Question

Columbia Pizza has one oven that can make a whole pie in about 10 minutes. Their late-night business among hungry college students is booming, and from 9 pm – 11 pm, they average 5 pie orders per hour. However, from 5 pm – 9 pm, they average only 3 pie orders per hour. Both order rates can be assumed arrive according to a Poisson distribution.

The owner of Columbia Pizza has discovered that if people have to wait for more than 20 minutes from when they get in the door to when they receive their pie, they are likely to simply go next door to University Falafel, meaning that Columbia Pizza would lose out on their $20 sale.

The owner is thinking of buying a second pizza oven and register (e.g. a second channel) that he thinks would cost $75 per night to operate (for the sake of our example, that is both the amortized cost of the oven and the pay of the person who is taking orders for the six hours that the pizza shop is open)

1. Calculate the following for the 5-9 shift and the 9-11 shift
   1. Average # of orders in the system
   2. Average time order spends in the system (minutes)
   3. Average # of orders in the queue
   4. Average waiting time in the queue (minutes)
   5. Utilization

Answer 1

Single server system

	5-9 shift	9-11 shift
Average arrival rate (/ hr.)	3	5
Average service rate (/ hr.)	6	6
Average # of orders in the system, Ls	1	5
Average time order spends in the system (minutes), Ws	20	60
Average # of orders in the queue, Lq	0.5	4.17
Average waiting time in the queue (minutes), Wq	10	50
Utilization	50.00%	83.33%

Calculations:

Two-server system

	5-9 shift	9-11 shift
Average arrival rate (/ hr.), λ	3	5
Average service rate (/ hr.), μ	6	6
P0	0.60	0.41	<-- use tables or QM software
Average # of orders in the queue, Lq	0.0333	0.175	<-- use tables or QM software
Average # of orders in the system, Ls	0.533	1.008
Average time order spends in the system (minutes), Ws	10.666	12.1012
Average waiting time in the queue (minutes), Wq	0.666	2.1012
Utilization	25.00%	41.67%