Page 414 - A First Course In Stochastic Models

FINITE-CAPACITY QUEUES 409

Table 9.8.1 Numerical results for P rej in the M/G/c/c + N queue (c = 5).
ρ = 0.5 ρ = 0.8 ρ = 1.5
c 2 N = 1 N = 5 N = 1 N = 5 N = 1 N = 5
S
0 app 0.0286 0.00036 0.1221 0.0179 0.3858 0.3348
exa [0.0281– [0.00032– [0.1212– [0.0168– [0.3854– [0.3332–
0.0293] 0.00038] 0.1236] 0.0182] 0.3886] 0.3372]
1
2 app 0.0311 0.0010 0.1306 0.0308 0.3975 0.3395
exa 0.0314 0.0010 0.1318 0.0314 0.4000 0.3400
2 app 0.0370 0.0046 0.1450 0.0603 0.4114 0.3555
exa 0.0366 0.0044 0.1435 0.0587 0.4092 0.3537

N+c−1
app app app
p = ρp − (1 − ρ) p , j = N + c,
j c−1 k
k=c
where ρ = λE(S)/c and the constants a n and b n are the same as in Theorem 9.6.1.

Proof The proof of the theorem is a minor modification of the proof of Theo-
rem 9.6.1. The details are left to the reader.

The result of Theorem 9.8.1 is exact for both the case of multiple servers with
exponential service times and the case of a single server with general service
times, since for these two special cases the approximation assumption holds exactly.
Further support for the approximate result of the theorem is provided by the fact
that the approximation is exact for the case of no waiting room (N = 0).
Numerical investigations indicate that the approximation for the state proba-
bilities is accurate enough for practical purposes. Table 9.8.1 gives the exact and
approximate values of the rejection probability P rej for several examples. The prob-
ability P rej denotes the long-run fraction of customers who are rejected. By the
PASTA property,
P rej = p N+c .
2
In all examples we take c = 5 servers. Deterministic services (c = 0), E 2 services
S
2
1
2
(c = ) and H 2 services with gamma normalization (c = 2) are considered. For
S 2 S
the latter two services, the exact values of P rej are taken from the tabulations of
Seelen et al. (1985). For deterministic services, computer simulation was used to
find P rej . In the table we give the 95% confidence intervals. It is interesting to
point out that the results in Table 9.8.1 support the long-standing conjecture for the
GI/G/c/c + N queue that P rej → 1 − 1/ρ as N → ∞ when ρ > 1.
A proportionality relation
For the case of ρ < 1 the computational work can be considerably reduced when
the approximation to P rej must be computed for several values of N. Denote by