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QUEUING MODELS WITH FINITE CALLING POPULATIONS 477

3 The average number of units in the system:

L ¼ L q þð1 P 0 Þ (11:35)

4 The average time a unit spends in the queue:

L q
W q ¼ (11:36)
ðN LÞ

5 The average time a unit spends in the system:

1
W ¼ W q þ (11:37)

6 The probability an arriving unit has to wait for service:

P W ¼ 1 P 0 (11:38)

7 The probability of n units in the system:

n
N!
P n ¼ P 0 for n ¼ 0; 1; ... ; N (11:39)
ðN nÞ!

One of the primary applications of the M/M/1 model with a finite calling population is
referred to as the machine repair problem. Inthisproblem,a groupofmachinesis
considered to be the finite population of ‘customers’ that may request repair service.
Whenever a machine breaks down, an arrival occurs in the sense that a new repair
request is initiated. If another machine breaks down before the repair work has been
completed on the first machine, the second machine begins to form a ‘queue’ for repair
service. Additional breakdowns by other machines will add to the length of the waiting
line. The assumption of first-come, first-served indicates that machines are repaired in
the order they break down. The M/M/1 model shows that one person or one channel is
available to perform the repair service. To return the machine to operation, each
machine with a breakdown must be repaired by the single-channel operation.

An Example The Kolkmeyer Manufacturing Company uses a group of six identical
machines; each machine operates an average of 20 hours between breakdowns. Thus,
1
the mean arrival rate or request for repair service for each machine is ¼ / 20 ¼ 0:05
per hour. With randomly occurring breakdowns, the Poisson probability distribution is
used to describe the machine breakdown arrival process. One person from the
maintenance department provides the single-channel repair service for the six
machines. The exponentially distributed service times have a mean of two hours per
machine or a mean service rate of ¼ ½ ¼ 0:50 machines per hour.
With l ¼ 0.05 and ¼ 0.50, we use Equations (11.33) through (11.38) to calcu-
late the operating characteristics for this system. Note that the use of Equation

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