Instruction:
McDonald's vs. Burger King - these two fast food chains use different waiting line design: Independent queue vs. pooled queue. To compare the two different queue systems on equal footing, let's assume that we pick a McDonald's store as an experiment site. Assume that the customer inter-arrival time has a mean value a= 2 min. which is also equal to its std.dev. (so CVa=1). It has 2 registers operated by 2 cashers and supported by a team of kitchen staff. The order processing time (from talking to the casher until receiving the food) is on average 3 minutes with a CVp=1. In the first experiment, suppose we set up the rails in advance (as in a BK store) and ask all customers who walked into the door to form a single queue. Then the first customer in the queue can go to any vacant register to order food. In the second experiment, suppose the customer arrival to the store stays the same as the above. However, we take away the rails in advance and ask customers to choose either register A or B upon their entry to the restaurant front door. Thus, there are two independent waiting lines. Suppose that all customers agree that line-hopping is not allowed after a customer chooses the register to join the waiting line. You can also assume that the CVa=1 in the second system. Please compare the mean waiting times (Tq) between two systems.
Answer:
1st case:
a = 2 min, CVa = 1, p = 3, CVp = 1, m = 2
u = (1/a)/(m/p) = (1/2)(2/3) = 0.75
Waiting time = (p/m)(u^[{2(m+1)}1/2 - 1]/1-u)[CVa2 + CVp2/2]
= (1.5) (0.75^ {(6)1/2 - 1}/0.25) (1 + 1/2)
= (1.5)(0.66)/0.25 = 3.96 minutes
2nd case:
a = 4 min, CVa = 1, p = 3, CVp = 1, m = 2
u = (1/a)/(m/p) = (1/4)(2/3) = 0.375
Waiting time = (p/m)(u^[{2(m+1)}1/2 - 1]/1-u)[CVa2 + CVp2/2]
= (1.5) (0.375^ {(6)1/2 - 1}/0.625) (1 + 1/2)
= (1.5)(0.24)/0.625 = 0.576 minutes