Page 296 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 91 QUEUEING THEORY 289

(b) Suppose that the arrival rate of the customers increases 10 percent. Find the corresponding
changes in L, W, and W,.
(a) The watch repair shop service can be modeled as an M/M/1 queueing system with 1 = &, p = 4. Thus,
from Eqs. (9.1 5), (9.1 6), and (9.43), we have

1
w=-=-- 1 - 40 minutes
- A Q-iij
W, = W - W, = 40 - 8 = 32 minutes
(b) Now 1 = 4, p = g. Then

1
w=- --= 1 72 minutes
-
p-a +-g
W, = W - W, = 72 - 8 = 64 minutes
It can be seen that an increase of 10 percent in the customer arrival rate doubles the average number
of customers in the system. The average time a customer spends in queue is also doubled.

9.8. A drive-in banking service is modeled as an M/M/1 queueing system with customer arrival rate
of 2 per minute. It is desired to have fewer than 5 customers line up 99 percent of the time. How
fast should the service rate be?
From Eq. (9.14),
a, a, 1
P(5 or more customers in the system} = C pn = C (1 - p)pn = p5 p = -
n=5 n=5 P
In order to have fewer than 5 customers line up 99 percent of the time, we require that this probability be
less than 0.01. Thus,

from which we obtain

Thus, to meet the requirements, the average service rate must be at least 5.024 customers per minute.

9.9. People arrive at a telephone booth according to a Poisson process at an average rate of 12 per
hour, and the average time for each call is an exponential r.v. with mean 2 minutes.
(a) What is the probability that an arriving customer will find the telephone booth occupied?
(b) It is the policy of the telephone company to install additional booths if customers wait an
average of 3 or more minutes for the phone. Find the average arrival rate needed to justify a
second booth.

(a) The telephone service can be modeled as an M/M/1 queueing system with 1 = 4, p = 3, and p =
1/p = 5. The probability that an arriving customer will find the telephone occupied is P(L > 0), where
L is the average number of customers in the system. Thus, from Eq. (9.1 3),