6. Suppose we want to find the covariance between the times spent in the system by the first two customers in an M/M/1 queueing system. To obtain this covariance, let Si be the service time of customer i, i = 1, 2, and let Y be the time between the two arrivals.

(a) Argue that (S1 − Y)

+ + S2 is the amount of time that customer 2 spends in the system, where x+ = max(x, 0).

(b) Find Cov(S1, (S1 − Y)

+ + S2).

Hint: Compute both E[(S−Y)

+] and E[S1(S1 −Y)

+] by conditioning on whether S1 > Y.