.

(a) Consider an M/G/1 finite queue with total space capacity n. Let To be the delay with DF H(t) and LST H*(s), let u be the service time with DF B(t) and LST B*(s), and let A ,, (s) be the LST of ordinary busy period T and Ad(s) be the LST of delay busy period Ta.

Denote 4(s) = TO Pre-CRASH B(1)

00

(Ar)k Ux(s) = 1

- e-(2+s)! dH(t).

k!

Show that Ho(s)

A„(s) =

[1-EK=]uz(s) ]=h-k+] A,(s) - Et =, "(s) ]=]A;(s)]

and A,(3) = Do(S) + EDR(s) [] A,(s) + Eve(s) []a,(s).

j=n-k+1 kan j=1 For an ordinary busy period T, writing P = 1 - >

Px = wx(0), Pk, and k=0

b, = mean busy period of M/G/1, one gets

(b) For an M/M/1 queue, deduce, that 11-p"+1 bn = -
u 1-p n+1 = , A = u.

E(v) b = Po [bn-1 bn Po n 2. Assume that p < 1, then lim, b, exists. The generating function B(z) of (b) is given by B(z) = bnz" ZE(v) and B*(-z) z - E(T) = lim b,, = lim (12) B(z). 11- z 1 (Miller, 1975; see also Harris, 1971)