Question

Assume that for a gas and car wash station one car can be serviced at a time. The arrivals follow a Poisson probability distribution, with an arrival rate of 1 car every 10 minutes and the service times follow an exponential probability distribution, with a service rate of 8 cars per hour.

1. What is the probability that the station will be idle?
2. What is the average number of cars that will be waiting for service?
3. What is the average time a car will be waiting for service?
4. What is the average time a car will be at the gas and wash station?

Answer 1

Arrival time = 10 minutes

Arrival Rate = 1/ Arrival time

Arrival Rate ($\lambda$) = 1/10 cars per minute

Arrival Rate ($\lambda$) = 60/10 cars per hour

Arrival Rate ($\lambda$) = 6 cars per hour

Service Rate ($\mu$) = 8 cars per hour

(A)

Probability that the station will be idle = P

P=1-

P = 1 - (6/8)

P = 0.25

(B)

Average number of cars waiting for service, Lq

(Y — 1) b V

Lq = 2.25

Therefore, average number of cars waiting for service = 2.25 customers

(C)

Average time a car will be waiting for service, Wq

Wq= LO

Wq = 2.25/6 hours

Wq = (2.25 × 60)/6 minutes

Wq = 22.5 minutes

Average time a car will be waiting for service = 22.5 minutes

(D)

Average time a car spends in the system, W

W = 1/(8-6)

W = 0.5 hours

W = 0.5 × 60 minutes

W = 30 minutes

Average time a car waits at the gas and wash station = 30 minutes

Answer 2