Page 477 -

SINGLE-CHANNEL QUEUING MODEL WITH POISSON ARRIVALS AND EXPONENTIAL SERVICE TIMES 457

The mathematical methodology used to derive the formulas for the operating
characteristics of queues is rather complex. However, our purpose in this chapter is
not to provide the theoretical development of queuing models, but rather to show
how the formulas that have been developed can provide information about operat-
ing characteristics of the queue. Readers interested in the mathematical develop-
ment of the formulas can consult the specialized texts listed in Appendix C at the
end of the text.

Operating Characteristics

The following formulas can be used to calculate the steady-state operating charac-
teristics for a single-channel queuing system with Poisson arrivals and exponential
service times, where:
¼ the mean number of arrivals per time period (the mean arrival rate)
¼ the mean number of services per time period (the mean service rate)
1 The probability that no units are in the system:
Equations (11.4) through
(11.10) do not provide
formulae for optimal P 0 ¼ 1 (11:4)
conditions. Rather, these
equations provide
information about the
steady-state operating 2 The average number of units in the queue:
characteristics of a
waiting line.
2
L q ¼ (11:5)
ð Þ

3 The average number of units in the system:

L ¼ L q þ (11:6)

4 The average time a unit spends in the queue:

L q
W q ¼ (11:7)

5 The average time a unit spends in the system:

1
W ¼ W q þ (11:8)

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

P w ¼ (11:9)

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.