Page 302 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 91 QUEUEING THEORY

rate is 3 aircraft per hour. The average service time per aircraft is 2 hours on the main concourse
and 3 hours on the back concourse.
Find the percentage of the arriving aircraft that are diverted to the back concourse.
If a holding area which can accommodate up to 8 aircraft is added to the main concourse,
find the percentage of the arriving aircraft that are diverted to the back concourse and the
expected delay time awaiting service.
The service system at the main concourse can be modeled as an M/M/s/s queueing system with s = 4,
1 = 3, p = 3, and 1/p = 6. The percentage of the arriving aircraft that are diverted to the back con-
course is
100 x P(al1 docks on the main concourse are full)
From Eq. (9.64),
64/4 !
--
54
0.47
P(al1 docks on the main concourse are full) = p, = 7
P
-
115
C 9
n=O
Thus, the percentage of the arriving aircraft that are diverted to the back concourse is about 47
percent.
With the addition of a holding area for 8 aircraft, the service system at the main concourse can now be
modeled as an M/M/s/K queueing system with s = 4, K = 12, and p = A/(sp) = 1.5. NOW, from Eqs.
(9.35) and (9.36),

Thus, about 33.2 percent of the arriving aircraft will still be diverted to the back concourse.
Next, from Eq. (9.37), the average number of aircraft in the queue is

Then, from Eq. (9.40), the expected delay time waiting for service is

W, = Lq = 6'0565 P 3.022 hours
1 - ) 3(1 - 0.332)
Note that when the 2-hour service time is added, the total expected processing time at the main
concourse will be 5.022 hours compared to the 3-hour service time at the back concourse.

Supplementary Problems

9.21. Customers arrive at the express checkout lane in a supermarket in a Poisson process with a rate of 15 per
hour. The time to check out a customer is an exponential r.v. with mean of 2 minutes.
(a) Find the average number of customers present.
(b) What is the expected idle delay time experienced by a customer?
(c) What is the expected time for a customer to clear a system?
Ans. (a) 1 ; (h) 2 min; (c) 4 min

9.22. Consider an M/M/1 queueing system. Find the probability of finding at least k customers in the system.
Ans. pk =