The following result is immediate.
derive function          from the desired equilibrium [see (24)]        Theorem 13: Suppose                                             .
from which we derive the marginal utility          or equivalently   Then, the system described by (28) converges to the unique and
       . We then obtain the utility function by integration.         nontrivial Nash equilibrium of random access game            if, for
                                                                     any         , the stepsize                .
A. Nash Equilibrium                                                       Proof: The result follows directly from Theorem 8.
  Now let us study the equilibrium of random access game .              The condition                                       is a mild as-
  Lemma 11: Suppose                       . Then, random ac-         sumption and admits a very large region in the parameter space.
cess game    has a unique nontrivial Nash equilibrium.               The Nash equilibrium can be easily calculated by numerically
                  IE


     Proof: Suppose there are      class- nodes, each with a         solving fixed point (10). If we consider a system of homogenous
                                                                     users, we choose the same utility function, i.e., the same weight
channel access probability . Let                       . Note
                                                                        for all users. If we want to provide differentiated services, we
that when            ,                                , and when     can choose larger value of       for the users of a higher priority
                                ,                                    class. For example, in wireless access network, we can assign
                                                                     a large     value to the access point, because usually downlink
if                 . Since                              and are      traffic is greater than the traffic of mobile nodes.
continuous functions of , there exists a solution      to the fol-
               Pr



lowing equilibrium equation [i.e., (11)]:                            C. Medium Access Control Design
                                                                        We design a medium access method according to channel
                                                                     access probability and contention window update mechanism
                                                                     (28)–(29) by modifying a CSMA/CA access method such as
                                                                     802.11 DCF [2]. The basic access mechanism in DCF works
if                . Thus, there exists a nontrivial Nash equi-       as follows. A node wishing to transmit senses the channel for a
librium                     for random access game . Fur-            period of time equal to the distributed interframe space (DIFS)
thermore,      defined by (24) satisfies assumption A2. It fol-        to check if it is idle. If the channel is determined to be idle, the
lows from Theorem 4 that the nontrivial Nash equilibrium must        node starts to transmit a DATA frame. If the channel is consid-
be unique.                                                           ered to be busy, the node waits for a random backoff time ,
CHEN et al.: RANDOM ACCESS GAME AND MEDIUM ACCESS CONTROL DESIGN                                                                         9



an integer uniformly distributed in the window                     be-                               TABLE II
fore attempting to transmit. Upon successful reception of the                       PARAMETERS USED TO OBTAIN NUMERICAL RESULTS
DATA frame, the receiving node waits for a short interframe
space (SIFS) interval and then sends an ACK frame. When the
node detects a failed transmission, it doubles the contention
window          (exponential backoff). In order to avoid channel
capture, a node must wait for a random backoff time, in the
same way as if the channel is sensed busy, between two con-
secutive new packet transmissions. Note that DCF employs a
discrete-time backoff scale. The time immediately following an
idle DIFS is slotted with slot-time size . The backoff time
counter is decremented as long as the channel is sensed idle
during a time slot, and frozen when the channel is sensed busy,
and reactivated when the channel is sensed idle again for a DIFS.
                                                                          where              . If is small, we weight history less, and if




                          rs f
The node transmits when the backoff time counter reaches zero.
                                                                          is large, we weight history more. The running average works as
See [2] for more details.
                                                                          a low-pass filter, and by choosing appropriate value, it gives




                       Ve o
   As described in Section III-C, our medium access method
                                                                          better estimate than the “naive” estimator                 .
makes two key modifications to 802.11 DCF. Instead of ad-




                            ion
                                                                             By our access method, the system is designed to reach and op-
justing contention window             to a binary feedback signal
                                                                          erate around the Nash equilibrium of random access game .
(packet loss or successful transmission) and using exponential
                                                                          Thus, its performance is determined by the Nash equilibrium of




                            ro
backoff algorithm, each node estimates its conditional colli-
                                                                              . Consider a system of greedy nodes that always have packets
sion probability , which is a continuous feedback, and adjusts
                                                                          to transmit. Denote the equilibrium channel access probability
     according to algorithm (28)–(29).
                                                                          of node by ; we can calculate its throughput and condi-
   There are several parameters in our medium access method.
                                                                          tional collision probability as follows:
The parameter depends only on the protocol parameters such
as the duration of an idle slot that are specified by the system
                   int P
designer. The maximal channel access probability affects the
number of the equilibria of random access game                but not
the location of the equilibrium (i.e., the equilibrium proper-
                                                                                                                                     (31)
ties of our design) once it satisfies the condition specified in
Section IV-A. The weight            determines the fairness or ser-                                                                  (32)
vice differentiation. The system designer will specify a set of
weights, according to the levels of quality of services he wants
                       EE

to provide, and each wireless node will choose a weight that              where idle probability                           and
corresponds to the specific level of service it desires. The pa-
rameters       and               determine the dynamic properties
such as stability and responsiveness. The stepsize affects the
convergence speed. In practice, we will choose a constant step-
size for all nodes. The number of transmissions,                  , for
each node before updating its channel access probability and              See Table II for other notations. Here, for simplicity, we have
contention window, affects the convergence speed and the ac-              assumed an equal payload size. The throughput for general pay-
curacy of the conditional collision probability estimation. Note          load size distribution can calculated in a similar way [23], and
                    IE


that in strategy update algorithm (28)–(29), is not real time,            the aggregate throughput is the summation of over all nodes .
but represents the stages at which the random access game is
played. In our design, each node repeatedly plays game every              D. Performance
              transmissions, and the channel access probability              We have conducted numerical experiments to evaluate the
and contention window are fixed between consecutive plays. If              performance of our medium access method. We develop a
              is too large, it will take a longer time to reach the       packet-level simulator that implements our method and the
Nash equilibrium, but if                  is too small, it will result    standard 802.11 DCF basic access method (i.e., no RTS/CTS).
in large estimation error in the average number of consecutive            The values for the parameters used to obtain numerical results
idle slots between transmissions and thus conditional collision
                 Pr



                                                                          are summarized in Table II. The system values are those speci-
probability. In order to achieve a good tradeoff between conver-          fied in the 802.11b standard with DSSS PHY layer [2]. Under
gence speed and estimation accuracy, we will choose a relatively          these specifications,               and                        .
small value for                and estimate average number of con-        We set             , which corresponds to a contention window
secutive idle slots between transmissions using an exponential            size of 16. In all simulations, we set the following values of
weighted running average                                                  the control parameters:                     ,            , and
                                                                                   . The simulation results reported aim to zoom in on
                                                                  (30)    specific properties of our access method.
10                                                                                                  IEEE/ACM TRANSACTIONS ON NETWORKING




                         rs f
                                                                     Fig. 2. Conditional collision probability comparison.
Fig. 1. Throughput comparison.




                      Ve o
                           ion
   1) Throughput and Collision Overhead: We consider a net-
work of homogeneous users with perfect channel (i.e., there is
no corrupted frame that is due to channel error) and compare the




                           ro
throughput achieved by our access method and 802.11b DCF as
well as the maximal achievable throughput as our method is in-
tended to achieve optimal throughput. In our design, each node
has the same weight of 1, i.e.,           , since we consider ho-
mogenous users. In our numerical experiments with DCF, after
a packet’s          th failed transmission, the contention window
                  int P
resets to the base contention window, where denotes the max-
imum backoff stage and is set to 5 for 802.11b. This is also
equivalent to the packet being discarded after failed retrans-
missions.
   Fig. 1 shows the aggregate throughput achieved by our design
and DCF and the maximal achievable throughput. Our design al-        Fig. 3. Fairness comparison for a network of 40 competing nodes.
                      EE

most achieves the optimal throughout, with only a slightly lower
than optimal throughput for a network of a very small number
of competing nodes. This also confirms that the approximations        aspects such as lower energy usage and less interference to the
(19) and (24) made in deriving the utility functions (25) are very   wireless nodes of neighboring cells.
accurate approximations, and the contention measure estimator           2) Fairness: It is well known that 802.11 DCF has a
(18) and (30) give fairly accurate estimation of the conditional     short-term fairness problem due to binary exponential backoff
collision probability.                                               process. In our access method for a system of homogeneous
   Compared to DCF, for a network of a very small number             users, wireless nodes have the same contention window size,
of wireless nodes, DCF provides a slightly higher throughput         specified by the symmetric Nash equilibrium of random ac-
than our design. However, as the number of nodes increases,          cess game . Thus, it is expected to have a better short-term
                   IE


our access method achieves much higher throughput. With DCF,         fairness. Fig. 3 compares short-term fairness of our access
each new transmission starts with the base contention window         method and DCF using Jain fairness index for the window
and executes binary exponential backoff upon collisions, while,      sizes that are multiples of the number of wireless nodes
with our access method, nodes choose a constant contention           [26]. Denote by        the number of transmissions of node
window determined by the Nash equilibrium, which is “op-             over a certain time window. Jain fairness index is defined
timal” for the current contention level in the network. Thus,        by                           , and higher index means better
for a system of many competing nodes where the contention in         fairness. We can see that our method provides much better
the network is heavy, DCF will incur much more packet col-           short-term fairness than 802.11 DCF.
                Pr



lisions than our access method, which results in much lower             3) Dynamic Scenario: To evaluate the responsiveness and
throughput, as shown in Fig. 1. This is further confirmed by          the convergence of our access method, we consider a dynamic
the comparison of collision overhead between DCF and our ac-         scenario as follows. In the beginning, there are five greedy
cess method, as shown in Fig. 2. Our access method achieves a        nodes in the network, which has converged to the equilibrium
very small, almost constant collision probability, better tradeoff   or the steady operating point. After 1004 more transmissions,
between channel access and collision avoidance, and hence a          five more nodes join in to compete for channel access. The five
higher throughput that is sustainable over a large range of num-     nodes then leave after 3000 transmissions. Fig. 4 shows the
bers of competing nodes. Practically, this means that our access     evolution of channel access probability and the corresponding
method can achieve higher throughput, but with fewer transmis-       contention window size of a long-stay node and a short-stay
sions than DCF, which will benefit the whole system in many           node, respectively. We see that our access method reaches the
CHEN et al.: RANDOM ACCESS GAME AND MEDIUM ACCESS CONTROL DESIGN                                                                                    11




                          rs f
                       Ve o
                            ion
                            ro
                   int P
Fig. 4. Evolution of channel access probability and the corresponding con-
                                                                             Fig. 5. Throughput comparison of our design and 802.11b DCF with different
tention window.
                                                                             frame error rates.
                       EE

new equilibrium or steady operating point after only about                   throughput achieved by our design and 802.11b DCF, with zero,
totally 65 transmissions. That is about seven transmissions for              10%, 20%, and 40% frame error rates, respectively. We see that,
each wireless node. The evolutions of different wireless nodes               as expected, our contention resolution algorithm is insensitive
follows roughly the same trend. This is because they observe                 to channel error, and the throughput degradation comes solely
the same signal to estimate their contention measure, though the             from the corrupted frames due to channel error.
times when the long-stay node and the short-stay node update                     The impact of channel error on DCF is a bit “twisted.” When
their channel access probabilities and contention windows are                the number of nodes is very small, the channel error leads
four transmissions away.                                                     to larger degradation in throughput. This is because packet
                    IE


   In order to respond quickly to the channel contention, in                 collision is rare in this situation, and channel error results in
our access method when a wireless node enters the system                     more backoff and hence fewer transmissions. However, when
or starts to have data to send after being idle, it will monitor             the number of nodes is large, the channel error actually helps.
the channel for a certain amount of time before transmitting                 This is because packet collision is frequent in this situation, and
to estimate current channel contention . Based on this es-                   additional backoff due to the corrupted frames greatly reduces
timation, it will set its starting channel access probability                the chance of packet collision. Note that we have assumed an
to                                  . In our simulation for the              i.i.d channel error model. If the channel error comes in bursts,
dynamic scenario, after entering the network, the short-stay                 it will lead to greater throughput degradation of DCF and make
                 Pr



nodes monitor the channel for the duration of three consecutive              DCF interact adversely with higher layer protocols such as TCP
transmissions to estimate current channel contention to set                  congestion control. Our access method has no such problems
the starting channel access probability and the corresponding                since its contention resolution algorithm is insensitive to the
contention window. This reflects in Fig. 4 where the starting                 channel error.
channel access probability and contention window of the                          5) Service Differentiation: As discussed before, we can
short-stay node are close to the channel access probability and              provide service differentiation by choosing different utility
contention window of the long-stay node.                                     functions for different classes of users. Regarding the concrete
   4) Throughput Under Unreliable Channel: We now con-                       medium access method we consider, each node will receive
sider the network with unreliable channel, i.e., there exist cor-            different services by choosing different weights . For the
rupted frames due to channel error. Fig. 5 shows the aggregate               simplicity of presentation, we consider two classes of users.
12                                                                                                           IEEE/ACM TRANSACTIONS ON NETWORKING



                                                                                  matrix. Nonetheless, we can still establish the uniqueness
                                                                                  and convergence of the Nash equilibrium by generalizing the
                                                                                  aforementioned conditions.
                                                                                     It is straightforward to verify that Theorem 2 holds for the
                                                                                  multicell network, i.e., under assumption A0 there exists at least
                                                                                  one Nash equilibrium for random access games . Let be the
                                                                                  Jacobi matrix of the contention measure vector       , i.e.,
                                                                                          , and denote by the smallest eigenvalue of matrix
                                                                                            over .
                                                                                     Theorem 14: Suppose assumption A0 holds and
                                                                                   . Then, random access game has a unique Nash equilibrium.
                                                                                        Proof: Since A0 holds, there exists at least one Nash
                                                                                  equilibrium, denoted by         . Define function




                           rs f
                                                                                                     . Note that                . Let
Fig. 6. The throughput ratio of a class–1 node with weight    = 1 to a class–2                                       . By Taylor expansion, we
node with weight    = 05: .




