Question: A gas station has a single automated car wash. Cars arrive to the gas station according to a Poisson process with an average of 30 cars/h. One-third of the customers also want a car wash, that is, for every customer there is a 33.33% chance it needs a car wash.
The mean service rate is 10 cars/h. Although, there are several different washing programs to choose from, it is a bit of a stretch to say that the service times follow an exponential distribution. Still, as a first-cut analysis, management has decided to make this assumption and model the queuing system as a birth-and-death process. It has been observed that customers balk from the car wash queue when it increases in length. More precisely, the probability that a customer will balk is n/3 for n = 1, 2, 3 customers in the car wash system. If there are more than 3 customers in the system, no customers will join the queue.
a. Construct a rate diagram for this queuing process.
b. Formulate the balance equations and solve them to determine the steady-state probability distribution for the number of cars in the car wash.
c. What is the expected utilization of the car wash?
d. Determine the expected waiting time in line for those customers who join the queue.