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 = P^{n} =
(λ/μ)^{n}*P_{o} = 0.6^{n}0.4

(E) Hence, Probability of more than 3 people in the system = 1 -
(P_{0} + P_{1} + P_{2} + P_{3})

= 1 - (0.4 + 0.6*0.4 + 0.6^{2}*0.4 +
0.6^{3}*0.4) = 0.129

(F) Probability of more than 4 people in the system = 1 -
(P_{0} + P_{1} + P_{2} + P_{3} +
P_{4})

= 1 - (0.4 + 0.6*0.4 + 0.6^{2}*0.4 + 0.6^{3}*0.4
+ 0.6^{3}*0.4) = 0.078

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