Question $6$

Consider a car workshop $3$ mechanics. A customer will bring their car for maintenance service or for inspection by the mechanics. However, when a customer arrives when the workshop is full, the customer will leave without entering the queue. The workshop has enough space to accommodate $15$ cars at one time including the car being repaired. Customers arrival follows Poisson distribution with mean of one in every $10$ minutes. Each mechanics require an average time of $60$ minutes to perform an inspection or service the car. The actual time is found to be approximately exponentially distributed around this average.

a$)$ Find the average number of idle mechanics.

b$)$ Find the total number of loss of customers due to restricted waiting space in a day $(10$ hours operations$).$

c$)$ Find the probability that the arriving customer will get their machine repaired immediately.

d$)$ Find the probability that at least one mechanics is idle.

e$)$ Find the average number of machines waiting to be repaired.