A Closed-Form Expression for Queuing Delay in Rayleigh Fading Channels Using Stochastic Network Calculus

A Closed-Form Expression for Queuing Delay in
Rayleigh Fading Channels Using Stochastic Network
Calculus

Giacomo Verticale

Politecnico di Milano
Italy

October 2009

Introduction

Stochastic Network Calculus is a modern theory for studying
QoS in wireline
in this paper, we use it to model of the wireless channel subject
to Rayleigh fading
then we obtain an approximate closed-form expression for the
probability tail of the queuing delay
ﬁnally, we compare our results to simulations

G. Verticale (Politecnico di Milano) An Expression for Queuing Delay . . . October 2009 2 / 21

Outline

1 Background
Markovian Model of the Rayleigh Fading Channel
Spectral Gap in Markov Chains
Stochastic Network Calculus

2 Service Envelope of the Fading Channel

3 Comparison with Simulations

4 Conclusion


Assumptions

Assumptions
1 the channel is frequency ﬂat
2 the channel coherence time is longer than the frame duration
3 perfect Channel State Information (CSI) is available
4 pathloss and shadowing do not change over time

Then, the Signal-to-Noise Ratio (SINR) at the receiver is

γ ∼ NegExp γ

where γ is the average SINR and depends only on pathloss and
shadowing


Multipath Fading

SINR evolves over time. Given a SINR threshold γn , the
level-crossing rate is:
3
2
γ γ
−γ
Nn = 2π fd e
γ

where fd is the Doppler spread.
Frame-by-frame the transmitter uses the most eﬃcient available
Modulation and Coding Schemes (MCS) to achieve target BER.
We model the channel as having a number of states
corresponding to the available MCSes plus a bad state.


States and Stationary Probabilities

The SINR range is partitioned using boundary points γn .
The BS uses the n-th MCS if γn ≤ γ < γn+1 .
We model the channel as a Finite States Markov Chain.
The stationary probability of the state n is
pn = Pr{γn ≤ γ < γn+1 }

fγ (γ)

N1 N2 Nn

p0 p1 ... pn
SINR, γ
γ1 γ2 γn


Transition Probabilities

Transitions are allowed only between adjacent states
Q is the matrix of transition rates
Nn the level crossing rate for threshold γn

Nn Nn+1
Qn−1,n = pn−1 Qn,n+1 = pn

n−1 n n+1

Nn Nn+1
Qn,n−1 = pn Qn+1,n = pn+1



Example (Modulation and Coding Schemes in WiMAX)
State, n MCS slot rate, c(γn ) γn @ BER=10−4
(kbit/s) (dB)
0 — 0 —
1 QPSK 1/2 (2×) 4.8 −0.06
2 QPSK 1/2 9.6 3.22
3 QPSK 3/4 14.4 5.64
4 16QAM 1/2 19.2 8.42
5 16QAM 3/4 28.8 11.91
6 64QAM 1/2 28.8 12.37
7 64QAM 2/3 38.4 15.25
8 64QAM 3/4 43.2 17.11

Approximated as
c(γ) = kγ h
with k=9.6 kbit/s and h=0.6


Theorem (Stroock, 2005; Levin et al, 2009)
Consider:
a Markov Chain with transition rates, Q
the steady state probability vector, p
the spectral gap, λ, of the chain
(the diﬀerence between the two largest eigenvalues of Q)

λ = λ1 − λ2 = −λ2 (λ1 = 0)

a function of the chain state, f (X )
Then
Varp eQτ f ≤ e−2λτ Varp f

Informally, the function of random variable f (X ) converges to its
steady state distribution exponentially quickly with rate λ.

Arrival and Service Curves

Traffic / Service

Arrived traffic

Served traffic

Time, t



Traffic / Service

Arrived traffic

Delay, d
Served traffic

Time, t



)
, B (t
pe
velo
en
ffic
Traﬃc / Service

Tra est
ima
ted
Delay, d

)
S(t
e,
l op
ve
en
v ice
r
Se
Time, t


Queuing Delay

Theorem (Knightly, 1998; Shroﬀ, 1998)
Under the assumption that B(t) and S(t) are Gaussian:
  2
1 E B(t) − S(t + d) 
log Pr{D > d} ≤ − min −
2 t≥0 Var B(t) − S(t + d)

Expressions for B(t) are known for some classes of traﬃc.
Expressions for S(t) are known for some wireline schedulers.


Outline

1 Background



4 Conclusion


Service Envelope of the Fading Channel
Mean and Variance of Service Curve in the Markovian Channel

Theorem
If the channel can be modeled as a reversible Markov Chain, then

λt + e−λt − 1
Var S(t) ≤ 2 Var[c]
λ2

We also provide a linear approximation:
t
Var S(t) ≤ 2 Var[c]
λ

The mean is trivial: E S(t) = E[c]t


Mean and Variance of Service Curve in the Markovian Channel

Discussing the Rayleigh channel, we wrote that:

c(γ) kγ h γ ∼ NegExp(γ)

If h = 1/2

πγ
E S(t) = k t
2
(4 − π )γ
Var S(t) ≤ k2 t
2λ

If h > 1/2, formulas slightly more complicated (have Γ -functions)


Queuing Delay with the CBR Source

Consider a Constant Bit Rate (CBR) traﬃc ﬂow with rate r.

Expression for the Queuing Delay
2λr(2r − k π γ)
log Pr{D > d} ≤ d
k 2 (4 − π )γ

The tail of the queuing delay distribution is exponentially
distributed and depends on:
the source rate
the channel mean rate and rate variance
the spectral gap of the channel.


Outline

1 Background



4 Conclusion


Comparison with Simulations
Simulation Scenario

We study the delay performance of a CBR ﬂuid traﬃc source over a
fading channel and compare with simulation results.

Reference scenario
a single source with rate r = 11.55 kbit/s
a single slot in every 2-ms frame
a single WiMAX subchannel using the Band-AMC permutation.


Queuing delay tail

100
Simulations
Analysis
10−1 Parameters
Doppler spread 10 Hz
10−2 (coherence time 10 ms)
Pr D > d

(user speed 20 km/h)
Average SINR γ = 5 dB
10−3
The delay probability
drops exponentially
10−4 quickly. The analytical
model captures well this
10−5 behavior.
0 50 100 150 200
Delay (ms)

Delay decay rate vs Doppler spread

0

Parameters
−10
s−1

Average SINR γ = 5 dB

The decay rate is
(log p0 )/d

−20 captured well for a wide
range of the Doppler
Spread.
−30 fd = 100 Hz corresponds
Simulations to a coherence time of
Analysis about 2 ms frame
−40 duration.
100 101 102
Doppler Spread (Hz)

Delay decay rate vs average SINR

0
Simulations
Analysis
−2
Parameters
s−1

−4 Doppler Spread = 10 Hz
(log p0 )/d

SINR ≤ 2 dB queue
−6 diverges.
SINR ≥ 5.5 dB almost no
delay.
−8
In the internal region the
model behaves well.
−10
2 3 4 5
Average SINR (dB)


Conclusion

Stochastic Network Calculus has been successful in the wireline
we provide a way to use it to study the wireless channel
we were able to obtain an expression for the probability tail of
the queuing delay by using results from the spectral graph
theory
there is a good match between our analytical results and
simulations


A Closed-Form Expression for Queuing Delay in Rayleigh Fading Channels Using Stochastic Network Calculus

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