Problem set: Queueing Theory
Question 1
A car wash facility operates with one bay. Cars arrive according to a Poisson
distribution with a mean of 4 cars per hour. Cars may wait in the facility's parking lot
if the bay is busy. The wash and clean time of the car is exponentially distributed with
a mean of 10 minutes. Assume that cars that do not find space in the parking lot can
also wait in the street bordering the wash facility so that there is no limit on the size of
waiting cars.
1. Determine the percent utilization of the wash bay.
2. Determine the probability that an arriving car should wait in the parking lot
prior to entering the wash bay.
3. If there are 7 parking spaces, determine the probability that an arriving car will
find an empty parking space.
4. What should be the minimum number of parking spaces so that an arriving car
will find an empty parking space 90% of the time?
5. The current number of parking spaces is four. Cars that do not find an empty
parking space will leave to another facility. How many cars are lost in the
average per 24 hours?
6. What is the chance that an arriving car will be immediately washed without
waiting?
7. How long in the average a car waits in the parking space and in the facility?
8. What is the expected number of empty parking spaces
9. What is the probability that all parking spaces are occupied?
Question 2
Dr. Ali is a famous physician. He receives patients in his clinic at a constant rate of 5
per hour. He spends an exponential time with an average of 18 minutes per patient.
Can he manage to serve all his patients? He is considering having no more than 3
patients per hour. Why?
He wants to determine how many seats to place in the waiting room, so that any
patient who does not find a free seat would leave to another clinic. Could you help
him? If so, how many patients would be on the average at his waiting room?
His children who like him a lot are asking him to call them every hour? How much
time he can afford per hour to call them?
Each patient pays a 40 Dinars for Dr. Ali Service. How much Dr. Ali makes on the
average every day if he spends in his clinic 10 hours a day?
Question 3
Cars arrive to a check point at a rate of 3 per minute. An average checking time of 0.2 minute
is made before cars can continue their ways. Check time is considered as exponential. The
check point uses one or two policemen for inspection. When a single policeman is in service,
determine the following:
1. The utilization rate
2. The average total waiting time of a car at the check point.
3. The average length of the queue
4. The average time each hour, the check point is empty.
Determine the same performance indicators when two policemen are operating
simultaneously in two different lanes.
Question 4
A publishing company is in the process of purchasing a high-speed commercial
copier. Four models whose specifications are summarized below are proposed by the
vendors. Jobs arrive to the company according to a Poisson process with mean of 4
jobs per 24-hr day. Job size is random but averages about 10,000 sheets per job.
Contracts with the customers specify a penalty cost for late delivery of $80 per job per
day. Which copier should the company purchase to minimize its expected total cost
(i.e., expected cost of operating the copier per unit time plus the expected penalty
cost).
Copier model
1
2
3
4
Operating cost ($/hr)
15
20
24
27
Speed (sheets/min)
30
36
50
66
Question 5
H&I Industry produces a special machine with different production rates (pieces per
hour) to meet customer specifications. The production cost is estimated at $ 0.10 per
unit increase in production rate. A shop owner is considering buying one of these
machines and wants to decide on the most economical speed (in pieces per hour) to be
ordered. From past experience, the owner estimates that orders from customers arrive
at the shop according to a Poisson process with rate of 3 orders per hour. Each order
averages about 500 pieces. Contracts between the owner and the customers specify a
penalty of $100 per late order per hour.
1. Assuming that the actual production time is exponential, develop a general
cost model as a function of the production rate, .
2. Develop an expression for the optimal production rate using the cost model
obtained in 1.
3. Using the above data, determine the optimal production rate the owner should
request from H&I.
Question 6
Mr. Nawaf owns a parking facility in the heart of the downtown that could
accommodate 20 cars at a time. Cars arrive at a constant rate of 15 per hr. and spend
an exponential time in the parking facility with an average of 1.5 hr. Cars that find no
free parking lots leave elsewhere without paying fees. The hourly rate of the parking
fee is of 2 DT.
1. What type of model this parking facility has?
2. Provide a formula (without calculation) to determine the average number of
cars parking at the facility
3. Provide a formula to determine the hourly income of the facility.
4. Provide a formula to determine the number of cars that leave the facility
without paying fees
5. Provide a formula to determine the opportunity losses per hour due to the
space limitation in the parking facility
6. Mr. Nawaf can rent the neighboring space at a charge of 1 DT per hour. This
space can accommodate 5 more cars. How could Mr. Nawaf evaluate whether
it is worth going for such an alternative?
Question 7
Jobs arrive to a machine shop at a rate of 80 per week. An automatic machine
represents the bottleneck in the shop. It is estimated that a unit increase in the
production rate in the machine will cost 250 $ per week. Delayed jobs normally result
in lost business, which is estimated at 500 $ per job per week. Determine the
optimum production rate for the automatic machine.
Question 8
A Co. is in the process of hiring a repairperson for a 10-machine shop. Two
candidates are under consideration. The first one can carry out repairs at the constant
rate of 5 machines per hour and earns 50 $ per hour. The second candidate, being
more skilled, receives 65 $ per hour and can repair 7 machines per hour. Assuming
that machines breakdown according to a Poisson process with mean of 3 per hour,
which candidate should be chosen?