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558
8
Queueing Theory
Exercises
1.
For the
M/M/
1 queue, compute
(a) the expected number of arrivals during a service period and
(b) the probability that no customers arrive during a service period.
Hint:
“Condition.”
*2.
Machines in a factory break down at an exponential rate of six per hour.
There is a single repairman who ±xes machines at an exponential rate of eight
per hour. The cost incurred in lost production when machines are out of service
is $10 per hour per machine. What is the average cost rate incurred due to failed
machines?
3.
The manager of a market can hire either Mary or Alice. Mary, who gives
service at an exponential rate of 20 customers per hour, can be hired at a rate of
$3 per hour. Alice, who gives service at an exponential rate of 30 customers per
hour, can be hired at a rate of $
C
per hour. The manager estimates that, on the
average, each customer’s time is worth $1 per hour and should be accounted for
in the model. If customers arrive at a Poisson rate of 10 per hour, then
(a) what is the average cost per hour if Mary is hired? if Alice is hired?
(b) ±nd
C
if the average cost per hour is the same for Mary and Alice.
4.
Suppose that a customer of the
1 system spends the amount of time
x>
0 waiting in queue before entering service.
(a) Show that, conditional on the preceding, the number of other customers that
were in the system when the customer arrived is distributed as 1
+
P
, where
P
is a Poisson random variable with mean
λ
.
(b) Let
W
∗
Q
denote the amount of time that an
1 customer spends in
queue. As a byproduct of your analysis in part (a), show that
P
{
W
∗
Q
6
x
}=
(
1
−
λ
μ
if
x
=
0
1
−
λ
μ
+
λ
μ
(
1
−
e
−
(μ
−
λ)x
)
if
0
5.
It follows from Exercise 4 that if, in the
1 model,
W
∗
Q
is the amount
of time that a customer spends waiting in queue, then
W
∗
Q
=
½
0
,
with probability 1
−
λ/μ
Exp
(μ
−
λ),
with probability
λ/μ
where Exp
(μ
−
λ)
is an exponential random variable with rate
μ
−
λ
. Using this,
±nd Var
(W
∗
Q
)
.
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559
6.
Two customers move about among three servers. Upon completion of service
at server
i
, the customer leaves that server and enters service at whichever of the
other two servers is free. (Therefore, there are always two busy servers.) If the
service times at server
i
are exponential with rate
μ
i
,i
=
1
,
2
,
3, what proportion
of time is server
i
idle?
*7.
Show that
W
is smaller in an
M/M/
1 model having arrivals at rate
λ
and
service at rate 2
μ
than it is in a twoserver
2 model with arrivals at rate
λ
and with each server at rate
μ
. Can you give an intuitive explanation for this
result?Woulditalsobetruefor
W
Q
?
8.
A group of
n
customers moves around among two servers. Upon completion
of service, the served customer then joins the queue (or enters service if the server
is free) at the other server. All service times are exponential with rate
μ
.Findthe
proportion of time that there are
j
customers at server 1,
j
=
0
,...,n
.
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This note was uploaded on 03/17/2011 for the course IEOR 161 taught by Professor Lim during the Spring '08 term at University of California, Berkeley.
 Spring '08
 Lim
 Operations Research

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