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QUEUING MODELS WITH FINITE CALLING POPULATIONS 483

9 Consider a two-channel waiting line with Poisson arrivals and exponential service times.
The mean arrival rate is 14 units per hour, and the mean service rate is ten units per hour
for each channel.
a. What is the probability that no units are in the system?
b. What is the average number of units in the system?
c. What is the average time a unit waits for service?
d. What is the average time a unit is in the system?
e. What is the probability of having to wait for service?
10 Refer to the Agan Interior Design situation in Problem 8. Agan’s management would like to
evaluate two alternatives:
l Use one consultant with an average service time of eight minutes per customer.
l Expand to two consultants, each of whom has an average service time of ten minutes per
customer.
If the consultants are paid $16 per hour and the customer waiting time is valued at $25 per
hour for waiting time prior to service, should Agan expand to the two-consultant system?
Explain.
11 A fast-food franchise is considering operating a drive-up window food-service operation.
Assume that customer arrivals follow a Poisson probability distribution, with a mean arrival
rate of 24 cars per hour, and that service times follow an exponential probability distribution.
Arriving customers place orders at an intercom station at the back of the parking area and
then drive to the service window to pay for and receive their orders. The following three
service alternatives are being considered.
l A single-channel operation in which one employee fills the order and takes the money
from the customer. The average service time for this alternative is two minutes.
l A single-channel operation in which one employee fills the order while a second
employee takes the money from the customer. The average service time for this
alternative is 1.25 minutes.
l A two-channel operation with two service windows and two employees. The employee
stationed at each window fills the order and takes the money for customers arriving at the
window. The average service time for this alternative is two minutes for each channel.
Answer the following questions and recommend an alternative design for the fast-food
franchise.
a. What is the probability that no cars are in the system?
b. What is the average number of cars waiting for service?
c. What is the average number of cars in the system?
d. What is the average time a car waits for service?
e. What is the average time in the system?
f. What is the probability that an arriving car will have to wait for service?
12 The following cost information is available for the fast-food franchise in Problem 11.
l Customer waiting time is valued at E25 per hour to reflect the fact that waiting time is
costly to the fast-food business.
l The cost of each employee is E6.50 per hour.
l To account for equipment and space, an additional cost of E20 per hour is attributable
to each channel.
What is the lowest-cost design for the fast-food business?

13 Patients arrive at a dentist’s office at a mean rate of 2.8 patients per hour. The dentist
can treat patients at the mean rate of three patients per hour. A study of patient waiting
times shows that, on average, a patient waits 30 minutes before seeing the dentist.

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