                        Ve o
                                                                                  have




                             ion
                             ro
                    int P
                                                                                  where                                       . If            ,
                                                                                              for any        . Now suppose there exists another
                                                                                  Nash equilibrium, denoted by , then               . However,
                                                                                  by equilibrium condition (7)
                        EE

Fig. 7. The aggregate throughput versus the number of nodes.


Assume that class 1 has       users with weight          , corre-
sponding to a higher priority of service, and class 2 has users
with weight              , corresponding to a lower priority of
service. Fig. 6 shows the throughput ratio of a class-1 node to a
class-2 node versus the total number of nodes for two different                   which contradicts the relation            . Thus, random access
scenarios: two classes have equal number of users, and class 1                    game has a unique Nash equilibrium.
                     IE


has fixed number of users. We see that the throughput ratio is                        The condition                 is a generalization of assump-
almost exactly           . This also confirms that the condition                   tion A1. As far as we know, our proof method for the unique-
(21) is a very accurate approximate requirement for achieving                     ness of the equilibrium is new. It only uses basic property of
the desired service differentiation.                                              convexity, and is expected to provide a general technique to es-
   Fig. 7 shows the corresponding aggregate throughput by our                     tablish the uniqueness result.
access method. We see that our design almost achieves the max-                       Theorem 15: Assume the same conditions as in Theorem 14
imal achievable throughput. This again confirms that the ap-                       and the same constant stepsize for the gradient play for all
proximations (19) and (24) made in deriving the utility func-                     wireless nodes. The gradient play (13) converges to the unique
tions (25) are very accurate approximations.                                      Nash equilibrium of random access game if the stepsize
                  Pr



                                                                                                 .
            V. EXTENSION TO MULTICELL NETWORKS                                         Proof: First note that the Nash equilibrium of random ac-
                                                                                  cess game is the fixed point of the gradient play (13), and vice
   We now extend the above development to multicell wire-                         versa. Define a mapping
less LANs in which some wireless nodes (e.g., those at the
boundaries of the cells) may belong to more than one cell.
Multicell networks are difficult to analyze because of the
asymmetric information involved at different nodes. Mathemat-                     The gradient play can be written as                         .
ically, this reflects in the fact that the Jacobi matrix of the user               Note that                                                   .
payoff functions                              is not a symmetric                  If we can show that        is a contraction mapping, then the
CHEN et al.: RANDOM ACCESS GAME AND MEDIUM ACCESS CONTROL DESIGN                                                                        13



gradient play is a contraction mapping, too. Now, assume a con-       the channel access probability vector          can be modeled by
stant stepsize for all nodes. For any two points and in the           stochastic function
strategy space, we have                                                                      with probability
                                                                                             with probability

                                                                      It is well established that for a single-cell wireless LAN at steady
                                                                      state, the average channel access probability relates to the con-
                                                                      ditional collision probability                as follows [23]:



                                                                      where               is the base contention window and       is the
                                                                      maximum backoff stage. Following the procedure (4)–(5) to de-
                                                                      rive a utility function      , we can define a random access game




                         rs f
                                                                      with payoff                    and interpret DCF as a distributed
                                                                      strategy update algorithm to achieve the corresponding Nash




                      Ve o
                                                                      equilibrium. Note that the dynamics of DCF cannot be described




                           ion
                                                                      by (stochastic) gradient play. The convergence of DCF to the
                                                                      Nash equilibrium is guaranteed by the stability of the underlying
                                                                      Markov chain that describes the dynamics of DCF. However, we




                           ro
where                                                    , and in     can design a medium access method according to the gradient
the third inequality we apply the same trick as in the proof of       play of this game, which will achieve the same throughput as
Theorem 14. We see that if the stepsize                          ,    that by DCF, but with better short-term fairness.
                                                                         One of the key aspects of our reverse-engineering is to re-
                                   . Thus        is a contraction
                                                                      verse-engineer the equilibrium (or steady operating point) of
mapping, and so is the gradient play. By the contraction map-
                                                                      the contention control protocol, i.e., the equilibrium of the re-
ping theorem [18], the gradient play will converge to the unique
                  int P
                                                                      sulting game should recover the equilibrium (or steady oper-
fixed point of (13), i.e., the unique Nash equilibrium of random
                                                                      ating point) of the contention control protocol; see (3) and (33).
access game .
                                                                      Our random access game framework applies to any dynamic
   The above condition on the stepsize looks strong, but seen
                                                                      contention-based medium access methods that have a steady op-
from the proof, it is overly conservative. We are seeking other
                                                                      erating point, such as the stabilized Aloha [25].
techniques to better bound the matrix norm to weaken this con-
dition. If the medium access method designed according to the
                                                                                             VII. CONCLUSION
                      EE

gradient play (13) of random access game is used in multi-
cell wireless LANs, Theorems 15 guarantees that it will operate          We have developed a game-theoretic model for contention
around the equilibrium of random access game and achieve              control and propose a novel medium access method derived
the performance specified by the equilibrium.                          from CSMA/CA, in which each node estimates its conditional
                                                                      collision probability and adjusts its persistence probability or
      VI. SOME COMMENTS ON REVERSE- ENGINEERING                       contention window according to a distributed strategy update
                                                                      mechanism achieving the Nash equilibrium. This results in
   The dynamic model (1)–(2) of contention control is very gen-
                                                                      controllable performance objectives through the specification
eral. The function       is usually a deterministic function, while
                                                                      of per-node utility functions. As wireless nodes can estimate
the function may be deterministic if the contention measure
                                                                      conditional collision probabilities by observing consecutive
                   IE


is a continuous variable or stochastic if the contention measure
                                                                      idle slots between transmissions, we can decouple contention
is a discrete variable. In the derivation of the utility function
                                                                      control from handling failed transmissions. As a case study of
in Section III, we have implicitly assumed a continuous con-
                                                                      medium access control design in the proposed game-theoretic
tention measure. If a discrete contention measure is used, the
                                                                      framework, we present a concrete medium access method that
fixed point equation is usually not dictated directly by function
                                                                      achieves optimal throughput, low collision, and good short-term
    as in (3). Instead, the fixed point may be dictated by a certain
                                                                      fairness and can provide flexible service differentiations among
function that relates the channel access probability        and the
                                                                      wireless nodes. In addition to guiding medium access control
average contention measure
                                                                      design, the random access game model also provides an an-
                Pr



                                                              (33)    alytical framework to understand equilibrium properties and
                                                                      dynamic properties of different medium access protocols.
Starting with this fixed point equation and following the same            This paper has been focusing on laying out a theoretical
procedure as in (4)–(5), we can define the utility function and        framework. Much work remains to take it from a promising
a random access game and interpret the contention control pro-        design framework to a full-fledged medium access control
tocol as a distributed strategy update algorithm to achieve the       protocol. We are extending the framework to the network
equilibrium of this game.                                             where wireless nodes have arbitrary traffic patterns (e.g., bursty
   Take the IEEE 802.11 DCF as an example. DCF responds to            packet arrival), which will be reported elsewhere. We will
whether there is a collision, and hence the measure of contention     search for other contention resolution algorithms that could
      in DCF is a binary feedback signal whose dependence on          achieve Nash equilibria of random access games and other
Ramac

Más contenido relacionado

La actualidad más candente

ACHIEVING ENHANCED THROUGHPUT IN MOBILE ADHOC NETWORK USING COLLISION AWARE M...
ACHIEVING ENHANCED THROUGHPUT IN MOBILE ADHOC NETWORK USING COLLISION AWARE M...ACHIEVING ENHANCED THROUGHPUT IN MOBILE ADHOC NETWORK USING COLLISION AWARE M...
ACHIEVING ENHANCED THROUGHPUT IN MOBILE ADHOC NETWORK USING COLLISION AWARE M...ijasuc
 
A CELLULAR BONDING AND ADAPTIVE LOAD BALANCING BASED MULTI-SIM GATEWAY FOR MO...
A CELLULAR BONDING AND ADAPTIVE LOAD BALANCING BASED MULTI-SIM GATEWAY FOR MO...A CELLULAR BONDING AND ADAPTIVE LOAD BALANCING BASED MULTI-SIM GATEWAY FOR MO...
A CELLULAR BONDING AND ADAPTIVE LOAD BALANCING BASED MULTI-SIM GATEWAY FOR MO...pijans
 
IEEE 2015 NS2 Projects
IEEE 2015 NS2 ProjectsIEEE 2015 NS2 Projects
IEEE 2015 NS2 ProjectsVijay Karan
 
IEEE 2015 NS2 Projects
IEEE 2015 NS2 ProjectsIEEE 2015 NS2 Projects
IEEE 2015 NS2 ProjectsVijay Karan
 
M phil-computer-science-network-security-projects
M phil-computer-science-network-security-projectsM phil-computer-science-network-security-projects
M phil-computer-science-network-security-projectsVijay Karan
 
2011 & 2012 ieee projects
2011 & 2012 ieee projects2011 & 2012 ieee projects
2011 & 2012 ieee projectsColege Buz
 
5113jgraph01
5113jgraph015113jgraph01
5113jgraph01graphhoc
 
M.Phil Computer Science Mobile Computing Projects
M.Phil Computer Science Mobile Computing ProjectsM.Phil Computer Science Mobile Computing Projects
M.Phil Computer Science Mobile Computing ProjectsVijay Karan
 
Multi hop wireless-networks
Multi hop wireless-networksMulti hop wireless-networks
Multi hop wireless-networksambitlick
 
Ieee projects 2011 ns 2 SBGC ( Trichy, Madurai, Chennai, Dindigul, Natham, Pu...
Ieee projects 2011 ns 2 SBGC ( Trichy, Madurai, Chennai, Dindigul, Natham, Pu...Ieee projects 2011 ns 2 SBGC ( Trichy, Madurai, Chennai, Dindigul, Natham, Pu...
Ieee projects 2011 ns 2 SBGC ( Trichy, Madurai, Chennai, Dindigul, Natham, Pu...SBGC
 
International Journal of Computer Science and Security Volume (2) Issue (4)
International Journal of Computer Science and Security Volume (2) Issue (4)International Journal of Computer Science and Security Volume (2) Issue (4)
International Journal of Computer Science and Security Volume (2) Issue (4)CSCJournals
 
Performance Evaluation using STP Across Layer 2 VLANs
Performance Evaluation using STP Across Layer 2 VLANsPerformance Evaluation using STP Across Layer 2 VLANs
Performance Evaluation using STP Across Layer 2 VLANsijcnesiir
 
Hexagonal based Clustering for Reducing Rebroadcasts in Mobile Ad Hoc Networks
Hexagonal based Clustering for Reducing Rebroadcasts in Mobile Ad Hoc NetworksHexagonal based Clustering for Reducing Rebroadcasts in Mobile Ad Hoc Networks
Hexagonal based Clustering for Reducing Rebroadcasts in Mobile Ad Hoc NetworksIJTET Journal
 
M.E Computer Science Network Security Projects
M.E Computer Science Network Security ProjectsM.E Computer Science Network Security Projects
M.E Computer Science Network Security ProjectsVijay Karan
 
Jamming Anticipation and Convolution through Immaculate Hiding Process of Pac...
Jamming Anticipation and Convolution through Immaculate Hiding Process of Pac...Jamming Anticipation and Convolution through Immaculate Hiding Process of Pac...
Jamming Anticipation and Convolution through Immaculate Hiding Process of Pac...IOSR Journals
 
System Structure for Dependable Software Systems
System Structure for Dependable Software SystemsSystem Structure for Dependable Software Systems
System Structure for Dependable Software SystemsVincenzo De Florio
 
Intelligent Approach for Seamless Mobility in Multi Network Environment
Intelligent Approach for Seamless Mobility in Multi Network EnvironmentIntelligent Approach for Seamless Mobility in Multi Network Environment
Intelligent Approach for Seamless Mobility in Multi Network EnvironmentIDES Editor
 

La actualidad más candente (19)

ACHIEVING ENHANCED THROUGHPUT IN MOBILE ADHOC NETWORK USING COLLISION AWARE M...
ACHIEVING ENHANCED THROUGHPUT IN MOBILE ADHOC NETWORK USING COLLISION AWARE M...ACHIEVING ENHANCED THROUGHPUT IN MOBILE ADHOC NETWORK USING COLLISION AWARE M...
ACHIEVING ENHANCED THROUGHPUT IN MOBILE ADHOC NETWORK USING COLLISION AWARE M...
 
