What is the probability that thereBeate Klingenberg manages a Poughkeepsie, New

Question

What is the probability that thereBeate 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 8 movies started at the same time. The theater has a single ticket booth and a cashier who can maintain an average service rate of 300 patrons per hour. Service times are assumed to follow an exponential distribution. Arrivals on a normally active day are Poisson distributed and average 210 per hour. To determine the efficiency of the current ticket operation, Beate wishes to examine several queue- 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?

What is the probability that there are more than three people in the system ?

What is the probability that there are More than four people in the system?

Explanation / Answer

Answer: Lambda = 210 patrons per hour

mu = 300 patrons per hour

Ls= average number of units (customers) in the system = 210/(300-210) = 210/90 = 2.33

Ws= average time a unit spends in the system = 1/90 = 0.011

Lq= average number of units waiting in the queue = 210^2 /300*(300-210) = 0.0155

Wq= average time a unit spends waiting in the queue = 210 /300*(300-210) = 0.0077

= utilization factor for the system = 210/300 = 0.7

Pn>k= probability of more than 2units in the system = (210/300)^2 = 0.49

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