Question

# Beate Klingenberg managers a Poughkeepsie, New York, movie theater complex called Cinema 8. Each of the...

Beate Klingenberg managers 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 300 patrons per hour. Service times are assumed to follow an exponential distribution. Arrivals on a normally active day are Poisson distribution and average 180 per hour.

E) Find the probability that there are more than three people in the system=_____ (round your response to three decimal places)

F) Find the probability that there are more than four people in the system=_____ (round your response to three decimal places)

Arrival rate = λ = 180 / hour

Service Rate = μ = 300 / hour

Probability of no customers in the system = Po = 1 - λ/μ = 1 - 180/300 = 0.4

Probability of n customers in the system = Pn = (λ/μ)n*Po = 0.6n0.4

(E) Hence, Probability of more than 3 people in the system = 1 - (P0 + P1 + P2 + P3)

= 1 - (0.4 + 0.6*0.4 + 0.62*0.4 + 0.63*0.4) = 0.129

(F) Probability of more than 4 people in the system = 1 - (P0 + P1 + P2 + P3 + P4)

= 1 - (0.4 + 0.6*0.4 + 0.62*0.4 + 0.63*0.4 + 0.63*0.4) = 0.078

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