Page 483 -

MULTIPLE-CHANNEL QUEUING MODEL WITH POISSON ARRIVALS AND EXPONENTIAL SERVICE TIMES 463

2 The average number of units in the queue:

k
ð = Þ
L q ¼ P 0 (11:12)
ðk 1Þ!ðk Þ 2

3 The average number of units in the system:

L ¼ L q þ (11:13)

4 The average time a unit spends in the queue:

L q
W q ¼ (11:14)

5 The average time a unit spends in the system:

1
W ¼ W q þ (11:15)

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

k
1 k
P w ¼ P 0 (11:16)
k! k

7 The probability of n units in the system:

ð = Þ n
P n ¼ P 0 for n k (11:17)
n!

ð = Þ n
P n ¼ P 0 for n > k (11:18)
k!k ðn kÞ

Because is the mean service rate for each channel, k is the mean service rate for
the multiple-channel system. As was true for the single-channel queues model, the
formulas for the operating characteristics of multiple-channel queues can be applied
only in situations where the mean service rate for the system is greater than the
mean arrival rate for the system; in other words, the formulas are applicable only if
k is greater than l.
Some expressions for the operating characteristics of multiple-channel queues are
more complex than their single-channel counterparts. However, Equations (11.11)
through (11.18) provide the same information as provided by the single-channel
model. To help simplify the use of the multiple-channel equations, Table 11.4

Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.