. M/M/I-PS: conditional delay Let F(x; t) = Pr {D ≤ x| S ={} be the DF of the conditional delay, given that total service requirement is of length t, and let F* (s; t) be its LST. Then, when p < 1,

(1-p)(1- pr2) exp{-À(1-r){}

F*(s; t) =

{(1 - pr2} - p(1 - r)2 exp { ="(1-pr2)t ]

where r is the (smaller) root of xx2-(Q++s)x + p = 0.

Deduce that pt E{DIS = = 1-P and 2pt 20 var{ D| S = t} =

x [1 - expl-(1 - p)ut].

Lu(1-p)3 u(1-p)4 Compare E{D|S = t} for M/M/1-PS with average queueing time E { Wq} for the M/ M/1-FIFO discipline.

Show that for t < 1/u, E{D|S={} < E{Wq}, that is, for arrivals whose service-time requirement t is less than the average service time 1 /u, the mean delay under the PS discipline is less than the mean delay (queueing time) under the FIFO discipline.

Show that, for p < 1, pt

[n(1- p) - p)] []- expl-(1-p)pt}]

E{D| S =t, N=n}=

1- p u(1 - p)2 where N is the number in the system in an M/ M/1-PS system.
Verify that 00 5 E{D|S= t,N=n)Pr{N=n} = E{D|S =t).
R=0 Show further that lim E{D\ S = t, N = n} = nt +
2 p-+ 1 (Coffman et al ., 1970)