QMB Question?

Question

At Quality Made Beemers a clerk who is paid $15 an hour can process 80 orders an hour. On average the clerk receives 50 orders an hour. There is a waiting cost of $20 for orders held up in the process. Assume the arrival of orders follows the Poisson distribution and service times follow the exponential distribution. What is the total hourly cost?

Answer 1

From your question follows:

Arrival rate lambda = 50 orders/hour.
service rate mu = 1/Ts = 80 orders/hour (Ts is service time of one order).


From Queueing Theory (see Leonard Kleinrock book I) follows:

clerk utilization ro = lambda/mu = 5/8 = 0.625
average (soujourn) time in the system = Ts/(1-ro) = 1/30 hours
if the clerk is paid only while working (processing orders) he works ro hours/hour = 0.625. Therefore total cost per hour = clerk cost per hour + waiting cost in the process per hour = 0.625*15 + (1/30)*20 = 10.42 $/hour.

This solution is conditioned upon clarification from your question:
1. is the clerk paid only while processing orders (my case) or all the time?
2. is the waiting in the process only waiting to be processed or total time in the system: waiting (ro*Ts/(1-ro))+ processing (my case).