A tailor specialises in ladies' dresses. The number of customers approaching the tailor appear to be Poisson distributed with a mean of 6 customers per hour. The tailor attends the customers on a first-come-first-served basis and the customers wait if the need be. The tailor can attend the customers at an average rate of 10 customers per hour with the service time exponentially distributed.

Required,

(a) Find the probability of the number of arrivals (0 through 5) during (i) a 15-minute interval, and (ii) a 30-minute interval.

(b) The utilisation parameter.

(c) The probability that the queuing system is idle.

(d) The average time that the tailor is free on a 10-hour working day.

(e) The probability associated with the number of customers (0 through 5) in the queuing system.

(f) What is the expected number of customers in the tailor shop?

(g) What is the expected number of customers waiting for tailor's services?

(h) What is the average length of queues that have at least one customer?

(i) How much time should a customer expect to spend in the queue?

0) What is the expected time a customer would spend in the tailor's shop?

(k) Assuming that n > 0 (i.e. customers are in the system) what is the probability that the waiting time

(excluding service time) of a customer in the queue shall be more than 10 minutes?

(I) Assuming that the customers are in the system, what is the probability that a customer shall be in the shop for more than 15 minutes?