A CELLULAR BONDING AND ADAPTIVE LOAD BALANCING BASED MULTI-SIM GATEWAY FOR MO...
A CELLULAR BONDING AND ADAPTIVE LOAD BALANCING BASED MULTI-SIM GATEWAY FOR MO...A CELLULAR BONDING AND ADAPTIVE LOAD BALANCING BASED MULTI-SIM GATEWAY FOR MO...
A CELLULAR BONDING AND ADAPTIVE LOAD BALANCING BASED MULTI-SIM GATEWAY FOR MO...
 
IEEE 2015 NS2 Projects
IEEE 2015 NS2 ProjectsIEEE 2015 NS2 Projects
IEEE 2015 NS2 Projects
 
Seamless and Secured wide Fidelity enhancement in moving vehicles Using Eeack...
Seamless and Secured wide Fidelity enhancement in moving vehicles Using Eeack...Seamless and Secured wide Fidelity enhancement in moving vehicles Using Eeack...
Seamless and Secured wide Fidelity enhancement in moving vehicles Using Eeack...
 
IEEE 2015 NS2 Projects
IEEE 2015 NS2 ProjectsIEEE 2015 NS2 Projects
IEEE 2015 NS2 Projects
 
M phil-computer-science-network-security-projects
M phil-computer-science-network-security-projectsM phil-computer-science-network-security-projects
M phil-computer-science-network-security-projects
 
2011 & 2012 ieee projects
2011 & 2012 ieee projects2011 & 2012 ieee projects
2011 & 2012 ieee projects
 
5113jgraph01
5113jgraph015113jgraph01
5113jgraph01
 
M.Phil Computer Science Mobile Computing Projects
M.Phil Computer Science Mobile Computing ProjectsM.Phil Computer Science Mobile Computing Projects
M.Phil Computer Science Mobile Computing Projects
 
Multi hop wireless-networks
Multi hop wireless-networksMulti hop wireless-networks
Multi hop wireless-networks
 
Ieee projects 2011 ns 2 SBGC ( Trichy, Madurai, Chennai, Dindigul, Natham, Pu...
Ieee projects 2011 ns 2 SBGC ( Trichy, Madurai, Chennai, Dindigul, Natham, Pu...Ieee projects 2011 ns 2 SBGC ( Trichy, Madurai, Chennai, Dindigul, Natham, Pu...
Ieee projects 2011 ns 2 SBGC ( Trichy, Madurai, Chennai, Dindigul, Natham, Pu...
 
International Journal of Computer Science and Security Volume (2) Issue (4)
International Journal of Computer Science and Security Volume (2) Issue (4)International Journal of Computer Science and Security Volume (2) Issue (4)
International Journal of Computer Science and Security Volume (2) Issue (4)
 
Performance Evaluation using STP Across Layer 2 VLANs
Performance Evaluation using STP Across Layer 2 VLANsPerformance Evaluation using STP Across Layer 2 VLANs
Performance Evaluation using STP Across Layer 2 VLANs
 
Hexagonal based Clustering for Reducing Rebroadcasts in Mobile Ad Hoc Networks
Hexagonal based Clustering for Reducing Rebroadcasts in Mobile Ad Hoc NetworksHexagonal based Clustering for Reducing Rebroadcasts in Mobile Ad Hoc Networks
Hexagonal based Clustering for Reducing Rebroadcasts in Mobile Ad Hoc Networks
 
M.E Computer Science Network Security Projects
M.E Computer Science Network Security ProjectsM.E Computer Science Network Security Projects
M.E Computer Science Network Security Projects
 
Jamming Anticipation and Convolution through Immaculate Hiding Process of Pac...
Jamming Anticipation and Convolution through Immaculate Hiding Process of Pac...Jamming Anticipation and Convolution through Immaculate Hiding Process of Pac...
Jamming Anticipation and Convolution through Immaculate Hiding Process of Pac...
 
Tmac
TmacTmac
Tmac
 
System Structure for Dependable Software Systems
System Structure for Dependable Software SystemsSystem Structure for Dependable Software Systems
System Structure for Dependable Software Systems
 
Intelligent Approach for Seamless Mobility in Multi Network Environment
Intelligent Approach for Seamless Mobility in Multi Network EnvironmentIntelligent Approach for Seamless Mobility in Multi Network Environment
Intelligent Approach for Seamless Mobility in Multi Network Environment
 

Destacado

基于云计算平台的移动Iptv系统设计及负载均衡技术研究
基于云计算平台的移动Iptv系统设计及负载均衡技术研究基于云计算平台的移动Iptv系统设计及负载均衡技术研究
基于云计算平台的移动Iptv系统设计及负载均衡技术研究liangxiao0315
 
Eucalyptus安装及实例映像制作
Eucalyptus安装及实例映像制作Eucalyptus安装及实例映像制作
Eucalyptus安装及实例映像制作liangxiao0315
 
Napred.bg - българската търсачка
Napred.bg - българската търсачкаNapred.bg - българската търсачка
Napred.bg - българската търсачкаVesela Gavrailova
 
Hadoop开发者入门专刊
Hadoop开发者入门专刊Hadoop开发者入门专刊
Hadoop开发者入门专刊liangxiao0315
 
Collective health nursing. the construction of critical thinking about the re...
Collective health nursing. the construction of critical thinking about the re...Collective health nursing. the construction of critical thinking about the re...
Collective health nursing. the construction of critical thinking about the re...Yese Correa
 
构建私有云计算平台的Eucalyptus架构分析
构建私有云计算平台的Eucalyptus架构分析构建私有云计算平台的Eucalyptus架构分析
构建私有云计算平台的Eucalyptus架构分析liangxiao0315
 
基于Eucalyptus的教育知识服务体系模型研究(1)
基于Eucalyptus的教育知识服务体系模型研究(1)基于Eucalyptus的教育知识服务体系模型研究(1)
基于Eucalyptus的教育知识服务体系模型研究(1)liangxiao0315
 
基于Eucalyptus的教育知识服务体系模型研究
基于Eucalyptus的教育知识服务体系模型研究基于Eucalyptus的教育知识服务体系模型研究
基于Eucalyptus的教育知识服务体系模型研究liangxiao0315
 
التخطيط الاستراتيجى أ.د. سامية العزب
التخطيط الاستراتيجى أ.د. سامية العزبالتخطيط الاستراتيجى أ.د. سامية العزب
التخطيط الاستراتيجى أ.د. سامية العزبProf_Samia
 
Resume Alan Feigenbaum
Resume   Alan FeigenbaumResume   Alan Feigenbaum
Resume Alan FeigenbaumAlanFeigenbaum
 

Destacado (11)

基于云计算平台的移动Iptv系统设计及负载均衡技术研究
基于云计算平台的移动Iptv系统设计及负载均衡技术研究基于云计算平台的移动Iptv系统设计及负载均衡技术研究
基于云计算平台的移动Iptv系统设计及负载均衡技术研究
 
Eucalyptus安装及实例映像制作
Eucalyptus安装及实例映像制作Eucalyptus安装及实例映像制作
Eucalyptus安装及实例映像制作
 
Napred.bg - българската търсачка
Napred.bg - българската търсачкаNapred.bg - българската търсачка
Napred.bg - българската търсачка
 
Hadoop开发者入门专刊
Hadoop开发者入门专刊Hadoop开发者入门专刊
Hadoop开发者入门专刊
 
Collective health nursing. the construction of critical thinking about the re...
Collective health nursing. the construction of critical thinking about the re...Collective health nursing. the construction of critical thinking about the re...
Collective health nursing. the construction of critical thinking about the re...
 
构建私有云计算平台的Eucalyptus架构分析
构建私有云计算平台的Eucalyptus架构分析构建私有云计算平台的Eucalyptus架构分析
构建私有云计算平台的Eucalyptus架构分析
 
基于Eucalyptus的教育知识服务体系模型研究(1)
基于Eucalyptus的教育知识服务体系模型研究(1)基于Eucalyptus的教育知识服务体系模型研究(1)
基于Eucalyptus的教育知识服务体系模型研究(1)
 
¿que es un blog?
¿que es un blog?¿que es un blog?
¿que es un blog?
 
基于Eucalyptus的教育知识服务体系模型研究
基于Eucalyptus的教育知识服务体系模型研究基于Eucalyptus的教育知识服务体系模型研究
基于Eucalyptus的教育知识服务体系模型研究
 
التخطيط الاستراتيجى أ.د. سامية العزب
التخطيط الاستراتيجى أ.د. سامية العزبالتخطيط الاستراتيجى أ.د. سامية العزب
التخطيط الاستراتيجى أ.د. سامية العزب
 
Resume Alan Feigenbaum
Resume   Alan FeigenbaumResume   Alan Feigenbaum
Resume Alan Feigenbaum
 

Similar a Ramac

2011 ieee projects
2011 ieee projects2011 ieee projects
2011 ieee projectsColege Buz
 
A New Approach to Improve the Efficiency of Distributed Scheduling in IEEE 80...
A New Approach to Improve the Efficiency of Distributed Scheduling in IEEE 80...A New Approach to Improve the Efficiency of Distributed Scheduling in IEEE 80...
A New Approach to Improve the Efficiency of Distributed Scheduling in IEEE 80...IDES Editor
 
A novel pause count backoff algorithm for channel access
A novel pause count backoff algorithm for channel accessA novel pause count backoff algorithm for channel access
A novel pause count backoff algorithm for channel accessambitlick
 
IEEE Networking 2016 Title and Abstract
IEEE Networking 2016 Title and AbstractIEEE Networking 2016 Title and Abstract
IEEE Networking 2016 Title and Abstracttsysglobalsolutions
 
Soft Real-Time Guarantee for Control Applications Using Both Measurement and ...
Soft Real-Time Guarantee for Control Applications Using Both Measurement and ...Soft Real-Time Guarantee for Control Applications Using Both Measurement and ...
Soft Real-Time Guarantee for Control Applications Using Both Measurement and ...CSCJournals
 
Mobile computing java ieee projects 2012 Seabirds ( Chennai, Pondicherry, Vel...
Mobile computing java ieee projects 2012 Seabirds ( Chennai, Pondicherry, Vel...Mobile computing java ieee projects 2012 Seabirds ( Chennai, Pondicherry, Vel...
Mobile computing java ieee projects 2012 Seabirds ( Chennai, Pondicherry, Vel...SBGC
 
2013 2014 ieee projects titles with abstract
2013 2014 ieee projects titles with abstract2013 2014 ieee projects titles with abstract
2013 2014 ieee projects titles with abstractMukundhan Elumalai
 
Ns2 2015 2016 ieee project list-(v)_with abstract(S3 Infotech:9884848198)
Ns2 2015 2016 ieee project list-(v)_with abstract(S3 Infotech:9884848198)Ns2 2015 2016 ieee project list-(v)_with abstract(S3 Infotech:9884848198)
Ns2 2015 2016 ieee project list-(v)_with abstract(S3 Infotech:9884848198)S3 Infotech IEEE Projects
 
Modified PREQ in HWMP for Congestion Avoidance in Wireless Mesh Network
Modified PREQ in HWMP for Congestion Avoidance in Wireless Mesh NetworkModified PREQ in HWMP for Congestion Avoidance in Wireless Mesh Network
Modified PREQ in HWMP for Congestion Avoidance in Wireless Mesh NetworkIRJET Journal
 
Java and .net IEEE 2012
Java and .net IEEE 2012Java and .net IEEE 2012
Java and .net IEEE 2012Vipin Jacob
 
An Accurate Performance Analysis of Hybrid Efficient and Reliable MAC Protoco...
An Accurate Performance Analysis of Hybrid Efficient and Reliable MAC Protoco...An Accurate Performance Analysis of Hybrid Efficient and Reliable MAC Protoco...
An Accurate Performance Analysis of Hybrid Efficient and Reliable MAC Protoco...IJECEIAES
 
