In: Operations Management
Duque Vergere manages a Do or Die Theater complex
called Cinema I, II, III, and IV. Each of the four auditoriums
plays a different film; the schedule staggers starting times to
avoid the large crowds that would occur if all four 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 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, Duque Vergere wishes to examine
several queue-operating characteristics.
e.) What is the probability that there are more than two people in
the system? More than three people? More than four?
Given:
Arrival Rate,
= 210 / Hour
Service Rate,
= 280 /hour
For Single Server System:
Probability that there are zero customers in the system,
P0 = 1 - (
/
)
P0 = 1 - (210 / 280)
= 0.25
Probability that there are "n" customers in the system,
Pn = (
/
)n P0
P(1) = (210/280)1 (0.25) = 0.1875
P(2) = (210/280)2 (0.25) = 0.14
P(3) = (210/280)3 (0.25) = 0.105
P(4) = (210/280)4 (0.25) = 0.079
P(There are more than 2 customers in the system) = P( n >2 ) = 1 - P( n <= 2 )
= 1 - P(n=0) - P(n=1) - P(n = 2)
= 1 - 0.25 - 0.1875 - 0.14
= 0.4225
P(There are more than 3 customers in the system) = P( n >3 ) = 1 - P( n <= 3 )
= 1 - P(n=0) - P(n=1) - P(n = 2) - P(n=3)
= 1 - 0.25 - 0.1875 - 0.14 - 0.105
= 0.3175
P(There are more than 4 customers in the system) = P( n >4 ) = 1 - P( n <= 4 )
= 1 - P(n=0) - P(n=1) - P(n = 2) - P(n=3) - P(n=4)
= 1 - 0.25 - 0.1875 - 0.14 - 0.105 - 0.079
= 0.2385