Page 292 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 91 QUEUEING THEORY

9.6 THE M/M/l/K QUEUEING SYSTEM
In the M/M/l/K queueing system, the capacity of the system is limited to K customers. When the
system reaches its capacity, the effect is to reduce the arrival rate to zero until such time as a cus-
tomer is served to again make queue space available. Thus, the M/M/l/K queueing system can be
modeled as a birth-death process with the following parameters :

Then, from Eqs. (9.1 0) and (9.1 I), we obtain (Pro b. 9.14)

where p = Alp. It is important to note that it is no longer necessary that the traffic intensity p = R/p
be less than 1. Customers are denied service when the system is in state K. Since the fraction of
arrivals that actually enter the system is 1 - p,, the effective arrival rate is given by
fe = 41 - PK)
The average number of customers in the system is given by (Prob. 9.15)

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

9.7 THE M/M/s/K QUEUEING SYSTEM

Similarly, the M/M/s/K queueing system can be modeled as a birth-death process with the fol-
lowing parameters :

Then, from Eqs. (9.1 0) and (9.1 I), we obtain (Prob. 9.17)

where p = A&). Note that the expression for p, is identical in form to that for the M/M/s system,
Eq. (9.21). They differ only in the po term. Again, it is not necessary that p = A/(sp) be less than 1. The