##### Question

In: Statistics and Probability

# 18-54. Have you ever heard someone repeat the contradictory statement, "The place is so crowded no...

18-54. Have you ever heard someone repeat the contradictory statement, "The place is so crowded no one goes there any more"? This statement can be interpreted as saying that the opportunity for balking increases with the increase in the number of customers seeking service. A possible platform for modeling this situation is to say that the arrival rate at the system decreases as the number of customers in the system increases. More specifically, we consider the simplified case of M&M Pool Club, where customers usually arrive in pairs to "shoot pool." The normal arrival rate is 6 pairs (of people) per hour. However, once the number of pairs in the pool hall exceeds 8, the arrival rate drops to 5 pairs per hour. The arrival process is assumed to follow the Poisson distribution. Each pair shoots pool for an exponential time with mean 30 minutes. The pool hall has a total of 5 tables and can accommodate no more than 12 pairs at any one time. Determine the following:

(a) The probability that customers will balk.

(b) The probability that all tables are in use.

(c) The average number of tables in use.

(d) The average number of pairs waiting for a pool table to be available.

## Solutions

##### Expert Solution

(a). 0.00135

(b). 0.2385

(c). 2.9768

(d). 0.2935

At first, we will find all probabilities were p0 is unknown. From sum=1 we will find p0. then one by one we will find all probabilities.

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