Page 363 - A First Course In Stochastic Models

358 ALGORITHMIC ANALYSIS OF QUEUEING MODELS

cycle is continuously distributed with a finite expectation. Thus, by Theorem 2.2.4,
lim t→∞ P {I (t) = 1} exists and equals E(D 1 )/E(L 1 ), where L 1 is the length of
one cycle and D 1 is the total amount of time in one cycle that a service is in
progress with a remaining service time larger than u. Denoting by N the number
of customers served in one cycle and using Wald’s equation, we find

∞ ∞
E(D 1 ) = E(N) (y − u)b(y) dy = E(N) {1 − B(y)} dy.
u u
By (9.2.27) and (9.2.28), E(N) = 1/(1 − ρ) and E(L 1 ) = 1/λ + E(S)/(1 − ρ).
This gives
lim P {the server is busy at time t and the remaining service time
t→∞
of the service in progress is larger than u}
∞

= λ {1 − B(y)} dy.
u
Noting that lim t→∞ P {the server is busy at time t} exists and equals ρ = λE(S),
the result (9.2.32) follows.

9.2.4 Work in System
Let the random variable V t be defined by
V t = the total amount of work that remains to be done on all
customers in the system at time t.

In other words, V t is the sum of the remaining service times of the customers in the
system at time t. The stochastic process {V t , t ≥ 0} is called the work-in-system
process or the virtual-delay process. Let
V ∞ (x) = lim P {V t ≤ x}, x ≥ 0.
t→∞
Also, V ∞ (x) is the long-run fraction of time that the work in system is no more
than x. By the PASTA property, it holds that V ∞ (x) is identical to the limiting
distribution function W q (x) of the waiting time of a customer when service is in
order of arrival. In particular, by (2.5.13),
∞ ρs − λ + λb (s)

∗
−sx
e {1 − V ∞ (x)} dx = , (9.2.33)
∗
0 s(s − λ + λb (s))
where b (s) is the Laplace transform of the service-time density b(x). For later
∗
purposes, we mention here the following additional relations for V ∞ (x):
x
′
V (x) = λV ∞ (x) − λ V ∞ (x − y)b(y) dy, x > 0, (9.2.34)
∞
0