This document provides instructions for homework assignments in an operations research course. It includes 6 problems covering queueing models like M/M/1, M/M/c, M/M/∞ queues. For each problem, students are asked to define the queueing system, derive its stationary distribution, and compute related metrics like average queue length and waiting time.
IE 425 Homework 10Submit on Tuesday, 12101.(20 pts) C.docx
1. IE 425 Homework 10
Submit on Tuesday, 12/10
1.(20 pts) Consider the M/M/1/∞ queuing system descried in
Problem 5 in Homework 9. Show
that:
(a) (11 pts) The average number of customers in the system is:
L =
λ
µ−λ
Hint:
L =
∞∑
n=0
nπn,
∞∑
n=0
nρn−1 =
d
dρ
∞∑
2. n=0
ρn =
d
dρ
(
1
1 −ρ
)
=
1
(1 −ρ)2
(b) (3 pts) The average waiting time in the system (from
entrance to exist) is:
W =
1
µ−λ
(c) (3 pts) The average waiting time in the queue, not including
service, is:
W0 =
λ
µ(µ−λ)
(d) (3 pts) The average number of customers in the queue, not
3. including service, is:
L0 =
λ2
µ(µ−λ)
2.(15 pts) Consider the M/M/c/∞ queuing system descried in
Problem 6 in Homework 9. Show
that:
L0 =
π0
c!
(
λ
µ
)c (
λ
cµ
)(
1 −
λ
cµ
)−2
Then, using Little’s law, we can compute:
W0 =
5. mρm =
ρ
(1 −ρ)2
3.(15 pts) Consider the M/M/∞/∞ queuing system descried in
Problem 7 in Homework 9. Show
that:
L =
λ
µ
, W =
1
µ
, W0 = 0, L0 = 0
4.(15 pts) Consider the M/M/c/c queuing system descried in
Problem 8 in Homework 9.
(a) (3 pts) Explain why L0 = 0 and W0 = 0.
(b) (3 pts) Explain why W = 1
µ
.
1
(c) (5 pts) Explain why the mean arrival rate to the system is
λ(1 −πc)
6. (d) (4 pts) Show that:
L =
λ
µ
(1 −πc)
5. (20 pts) Consider the M/M/c/k queuing system descried in
Problem 9 in Homework 9. Show
that:
(a) (14 pts)
L0 =
π0
c!
(
λ
µ
)c (
λ
cµ
)(
1 −
λ
cµ
8. dρ
M∑
m=0
ρm
(b) (2 pts) Explain why:
L = L0 +
c−1∑
n=0
nπn + c
(
1 −
c−1∑
n=0
πn
)
(c) (2 pts) Explain why the mean arrival rate to the system is
λ(1 −πk).
(d) (2 pts) Show that:
W0 =
L0
λ(1 −πk)
W =
L0
9. λ(1 −πk)
+
1
µ
6. (15 pts) For the M/M/c/∞ queuing system with a finite calling
population N descried in
Problem 10 in Homework 9, it is more convenient to use the
generic formulas to compute the queue
length and the number of customers in the system :
L0 =
N∑
n=c+1
(n− c)πn
L = L0 +
c−1∑
n=0
nπn + c
(
1 −
c−1∑
n=0
πn
10. )
(a) (10 pts) Show that the mean arrival rate to the system is:
N∑
n=0
(N −n)λπn = · · · = λ(N −L)
(b) (5 pts) Show that:
W0 =
L0
λ(N −L)
W =
L0
λ(N −L)
+
1
µ
2
IE 425 Homework 9
Submit on Tuesday, 12/3
1. Report your notebook score for Midterm Exam 2 along with a
picture as the proof.
11. 2. (11 pts) Consider a Discrete State Continuous Time Markov
Chain (DSCTMC) defined on
Ω = {1, 2, 3} with generator matrix G:
G =
3 1 −4
Suppose the DSCTMC is in state 1.
(a) What is the expected time until the DSCTMC leaves state 1?
(b) What is the probability that the DSCTMC will jump to state
2 after it leaves state 1?
In Problem 3 ∼ Problem 10, model the systems as DSCTMCs.
For each DSCTMC:
(a) Define the states of the DSCTMC and write down their
holding time distributions.
(b) Write down the transition probability matrix P of the jump
chain of the DSCTMC.
(c) Write down the generator matrix G of the DSCTMC.
(d) Draw the transition rate diagram of the DSCTMC.
