Question

1. During peak times the entry gate at a large amusement park experiences an average arrival of 30 customers per minute, according to a Poisson distribution. The average customer requires 12 seconds to be processed through the entry gate (exponential distribution). The park’s goal is to keep the time in the system (waiting time plus time through the gate) to less than 90 seconds. What is the minimum number of gates that are necessary to meet this goal? (Note that the overall arrivals will be evenly spread among the gates, generating a separate line behind each gate.)

Answer 1

arrival rate lambda = 30 per min = 30x60 =1800 per hour

service rate mu = 60/12 =5 per min = 5x60 =300 per hour

rho =lambda /mu =1800/300 =6

Since each gate will have a seperate line, the incoming crowd will divide evenly forming a line ( each one has lambda = total arrivals /number of gates)

waiting time = 1 / ( mu-lambda)

= l / ( 300-lambda)

Max waiting time = 90 sec = 1.5 min = 1.5/60 =0.025 hour

0.025 = l / (300-lambda)

(300-lambda) = 40

which gives lambda = 260

Number of gates = 1800/260 =6.92 =7