A queueing system has two servers whose service times are independent random variables with an exponential distribution with a mean of 15 minutes. Customer X arrives when both servers are idle. Five minutes later, customer Y arrives and customer X still is being served. Another 10 minutes later, customer Z arrives and both customers X and Y still are being served. No other customers arrived during this 15-minute interval.

(a) What is the probability that customer X will complete service before customer Y?

(b) What is the probability that customer Z will complete service before customer X?

(c) What is the probability that customer Z will complete service before customer Y?

(d) Determine the cumulative distribution function of the waiting time in the system for customer X. Also determine the mean and standard deviation.

(e) Repeat part

(d) for customer Y.

(f) Determine the expected value and standard deviation of the waiting time in the system for customer Z.

(g) Determine the probability of exactly 2 more customers arriving during the next 15-minute interval.