Beate Klingenberg manages a Poughkeepsie, New York, movie theater complex called Cinema 8. Each of the eight auditoriums plays a different film; the schedule staggers starting times to avoid the large crowds that would occur if all eight movies started at the same time. The theater has a single ticket booth and a cashier who can maintain an average service rate of 280 patrons per hour. Service times are assumed to follow a negative exponential distribution. Arrivals on a normally active day are Poisson distributed and average 200 per hour.
To determine the efficiency of the current ticket operation, Beate wishes to examine several queues- operating characteristics.
a. Find the average number of moviegoers waiting in line to purchase a ticket.
b. What percentage of the time is the cashier busy?
c. What is the average time that a customer spends in the system?
d. What is the average time spent waiting in line to get to the ticket window?
e. What is the probability that there are more than two people in the system? More than three people? More than four?

A) 1.79

b) 71%

c) 0.75 minutes

d) 0.537 minutes

e) 0.343, 0.240 , 0.1681

Explanation:

L = average number of customers in the system ( 200 / 80 ) = 2.5

a = poission distribution per hour = 200

b = service rate of cashier = 280

A) average number of moviegoers waiting in line to purchase ticket

Lq = L - = 2.5 - (200/280) = 2.5 - 0.71 = 1.79

B) percentage of cashier been busy

p = a/b = 0.71 = 71%

C) average time spent by a customer in the system

w = L / a = 2.5 / 200 = 0.0125 hours = 0.75 minutes

D) average time spent waiting in line to get to the ticket window ?

W2 = Lq / a = 1.79 / 200 = 0.00895 hours = 0.537 minutes

E) probabilities of people in the system

i) more than two people

p ( x ≥ 2 ) = 1 - ( p0 + p1 + p2 ) = 1 - 0.657 = 0.343

more than three people

ii) p ( x ≥ 3 ) = 1 - (p0 + p1 + p2 + p3 ) = 1 - 0.7599 = 0.240

iii) more than four people

p ( x ≥ 4 ) = 1 - ( p0 - p1 + p2 + p3 + p4 ) = 1 - 0.8319 = 0.1681

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