Question

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?

Answer 1

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