SSOC032 Tut 1 Problems: Waiting-Line Models
Problem 1:
A new shopping mall is considering setting up an information desk manned by one employee. Based upon
information obtained from similar information desks, it is believed that people will arrive at the desk at a
rate of 20 per hour. It takes an average of 2 minutes to answer a question. It is assumed that the arrivals
follow a Poisson distribution and answer times are exponentially distributed.
(a) Find the probability that the employee is idle.
(b) Find the proportion of the time that the employee is busy.
(c) Find the average number of people receiving and waiting to receive some information.
(d) Find the average number of people waiting in line to get some information.
(e) Find the average time a person seeking information spends in the system.
(f) Find the expected time a person spends just waiting in line to have a question answered (time in the
queue).
Problem 2:
Assume that the information desk employee in Problem 1 earns $5 per hour. The cost of waiting time, in
terms of customer unhappiness with the mall, is $12 per hour of time spent waiting in line. Find the total
expected costs over an 8-hour day.
Problem 3:
The shopping mall has decided to investigate the use of two employees on the information desk.
(a) Find the proportion of the time the employees will be idle.
(b) Find the average number of people waiting in this system.
(c) Find the expected time a person spends waiting in this system.
(d) Assuming the same salary level and waiting costs as in Problem 2, find the total expected costs over an
8-hour day.
Problem 4:
A waiting line meeting the M/M/1 assumptions has an arrival rate of 4 per hour and a service rate of 12 per
hour. What is the probability that the waiting line is empty?
1
Problem 5:
A waiting line meeting the M/M/1 assumptions has an arrival rate of 4 per hour and a service rate of 12 per
hour. What is the average time a unit spends in the system and the average time a unit spends waiting?
Problem 6:
A waiting line meeting the M/M/1 assumptions has an arrival rate of 10 per hour and a service rate of 12 per
hour. What is the average time a unit spends in the system and the average time a unit spends waiting?
Problem 7:
A waiting line meeting the M/M/1 assumptions has an arrival rate of 10 per hour and a service rate of 12 per
hour. What is the probability that the waiting line is empty?
Problem 8:
A crew of mechanics at the Highway Department garage repair vehicles that break down at an average of 
= 7.5 vehicles per day (approximately Poisson in nature). The mechanic crew can service an average of  =
10 vehicles per day with a repair time distribution that approximates an exponential distribution.
a. What is the utilization rate for this service system?
b. What is the average time before the facility can return a breakdown to service?
c. How much of that time is spent waiting for service?
d. How many vehicles are likely to be in the system at any one time?
Problem 9:
A crew of mechanics at the Highway Department garage repair vehicles that break down at an average of 
= 7 vehicles per day (approximately Poisson in nature). The mechanic crew can service an average of  = 11
vehicles per day with a repair time distribution that approximates an exponential distribution.
a. What is the utilization rate for this service system?
b. What is the average time before the facility can return a breakdown to service?
c. How much of that time is spent waiting for service?
d. How many vehicles are likely to be waiting for service at any one time?
Problem 10:
A crew of mechanics at the Highway Department garage repair vehicles which break down at an average of
 = 5 vehicles per day (approximately Poisson in nature). The mechanic crew can service an average of  =
10 vehicles per day with a repair time distribution that approximates an exponential distribution.
a. What is the probability that the system is empty?
b. What is the probability that there is precisely one vehicle in the system?
c. What is the probability that there is more than one vehicle in the system?
d. What is the probability of 5 or more vehicles in the system?
2