Page 300 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 91 QUEUEING THEORY 293

Using Eqs. (9.28) and (9.51), the average number of customers in the system is given by

9.16. Consider the M/M/l/K queueing system. Show that
Lq = L - (1 - p,)
1
wq=-L
P

In the M/M/l/K queueing system, the average number of customers in the system is
K K
p,
L = E(N) = np, and 1 = 1
n = 0 n = 0
The average number of customers in the queue is

A customer arriving with the queue in state S, has a wait time T, that is the sum of n independent exponen-
tial r.v.'s, each with parameter p. The expected value of this sum is n/p [Eq. (4.10811. Thus, the average
amount of time that a customer spends waiting in the queue is

Simililarly, the amount of time that a customer spends in the system is

Note that Eqs. (9.57) to (9.59) are equivalent to Eqs. (9.31) to (9.33) (Prob. 9.27).

9.17. Derive Eqs. (9.35) and (9.36).
As in Prob. 9.10, from Eqs. (9.34) and (9.10), we have

Let p = rl/(sp). Then Eqs. (9.60) and (9.61) can be rewritten as