Effective Streaming of Clustered Sensor Data in Harsh Environment
Effective Streaming of Clustered Sensor Data in Harsh EnvironmentEffective Streaming of Clustered Sensor Data in Harsh Environment
Effective Streaming of Clustered Sensor Data in Harsh EnvironmentCSCJournals
 
JPN1411 Secure Continuous Aggregation in Wireless Sensor Networks
JPN1411   Secure Continuous Aggregation in Wireless Sensor NetworksJPN1411   Secure Continuous Aggregation in Wireless Sensor Networks
JPN1411 Secure Continuous Aggregation in Wireless Sensor Networkschennaijp
 
Achieving Transmission Fairness in Distributed Medium Access Wireless Mesh Ne...
Achieving Transmission Fairness in Distributed Medium Access Wireless Mesh Ne...Achieving Transmission Fairness in Distributed Medium Access Wireless Mesh Ne...
Achieving Transmission Fairness in Distributed Medium Access Wireless Mesh Ne...ijwmn
 
Fuzzy Logic-based Efficient Message Route Selection Method to Prolong the Net...
Fuzzy Logic-based Efficient Message Route Selection Method to Prolong the Net...Fuzzy Logic-based Efficient Message Route Selection Method to Prolong the Net...
Fuzzy Logic-based Efficient Message Route Selection Method to Prolong the Net...IJCNCJournal
 
FUZZY LOGIC-BASED EFFICIENT MESSAGE ROUTE SELECTION METHOD TO PROLONG THE NET...
FUZZY LOGIC-BASED EFFICIENT MESSAGE ROUTE SELECTION METHOD TO PROLONG THE NET...FUZZY LOGIC-BASED EFFICIENT MESSAGE ROUTE SELECTION METHOD TO PROLONG THE NET...
FUZZY LOGIC-BASED EFFICIENT MESSAGE ROUTE SELECTION METHOD TO PROLONG THE NET...IJCNCJournal
 
A dynamic performance-based_flow_control
A dynamic performance-based_flow_controlA dynamic performance-based_flow_control
A dynamic performance-based_flow_controlingenioustech
 
Towards the Performance Analysis of IEEE 802.11 in Multi-hop Ad-Hoc Networks
Towards the Performance Analysis of IEEE 802.11 in Multi-hop Ad-Hoc NetworksTowards the Performance Analysis of IEEE 802.11 in Multi-hop Ad-Hoc Networks
Towards the Performance Analysis of IEEE 802.11 in Multi-hop Ad-Hoc Networksambitlick
 

Similar a Ramac (20)

2011 ieee projects
2011 ieee projects2011 ieee projects
2011 ieee projects
 
A New Approach to Improve the Efficiency of Distributed Scheduling in IEEE 80...
A New Approach to Improve the Efficiency of Distributed Scheduling in IEEE 80...A New Approach to Improve the Efficiency of Distributed Scheduling in IEEE 80...
A New Approach to Improve the Efficiency of Distributed Scheduling in IEEE 80...
 
A novel pause count backoff algorithm for channel access
A novel pause count backoff algorithm for channel accessA novel pause count backoff algorithm for channel access
A novel pause count backoff algorithm for channel access
 
IEEE Networking 2016 Title and Abstract
IEEE Networking 2016 Title and AbstractIEEE Networking 2016 Title and Abstract
IEEE Networking 2016 Title and Abstract
 
Soft Real-Time Guarantee for Control Applications Using Both Measurement and ...
Soft Real-Time Guarantee for Control Applications Using Both Measurement and ...Soft Real-Time Guarantee for Control Applications Using Both Measurement and ...
Soft Real-Time Guarantee for Control Applications Using Both Measurement and ...
 
Mobile computing java ieee projects 2012 Seabirds ( Chennai, Pondicherry, Vel...
Mobile computing java ieee projects 2012 Seabirds ( Chennai, Pondicherry, Vel...Mobile computing java ieee projects 2012 Seabirds ( Chennai, Pondicherry, Vel...
Mobile computing java ieee projects 2012 Seabirds ( Chennai, Pondicherry, Vel...
 
2013 2014 ieee projects titles with abstract
2013 2014 ieee projects titles with abstract2013 2014 ieee projects titles with abstract
2013 2014 ieee projects titles with abstract
 
Ns2 2015 2016 ieee project list-(v)_with abstract(S3 Infotech:9884848198)
Ns2 2015 2016 ieee project list-(v)_with abstract(S3 Infotech:9884848198)Ns2 2015 2016 ieee project list-(v)_with abstract(S3 Infotech:9884848198)
Ns2 2015 2016 ieee project list-(v)_with abstract(S3 Infotech:9884848198)
 
Modified PREQ in HWMP for Congestion Avoidance in Wireless Mesh Network
Modified PREQ in HWMP for Congestion Avoidance in Wireless Mesh NetworkModified PREQ in HWMP for Congestion Avoidance in Wireless Mesh Network
Modified PREQ in HWMP for Congestion Avoidance in Wireless Mesh Network
 
Java and .net IEEE 2012
Java and .net IEEE 2012Java and .net IEEE 2012
Java and .net IEEE 2012
 
C0941017
C0941017C0941017
C0941017
 
An Accurate Performance Analysis of Hybrid Efficient and Reliable MAC Protoco...
An Accurate Performance Analysis of Hybrid Efficient and Reliable MAC Protoco...An Accurate Performance Analysis of Hybrid Efficient and Reliable MAC Protoco...
An Accurate Performance Analysis of Hybrid Efficient and Reliable MAC Protoco...
 
Effective Streaming of Clustered Sensor Data in Harsh Environment
Effective Streaming of Clustered Sensor Data in Harsh EnvironmentEffective Streaming of Clustered Sensor Data in Harsh Environment
Effective Streaming of Clustered Sensor Data in Harsh Environment
 
JPN1411 Secure Continuous Aggregation in Wireless Sensor Networks
JPN1411   Secure Continuous Aggregation in Wireless Sensor NetworksJPN1411   Secure Continuous Aggregation in Wireless Sensor Networks
JPN1411 Secure Continuous Aggregation in Wireless Sensor Networks
 
A Systematic Review of Congestion Control in Ad Hoc Network
A Systematic Review of Congestion Control in Ad Hoc NetworkA Systematic Review of Congestion Control in Ad Hoc Network
A Systematic Review of Congestion Control in Ad Hoc Network
 
Achieving Transmission Fairness in Distributed Medium Access Wireless Mesh Ne...
Achieving Transmission Fairness in Distributed Medium Access Wireless Mesh Ne...Achieving Transmission Fairness in Distributed Medium Access Wireless Mesh Ne...
Achieving Transmission Fairness in Distributed Medium Access Wireless Mesh Ne...
 
Fuzzy Logic-based Efficient Message Route Selection Method to Prolong the Net...
Fuzzy Logic-based Efficient Message Route Selection Method to Prolong the Net...Fuzzy Logic-based Efficient Message Route Selection Method to Prolong the Net...
Fuzzy Logic-based Efficient Message Route Selection Method to Prolong the Net...
 
FUZZY LOGIC-BASED EFFICIENT MESSAGE ROUTE SELECTION METHOD TO PROLONG THE NET...
FUZZY LOGIC-BASED EFFICIENT MESSAGE ROUTE SELECTION METHOD TO PROLONG THE NET...FUZZY LOGIC-BASED EFFICIENT MESSAGE ROUTE SELECTION METHOD TO PROLONG THE NET...
FUZZY LOGIC-BASED EFFICIENT MESSAGE ROUTE SELECTION METHOD TO PROLONG THE NET...
 
A dynamic performance-based_flow_control
A dynamic performance-based_flow_controlA dynamic performance-based_flow_control
A dynamic performance-based_flow_control
 
Towards the Performance Analysis of IEEE 802.11 in Multi-hop Ad-Hoc Networks
Towards the Performance Analysis of IEEE 802.11 in Multi-hop Ad-Hoc NetworksTowards the Performance Analysis of IEEE 802.11 in Multi-hop Ad-Hoc Networks
Towards the Performance Analysis of IEEE 802.11 in Multi-hop Ad-Hoc Networks
 

Último

Basic Building Blocks of Internet of Things.
Basic Building Blocks of Internet of Things.Basic Building Blocks of Internet of Things.
Basic Building Blocks of Internet of Things.YounusS2
 
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve DecarbonizationUsing IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve DecarbonizationIES VE
 
Comparing Sidecar-less Service Mesh from Cilium and Istio
Comparing Sidecar-less Service Mesh from Cilium and IstioComparing Sidecar-less Service Mesh from Cilium and Istio
Comparing Sidecar-less Service Mesh from Cilium and IstioChristian Posta
 
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesAI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesMd Hossain Ali
 
20200723_insight_release_plan
20200723_insight_release_plan20200723_insight_release_plan
20200723_insight_release_planJamie (Taka) Wang
 
Bird eye's view on Camunda open source ecosystem
Bird eye's view on Camunda open source ecosystemBird eye's view on Camunda open source ecosystem
Bird eye's view on Camunda open source ecosystemAsko Soukka
 
Introduction to Quantum Computing
Introduction to Quantum ComputingIntroduction to Quantum Computing
Introduction to Quantum ComputingGDSC PJATK
 
Things you didn't know you can use in your Salesforce
Things you didn't know you can use in your SalesforceThings you didn't know you can use in your Salesforce
Things you didn't know you can use in your SalesforceMartin Humpolec
 
GenAI and AI GCC State of AI_Object Automation Inc
GenAI and AI GCC State of AI_Object Automation IncGenAI and AI GCC State of AI_Object Automation Inc
GenAI and AI GCC State of AI_Object Automation IncObject Automation
 
PicPay - GenAI Finance Assistant - ChatGPT for Customer Service
PicPay - GenAI Finance Assistant - ChatGPT for Customer ServicePicPay - GenAI Finance Assistant - ChatGPT for Customer Service
PicPay - GenAI Finance Assistant - ChatGPT for Customer ServiceRenan Moreira de Oliveira
 
COMPUTER 10: Lesson 7 - File Storage and Online Collaboration
COMPUTER 10: Lesson 7 - File Storage and Online CollaborationCOMPUTER 10: Lesson 7 - File Storage and Online Collaboration
COMPUTER 10: Lesson 7 - File Storage and Online Collaborationbruanjhuli
 
UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8DianaGray10
 
Introduction to Matsuo Laboratory (ENG).pptx
Introduction to Matsuo Laboratory (ENG).pptxIntroduction to Matsuo Laboratory (ENG).pptx
Introduction to Matsuo Laboratory (ENG).pptxMatsuo Lab
 
NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopBachir Benyammi
 
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCostKubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCostMatt Ray
 
Linked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond OntologiesLinked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond OntologiesDavid Newbury
 
Computer 10: Lesson 10 - Online Crimes and Hazards
Computer 10: Lesson 10 - Online Crimes and HazardsComputer 10: Lesson 10 - Online Crimes and Hazards
Computer 10: Lesson 10 - Online Crimes and HazardsSeth Reyes
 
Artificial Intelligence & SEO Trends for 2024
Artificial Intelligence & SEO Trends for 2024Artificial Intelligence & SEO Trends for 2024
Artificial Intelligence & SEO Trends for 2024D Cloud Solutions
 
Cloud Revolution: Exploring the New Wave of Serverless Spatial Data
Cloud Revolution: Exploring the New Wave of Serverless Spatial DataCloud Revolution: Exploring the New Wave of Serverless Spatial Data
Cloud Revolution: Exploring the New Wave of Serverless Spatial DataSafe Software
 
Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1DianaGray10
 

Último (20)

Basic Building Blocks of Internet of Things.
Basic Building Blocks of Internet of Things.Basic Building Blocks of Internet of Things.
Basic Building Blocks of Internet of Things.
 
