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Homework #2
IOE 416 - Queueing Systems
Homework #2 (2 pages)
Answers are due before start of class on Thursday, March 27, 2014
1. A store with one server has a queue of customers. The customers arrive in a Poisson process
with rate 27 customers per hour. The service times are independent and exponentially
distributed. The average time in system (queueing plus service) is 20 minutes.
(a) What is the service rate?
(b) What percentage of customers have a time in system less than 1 minute?
(c) You join this queue when there is only 1 customer in the system (the customer in
service). What is the probability that your time in system less than 1 minute?
Hint: Obtain distribution of time in system for fixed number of customers in system
(see Notes 4, page 8), and use integration by parts find the required probability.
2. Vehicles along a single highway lane form a queue at a traffic bottleneck.
The vehicles arrive at the queue in a Poisson process with rate 700 vehicles per hour, and
only one vehicle at a time can pass through the bottleneck.
The times for vehicles to pass through the bottleneck are independent and exponentially
distributed, with an average time of 5 seconds per vehicle.
The queue is made up of 40% cars and 60% trucks. The average length of road taken up
by a vehicle (vehicles length plus some gap space) is 20 feet per car and 60 feet per truck.
For 2% of the time, the queue (including the vehicle passing through the bottleneck) is
longer than than d miles.
(a) What is the distance d (in miles, approximately)? (1 mile = 5280 feet)
(b) How long is this queue on average (in miles, approximately)?
3. A manufacturing process utilizes one robot that processes items at 10 items per hour.
Items waiting to be processed form a queue that operates as an M/M/1 queue. Due to an
increase in demand, the current arrival rate of items is 16 items per hour. The supervisor
for the process realizes that the processing rate must be increased to prevent indefinite
queue buildup, and considers two options:
Option 1: Add a second robot in parallel with same processing rate, so that the queue
becomes an M/M/2 queue;
Option 2: Keep operating as an M/M/1 queue, and install additional hardware and
software on the robot so that its processing rate is doubled.
(a) Compare the average performance measures (L, L
q
, W, W
q
) for the two options.
(b) The supervisor seeks the advice of a learned IOE student before making a decision.
Which option do you recommend for the supervisor? Explain briefly your reasoning.
(Assume there is no cost advantage of one option over the other.)
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4. A busy bank has 5 tellers to serve customers, and the system operates as an M/M/5
queue. Customers arrive at a rate of 90 customers per hour and the average service time
at a teller is 3 minutes.
(a) Show that there is probability that the system is empty is 0.005
(b) Calculate the average queue length.
(c) Compare result (b) with the estimate of average queue length from Sakasegawas
approximation.
(d) Calculate the probability that the number of customers in the system is less than 20.
Note: No need to calculate individual probabilities , , ,
2 1 0
p p p etc.
5. An M/M/1 queue has a service rate of 10 customers per hour. The arrival rate is 8
customers per hour as long as there are no more than 4 customers in the system. If there
are more than 4 customers in the system, new customers are discouraged from joining
the queue, and the arrival rate drops to 5 customers per hour.
(a) Draw the state transition rate diagram for states 0 to 7.
Let p
n
= probability that the number of customers in the system is n,
and p
0
= probability that the system is empty.
Using the basic solution to the general birth-death process for p
n
, obtain
(b) the probability that the number of customers in the system is 1.
(c) the probability that the number of customers in the system is 6.
Give your answers for (b) and (c) in terms of p
0
(i.e, without calculating p
0
).
