Page 291 - Schaum's Outlines - Probability, Random Variables And Random Processes

284 QUEUEING THEORY [CHAP. 9

system. The traffic intensity of the system is defined as
mean service time mean arrival rate
-
Traffic intensity = -
mean interarrival time mean service rate
The average number of customers in the system is given by (Prob. 9.4)

Then, setting A, = 3, in Eqs. (9.2) to (9.4), we obtain (Prob. 9.5)

9.5 THE M/M/s QUEUEING SYSTEM
In the M/M/s queueing system, the arrival process is the Poisson process with rate A and each of
the s servers has an exponential service time with parameter p. In this case, the process N(t) describ-
ing the state of the M/M/s queueing system at time t is a birth-death process with the following
parameters :

Note that the departure parameter d, is state dependent. Then, from Eqs. (9.10) and (9.1 I), we obtain
(Prob. 9.10)

where p = A/(+) < 1. Note that t :he ratio p = is the traffic intensity of the M/M/s queueing
system. The average number of customers in the system and the average number of customers in the
queue are given, respectively, by (Prob. 9.12)

By Eqs. (9.2) and (9.3), the quantities W and W, are given by