Page 415 - A First Course In Stochastic Models

410 ALGORITHMIC ANALYSIS OF QUEUEING MODELS

(∞) (∞)
p (app) the approximation given in Theorem 9.6.1 to the state probability p
j j
in the infinite-capacity M/G/c queue. This approximation requires that ρ < 1. An
inspection of the recursion schemes in Theorems 9.6.1 and 9.8.1 reveals that, for
some constant γ ,

app (∞)
p = γp (app), j = 0, 1, . . . , N + c − 1. (9.8.1)
j j
(∞) −1

The constant γ is given by γ = [1 − ρ ∞ p (app)] . In the next section
j=N+c j
it will be seen that this proportionality relation implies
∞
(∞)
(1 − ρ) p (app)
j
app j=N+c
P = , (9.8.2)
rej ∞
(∞)
1 − ρ p (app)
j
j=N+c
app app
where P = p denotes the approximation to P rej . The computation of the
rej N+c
(∞)
probabilities p (app) was discussed in Section 9.6.2.
j
app (∞)
The approximations p and p (app) are exact both for the case of multiple
j j
servers with exponential service times and for the case of a single server with
general service times. Therefore relations (9.8.1) and (9.8.2) hold exactly for the
M/M/c/c +N queue and the M/G/1/N +1 queue. For these particular queueing
models the proportionality relation (9.8.1) can be directly explained by a simple
probabilistic argument. This will be done in the next subsection. It is noted that for
the general M/G/c/c + N queue the proportionality relation is not satisfied when
(∞)
the exact values of p j and p are taken instead of the approximate values.
j
9.8.2 A Basic Relation for the Rejection Probability
In this section a structural form will be revealed for the rejection probability. In
many situations the rejection probability can be expressed in terms of the state
(∞)
probabilities in the infinite-capacity model. In the following, p j and p denote
j
the time-average state probabilities for the finite-capacity model and the infinite-
(∞)
capacity model. To ensure the existence of the probabilities p , it is assumed
j
that the server utilization ρ is smaller than 1.
Theorem 9.8.2 Both for the M/M/c/c + N queue and the M/G/1/N + 1 queue
it holds that
(∞)
p j = γp , j = 0, 1, . . . , N + c − 1 (9.8.3)
j
∞ (∞) −1
for some constant γ > 0. The constant γ is given by γ = [1 − ρ j=N+c p j ]
and the rejection probability is given by