Question

A fast food franchise is considering a drive-up window food-service operation. Assume that

customer arrivals follow a Poisson probability distribution with a mean arrival rate of 24 cars

per hour, and that service times follow an exponential probability distribution. Arriving

customers place orders at an intercom station at the back of the parking lot and then drive up

to the service window to pay for and receive their order. The following three service

alternatives are being considered:

a) A single-channel operation where one employee fills the order and takes the money from

the customer. The average service time for this alternative is 2 minutes.

b) A single-channel operation where one employee fills the order while a second employee

takes the money from the customer. The average service time for this alternative is 1.25

minutes.

c) A two-channel operation with two service windows and a single employee at each

window. The employee stationed at each window fills the order and takes the money

from the customer. The average service time for this alternative is 2 minutes for each

channel.

Compute the following operating characteristics for each alternative and recommend an

alternative design for the fast food franchise.

(i) What is the probability that there are no customers in the system?

(ii) What is the average number of cars waiting for service?

(iii)What is the average time a car waits for service?

(iv)What is the average time in the system?

(v) What is the average number of cars in the system?

(vi)What is the probability an arriving car will have to wait for service?

Answer 1

	Alternative (a)	Alternative (b)	Alternative (c)
Queue type	M/M/1	M/M/1	M/M/2
Average arrival rate, λ	24	24	24
Average service rate, μ	30	48	30
Number of servers, M	1	1	2
Probability of having no customer (P0)	0.2	0.5	0.429 **
Average number of cars waiting (Lq)	3.2	0.5	0.152 **
Average time a car waits for service (Wq) in minutes	8	1.25	0.38
Average time in the system (Ws) in minutes	10	2.5	2.38
Average number of cars in the system (Ls)	4	1	0.952
Probability an arriving car will have to wait (Pw)	0.8	0.5	0.228

^** Use infinite-source values of Lq and P0 from tables. The value of λ/μ = 24/30 = 0.80 and M=2. The excerpt is shown as follows:

Χμ Μ 0.80 3.200 200 2 0.152 429 0019 447 3

Calculations:

A. E Alternative (a) Alternative (c) Alternative (b) 4 м/м/1 м/м/1 M/M/2 Queue type 5 Average arrival rate, 6 24 24 24 60/2 60/1.25 60/2 7 Average service rate, u Number of servers, m 1 1 2. Probability of having no customer (P0) |-1-(D6/D7) 31-(С6/с7) 0.429 Average number of cars waiting (Lq) C6A2/(C7*(C7-C6)) D6A2/(D7* (D7-D6)) 10 0.152 Average time a car waits for service E10*60/E6 =C10*60/C6 D10*60/D6 11 Average time in the system (Ws) in minutes C13 60/C6 E13*60/E6 D13 60/D6 12. Average number of cars in the system (Ls) C10+(C6/C7) D10+(D6/D7) -E10+(E6/E7) 13 Probability an arriving car will have to wait (Pw) =C10*((C8 C7/C6)-1) =D10*((D8*D7/D6)-1) =E10*((E8*E7/E6)-1) 14 CO LO CO

Based on the operating characteristics, the third option is the best as far as the customer service is concerned.