Page 303 - Schaum's Outlines - Probability, Random Variables And Random Processes

QUEUEING THEORY [CHAP. 9

9.23. In a university computer center, 80 jobs an hour are submitted on the average. Assuming that the computer
service is modeled as an M/M/1 queueing system, what should the service rate be if the average turnaround
time (time at submission to time of getting job back) is to be less than 10 minutes?
Ans. 1.43 jobs per minute

9.24. The capacity of a communication line is 2000 bits per second. The line is used to transmit 8-bit characters,
and the total volume of expected calls for transmission from many devices to be sent on the line is 12,000
characters per minute. Find (a) the traffic intensity, (h) the average number of characters waiting to be
transmitted, and (c) the average transmission (including queueing delay) time per character.
Ans. (a) 0.8; (h) 3.2; (c) 20 ms
9.25. A bank counter is currently served by two tellers. Customers entering the bank join a single queue and go
to the next available teller when they reach the head of the line. On the average, the service time for a
customer is 3 minutes, and 15 customers enter the bank per hour. Assuming that the arrivals process is
Poisson and the service time is an exponential r.v., find the probability that a customer entering the bank
will have to wait for service.
Ans. 0.205
9.26. A post office has three clerks serving at the counter. Customers arrive on the average at the rate of 30 per
hour, and arriving customers are asked to form a single queue. The average service time for each customer
is 3 minutes. Assuming that the arrivals process is Poisson and the service time is an exponential r.v., find
(a) the probability that all the clerks will be busy, (b) the average number of customers in the queue, and (c)
the average length of time customers have to spend in the post office.
Ans. (a) 0.237; (6) 0.237; (c) 3.947 min
9.27. Show that Eqs. (9.57) to (9.59) and Eqs. (9.31) to (9.33) are equivalent.
Hint: Use Eq. (9.29).

9.28. Find the average number of customers L in the M/M/l/K queueing system when iZ = p.
Ans. K/2

9.29. A gas station has one diesel fuel pump for trucks only and has room for three trucks (including one at the
pump). On the average trucks arrive at the rate of 4 per hour, and each truck takes 10 minutes to service.
Assume that the arrivals process is Poisson and the service time is an exponential r.v.
(a) What is the average time for a truck from entering to leaving the station?
(b) What is the average time for a truck to wait for service?
(c) What percentage of the truck trafic is being turned away?
Ans. (a) 20.1 5 min; (h) 10.14 min; (c) 12.3 percent

9.30. Consider the air freight terminal service of Prob. 9.20. How many additional docks are needed so that at
least 80 percent of the arriving aircraft can be served in the main concourse with the addition of holding
area?
Ans. 4