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve DecarbonizationUsing IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
 
Comparing Sidecar-less Service Mesh from Cilium and Istio
Comparing Sidecar-less Service Mesh from Cilium and IstioComparing Sidecar-less Service Mesh from Cilium and Istio
Comparing Sidecar-less Service Mesh from Cilium and Istio
 
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesAI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
 
20200723_insight_release_plan
20200723_insight_release_plan20200723_insight_release_plan
20200723_insight_release_plan
 
Bird eye's view on Camunda open source ecosystem
Bird eye's view on Camunda open source ecosystemBird eye's view on Camunda open source ecosystem
Bird eye's view on Camunda open source ecosystem
 
Introduction to Quantum Computing
Introduction to Quantum ComputingIntroduction to Quantum Computing
Introduction to Quantum Computing
 
Things you didn't know you can use in your Salesforce
Things you didn't know you can use in your SalesforceThings you didn't know you can use in your Salesforce
Things you didn't know you can use in your Salesforce
 
GenAI and AI GCC State of AI_Object Automation Inc
GenAI and AI GCC State of AI_Object Automation IncGenAI and AI GCC State of AI_Object Automation Inc
GenAI and AI GCC State of AI_Object Automation Inc
 
PicPay - GenAI Finance Assistant - ChatGPT for Customer Service
PicPay - GenAI Finance Assistant - ChatGPT for Customer ServicePicPay - GenAI Finance Assistant - ChatGPT for Customer Service
PicPay - GenAI Finance Assistant - ChatGPT for Customer Service
 
COMPUTER 10: Lesson 7 - File Storage and Online Collaboration
COMPUTER 10: Lesson 7 - File Storage and Online CollaborationCOMPUTER 10: Lesson 7 - File Storage and Online Collaboration
COMPUTER 10: Lesson 7 - File Storage and Online Collaboration
 
UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8
 
Introduction to Matsuo Laboratory (ENG).pptx
Introduction to Matsuo Laboratory (ENG).pptxIntroduction to Matsuo Laboratory (ENG).pptx
Introduction to Matsuo Laboratory (ENG).pptx
 
NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 Workshop
 
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCostKubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
 
Linked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond OntologiesLinked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond Ontologies
 
Computer 10: Lesson 10 - Online Crimes and Hazards
Computer 10: Lesson 10 - Online Crimes and HazardsComputer 10: Lesson 10 - Online Crimes and Hazards
Computer 10: Lesson 10 - Online Crimes and Hazards
 
Artificial Intelligence & SEO Trends for 2024
Artificial Intelligence & SEO Trends for 2024Artificial Intelligence & SEO Trends for 2024
Artificial Intelligence & SEO Trends for 2024
 
Cloud Revolution: Exploring the New Wave of Serverless Spatial Data
Cloud Revolution: Exploring the New Wave of Serverless Spatial DataCloud Revolution: Exploring the New Wave of Serverless Spatial Data
Cloud Revolution: Exploring the New Wave of Serverless Spatial Data
 
Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1Secure your environment with UiPath and CyberArk technologies - Session 1
Secure your environment with UiPath and CyberArk technologies - Session 1
 

