Practice Problems: Module D, WaitingLine 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 are Poisson 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
at the desk.
(f) Find the expected time a person spends just waiting in line to
have a question answered.
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 mall has decided to investigate the use of two employees
on the information desk.
(a) Find the proportion of the time that the employees are idle.
(b) Find the average number of people waiting in this system.
(c) Find the expected time a person spends waiting in the
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:
Three students arrive per minute at a coffee machine that
dispenses exactly 4 cups per minute at a constant rate.
Describe the system parameters.
Problem 5.
A repairman services five drill presses. Service time averages
10 minutes, and is exponential. Machines breakdown after an
average of 70 minutes following a Poisson distribution. Describe
the major system characteristics.
ANSWERS:
Problem 1:
(a)
(b)
(c)
(d)
(e)
(f)
Problem 2:
From the solution to Problem 1:
The average person waits 0.0667 hours and here are 160
arrivals per day.
Total waiting time = 160 * 0.0667 = 10.67 hours
Total cost for waiting = total waiting time * cost per hour = 10.67
* 12 = $128 per day.
Salary cost = 8*5 = $40
Total cost = Salary cost + waiting cost = 40 + 128 = $168 per
day.
Problem 3:
 = 20 per hour  = 30 per hour M = 2 open channels (servers)
(a)
(b)
(c)
Problem 4:
Problem 5:
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