Page 295 - Schaum's Outlines - Probability, Random Variables And Random Processes

QUEUEING THEORY [CHAP. 9

which is Eq. (9.1 6). Next, by definition,
wq = w - w,
where W, = 1/p, that is, the average service time. Thus,

which is Eq. (9.1 7). Finally, by Eq. (9.31,

which is Eq. (9.1 8).

9.6. Let W, denote the amount of time an arbitrary customer spends in the M/M/1 queueing system.
Find the distribution of W,.
We have
03
P{ W, I a} = P( W, I a 1 n in the system when the customer arrives}
n = 0
x P{n in the system when the customer arrives} (9.44)
where n is the number of customers in the system. Now consider the amount of time W, that this customer
will spend in the system when there are already n customers when he or she arrives. When n = 0, then
W, = W,,,,, that is, the service time. When n 2 1, there will be one customer in service and n - 1 customers
waiting in line ahead of this customer's arrival. The customer in service might have been in service for some
time, but because of the memoryless property of the exponential distribution of the service time, it follows
that (see Prob. 2.57) the arriving customer would have to wait an exponential amount of time with param-
eter p for this customer to complete service. In addition, the customer also would have to wait an exponen-
tial amount of time for each of the other n - 1 customers in line. Thus, adding his or her own service time,
the amount of time W, that the customer will spend in the system when there are already n customers when
he or she arrives is the sum of n + 1 iid exponential r.v.'s with parameter p. Then by Prob. 4.33, we see that
this r.v. is a gamma r.v. with parameters (n + 1, p). Thus, by Eq. (2.83),

P{Wa 5 a I n in the system when customer arrives) =
From Eq. (9.1 4),
( XY
P{n in the system when customer arrives) = pn = 1 - -
Hence, by Eq. (9.44),

Thus, by Eq. (2.79), W, is an exponential r-v. with parameter p - 1. Note that from Eq. (2.99), E(W,) =
1/(p - A), which agrees with Eq. (9.16), since W = E( W,).

9.7. Customers arrive at a watch repair shop according to a Poisson process at a rate of one per
every 10 minutes, and the service time is an exponential r.v. with mean 8 minutes.
(a) Find the average number of customers L, the average time a customer spends in the shop
W, and the average time a customer spends in waiting for service W,.