The Grand Movie Theater has one box office clerk. On average, each customer that

Question

The Grand Movie Theater has one box office clerk. On average, each customer that comes to see a
movie can be sold its ticket at the rate of 6 per minute. For the theater's normal offerings of older
movies, customers arrive at the rate of 3 per minute. Assume arrivals follow the Poisson distribution
and service times follow the exponential distribution.
a. What is the average number of customers waiting in line?
b. What is the average time a customer spends in the waiting line?
c. What is the average number of customers in the system?
d. What is a customer's average time in the system?
e. What is the probability that someone will be buying tickets when an arrival occurs?

Explanation / Answer

= average number of arrivals per minute = 3
= average number of people served per minute = 6

(a)The average number of customers waiting in line is

Lq = 2 / ( - )

= 9 / 18 = 0.5

(b) The average time a customer spends in the waiting line is

Wq = Lq / = 0.5 / 3 = 0.1667

(c) The average number of customers in the system is

L = Lq + / = 0.5 + 0.5 = 1

(d) A customer's average time in the system is

W = Wq + 1 / = 0.1667 + 0.1667 = 0.3334

(e) The probability that noone will be buying tickets when an arrival occurs is

P0 = 1 – / = 1 - 3/6 = 0.5

The probability that someone will be buying tickets when an arrival occurs is

1 - P0 = 0.5

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