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482 CHAPTER 11 QUEUING MODELS

a. What is the probability the service time is one minute or less?
b. What is the probability the service time is two minutes or less?
c. What is the probability the service time is more than two minutes?
3 Use the single-channel drive-up bank teller operation referred to in Problems 1 and 2 to
determine the following operating characteristics for the system.
a. The probability that no customers are in the system.
b. The average number of customers waiting.
c. The average number of customers in the system.
d. The average time a customer spends waiting.
e. The average time a customer spends in the system.
f. The probability that arriving customers will have to wait for service.
4 Use the single-channel drive-up bank teller operation referred to in Problems 1–3 to determine
the probabilities of zero, one, two and three customers in the system. What is the probability that
more than three customers will be in the drive-up teller system at the same time?

5 The reference desk of a university library receives requests for assistance. Assume that a
Poisson probability distribution with a mean rate of ten requests per hour can be used to
describe the arrival pattern and that service times follow an exponential probability
distribution with a mean service rate of 12 requests per hour.
a. What is the probability that no requests for assistance are in the system?
b. What is the average number of requests that will be waiting for service?
c. What is the average waiting time in minutes before service begins?
d. What is the average time at the reference desk in minutes (waiting time plus service time)?
e. What is the probability that a new arrival has to wait for service?
6 Films Tonight is a typical DVD movie rental outlet for home viewing customers. During the
weeknight evenings, customers arrive at Films Tonight at the average rate of 1.25
customers per minute. The cashier can serve an average of two customers per minute.
Assume Poisson arrivals and exponential service times.
a. What is the probability that no customers are in the system?
b. What is the average number of customers waiting for service?
c. What is the average time a customer waits for service to begin?
d. What is the probability that an arriving customer will have to wait for service?
e. Do the operating characteristics indicate that the one-cashier checkout system provides
an acceptable level of service?
7 For the Dome single-channel queuing system in Section 11.2, assume that the arrival rate
is increased to one customer per minute and that the mean service rate is increased to 1.25
customers per minute. Calculate the following operating characteristics for the new
system: P 0 , L q , L, W q , W and P w . Does this system provide better or poorer service
compared to the original system? Discuss any differences and the reason for these
differences.
8 Agan Interior Design provides home and office decorating assistance to its customers. In
normal operation, an average of 2.5 customers arrive each hour. One design consultant is
available to answer customer questions and make product recommendations. The
consultant averages ten minutes with each customer.
a. Calculate the operating characteristics of the customer waiting line, assuming Poisson
arrivals and exponential service times.
b. Service goals dictate that an arriving customer should not wait for service more than an
average of five minutes. Is this goal being met? If not, what action do you recommend?
c. If the consultant can reduce the average time spent per customer to eight minutes, what
is the mean service rate? Will the service goal be met?

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