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# Queuing Problem and Poisson Distribution

Help is needed with following:

A) Sal's International is a popular haircutting and styling salon near the campus of the University of New Orleans. Four barbers work full-time and spend an average of 15 minutes per customer. Customers arrive throughout the day at an average rate of 12 each hour. All arriving customers are assigned a waiting number. Arrivals tend to follow the Poisson distribution, while service time is exponentially distributed. Assuming an infinite population source, determine the following:
1. What is the average number of customers in the salon?
2. What is the average time that a customer spends in the salon?
3. What is the average time a customer spends waiting to be attended?
4. What is the average number of customers waiting to be attended?

(B) Sal is now considering changing the queuing characteristics of his salon. Upon arrival, instead of being assigned waiting numbers, customers will be able to choose the barbers they prefer. Assuming this selection does not change while the customers are waiting for their barbers to become available and the requests for each of the four barbers are evenly distributed, answer the following:

What is the average number of customers in the salon?
What is the average time that a customer spends in the salon?
What is the average time a customer spends waiting to be attended?
What is the average number of customers waiting to be attended?
(C) Explain why the results from parts (A) and (B) are different.

#### Solution Summary

In this solution a practical queuing situation is analyzed by using two queuing models (M/M/c and M/M/1). The general formulae for the various queue characteristics such as average number of customers, average waiting time etc are given and their values are computed for the present problem. Finally the performance of the process is compared by using the models.

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