CS 457/657 – Winter 2012
Assignment 2
Due: February 6, 2012 at 5 pm
Late assignments will not be accepted
Question 1
Consider a cyclic queue model with M servers and N circulating jobs (see Figure below).
N circulating jobs
1
2
…
M
Let !! be the mean response time at server i. i = 1, 2, …, M. Also let ! be the total arrival rate at server 1 (or
at any of the other servers). Note that the total arrival rates at the M servers are the same. Obtain a formula
that relates !, N, and !! , i = 1, 2, …, M.
Question 2
Consider a single server queue with a "vacation" feature. This model can be described as follows:
Let
•
At time = 0, the system is empty, and the server starts a vacation period immediately.
•
At the end of each vacation period, the server checks the queue. If the queue is empty, the server
takes another vacation immediately, otherwise the server starts serving jobs and continues to serve
jobs until the system is empty. As soon as the system becomes empty, the server takes its next
vacation.
L = length of an observation interval (from 0 to L)
n = number of jobs arrived and served in (0, L)
!! = service time of the i-th job, i = 1, 2, …, n
What is the percentage of time that the server is on vacation? Express your answer as a function of the arrival
rate ! and the mean service time S. Explain your answer.
Question 3
Suppose there are n arrivals in a time period of length L (from 0 to L), and the nth arrival (or last arrival)
occurs at time L. Let !! be the time between 0 and the time of the first arrival, and !! be the interarrival time
between the (i-1)st and ith arrivals, i = 2, 3, …, n.
(a) Let X be the time until the next arrival. Plot X as a function of time t for 0 ≤ t ≤ L.
! !
! !
(b) Let ! ! =
! and ! ! ! =
! ! be the mean and second moment of the interarrival time,
! !!! !
! !!! !
respectively. Using the plot in part (a), obtain an expression for ![!], the mean of X, as a function of
![!] and ![! ! ].
Question 4
Consider the tandem queue model shown in the Figure below. Packets sent by the sender are transmitted
along a path consisting of links 1, 2, 3 and 4, and then delivered to the receiver. Each link is modeled by a
single server.
Sender
link
1
link
2
link
3
link
4
Receiver
Blocking probability !!
Suppose link 3 is under heavy load and the blocking probability at link 3 is !! . The blocking probabilities at
the other three links are zero. Let
Y be the system throughput = the rate at which packets are delivered to the receiver
R be the mean end-to-end delay from sender to receiver
Q be the mean number of packets in the network
It has been suggested that according to Little’s Law, we can write YR = Q. Do you agree? Explain your
answer.
Question 5
Consider a finite-population model with one user (N = 1). Let Z be the mean think time and S be the mean
service time. Suppose the user starts his/her first think time at time 0 and the nth job submitted by this user
completes service at time L. Derive an expression for U, the utilization of the server, as a function of Z and S.
You must show details of your derivation.
Question 6
Consider the closed queueing network model shown in Figure 4 (on page 7) of the lecture notes entitled
“Queueing Models – Analytic Results”. Each job, after completing service at the CPU, has probability 0.3 of
visiting Disk A, 0.65 of visiting Disk B, and 0.05 of following the out-to-in transition. Suppose the mean
service times per visit to the CPU, Disk A, and Disk B are 0.03, 0.05, and 0.04 seconds, respectively. Derive
an upper bound for the system throughput Y as a function of N, the number of jobs in the network.
Question 7
Consider the following open queueing network model. Suppose the mean service times per visit at the four
servers, denoted by !! , !! , !! , and !! , are 0.02, 0.03, 0.05, and 0.02 seconds, respectively. What is the
largest value of !!∗ such that the system is stable when !! < !!∗ ?
p21 = p31 = p41 = 1.0
!1 jobs
per second
server
1
p12 = 0.4
server
2
p13 = 0.35
server
3
p14 = 0.2
server
4
p15 = 0.05
Problem 8
Consider the interactive system model shown in Figure 5 (on page 9) of the lecture notes entitled “Queueing
Models – Analytic Results” (see Figure below).
N User Workstation
Disk A
CPU
Disk B
Each job, after completing service at the CPU, has probability 0.6 of visiting Disk A, 0.35 of visiting Disk B,
and 0.05 of returning to the user workstation. The mean think time is 15 seconds, and the mean service times
per visit to the CPU, Disk A, and Disk B are 0.05, 0.07, and 0.02 seconds, respectively.
(a) Suppose the utilization of Disk A is 0.7 when N = 25. What is the mean response time R?
(b) Is it feasible for the mean response time to be 12 seconds when N = 30? Explain your answer.
(c) If the answer to Part (b) is no, what percentage increase in the speed of the bottleneck server will make it
feasible for the mean response time to be approximately 12 seconds when N = 30?