Page 361 - A First Course In Stochastic Models

356 ALGORITHMIC ANALYSIS OF QUEUEING MODELS

Transient emptiness probability
The distribution of the length of the busy period is closely related to the transient
emptiness probability p 00 (t) defined by
p 00 (t) = P {no customers will be present at time t when
at the current epoch 0 the system is empty}
∗
for t ≥ 0. Defining the Laplace transform p (s) by
00

∞
∗ −st
p (s) = e p 00 (t) dt,
00
0
it holds that
1
∗
p (s) = . (9.2.30)
00
∗
λ + s − λβ (s)
The derivation is simple. By conditioning on the epoch of the first arrival and on
the length of the subsequent busy period, it is readily seen that
t
p 00 (t) = e −λt + h(t − x)λe −λx dx, t ≥ 0,
0
where
u
h(u) = p 00 (u − v)β(v) dv.
0
Taking the Laplace transform of both sides of the integral equation for p 00 (t) and
using the convolution formula (E.6) in Appendix E, we obtain
1 λ
∗ ∗ ∗
p (s) = + p (s)β (s).
00 00
s + λ s + λ
Solving this equation gives the desired result (9.2.30).

Waiting-time probabilities for LCFS service
Under the last-come first-served discipline (LCFS) the latest arrived customer enters
service when the server is free to start a new service. The LCFS discipline was
in fact used in the derivation of the Laplace transform of the busy period. It will
therefore be no surprise that under this service discipline the limiting distribution
of the waiting time of a customer can be related to the distribution of the length
of a busy period. Assuming the LCFS discipline, let D n be the delay in queue of
the nth arriving customer and let W q (x) = lim n→∞ P {D n ≤ x}. Then

∞ −sx 1 λ(1 − β (s))
∗

e {1 − W q (x)} dx = ρ − . (9.2.31)
∗
0 s s + λ − λβ (s)