Ramac

  • 1. IEEE/ACM TRANSACTIONS ON NETWORKING 1 Random Access Game and Medium Access Control Design Lijun Chen, Member, IEEE, Steven H. Low, Fellow, IEEE, and John C. Doyle Abstract—Motivated partially by a control-theoretic viewpoint, channel. In the backoff mechanism, channel access probability we propose a game-theoretic model, called random access game, is implemented through a backoff algorithm, and each wireless for contention control. We characterize Nash equilibria of random node maintains a contention window and waits for a random access games, study their dynamics, and propose distributed algo- amount of time bounded by the contention window before a rithms (strategy evolutions) to achieve Nash equilibria. This pro- vides a general analytical framework that is capable of modeling a transmission. When simultaneous accesses to the channel by rs f large class of system-wide quality-of-service (QoS) models via the different nodes cause contention, the persistence probability or specification of per-node utility functions, in which system-wide contention window is adjusted appropriately so that contention Ve o fairness or service differentiation can be achieved in a distributed is reduced. Different medium access control methods differ manner as long as each node executes a contention resolution al- in terms of how they adjust these parameters in response to ion gorithm that is designed to achieve the Nash equilibrium. We thus contention and what contention measure they use. For example, propose a novel medium access method derived from carrier sense the standard IEEE 802.11 DCF uses a backoff mechanism and a multiple access/collision avoidance (CSMA/CA) according to dis- ro tributed strategy update mechanism achieving the Nash equilib- binary contention signal—packet collision or successful trans- rium of random access game. We present a concrete medium access mission—in which each wireless node doubles its contention method that adapts to a continuous contention measure called con- window upon a collision (binary exponential backoff) and sets ditional collision probability, stabilizes the network into a steady it to the base value upon a successful transmission [2]. state that achieves optimal throughput with targeted fairness (or The choice of contention measure and contention resolution service differentiation), and can decouple contention control from algorithm is key to the performance of medium access methods. handling failed transmissions. In addition to guiding medium ac- int P cess control design, the random access game model also provides “Inappropriate” choice of these two components will result in an analytical framework to understand equilibrium and dynamic poor performance. For example, in high-load scenarios, 802.11 properties of different medium access protocols. DCF results in excessive collisions and, hence, low throughput because setting to the base contention window upon successful Index Terms—Contention-based medium access, control-the- oretic analysis, game theory, Nash equilibrium, strategy update transmission is too drastic and each new transmission starts with mechanism. the base contention window independent of the contention level in the network. It also has short-term unfairness problem due EE to oscillation in the contention window. The binary exponential backoff directly causes short-term unfairness. However, this os- cillation in the contention window is unavoidable because DCF I. INTRODUCTION uses a binary contention signal. In order to achieve high effi- ciency (high throughput and low collision) and better fairness, IRELESS channel is a shared medium that is interfer- we need to stabilize the network into a steady state that sustains W ence-limited. Contention-based medium access control (contention control) is a distributed strategy to access and share an appropriate contention window size (or equivalently, channel access probability) for each node. Furthermore, how we can es- wireless channel among contending wireless nodes. From a timate and implement the contention measure is important. Al- IE control-theoretic point of view, it consists of two components: most all medium access methods, including 802.11 DCF, adapt a contention resolution algorithm that dynamically adjusts to packet collisions. However, they cannot distinguish collisions channel access probability in response to contention in the from corrupted frames that are common in wireless networks. network and a feedback mechanism that updates a contention This leads to increased unfairness and lower throughput. To en- measure and sends it back to wireless nodes. Contention sure good performance, we need to use a contention measure resolution is usually achieved through two mechanisms: per- whose estimation is not based on packet collisions and decouple sistence and backoff [1]. In the persistence mechanism, each contention control from handling failed transmissions. wireless node maintains a persistence probability and accesses The main motivation of this work is to provide an analytical framework to systematically study contention control and de- Pr the channel with this probability when it perceives an idle sign medium access methods that could stabilize the network Manuscript received August 20, 2008; revised May 21, 2009; approved by around a steady state with a target fairness (or service differ- IEEE/ACM TRANSACTIONS ON NETWORKING Editor R. Mazumdar. entiation) and high efficiency. To this end, we define a gen- The authors are with the Engineering and Applied Science Division, Cali- eral game-theoretic model, called random access game, to cap- fornia Institute of Technology, Pasadena, CA 91125 USA (e-mail: chen@cds. ture the interaction among wireless nodes in wireless networks caltech.edu; slow@caltech.edu; doyle@cds.caltech.edu). Color versions of one or more of the figures in this paper are available online with contention-based medium access. The basic idea is to re- at http://ieeexplore.ieee.org. gard the process of contention control as carrying out a dis- Digital Object Identifier 10.1109/TNET.2010.2041066 tributed strategy update algorithm to achieve the equilibrium of 1063-6692/$26.00 © 2010 IEEE
  • 2. 2 IEEE/ACM TRANSACTIONS ON NETWORKING random access game. Hence the equilibrium (or steady state) tributed strategy update algorithms derived from the random ac- and dynamic properties of a MAC can be understood or de- cess game as prescriptive control approach for channel access.1 signed through the specification of the underlying random ac- cess game. II. RELATED WORK Specifically, in random access games, a wireless node’s Game-theoretic approach has been applied extensively to strategy is its channel access probability, and its payoff func- study random access (see, e.g., [4]–[10]). The work closest to tion includes both utility gain from channel access and cost ours is Jin et al. [4], which studies noncooperative equilibrium from packet collision. Through the specification of per-node of Aloha networks and their local convergence, and Altman utility function, we can model a large class of system-wide et al. [6] and Borkar et al. [7], which study distributed scheme quality-of-service (QoS) models, similar to that in utility for adapting random access. Our motivation, model, and results framework for network flows [3]. We characterize the Nash are different from those works. Our random access game model equilibrium of random access games, study their dynamics, and is intended as an analytical framework to reverse-engineer con- propose algorithms (strategy evolutions) to achieve the Nash tention control as well as to guide the design of new medium equilibrium. We show that system-wide fairness or service access methods to achieve system-wide performance objec- rs f differentiation can be achieved in a distributed manner as long tives. The specific structure of random access game is derived as each node executes a contention resolution algorithm that from a control-theoretic viewpoint of contention control, and Ve o is designed to achieve the Nash equilibrium. We thus propose the utility functions are derived from the steady operating a novel medium access method derived from carrier sense ion points of existing protocols or the desired operating points we multiple access/collision avoidance (CSMA/CA) in which want medium access control to achieve. Inaltekin et al. [10] each node estimates its conditional collision probability and analyze Nash equilibria in multiple access with selfish nodes. ro adjusts its channel access probability accordingly. Our method ˇ Cagalj et al. [8] study selfish behavior in CSMA/CA networks adapts to continuous feedback signal (conditional collision and propose a distributed protocol to guide multiple selfish probability) rather than binary contention signal, and each node nodes to a Pareto-optimal Nash equilibrium. We do not con- tries to keep a fixed channel access probability (persistence sider such selfish behaviors of wireless users that tamper with probability or equivalently contention window size) specified wireless interfaces to increase their share of channel access by the Nash equilibrium of random access game. In addition as in [8]. In contrast, we use game-theoretic model to capture int P to controllable performance objectives via the specification of the information and implementation constraints encountered in per-node utility functions, as wireless nodes can estimate con- real networks and design games to guide distributed users to ditional collision probabilities by observing consecutive idle achieve system-wide performance objectives. slots between transmissions, our access method can decouple There are lots of works on various enhancements and im- contention control from handling failed transmissions. As a provements to 802.11 DCF. We will only briefly discuss some case study of medium access control design in the proposed designs that propose better contention resolution algorithms and game-theoretic framework, we present a concrete medium that improve throughput by tuning the contention window ac- EE access method and show that it achieves optimal throughput, cording to the number of contending nodes. Aad et al. [11] in- low collision, and good short-term fairness and can provide troduce a slow decrease method to improve efficiency and fair- flexible service differentiations among wireless nodes. ness. Kwon et al. [12] propose a fast collision resolution al- Game-theoretic approach has recently been applied to net- gorithm for throughput improvement. Our design is different work design and control. There are two complementary per- in terms of both contention measure and contention resolution spectives to the game-theoretic approach: economic perspec- algorithm. Bianchi et al. [13] and Cali et al. [14] propose to tive and engineering perspective. The economic perspective as- choose and compute an optimal contention window to maxi- sumes network components or users are selfish and try to thwart mize the throughput. They need sophisticated methods to es- selfish behaviors or induce cooperation using external mecha- timate the number of contending nodes in the system, while our IE nisms such as pricing. The engineering perspective envisions a method does not require that information. There also exists ex- scenario where network components or users are willing to co- tensive work on 802.11 QoS provisioning (see, e.g., [15]). Our operate but only have limited information about network states access method can provide more flexible service differentiations due to various practical constraints in real networks. In such a through the specification of per-node utility functions, except situation, the best an agent can do is to optimize some local or for manipulating the length of interframe space. private objective and adjust its action based on limited informa- Related work also includes [16] and [17]. Heusse et al. [16] tion about the network state. We use noncooperative game to propose a novel idle sense access method for a single-cell wire- “model” such a situation (especially to capture the information less LAN, which compares the mean number of idle slots be- structure of the system) and let network agents behave “self- Pr tween transmission attempts to the optimal value and adopts an ishly” according to the game that is designed to guide individual additive increase and multiplicative decrease algorithm to dy- agents to seek an equilibrium achieving some system-wide per- namically control the contention window in order to improve formance objective. Thus, from the engineering perspective, the throughput and short-term fairness. In our access method, wire- focus is not on incentive issues, but on the implementation in less nodes estimate conditional collision probabilities by ob- practical networks. There are some controversies around the serving consecutive idle slots between transmissions. Therefore, engineering perspective. Nonetheless, game-theoretic technique like idle sense access method, our access method can decouple and language provide a powerful framework to reason about the engineering systems and guide their design and control. In this 1However, our random access game model can also serve as a descriptive paper, we take the engineering perspective and implement dis- model of existing contention control protocols with selfish wireless nodes. This part is consistent with the economic perspective.
  • 3. CHEN et al.: RANDOM ACCESS GAME AND MEDIUM ACCESS CONTROL DESIGN 3 contention control from handling failed transmissions. Hu et al. Define the utility function of each node as [17] use a similar idea to propose a novel contention control method to maximize the bandwidth utilization and achieve pro- (5) portional bandwidth allocation. This method observes two ref- erence variables, the number of consecutive idle slots and the number of collisions between two consecutive successful trans- Being an integral, and since , is a con- missions, based on which to control the dequeueing rate of wire- tinuous and nondecreasing function. It is reasonable to assume less nodes instead of dynamically tuning their contention win- that is a decreasing function: The larger the contention, the dows. Our access method dynamically adjusts the contention smaller the channel access probability. This implies that window and achieves higher throughput (almost no throughput is strictly concave. degradation compared to the optimal throughput) and provides With the above utility function, we define a random access a better guarantee of service differentiation. game as follows. A comparison with TCP congestion control is also in place. Definition 1: A random access game is defined as Contention control has a striking similarity with congestion con- a triple , where is a set of players (wireless nodes), player strategy rs f trol. They need to handle almost the same issues such as conges- tion/contention measure, load control (e.g., window update) al- with , and payoff function with utility function Ve o gorithm, and decoupling load control from handling failed trans- missions, etc. However, the interaction among wireless nodes is and contention measure . ion different from that among TCP flows, which means a different We constrain the strategy to be strictly less than 1 framework is needed to study contention control. Actually, one in order to prevent a node from exclusively occupying wire- less channel. Note that the throughput of node is proportional ro of the motivations of this work is to try to develop a parallel story for contention control to what has been done for TCP con- to if there is no collision, and can be seen as contention gestion control in the utility maximization framework (see, e.g., price for node . Thus, the payoff function has a nice eco- [3]). nomic interpretation: the net gain of utility from channel access, discounted by contention cost. The key to understanding con- tention control is to regard (1)–(2) as the strategy update algo- III. RANDOM ACCESS GAME rithm to achieve the equilibrium of the random access game. int P Consider a set of wireless nodes in a wireless LAN with Hence, the equilibrium (or steady-state) properties of a con- contention-based medium access. Associated with each wire- tention control protocol can be understood and designed through less node is its channel access probability 2 and a the specification of the utility function and the contention certain contention measure it can observe at time . measure (e.g., collision probability). Their specification de- Node can observe its own access probability and con- fines the underlying random access game whose equilibrium tention measure , but not those of other nodes. It adjusts its determines the steady-state properties such as throughput, fair- EE channel access probability based only on and ness, and collision of the contention control protocol. The adap- tation of channel access probability can be specified through (1) and corresponds to different strategies to approach the equilibrium of the game. The contention measure depends on the channel access Random access game is defined in a rather general probabilities chosen by the wireless manner. As we saw above, it can be reverse engineered from nodes given medium access control methods. Regarding the design of medium access control (i.e., forward-engineering), we can (2) choose to implement any contention measures and specify any utility functions we think appropriate. To make our presentation IE Here, models the contention resolution algorithms, and and theoretical development concrete, in this paper we choose models the contention measure update mechanisms. For conve- conditional collision probability as contention measure, i.e., nience, we will also denote by . We assume that (1) and (2) have an equilibrium . The (6) fixed point of (1) defines an implicit relation between the equi- librium channel access probability and contention measure where denotes the set of nodes that interfere with the trans- (3) mission of node . As will become clear later, such a choice of Pr contention measure has two nice properties, among others. First, Assume is continuously differentiable and in conditional collision probability is an accurate measure of con- . Then, by the implicit function theorem [18], there exists tention in the network. Each measurement of conditional colli- a unique continuously differentiable function such that sion probability provides multibit information of the contention. This makes it easier for an equation-based contention resolution (4) algorithm to stabilize the network into a steady state with a target fairness (or service differentiation) and high efficiency. Second, 2In our description of contention control, each wireless node’s control vari- able is its channel access probability. For the same variable, we may have dif- wireless nodes can estimate conditional collision probabilities ferent ways to implement it in practical networks. Window-based control is seen without using explicit feedback, which enables the decoupling as a way of “implementing” channel access probability. of contention control from handling failed transmissions.
