Question

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)

Answer 1

Arrival rate = λ = 180 / hour

Service Rate = μ = 300 / hour

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

Probability of n customers in the system = Pⁿ = (λ/μ)ⁿ*P_o = 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