3. (11 pts) A machine, once in production mode, operates
12. continuously until an alarm signal is
generated. The time up to the alarm signal is an exponential
random variable with λ1 = 1. Sub-
sequent to the alarm signal, the machine is tested for an
exponentially distributed amount of time
with λ2 = 5. The test results are positive, with probability 0.5,
in which case the machine returns
to production mode, or negative, with probability 0.5, in which
case the machine is taken for repair.
The duration of the repair is exponentially distributed with λ3 =
3. We assume that the above
mentioned random variables are all independent and also
independent of the test results. Does the
long-run convergence theorem apply to this DSCTMC? Why? If
so, what are the portions of time
that the DSCTMC spends in production mode, test mode, and
repair mode, respectively?
4. (11 pts) Consider two machines that are maintained by a
single repairman. Machine i functions
for an exponential time with rate µi before breaking down, i =
1, 2. The repair times (for either
machine) are exponential with rate λ.
5. (11 pts) M/M/1/∞ Queuing System Consider a food truck that
sells lunch on the outskirts
13. of a college campus. Customers arrive to the food truck
according to a Poisson process with rate
λ (customers arrive one at a time). Customers are served by one
cashier, service times follow an
exponential distribution with mean 1/µ. Assuming ρ = λ
µ
< 1, show that the stationary distribution
π of the M/M/1/∞ queuing system admits the following form:
πn =
(
λ
µ
)n (
1 −
λ
µ
)
for n = 0, 1, 2, · · ·
6. (11 pts) M/M/c/∞ Queuing System A bank has c tellers. When
a customer arrives, he goes to
an empty teller (if there is one) or joins a single queue.
Customers arrive following a Poisson process
with rate λ. Transaction times between a teller and a customer
follow an exponential distribution
14. 1
with mean 1/µ. Assuming ρ = λ
cµ
< 1, show that the stationary distribution π of the M/M/c/∞
queuing system admits the following form:
π0 =
[
1 +
c−1∑
i=1
1
i!
(
λ
µ
)i
+
1
c!
(
16. µ
)c (
λ
cµ
)n−c
π0 for n ≥ c
7. (11 pts) M/M/∞/∞ Queuing System Consider a self-service
system where an unlimited number
of servers are always available. Customers arrive following a
Poisson process with rate λ. All
customers in the system at any instant are simultaneously being
served, with each customer’s service
time following an exponential distribution with mean 1/µ. Show
that the stationary distribution π
of the M/M/∞/∞ queuing system admits the following form:
πn =
1
n!
(
λ
µ
)n
e
17. − λ
µ for n ≥ 0
Hint: Taylor expansion of exponential functions.
8. (11 pts) M/M/c/c Queuing System A telephone company owns
a limited number c of transatlantic
telephone lines. When a customer wants to call overseas, he is
assigned a line immediately provided
the lines are not all busy. If all lines are busy, customer is
denied service and asked to try again later.
Calls arrive according to a Poisson process with rate λ. Each
call has an exponentially distributed
length with mean 1/µ. Show that the stationary distribution π of
the M/M/c/c queuing system
admits the following form:
π0 =
[
1 +
c∑
i=1
1
i!
(
18. λ
µ
)i]−1
πn =
1
n!
(
λ
µ
)n
π0 for 1 ≤ n ≤ c
9. (11 pts) M/M/c/k Queuing System Consider a manufacturing
shop. Parts arrive according to
a Poisson process with rate λ. Shop contains c machines,
allowing up to c parts to be processed
simultaneously. Shop has queue space for up to (k−c) other
parts waiting in line when all machines
are busy. Time required to process a part follows an exponential
distribution with mean 1/µ. Show
that the stationary distribution π of the M/M/c/k queuing system
admits the following form:
π0 =
[
21. 10. (12 pts) M/M/c/∞ Queuing System with a Finite Calling
Population Suppose c maintenance
personnel is responsible for keeping a set of N machines in
operational order. Each maintainer
can repair a machine individually with an exponentially
distributed amount of time with mean
1/µ. For each machine, the elapsed time between when it is
returned to a serviceable condition
and when it next breaks down follows an exponential
distribution with mean 1/λ. (Each machine
is considered a customer in the queueing system when it is
down waiting to be repaired, when a
machine is operational it is outside the queuing system.) Show
that the stationary distribution π
of the M/M/c/∞ queuing system with a finite Calling Population
N(N > c) admits the following
form:
π0 =
[
1 +
c−1∑
i=1
N!
