HW 4: Queueing games
Guru: Lina
Assigned: 04/27/10
Due: 05/07/10, Raga’s mailbox, 1pm
We encourage you to discuss these problems with others, but you need to write up the actual solutions alone.
At the top of your homework sheet, list all the people with whom you discussed. Crediting help from other
classmates will not take away any credit from you. Start early and come to ofﬁce hours with your questions!
1
Warmup: ztransforms [8 points]
We’ll need to use ztransforms (also known as probabilitygenerating functions) soon in class and later
on this homework, so we want you to practice with them a bit beforehand. The ztransform of a discrete
function on the nonnegative integers is deﬁned as
G
p
(
z
) =
∞
∑
i
=0
p
(
i
)
z
i
When we write the ztransform of a random variable
X
having p.m.f.
p
(
i
)
we will often write
b
X
(
z
) :=
G
p
(
z
)
. (Similarly, for the Laplace transform of
X
we will often write
e
X
(
s
) :=
E
[
e
−
sX
]
.)
(a) Let
A
t
denote the number of arrivals by time
t
when arrivals occur according to a Poisson process with
rate
λ
. Derive
c
A
t
(
z
)
.
(b) Let
A
X
denote the number of arrivals during one service time
X
under a Poisson process with rate
λ
.
Prove that
d
A
X
(
z
) =
e
X
(
λ
(1

z
))
.
2
Beyond First Come First Served [20 points]
We have focused almost entirely on FCFS scheduling so far in the course. We will move to other scheduling
policies soon, so here is an intro to some other ones.
(a) Under Processor Sharing (PS) the server is divided evenly among the jobs in the system. If there are 3
jobs, each gets
1
3
of the total service rate. Derive
E
[
T
]
and
E
[
N
]
for the M/M/1/PS queue.
(b) Under Preemptive Last Come First Served (PLCFS) the server is always devoted entirely to the most
recent arrival present. So, whenever a job arrives, the job at the server is interrupted and returned to the
queue. (We will assume there is no overhead associated with this preemption.) Note that this is exactly
like pushing and popping from a stack. Derive
E
[
T
]
and
E
[
N
]
for the M/M/1/PLCFS queue.
(c) Comment on the similarity of the prior two analyses. How far can this similarity be pushed? Does it
hold in GI/M/1, M/GI/1, GI/GI/1?
(d) Now, let us consider a priority queue. Suppose there are two classes of jobs and that class 1 customers
have absolute priority over class 2 customers. That is, a class 2 customer cannot receive service while
a class 1 customer is in the system. (Again, assume there is no overhead associated with preemptions.)
Class 1 and 2 customers both have Exponential(
μ
) job sizes and arrive according to Poisson processes
with rates
λ
1
and
λ
2
respectively.
Derive
E
[
T
]
. Also derive and contrast
E
[
T
1
]
and
E
[
T
2
]
, the mean response time for class 1 and class 2
customers respectively.
1
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 Fall '09
 Poisson Distribution, Exponential distribution, Poisson process

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