Waiting-Line Models Problem 1: A new shopping mall is considering setting up an
ID: 362776 • Letter: W
Question
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 $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.
Problem 3:
The shopping mall has decided to investigate the use of two employees on the information desk.
(a) Find the probability of no people in the system.
(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:
Three students arrive per minute at a coffee machine that dispenses exactly four cups per minute at a constant rate. Describe the system parameters.
Problem 5:
A repairman at a local metal working shop services their five drill presses. Service time averages 10 minutes and is exponentially distributed. Machines breakdown after an average of 70 minutes operation (following a Poisson distribution). Describe the major system characteristics.
Explanation / Answer
1) Arrival rate, = 20 per hour
Service rate, = 60/2 = 30 per hour
Utilization rate, = / = 20/30 = 0.67
a) Probability that the employee is idle = 1- = 1-0.67 = 0.33 or 33%
b) Proportion of that the employee is busy = = 0.67 or 67%
c) Average number of people receiving and waiting to receive some information (L) = /(-) = 20/(30-20) = 2
d) Average number of people waiting in line to get some information (Lq) = 2/(*(-)) = 202/(30*(30-20)) = 1.33
e) Average time a person seeking information spends in the system (W) = L/ = 2/20 = 0.1 hour or 6 minutes
f) Average time in queue (Wq) = Lq/ = 1.33/20 = 0.067 hour or 4 minutes
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