The parts department of a large automobile dealership has a counter used exclusively for their own service mechanics requesting parts. The length of time between requests can be modelled by an Exponential distribution that has a mean of five minutes. A parts clerk can handle requests at an average rate of 15 per hour, and this can be modelled by a Poisson distribution. Suppose there are two parts clerks at the counter.
a. On average, how many mechanics would be at the counter, including those being served?
b. If a mechanic has to wait, how long would the average wait be?
c. What is the probability that a mechanic would have to wait for service?
d. What percentage of time is a clerk idle?
e. If clerks represent a cost of $20 per hour and mechanics represent a cost of $30 per hour, how many clerks would be optimal in terms of minimizing total cost?