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QUEUING MODELS WITH FINITE CALLING POPULATIONS 485

c. The company is considering purchasing a computer-based equipment repair system
that would enable a constant repair time of two hours. For practical purposes, the
standard deviation is zero. Because of the computer-based system, the company’s cost
of the new operation would be E32 per hour. The firm’s director of operations said no to
the request for the new system because the hourly cost is E4 higher and the mean
repair time is the same. Do you agree? What effect will the new system have on the
queuing characteristics of the repair service?
d. Does paying for the computer-based system to reduce the variation in service time
make economic sense? How much will the new system save the company during a 40-
hour work week?

18 A large insurance company maintains a central computing system that contains a variety of
information about customer accounts. Insurance agents in a six-state area use telephone
lines to access the customer information database. Currently, the company’s central
computer system allows three users to access the central computer simultaneously.
Agents who attempt to use the system when it is full are denied access; no waiting is
allowed. Management realizes that with its expanding business, more requests will be
made to the central information system. Being denied access to the system is inefficient as
well as annoying for agents. Access requests follow a Poisson probability distribution, with
a mean of 42 calls per hour. The mean service rate per line is 20 calls per hour.
a. What is the probability that zero, one, two and three access lines will be in use?
b. What is the probability that an agent will be denied access to the system?
c. What is the average number of access lines in use?
d. In planning for the future, management wants to be able to handle l ¼ 50 calls per
hour; in addition, the probability that an agent will be denied access to the system
should be no greater than the value computed in part (b). How many access lines
should this system have?

19 Cairo Cab, Inc., uses two dispatchers to handle requests for service and to dispatch the
cabs. The telephone calls that are made to Cairo Cab use a common telephone number.
When both dispatchers are busy, the caller hears a busy signal; no waiting is allowed.
Callers who receive a busy signal can call back later or call another cab service. Assume
that the arrival of calls follows a Poisson probability distribution, with a mean of 40 calls per
hour, and that each dispatcher can handle a mean of 30 calls per hour.
a. What percentage of time are both dispatchers idle?
b. What percentage of time are both dispatchers busy?
c. What is the probability callers will receive a busy signal if two, three or four dispatchers
are used?
d. If management wants no more than 12 per cent of the callers to receive a busy signal,
how many dispatchers should be used?
20 Five administrative assistants use an office copier. The average time between arrivals for
1
each assistant is 40 minutes, which is equivalent to a mean arrival rate of / 40 ¼ 0.025
arrival per minute. The mean time each assistant spends at the copier is five minutes, which
1
is equivalent to a mean service rate of / 5 ¼ 0:20 user per minute. Use the M/M/1 model with
a finite calling population to determine the following:
a. The probability that the copier is idle.
b. The average number of administrative assistants in the queue.
c. The average number of administrative assistants at the copier.
d. The average time an assistant spends waiting for the copier.
e. The average time an assistant spends at the copier.
f. During an eight-hour day, how many minutes does an assistant spend at the copier?
How much of this time is waiting time?
g. Should management consider purchasing a second copier? Explain.

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