Introduction to Operations Research
MSC 521
An Outstanding Set of Queuing Problems
Instructions: Assume the pure birth-death process for the following problems unless otherwise stated. Use
the computer where necessary.
1. Documents to be typed at an office arrive (Poisson) at the rate of 100 per day and
secretaries can type (Exponential) at the rate of 20 documents per secretary per day.
Each secretary costs the company $40 per hour (fully burdened) and the cost of holding
documents in the system is estimated at $20 per document per hour. How many
secretaries should operate in the typing pool?
2. Calls come into a customer service facility at the rate of 10 per hour. Service times
are exponential with the mean time to service a caller of 15 minutes. Customers are put
on hold if there are no service people available to take their call. On the average
customers will hang up after waiting 3 minutes or more. How many customer service
personnel are necessary to insure that the average waiting time is 2 minutes or less?
3. Two technicians have been hired to maintain 10 copying machines located throughout
the building. Because of their extremely heavy use, each machine breaks down or is in
need of adjustment at least once per day. It takes an average of 80 minutes (8 hour day)
for one technician to repair or adjust a machine. Describe the long-run average status of
the 10 copying machines.
4. The Q. S. Goode Company has 8 personal computers on its local area network (LAN).
The demand for these computers has averaged 14 per hour. Once a company associate
obtains a computer, it will be used for an average of 30 minutes (exponentially
distributed). If there are no terminals available, the associate takes a coffee break and try
again later. Management has observed this and is not very happy. A directive is issued
to the purchasing department to buy sufficient computers so that there is than one chance
in twenty that an associate will not have a computer immediately available. Compare the
system both before and after additional computers are purchased. Does the computer
utilization justify the purchase of more computers?
5. A fast food restaurant has two drive-up windows but only sufficient space to
accommodate four vehicles (two in service and two waiting). Service time has a mean of
two minutes. However, if there are one or two cars waiting, an extra clerk assists both
windows thus decreasing the mean service time to one minute. Cars arrive at the rate of
120 per hour unless both windows are busy; then balking results in an effective arrival
rate of 60 per hour. Find the queuing measures of effectiveness.
6. The local credit union has observed on high volume days a mean arrival rate of 45
customers per hour. The experienced teller will average 5 minutes per customer. A fully
burdened hourly rate for an experienced teller is $45. If the credit union places a $35 per
hour cost of waiting on its customers, what is the minimum cost number of tellers
required on busy days? If a customer is to wait no longer than 5 minutes for a teller, how
many tellers should be employed?
Introduction to Operations Research
MSC 521
7. The Help Desk for a large multinational company is to be established to handle an
average of 100 calls per hour. A typical service representative, costing $30 an hour, can
handle 8 calls per hour. The company estimates that the hourly cost of its employees
being without computer support is $15 an hour. How many service representatives
should be hired? Compute the performance measures for the optimum number of service
representatives.
8. Patients enter an emergency room on Friday nights at the rate of 8 per hour. It takes
the emergency room on-call physician 20 minutes on the average to treat a patient. If the
daily cost of the physician is $1200 (8 hour day) and the value placed upon a patient’s
waiting time by the insurance industry is $5 per hour, how many physicians should be
assigned on Friday nights? Compute the performance measures.