  • 4. 4 IEEE/ACM TRANSACTIONS ON NETWORKING In the rest of this section and Section IV, we focus on single- and cell wireless LANs,3 and will consider multicell wireless LANs in Section V. Before proceeding, we summarize the assump- tions that will be used in this paper as follows. Thus, under assumption A1 A0: The utility function is continuously differen- tiable, strictly concave, and with finite curvatures that are bounded away from zero, i.e., there exist some constants and such that . By second-order conditions [20], is a strictly concave A1: Let and denote the smallest function over the strategy space. Therefore, the optimization eigenvalue of over by . Then, problem (8) has a unique optimal, and hence the random access . game has a unique Nash equilibrium. A2: Functions are The equilibrium condition (7) implies that, at the Nash equi- all strictly increasing or all strictly decreasing. librium, either takes value at the boundaries of the strategy Assumption A0 is a standard assumption in economics and can space or satisfies rs f also be seen as derived from those assumptions on function . A1 guarantees the uniqueness of the Nash equilibrium of (10) Ve o the random access game. A2 guarantees the uniqueness of non- We call a Nash equilibrium a nontrivial equilibrium if, for all trivial Nash equilibrium, which we will define later. ion nodes , satisfies (10), and trivial equilibrium otherwise. In A. Nash Equilibrium the remainder of this section, we will mainly focus on nontrivial Nash equilibria. It follows from (10) that at nontrivial Nash equi- ro We now analyze the equilibrium of random ac- librium cess game. The solution concept we use is the Nash equilibrium [19]. Denote the strategy (channel (11) access probability) selection for all nodes but by and write Note that the right-hand side of the above equation is indepen- for the strategy profile .A dent of . Thus, for any . Theorem 4: Suppose assumption A2 holds. If the random int P vector of access probability is a Nash equilibrium if, for all nodes , for all . We access game has a nontrivial Nash equilibrium, it must be see that the Nash equilibrium is a set of strategies for which no unique. player has an incentive to change unilaterally. Proof: Suppose that there are two nontrivial Nash equi- Theorem 2: Under assumption A0, there exists a Nash equi- libria and . From (11), we require that there exist librium for random access game . such that, for all Proof: Since the strategy spaces are compact convex EE sets, and the payoff functions are continuous and concave in , there exists a Nash equilibrium [19]. Since payoff function is concave in , at the Nash equi- Since is one-to-one, . Without loss of generality, librium, satisfies assume that is increasing and . Thus, for all . By (10), , which contra- (7) dicts the fact that is a decreasing function. Thus, random access game has a unique nontrivial Nash equilibrium. Define function . Theorem 4 complements Theorem 3. Although A1 guaran- It is easy to verify that (7) is an optimality condition for the tees the uniqueness of nontrivial Nash equilibrium when there IE following optimization problem [20]: is one, it only gives a sufficient condition for the uniqueness. When there exist nontrivial Nash equilibria while A1 does not (8) hold, A2 guarantees the uniqueness of nontrivial equilibrium. Each node can choose any utility function it thinks ap- i.e., the Nash equilibria of the random access game are optimal propriate. If all nodes have the same utility functions, the system points of the problem (8). is said to have homogeneous users. If the nodes have different Theorem 3: Suppose additionally assumption A1 holds. utility functions, the system is said to have heterogeneous users. Then, random access game has a unique Nash equilibrium. The motivation for studying systems of heterogeneous users is Pr Proof: The Hessian of function is written as to provide differentiated services to different wireless nodes. To this end, we further differentiate among symmetric and asym- (9) metric equilibria as follows. Definition 5: A Nash equilibrium is said to be a symmetric Note that equilibrium if for all , and an asymmetric equilibrium otherwise. For a system of homogeneous users, both symmetric and asymmetric Nash equilibria are possible. For example, take 3Single-cell means that every wireless node can hear every other node in the with and . In network—for example, a single-cell wireless access network. addition to nontrivial symmetric Nash equilibrium that
  • 5. CHEN et al.: RANDOM ACCESS GAME AND MEDIUM ACCESS CONTROL DESIGN 5 satisfies , , when the number of Clearly, if the above dynamics reach a steady state, then this wireless nodes there exists a family of nontrivial asym- state is a Nash equilibrium. Nonetheless, there are no conver- metric Nash equilibria that satisfy , gence results for general games using these dynamics. We thus consider an alternative strategy update mechanism for all and . called gradient play [22]. Compared to “best response” strategy, Since by symmetry there must be multiple asymmetric Nash gradient play can be viewed as a “better response.” In gradient equilibria if there exists any, the following result follows directly play, every player adjusts a current channel access probability from Theorems 3 and 4. gradually in a gradient direction suggested by observations of Corollary 6: For a system of homogeneous users, suppose other player actions. Mathematically, each node updates assumption A1 or A2 holds. If the random access game has a its strategy according to nontrivial Nash equilibrium, it must be unique and symmetric. More generally, for a system with several classes of homoge- (13) neous users, under the same assumption, if the random access game has a nontrivial Nash equilibrium, it must be unique and where the stepsize can be a function of the strategy symmetric.4 5 of player and “ ” denotes the projection onto the player rs f Corollary 6 guarantees the uniqueness of nontrivial Nash strategy space. The gradient play admits a nice economic inter- equilibrium, and moreover, it guarantees fair sharing of wireless pretation if we interpret the conditional collision probability Ve o channel among the same class of wireless nodes and provides as contention price for node : If the marginal utility is ion service differentiation among different classes of wireless greater than contention price, we increase the access probability, nodes. This will facilitate the analysis of dynamic property of and if the marginal utility is less than contention price, we de- random access games and the design of medium access control. ro crease the access probability. The following result is immediate. Remark: Since some player takes a strategy at the boundary Lemma 7: By the equilibrium condition (7), the Nash equi- of the strategy space at trivial Nash equilibrium, a trivial Nash libria of random access game are fixed points of the gradient equilibrium usually has great unfairness or low payoff. There- play (13), and vice versa. fore, nontrivial Nash equilibrium is desired. If there does not Theorem 8: Suppose assumptions A0 and A1 hold. The gra- exist any nontrivial Nash equilibrium for a random access game, dient play (13) converges to the unique Nash equilibrium of we may need to look for an alternative solution other than the random access game if, for any , the stepsize int P Nash equilibrium. For example, we may use Nash bargaining . framework in cooperative game theory to derive a desired equi- Proof: Consider function librium solution. . Define a matrix . We have B. Dynamics of Random Access Game EE Since is symmetric, , and hence The dynamics of game studies how interacting players could converge to a Nash equilibrium. It is a difficult problem in gen- eral, as pointed out in [19] that “game theory lacks a general and convincing argument that a Nash outcome will occur.” In the setting of random access, players (wireless nodes) can ob- serve the outcome (e.g., packet collision or successful transmis- sion) of the actions of others, but do not have direct knowledge of other player actions and payoffs. We consider repeated play of random access game and look for an update mechanism in IE which players repeatedly adjust strategies in response to obser- By Taylor expansion, we have vations of other player actions so as to achieve the Nash equi- librium. The simplest strategy update mechanism is of best response sort: At each stage, every node chooses the best response to the actions of all the other nodes in the previous round. Mathemat- ically, at stage , node chooses a channel access probability Pr (12) where . Now, by the projection theorem [20] 4For a system with several classes of users, a Nash equilibrium is symmetric if the users of the same class choose the same strategy at equilibrium. 5There are many ways to give conditions on the existence of nontrivial Nash equilibrium ( see [21]) for one example condition. Here, we do not do that since from which we obtain those conditions intended for the game with arbitrary utility functions are re- strictive. It is more sensible to give the existence condition for a concrete game with concrete utility functions; see Section IV-A for an example.
  • 6. 6 IEEE/ACM TRANSACTIONS ON NETWORKING Thus, we have If is nonsingular, function satisfies the conditions of the implicit function theorem [18]. The result fol- lows directly from the implicit function theorem. Theorem 10: Assume the same conditions as in Theorem 8. For any , there exists a such that if , then the gradient play (15) with estimation errors converges into an Thus, if , . We -neighborhood of nontrivial Nash equilibrium. see that will keep increasing until the system reaches a Proof: By Theorem 9, for any , there exists a such fixed point of (13). By Theorem 3 and Lemma 7, under assump- that implies that . tions A0 and A1, (13) has a unique fixed point in strategy space. Now, under the gradient play (15), we have Hence, the gradient play (13) converges to the unique Nash equi- librium of random access game . rs f Theorem 8 guarantees the convergence of distributed gradient play to the Nash equilibrium. If a backoff mechanism is imple- Ve o mented, each node updates its contention window as ion follows: ro (14) Equation (14) follows from the relation that re- lates channel access probability to a constant contention window . This relation can be derived under the decou- pling approximation for a set of wireless nodes with constant int P contention windows (see, e.g., [23] and [24]). The decoupling If , approximation is an extremely accurate approximation, as will keep increasing. Therefore, the gradient play (15) will validated by extensive simulations reported in, e.g., [23] and enter at least once into a region defined by [24] . Take any In practice, there will be estimation error in the contention , and we have , i.e., there exits a such measure. Then, the gradient play can be written as that and . It follows EE that the gradient play (15) will enter at least once into a neigh- borhood of defined by . Once it enters into this neighborhood, the gradient play will (15) stay inside a larger neighborhood defined by . Theorem 10 is the robust verification of the gradient play to where denotes the estimation error in node ’s conditional the estimation error. It holds even under weaker condition that collision probability.6 We assume that the estimation errors are only requires the estimation errors to eventually remain within a bounded, i.e., there exists a constant such that -neighborhood of the origin. Being a dynamic feedback-based . We will show that under certain conditions, the gradient play protocol, contention control usually operates at a region where (15) with estimation error converges to a neighborhood of non- IE the equilibrium condition is approximately satisfied due to var- trivial Nash equilibrium, where the size of the neighborhood de- ious practical factors or constraints. Theorem 10 guarantees that pends on the accuracy of the contention measure estimation. this region is in a small neighborhood of the equilibrium, and Theorem 9: Let be a nontrivial Nash equilibrium. Suppose thus the performance of the medium access method derived that is nonsingular. Then, there exist a and a unique from the random access game is determined by the equilibrium. continuously differentiable function defined on a -neighborhood of the origin such that C. Medium Access Control Design (16) Our ultimate purpose for studying random access games is to Pr design medium access method with better performance. Corol- Proof: At nontrivial Nash equilibrium , we have lary 6 and Theorem 8 suggest that random access games provide a general analytical framework to model a large class of system- wide QoS models (mainly in terms of throughput) via the spec- (17) ification of per-node utility functions, and system-wide fairness 6For the simplicity of presentation, we choose the same stepsize for all wire- or service differentiation can be achieved in a distributed manner less nodes and neglect the projection operation. However, similar results as those as long as each node executes a contention resolution algorithm of Theorems 9 and 10 can be established for the general case with heteroge- neous stepsizes and the projection operation. The key is to note that function that is designed to achieve the nontrivial Nash equilibrium. [p + f 1 (rV (p) 0 e)] 0 p is one-to-one for fixed e, and then Based on this understanding of the equilibrium and dy- apply the implicit function theorem for nonsmooth functions. namics of random access games, we propose a novel medium
  • 7. CHEN et al.: RANDOM ACCESS GAME AND MEDIUM ACCESS CONTROL DESIGN 7 TABLE I With this estimation of the contention measure, our access MEDIUM ACCESS METHOD VIA GRADIENT PLAY method can decouple contention control from handling packet losses and is immune to those problems incurred in methods that infer channel contention from packet collisions. In Section IV, we will study a concrete random access game and the corresponding medium access control design as a case study for the proposed design methodology in game-theotic framework. We will discuss more on the design of our medium access method. IV. A CASE STUDY We have mostly ignored the design of utility functions in the previous section. In the following, we will derive utility functions and random access games from the desired operating rs f points and design medium access methods accordingly. Consider a single-cell wireless network with classes of Ve o users. Each class is associated with a weight , and without loss of generality, we assume . We ion want to achieve maximal throughput under the weighted fair- ness constraint ro access method derived from CSMA/CA7: instead of executing exponential backoff upon collisions, each node estimates its conditional collision probability and adjusts its channel access probability and contention window according to the gradient where is the throughput of a class- node. Let . play (13)–(14); see Table I for a formal description (see Under the assumption of the Poisson arrival process [25], the Section IV-C for more explanation on notations). Our method channel idle probability is approximately int P adapts to continuous feedback signal (conditional collision probability) rather than binary feedback (packet collision) (19) and stabilizes the network around a steady state specified by the Nash equilibrium of random access game. Therefore, it can achieve controllable performance objectives by choosing and the aggregate successful packet transmission probability is appropriate per-node utility functions. Our access method . Hence, the aggregate throughout can be written as is an equation-based control, and its performance (such as EE throughput, collision and fairness) is determined by the Nash equilibrium. Note that specifies how far the current state is from the equilibrium. The contention which achieves maximum at that solves window adjustment is small when the current state is close to the equilibrium, and large otherwise, independent of where (20) the equilibrium is. This is in sharp contrast to the approach taken by 802.11 DCF, where window adjustment depends on where is the packet payload, is the duration of an idle slot, just the current window size and is independent of where the and and are the durations the channel is sensed busy be- current state is with respect to the target equilibrium. Thus, our cause of a successful transmission and during a collision, re- IE access method can achieve better contention control and better spectively. short-term fairness. Now, consider how to achieve the weighted fairness. When Now, consider how wireless nodes can estimate conditional there is a large number of nodes accessing the channel, each user collision probabilities. Let denote the number of consecu- should sense approximately the same environment on average, tive idle slots between two transmissions. Here, “a transmis- and we can assume that each user has the same conditional col- sion” corresponds to a busy period in the channel when only a lision probability.8 Thus, with equal packet payload sizes the node transmits (i.e., a successful transmission) or multiple nodes throughput ratio between users of different classes is approxi- transmit simultaneously (i.e., a collision). Since has the geo- mately the ratio between their channel access probabilities, i.e., metric distribution with parameter , its mean is given by we require Pr . Thus, each node can estimate its condi- tional collision probability by observing the average number of (21) consecutive idle slots according to The users of each class are associated with the same utility (18) and thus the same function . Note that at the nontrivial Nash equilibrium, 7We can also design medium access methods based on the ordinary Aloha- for all . Thus, condition (21) can be achieved at a type access method [25]. One of the main differences would be that, without CSMA, wireless nodes have to estimate the contention measure based on ex- 8This assumption is similar to the decoupling approximation made in [23] plicit feedback (i.e., failed transmissions). and other works in performance analysis of 802.11 DCF.
  • 8. 8 IEEE/ACM TRANSACTIONS ON NETWORKING nontrivial Nash equilibrium if we design utility functions such Theorem 12: Suppose additionally . Then, that random access game has a unique and nontrivial Nash equi- librium. (22) Proof: We have By (19)–(20), a nontrivial equilibrium that achieves maximal throughput should satisfy (26) and (23) (27) Equations (22)–(23) plus assumption A2 are the constraints on the desired utility functions that achieve the maximal throughput where . Note that under the weighted fairness constraint at the nontrivial equilib- . is a positive semidefinite matrix of rank 1, and rs f rium. Note that when the number of wireless nodes is large, should be very small. A convenient choice that approximately thus . Therefore, if , . It follows from Theorem 3 that random access Ve o satisfies the above constraints is game has a unique Nash equilibrium. Since, by Lemma 11, ion has a nontrivial Nash equilibrium, the unique equilibrium (24) must be nontrivial. ro from which we derive a utility function B. Dynamics Assume that each node adjusts its strategy according (25) to gradient play With this utility function, we define a random access game int P where all players have the same strategy space with and different user classes are identified with different choices of (28) the utility functions (25) with different weights. (29) Remark: The way we derive utility function (25) follows the same idea as in reverse-engineering (4)-(5): to derive the utility function and game from the equilibrium. To see this, where the stepsize . Note that the Nash equilibria of note that the equilibrium condition (4) or equivalently (10) is random access game are the fixed points of (28), and vice EE encapsulated in (11). In the above utility function design, we versa. The following result is immediate. derive function from the desired equilibrium [see (24)] Theorem 13: Suppose . from which we derive the marginal utility or equivalently Then, the system described by (28) converges to the unique and . We then obtain the utility function by integration. nontrivial Nash equilibrium of random access game if, for any , the stepsize . A. Nash Equilibrium Proof: The result follows directly from Theorem 8. Now let us study the equilibrium of random access game . The condition is a mild as- Lemma 11: Suppose . Then, random ac- sumption and admits a very large region in the parameter space. cess game has a unique nontrivial Nash equilibrium. The Nash equilibrium can be easily calculated by numerically IE Proof: Suppose there are class- nodes, each with a solving fixed point (10). If we consider a system of homogenous users, we choose the same utility function, i.e., the same weight channel access probability . Let . Note for all users. If we want to provide differentiated services, we that when , , and when can choose larger value of for the users of a higher priority , class. For example, in wireless access network, we can assign a large value to the access point, because usually downlink if . Since and are traffic is greater than the traffic of mobile nodes. continuous functions of , there exists a solution to the fol- Pr lowing equilibrium equation [i.e., (11)]: C. Medium Access Control Design We design a medium access method according to channel access probability and contention window update mechanism (28)–(29) by modifying a CSMA/CA access method such as 802.11 DCF [2]. The basic access mechanism in DCF works if . Thus, there exists a nontrivial Nash equi- as follows. A node wishing to transmit senses the channel for a librium for random access game . Fur- period of time equal to the distributed interframe space (DIFS) thermore, defined by (24) satisfies assumption A2. It fol- to check if it is idle. If the channel is determined to be idle, the lows from Theorem 4 that the nontrivial Nash equilibrium must node starts to transmit a DATA frame. If the channel is consid- be unique. ered to be busy, the node waits for a random backoff time ,
