Module D – Waiting Line Models
Classroom Examples
Model A – Single Channel
1. A crew of mechanics at the Highway Department garage repair vehicles that breakdown at an average rate of 7.5 vehicles per day. The mechanic crew can service an average of 10 vehicles per day with a repair distribution that approximates exponential distribution.
a. What is the utilization rate for the service system?
b. What is the average time before a facility can return a breakdown to service?
c. How much time do vehicles wait for service once they arrive?
d. How many vehicles in the system at one time?
2. 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 on 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 queue).
3. Assume that the information desk employee in Problem 2 earns $10 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.
Model B – Multi Channel
1. Using information from Model A problem 2, the shopping mall has decided to investigate the use of two employees on the information desk. The is probability Po = 50%.
a. Find the average number of people waiting in this system.
b. Find the expected time a person spends waiting in this system.
2. Using information from Model A problem 1, the chief is interested in increasing the crew to two so that the vehicles get back to work faster. The probability of having no vehicles in the system is 50%.
a. Find the average number of vehicles in the system.
b. Find the expected time a car spends waiting to be fixed.
Comparison:
Single channel - 1 crew – vehicles take a total of .4 days through the system to get repaired
Multi channel – 2 crew – vehicles take a total of .12 days through the system to get repaired