Question

Question 2

A fast-food franchise is considering operating a drive-up window food-service operation. Assume that customer arrivals follow a Poisson probability distribution, with an 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 to the service window to pay for and receive their orders. The following three service alternatives are being considered:

• A single-channel operation in which one employee fills the order and takes the money from the customer. The average service time for this alternative is 2 minutes.
• A single-channel operation in which 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.
• A two-channel operation with two service windows and two employees. The employee stationed at each window fills the order and takes the money from customers arriving at the window. The average service time for this alternative is 2 minutes for each channel.

Answer the following questions and recommend an alternative design for the fast-food franchise:

1. What is the probability that no cars are in the system?
2. What is the average number of cars waiting for service?
3. What is the average number of cars in the system?
4. What is the average time a car waits for service?
5. What is the average time in the system?
6. What is the probability that an arriving car will have to wait for service?

The following cost information is available for the fast-food franchise above:

• Customer waiting time is valued at $25 per hour to reflect the fact that waiting time is costly to the fast-food business.
• The cost of each employee is $6.50 per hour.
• To account for equipment and space, an additional cost of $20 per hour is attributable to each channel.

What is the lowest-cost design for the fast-food business?

These are two question of queuing theory which I am not certain that I understand could you please work out and provide me with explanation as to how the answer was found.

Answer 1

Aveage Service "a -I tn ever a Minutes F 30 per hour. PY b bility are are no Cos-ton in 36 = (2n2)(30 36-24 = 3.2 (ii) Avease Eime a Gra uwoits fer Service, Service eime iAvrae tin in a sustem ws = (0 mins や Avevase no. ニ(CO/60/.aa Cs Scanned 너 CamScann

vi) -A evese Service Ode ,M-1 meu ery !、a5 minutes 8 per hour o.S Ci Averaou Gim a Core wts or Servite, wy La/A a- 24 hous

civ) +service - 1-3 (vI Average service roate, μ=t in every ?mnutes 20 Per ur o.c Service cs) C)Dro bability re are no stomers in CAA ystem

(1) -Averoe no.ct Core cwaitinm Lay Po 0 38) minuts = (1381760) 84 arnival = σ-2286 (e) Q2.86%