39. Consider an M/G/1 system in which the first customer in a busy period has the service distribution G1 and all others have distribution G2. Let C denote the number of customers in a busy period, and let S denote the service time of a customer chosen at random.

Argue that

(a) a0 = P0 = 1 − λE[S].

(b) E[S] = a0E[S1] + (1 − a0)E[S2] where Si has distribution Gi.

(c) Use

(a) and

(b) to show that E[B], the expected length of a busy period, is given by E[B] =

E[S1]

1 − λE[S2]

(d) Find E[C].