Page 417 - A First Course In Stochastic Models

412 ALGORITHMIC ANALYSIS OF QUEUEING MODELS

0≤ j ≤ N + c − 1. Next it follows from (9.8.5) that (9.8.3) holds with
E(T (∞) )
γ = .
E(T )
The proportionality relation (9.8.3) is the key to the proof of (9.8.4). We first
note that in the finite-capacity model the average number of busy servers equals
λ(1 − P rej )E(S) by Little’s formula. Writing λ(1 − P rej )E(S) as cρ(1 − P rej ), it
follows that
N+c c−1 c−1

cρ(1 − P rej ) = min(j, c)p j = jp j + c(1 − p j ).
j=0 j=0 j=0
Substituting (9.8.3) in this equation gives
c−1 c−1
(∞) (∞)

cρ(1 − P rej ) = γ jp + c(1 − γ p )
j j
j=0 j=0
c−1 ∞
(∞) (∞)
= γ jp + c[1 − γ (1 − p )]
j j
j=0 j=c
∞
(∞)

= γ min(j, c)p + c − cγ.
j
j=0
By Little’s formula, the average number of busy servers equals cρ in the infinite-
buffer model and so ∞ min(j, c)p (∞) = cρ. This leads to
j=0 j
cρ(1 − P rej ) = γ cρ + c − cγ.
Solving for γ gives
(1 − ρ)(γ − 1)
P rej = . (9.8.6)
ρ
Also, using the PASTA property,
N+c−1 N+c−1
(∞)
P rej = p N+c = 1 − p j = 1 − γ p
j
j=0 j=0
∞
(∞)
= 1 − γ [1 − p ]. (9.8.7)
j
j=N+c
]
By (9.8.6) and (9.8.7), γ = [1−ρ ∞ p (∞) −1 . Next the result (9.8.4) follows.
j=N+c j
It is important to point out that the assumption of a single server with general
service times or multiple servers with exponential service times was only used for
the proof of (9.8.3). The proof of (9.8.4) does not use this assumption, but is solely
based on the proportionality relation (9.8.3).