# Means & Probability: Queuing Theory

Q1

In the Willow Brook National Bank waiting line system (see Problem 1), assume that the service times for the drive-up teller follow an exponential probability distribution with a service rate of 36 customers per hour or 0.6 customer per minute. Use the exponential probability distribution to answer the following questions.

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?

Q2

Use the single-channel drive-up bank teller operation referred to in problems 1 and Q1 to determine the following operating characteristics for the system.

a. The probability that no customers are in the system

b. The average number of the customers waiting

c. The average number of the customers in the system

d. The average time a customer spend waiting

e. The average time a customer spends in the system

f. The probability that arriving customers will have to wait for service

Q3

Use the single-channel drive-up bank teller operation referred to in Problem1 and Q1 and Q2 to determine the probabilities of 0, 1, 2, and 3 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?

Use Problem 1 to solve questions Q1,Q2, and Q3 ( Problem 1 do not need to be solved)

Problem 1

A bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars. On weekday mornings, arrivals to the drive-up window occur at random, with a mean arrival rate of 24 customers per hour or 0.4 customers per minute.

1) What is the mean or expected number of customers that will arrive in a five-minute period?

2) Assume that the Poisson probability distribution can be used to describe the arrival process. Use the mean arrival rate in part 1 and compute the probabilities that exactly 0, 1, 2, and 3 customers will arrive during a five-minute period.

3) Delays are expected if more than three customers arrive during any five-minute period. What is the probability that delays will occur?

#### Solution Summary

Solutions to 3 queuing theory questions are provided in a Word attachment of 474 words.