Page 420 - A First Course In Stochastic Models

FINITE-CAPACITY QUEUES 415

As before, let P rej denote the long-run fraction of customers who are rejected in
X
the finite-capacity model. For the M /G/c/c + N queue with partial rejection,
we approximate P rej by
(∞)
(1 − ρ)
N+c
P rej ≈ (∞) . (9.8.12)
1 − ρ
N+c
X
The approximation (9.8.12) to P rej holds exactly for the M /G/1/N queue with
X
partial rejection and the M /M/c/c + N queue with partial rejection. It is left to
the reader to verify that the proportionality relation (9.8.3) remains valid for these
special cases. In the proof of Theorem 9.8.2 one needs only to modify formula
X
(9.8.7). In the M /G/c/c + N model with partial rejection,
N+c ∞
1
P rej = p k (k + s − N − c)β s .
β
k=0 s=N+c−k+1
This result follows by noting that the fraction of customers rejected is the ratio of
the average number of customers rejected per batch and the average batch size.

Complete rejection
X
In the M /G/c/c + N queue with complete rejection it is no longer true that the
proportionality relation (9.8.3) holds for the case of a single server with general
service times and for the case of multiple servers with exponential service times.
(∞)
However, one might make the heuristic assumption that p j ≈ γp for 0 ≤ j ≤
j
N + c − 1. Exercise 9.19 is to verify that this heuristic assumption leads to the
approximation
 
N+c−1
(∞)
(1 − ρ)  1 − u 
j
j=0
P rej ≈   , (9.8.13)
N+c−1
(∞)
1 − ρ 1 − u 

j
j=0
where
j N+c−k
1
(∞) (∞)
u = p β s .
j k
β
k=0 s=j−k+1
A remarkable result is that for the case of a constant batch size Q with Q ≤
X
N + 1 the approximation (9.8.13) is exact for both the M /G/1/N + 1 queue
X
with complete rejection and the M /M/c/c + N queue with complete rejection;
see Exercises 9.20 and 9.21. In these cases with a constant batch size Q it holds
(∞)
that p j ≈ γp for any 0 ≤ j ≤ N + c − Q.
j