The time required for a mechanic at a small auto shop to perform a state required safety and emissions car inspection follows an exponential distribution with an average of 12 minutes. Customers needing inspections arrive at the shop according to a Poisson process with an average of 3 per hour.
a. Use the M/M/1 model to determine (1) the average number of customers in the shop, (2) the average time that elapses from the time a customer arrives at the shop and the time he or she drives away with a new muffler, and (3) the fraction of time that there are no customers in the shop.
b. What is the probability that there are more than three customers in the muffler shop at any particular time?
c. Determine the average hourly cost for the shop to offer inspections if the cost of customer waiting time, in terms of customer dissatisfaction and lost goodwill, is $20 per hour of time spent waiting at the shop, and the mechanic is paid $12.50 per hour.
d. Determine whether or not it is worth replacing the mechanic with a faster mechanic who can do inspections at an average rate of 6 per hour, but requires a salary of $15 per hour.