Page 293 - Schaum's Outlines - Probability, Random Variables And Random Processes

QUEUEING THEORY [CHAP. 9

average number of customers in the queue is given by (Prob. 9.18)

The average number of customers in the system is

The quantities W and Wq are given by

Solved Problems

9.1. Deduce the basic cost identity (9.1).
Let T be a fixed large number. The amount of money earned by the system by time T can be com-
puted by multiplying the average rate at which the system earns by the length of time T. On the other
hand, it can also be computed by multiplying the average amount paid by an entering customer by the
average number of customers entering by time T, which is equal to AaT, where A, is the average arrival rate
of entering customers. Thus, we have
Average rate at which the system earns x T = average amount paid by an entering customer x (JOT)
Dividing both sides by T (and letting T -, co), we obtain Eq. (9.1).

9.2. Derive Eq. (9.6).
From properties 1 to 3 of the birth-death process N(t), we see that at time t + At, the system can be in
state S, in three ways:
1. By being in state S, at time t and no transition occurring in the time interval (t, t + At). This happens
with probability (1 - a, At)(l - d, At) = 1 - (a, + d,) At [by neglecting the second-order effect
a, dn(At)21.
2. By being in state S,-, at time t and a transition to S, occurring in the time interval (t, t + At). This
happens with probability a, -, At.
3. By being in state S,, , at time t and a transition to S, occurring in the time interval (t, t + At). This
happens with probability d, + , At.
Let pi(t) = P[N(t) = i]
Then, using the Markov property of N(t), we obtain