38. For the M/G/1 queue, let Xn denote the number in the system left behind by the nth departure.

(a) If Xn+1 =



Xn − 1 + Yn, if Xn 1 Yn, if Xn = 0 what does Yn represent?

(b) Rewrite the preceding as Xn+1 = Xn − 1 + Yn + δn (8.64)

where

δn =



1, if Xn = 0 0, if Xn 1 Take expectations and let n → ∞ in Equation (8.64) to obtain E[δ∞] = 1 − λE[S]

(c) Square both sides of Equation (8.64), take expectations, and then let n →

∞ to obtain E[X∞] =

λ2E[S2]

2(1 − λE[S])

+ λE[S]

(d) Argue that E[X∞], the average number as seen by a departure, is equal to L.