  • 9. CHEN et al.: RANDOM ACCESS GAME AND MEDIUM ACCESS CONTROL DESIGN 9 an integer uniformly distributed in the window be- TABLE II fore attempting to transmit. Upon successful reception of the PARAMETERS USED TO OBTAIN NUMERICAL RESULTS DATA frame, the receiving node waits for a short interframe space (SIFS) interval and then sends an ACK frame. When the node detects a failed transmission, it doubles the contention window (exponential backoff). In order to avoid channel capture, a node must wait for a random backoff time, in the same way as if the channel is sensed busy, between two con- secutive new packet transmissions. Note that DCF employs a discrete-time backoff scale. The time immediately following an idle DIFS is slotted with slot-time size . The backoff time counter is decremented as long as the channel is sensed idle during a time slot, and frozen when the channel is sensed busy, and reactivated when the channel is sensed idle again for a DIFS. where . If is small, we weight history less, and if rs f The node transmits when the backoff time counter reaches zero. is large, we weight history more. The running average works as See [2] for more details. a low-pass filter, and by choosing appropriate value, it gives Ve o As described in Section III-C, our medium access method better estimate than the “naive” estimator . makes two key modifications to 802.11 DCF. Instead of ad- ion By our access method, the system is designed to reach and op- justing contention window to a binary feedback signal erate around the Nash equilibrium of random access game . (packet loss or successful transmission) and using exponential Thus, its performance is determined by the Nash equilibrium of ro backoff algorithm, each node estimates its conditional colli- . Consider a system of greedy nodes that always have packets sion probability , which is a continuous feedback, and adjusts to transmit. Denote the equilibrium channel access probability according to algorithm (28)–(29). of node by ; we can calculate its throughput and condi- There are several parameters in our medium access method. tional collision probability as follows: The parameter depends only on the protocol parameters such as the duration of an idle slot that are specified by the system int P designer. The maximal channel access probability affects the number of the equilibria of random access game but not the location of the equilibrium (i.e., the equilibrium proper- (31) ties of our design) once it satisfies the condition specified in Section IV-A. The weight determines the fairness or ser- (32) vice differentiation. The system designer will specify a set of weights, according to the levels of quality of services he wants EE to provide, and each wireless node will choose a weight that where idle probability and corresponds to the specific level of service it desires. The pa- rameters and determine the dynamic properties such as stability and responsiveness. The stepsize affects the convergence speed. In practice, we will choose a constant step- size for all nodes. The number of transmissions, , for each node before updating its channel access probability and See Table II for other notations. Here, for simplicity, we have contention window, affects the convergence speed and the ac- assumed an equal payload size. The throughput for general pay- curacy of the conditional collision probability estimation. Note load size distribution can calculated in a similar way [23], and IE that in strategy update algorithm (28)–(29), is not real time, the aggregate throughput is the summation of over all nodes . but represents the stages at which the random access game is played. In our design, each node repeatedly plays game every D. Performance transmissions, and the channel access probability We have conducted numerical experiments to evaluate the and contention window are fixed between consecutive plays. If performance of our medium access method. We develop a is too large, it will take a longer time to reach the packet-level simulator that implements our method and the Nash equilibrium, but if is too small, it will result standard 802.11 DCF basic access method (i.e., no RTS/CTS). in large estimation error in the average number of consecutive The values for the parameters used to obtain numerical results idle slots between transmissions and thus conditional collision Pr are summarized in Table II. The system values are those speci- probability. In order to achieve a good tradeoff between conver- fied in the 802.11b standard with DSSS PHY layer [2]. Under gence speed and estimation accuracy, we will choose a relatively these specifications, and . small value for and estimate average number of con- We set , which corresponds to a contention window secutive idle slots between transmissions using an exponential size of 16. In all simulations, we set the following values of weighted running average the control parameters: , , and . The simulation results reported aim to zoom in on (30) specific properties of our access method.
  • 10. 10 IEEE/ACM TRANSACTIONS ON NETWORKING rs f Fig. 2. Conditional collision probability comparison. Fig. 1. Throughput comparison. Ve o ion 1) Throughput and Collision Overhead: We consider a net- work of homogeneous users with perfect channel (i.e., there is no corrupted frame that is due to channel error) and compare the ro throughput achieved by our access method and 802.11b DCF as well as the maximal achievable throughput as our method is in- tended to achieve optimal throughput. In our design, each node has the same weight of 1, i.e., , since we consider ho- mogenous users. In our numerical experiments with DCF, after a packet’s th failed transmission, the contention window int P resets to the base contention window, where denotes the max- imum backoff stage and is set to 5 for 802.11b. This is also equivalent to the packet being discarded after failed retrans- missions. Fig. 1 shows the aggregate throughput achieved by our design and DCF and the maximal achievable throughput. Our design al- Fig. 3. Fairness comparison for a network of 40 competing nodes. EE most achieves the optimal throughout, with only a slightly lower than optimal throughput for a network of a very small number of competing nodes. This also confirms that the approximations aspects such as lower energy usage and less interference to the (19) and (24) made in deriving the utility functions (25) are very wireless nodes of neighboring cells. accurate approximations, and the contention measure estimator 2) Fairness: It is well known that 802.11 DCF has a (18) and (30) give fairly accurate estimation of the conditional short-term fairness problem due to binary exponential backoff collision probability. process. In our access method for a system of homogeneous Compared to DCF, for a network of a very small number users, wireless nodes have the same contention window size, of wireless nodes, DCF provides a slightly higher throughput specified by the symmetric Nash equilibrium of random ac- than our design. However, as the number of nodes increases, cess game . Thus, it is expected to have a better short-term IE our access method achieves much higher throughput. With DCF, fairness. Fig. 3 compares short-term fairness of our access each new transmission starts with the base contention window method and DCF using Jain fairness index for the window and executes binary exponential backoff upon collisions, while, sizes that are multiples of the number of wireless nodes with our access method, nodes choose a constant contention [26]. Denote by the number of transmissions of node window determined by the Nash equilibrium, which is “op- over a certain time window. Jain fairness index is defined timal” for the current contention level in the network. Thus, by , and higher index means better for a system of many competing nodes where the contention in fairness. We can see that our method provides much better the network is heavy, DCF will incur much more packet col- short-term fairness than 802.11 DCF. Pr lisions than our access method, which results in much lower 3) Dynamic Scenario: To evaluate the responsiveness and throughput, as shown in Fig. 1. This is further confirmed by the convergence of our access method, we consider a dynamic the comparison of collision overhead between DCF and our ac- scenario as follows. In the beginning, there are five greedy cess method, as shown in Fig. 2. Our access method achieves a nodes in the network, which has converged to the equilibrium very small, almost constant collision probability, better tradeoff or the steady operating point. After 1004 more transmissions, between channel access and collision avoidance, and hence a five more nodes join in to compete for channel access. The five higher throughput that is sustainable over a large range of num- nodes then leave after 3000 transmissions. Fig. 4 shows the bers of competing nodes. Practically, this means that our access evolution of channel access probability and the corresponding method can achieve higher throughput, but with fewer transmis- contention window size of a long-stay node and a short-stay sions than DCF, which will benefit the whole system in many node, respectively. We see that our access method reaches the
  • 11. CHEN et al.: RANDOM ACCESS GAME AND MEDIUM ACCESS CONTROL DESIGN 11 rs f Ve o ion ro int P Fig. 4. Evolution of channel access probability and the corresponding con- Fig. 5. Throughput comparison of our design and 802.11b DCF with different tention window. frame error rates. EE new equilibrium or steady operating point after only about throughput achieved by our design and 802.11b DCF, with zero, totally 65 transmissions. That is about seven transmissions for 10%, 20%, and 40% frame error rates, respectively. We see that, each wireless node. The evolutions of different wireless nodes as expected, our contention resolution algorithm is insensitive follows roughly the same trend. This is because they observe to channel error, and the throughput degradation comes solely the same signal to estimate their contention measure, though the from the corrupted frames due to channel error. times when the long-stay node and the short-stay node update The impact of channel error on DCF is a bit “twisted.” When their channel access probabilities and contention windows are the number of nodes is very small, the channel error leads four transmissions away. to larger degradation in throughput. This is because packet IE In order to respond quickly to the channel contention, in collision is rare in this situation, and channel error results in our access method when a wireless node enters the system more backoff and hence fewer transmissions. However, when or starts to have data to send after being idle, it will monitor the number of nodes is large, the channel error actually helps. the channel for a certain amount of time before transmitting This is because packet collision is frequent in this situation, and to estimate current channel contention . Based on this es- additional backoff due to the corrupted frames greatly reduces timation, it will set its starting channel access probability the chance of packet collision. Note that we have assumed an to . In our simulation for the i.i.d channel error model. If the channel error comes in bursts, dynamic scenario, after entering the network, the short-stay it will lead to greater throughput degradation of DCF and make Pr nodes monitor the channel for the duration of three consecutive DCF interact adversely with higher layer protocols such as TCP transmissions to estimate current channel contention to set congestion control. Our access method has no such problems the starting channel access probability and the corresponding since its contention resolution algorithm is insensitive to the contention window. This reflects in Fig. 4 where the starting channel error. channel access probability and contention window of the 5) Service Differentiation: As discussed before, we can short-stay node are close to the channel access probability and provide service differentiation by choosing different utility contention window of the long-stay node. functions for different classes of users. Regarding the concrete 4) Throughput Under Unreliable Channel: We now con- medium access method we consider, each node will receive sider the network with unreliable channel, i.e., there exist cor- different services by choosing different weights . For the rupted frames due to channel error. Fig. 5 shows the aggregate simplicity of presentation, we consider two classes of users.
  • 12. 12 IEEE/ACM TRANSACTIONS ON NETWORKING matrix. Nonetheless, we can still establish the uniqueness and convergence of the Nash equilibrium by generalizing the aforementioned conditions. It is straightforward to verify that Theorem 2 holds for the multicell network, i.e., under assumption A0 there exists at least one Nash equilibrium for random access games . Let be the Jacobi matrix of the contention measure vector , i.e., , and denote by the smallest eigenvalue of matrix over . Theorem 14: Suppose assumption A0 holds and . Then, random access game has a unique Nash equilibrium. Proof: Since A0 holds, there exists at least one Nash equilibrium, denoted by . Define function rs f . Note that . Let Fig. 6. The throughput ratio of a class–1 node with weight = 1 to a class–2 . By Taylor expansion, we node with weight = 05: . Ve o have ion ro int P where . If , for any . Now suppose there exists another Nash equilibrium, denoted by , then . However, by equilibrium condition (7) EE Fig. 7. The aggregate throughput versus the number of nodes. Assume that class 1 has users with weight , corre- sponding to a higher priority of service, and class 2 has users with weight , corresponding to a lower priority of service. Fig. 6 shows the throughput ratio of a class-1 node to a class-2 node versus the total number of nodes for two different which contradicts the relation . Thus, random access scenarios: two classes have equal number of users, and class 1 game has a unique Nash equilibrium. IE has fixed number of users. We see that the throughput ratio is The condition is a generalization of assump- almost exactly . This also confirms that the condition tion A1. As far as we know, our proof method for the unique- (21) is a very accurate approximate requirement for achieving ness of the equilibrium is new. It only uses basic property of the desired service differentiation. convexity, and is expected to provide a general technique to es- Fig. 7 shows the corresponding aggregate throughput by our tablish the uniqueness result. access method. We see that our design almost achieves the max- Theorem 15: Assume the same conditions as in Theorem 14 imal achievable throughput. This again confirms that the ap- and the same constant stepsize for the gradient play for all proximations (19) and (24) made in deriving the utility func- wireless nodes. The gradient play (13) converges to the unique tions (25) are very accurate approximations. Nash equilibrium of random access game if the stepsize Pr . V. EXTENSION TO MULTICELL NETWORKS Proof: First note that the Nash equilibrium of random ac- cess game is the fixed point of the gradient play (13), and vice We now extend the above development to multicell wire- versa. Define a mapping less LANs in which some wireless nodes (e.g., those at the boundaries of the cells) may belong to more than one cell. Multicell networks are difficult to analyze because of the asymmetric information involved at different nodes. Mathemat- The gradient play can be written as . ically, this reflects in the fact that the Jacobi matrix of the user Note that . payoff functions is not a symmetric If we can show that is a contraction mapping, then the
  • 13. CHEN et al.: RANDOM ACCESS GAME AND MEDIUM ACCESS CONTROL DESIGN 13 gradient play is a contraction mapping, too. Now, assume a con- the channel access probability vector can be modeled by stant stepsize for all nodes. For any two points and in the stochastic function strategy space, we have with probability with probability It is well established that for a single-cell wireless LAN at steady state, the average channel access probability relates to the con- ditional collision probability as follows [23]: where is the base contention window and is the maximum backoff stage. Following the procedure (4)–(5) to de- rive a utility function , we can define a random access game rs f with payoff and interpret DCF as a distributed strategy update algorithm to achieve the corresponding Nash Ve o equilibrium. Note that the dynamics of DCF cannot be described ion by (stochastic) gradient play. The convergence of DCF to the Nash equilibrium is guaranteed by the stability of the underlying Markov chain that describes the dynamics of DCF. However, we ro where , and in can design a medium access method according to the gradient the third inequality we apply the same trick as in the proof of play of this game, which will achieve the same throughput as Theorem 14. We see that if the stepsize , that by DCF, but with better short-term fairness. One of the key aspects of our reverse-engineering is to re- . Thus is a contraction verse-engineer the equilibrium (or steady operating point) of mapping, and so is the gradient play. By the contraction map- the contention control protocol, i.e., the equilibrium of the re- ping theorem [18], the gradient play will converge to the unique int P sulting game should recover the equilibrium (or steady oper- fixed point of (13), i.e., the unique Nash equilibrium of random ating point) of the contention control protocol; see (3) and (33). access game . Our random access game framework applies to any dynamic The above condition on the stepsize looks strong, but seen contention-based medium access methods that have a steady op- from the proof, it is overly conservative. We are seeking other erating point, such as the stabilized Aloha [25]. techniques to better bound the matrix norm to weaken this con- dition. If the medium access method designed according to the VII. CONCLUSION EE gradient play (13) of random access game is used in multi- cell wireless LANs, Theorems 15 guarantees that it will operate We have developed a game-theoretic model for contention around the equilibrium of random access game and achieve control and propose a novel medium access method derived the performance specified by the equilibrium. from CSMA/CA, in which each node estimates its conditional collision probability and adjusts its persistence probability or VI. SOME COMMENTS ON REVERSE- ENGINEERING contention window according to a distributed strategy update mechanism achieving the Nash equilibrium. This results in The dynamic model (1)–(2) of contention control is very gen- controllable performance objectives through the specification eral. The function is usually a deterministic function, while of per-node utility functions. As wireless nodes can estimate the function may be deterministic if the contention measure conditional collision probabilities by observing consecutive IE is a continuous variable or stochastic if the contention measure idle slots between transmissions, we can decouple contention is a discrete variable. In the derivation of the utility function control from handling failed transmissions. As a case study of in Section III, we have implicitly assumed a continuous con- medium access control design in the proposed game-theoretic tention measure. If a discrete contention measure is used, the framework, we present a concrete medium access method that fixed point equation is usually not dictated directly by function achieves optimal throughput, low collision, and good short-term as in (3). Instead, the fixed point may be dictated by a certain fairness and can provide flexible service differentiations among function that relates the channel access probability and the wireless nodes. In addition to guiding medium access control average contention measure design, the random access game model also provides an an- Pr (33) alytical framework to understand equilibrium properties and dynamic properties of different medium access protocols. Starting with this fixed point equation and following the same This paper has been focusing on laying out a theoretical procedure as in (4)–(5), we can define the utility function and framework. Much work remains to take it from a promising a random access game and interpret the contention control pro- design framework to a full-fledged medium access control tocol as a distributed strategy update algorithm to achieve the protocol. We are extending the framework to the network equilibrium of this game. where wireless nodes have arbitrary traffic patterns (e.g., bursty Take the IEEE 802.11 DCF as an example. DCF responds to packet arrival), which will be reported elsewhere. We will whether there is a collision, and hence the measure of contention search for other contention resolution algorithms that could in DCF is a binary feedback signal whose dependence on achieve Nash equilibria of random access games and other