Question

The following are the sample data sets from your class project about waiting line theories: Arrival: 3, 4, 2, 2, 1, 2, 1, 4 Service Time: 67, 55, 66, 64, 42, 37, 49, 60 The arrival data were number of cars arrived per every 5-minute interval, and the service data were the number of seconds spent on taking the corresponding order. The management is not satisfied with the current total amount of time (in seconds) that a customer has to spend in the overall service process and would like to see a reduction of 6% in that total time. If the management's goal is to achieved, what would the minimum expected service rate (# cars served in a 5-min interval) has to be? Give your answer to two significant places.

Answer 1

Average arrival rate = (3+4+2+2+1+2+1+4)/8 = 2.375 cars every 5-minute interval

Average service time = (67*3+55*4+66*2+64*2+42*1+37*2+49*1+60*4)/(3+4+2+2+1+2+1+4)

= 57.16 seconds

Average Service rate per 5-minute interval = 5*60/57.16 = 5.25 cars serviced every 5-minute interval

Average time a customer has to spend in the system (W) = 1/(service rate - arrival rate)

= 1/(5.25-2.375)

= 0.3478 fraction of 5-minute interval or 104.35 seconds

With 6% reduction in it, W = 0.3478*(1-.06) = 0.327 fraction of 5-minute interval

So, Minimum Expected service rate should be = arrival rate + 1/W

= 2.375+1/.327

5.43 cars served in a 5-min interval