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# Waiting Line Models

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In a waiting line situation, arrivals occur at a rate of 2 per minute, and the service times average 18 seconds per customer. Assume the Poisson and exponential distributions. Carry all answers to at least 3 decimals.

a. What is lambda?

b. What is mu?

c. What is the probability of no units in the system?

d. What is the average number of units in the system?

e. What is the time in the waiting line?

f. What is the average time in the system?

g. What is the probability that there is one person waiting? h. What is the probability that an arrival will have to wait?

#### Solution Summary

Solution determines the average waiting time in queue, average number of customers in the system and desired probability values in the given case.

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## Waiting line model with a scenario

As it is, models have been developed to assist managers understand and make better decisions concerning the operation of waiting lines. In management, a waiting line is a queue, and the body of knowledge dealing with waiting lines is known as queueing theory. As I understand it, queueing theory has become more sophisticated with applications in a wide variety of waiting line situations. I believe waiting line models consist of mathematical formulas and relationships that can be used to determine the operating characteristics (performance measures) for a waiting line.

Operating characteristics of interest include the following:

The probability that no units are in the system
The average number of units in the waiting line
The average number of units in the system (the number of units in the waiting line plus the number of units being served)
The average time a unit spends in the waiting line
The average time a unit spends in the system (the waiting time plus the service time)
The probability that an arriving unit has to wait for service

Managers with such information are better able to make decisions that balance desirable service levels against the cost of providing the service.

The situation...

The New York City franchise of U.S. Citibank operates approximately 250 banking centers. Each center provides one or more automatic teller machines (ATMs) capable of performing a variety of banking transactions. At each center, a waiting line is formed by randomly arriving customers who seek service at one of the ATMs.

In order to make decisions on the number of ATMs to have at selected banking center locations, management needs information about potential waiting times and general customer service. Waiting line operating characteristics such as average number of customers in the waiting line, average time a customer spends waiting, and the probability that an arriving customer has to wait would help management determine the number of ATMs to recommend at each banking center.

For instance, one busy Midtown Manhattan center had a peak arrival rate of 172 customers per hour. A multiple-channel waiting line model with six ATMs showed that 88% of the customers would have to wait, with an average wait time between 6 and 7 minutes. This level of service was considered unacceptable. How did this information help management to know that they should change their ATM system? Is it a good model to use to provide guidelines for such a decision? What other information would you suggest that they monitor and why?

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