Question: . Show that the DF K(x) = P{X ,, x), X = Un - u ,, for the system G/M/1 is given by 00

. Show that the DF K(x) = P{X ,, ≤ x), X" = Un - u ,, for the system G/M/1 is given by 00 1

K(x) =

dA(u) {1 - e->(*+*), x <0, and 1-x

= 1 - ce -* x, x 2 0,

where c = v(0) = 0ex dA(x) is the IST of the interarrival-time distribution. Further show that K (x) has the partial lack of memory P[X] < y | X1 > x} = 1 - e-(y -* ), y > x > 0.
(Prabhu, 1980)

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