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458 CHAPTER 11 QUEUING MODELS

7 The probability of n units in the system:

n

P n ¼ P 0 (11:10)

The values of the mean arrival rate l and the mean service rate are clearly
important components in determining the operating characteristics. Equation
(11.9) shows that the ratio of the mean arrival rate to the mean service rate, l/ ,
provides the probability that an arriving unit has to wait because the service facility is
in use. Hence, l/ is often referred to as the utilization factor for the service facility.
The operating characteristics presented in Equations (11.4) through (11.10) are
applicable only when the mean service rate is greater than the mean arrival rate l –
in other words, when l/ < 1. If this condition does not exist, the queue will
continue to grow without limit because the service facility does not have sufficient
capacity to handle the arriving units. Thus, in using Equations (11.4) through
(11.10), we must have > l.

Operating Characteristics for the Dome Problem
Recall that for the Dome problem we had a mean arrival rate of l ¼ 0.75 customers
per minute and a mean service rate of ¼ one customer per minute. So, with > l,
Equations (11.4) through (11.10) can be used to provide operating characteristics for
the Dome’s single-channel queuing system:

0:75
P 0 ¼ 1 ¼ 1 ¼ 0:25
1
2 0:75 2
L q ¼ ¼ ¼ 2:25 customers
ð Þ 1ð1 0:75Þ
0:75
L ¼ L q ¼ þ 2:25 þ ¼ 3 customers
1
L q 2:25
W q ¼ ¼ ¼ 3 minutes
0:75
1 1
W ¼ W q þ ¼ 3 þ ¼ 4 minutes
1
0:75
P w ¼ ¼ ¼ 0:75
1
Problem 5 asks you to Equation (11.10) can be used to determine the probability of any number of customers
calculate the operating
characteristics for a in the system. Applying it provides the probability information in Table 11.2.
single-channel queuing
system application.
Managers’ Use of Queuing Models

The results of the single-channel queue for the Dome show several important things
about the operation of the queue. In particular, customers wait an average of three
minutes before beginning to place an order. In addition, the facts that the average
number of customers waiting is 2.25 and that 75 per cent of the arriving customers
have to wait for service are indicators that something should be done to improve the
queuing operation. Table 11.2 shows a 0.1335 probability that seven or more
customers are in the Dome system at one time. This condition indicates a fairly
high probability that the Dome will experience some long queues if it continues to
use the single-channel operation.

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