Page 418 - A First Course In Stochastic Models

FINITE-CAPACITY QUEUES 413

Table 9.8.2 Numerical results for the D/M/c/c + N queue
ρ = 0.8 ρ = 0.95
N = 0 N = 10 N = 25 N = 0 N = 50 N = 75
c = 5 app 1.02E-2 6.99E-4 6.59E-7 1.48E-1 1.39E-6 6.83E-9
exa 1.11E-2 7.49E-4 7.06E-7 1.59E-1 1.44E-6 6.74E-9
c = 25 app 1.59E-3 1.44E-4 1.37E-7 4.98E-2 7.54E-7 3.53E-9
exa 1.71E-3 1.55E-4 1.46E-7 5.23E-2 7.80E-7 3.65E-9
c = 100 app 2.16E-4 2.08E-6 1.97E-9 9.60E-3 1.94E-7 9.07E-10
exa 2.32E-4 2.23E-6 2.11E-9 9.96E-3 2.00E-7 9.39E-10

Interpretation of formula (9.8.4)
Define for the infinite-capacity M/G/c queue the tail probability
(∞)
= the long-run fraction of customers who find N + c or
N+c
more other customers present upon arrival.
(∞) ∞ (∞)
By the PASTA property = p , and so formula (9.8.4) can be
N+c j=N+c j
written in the more insightful form
(∞)
(1 − ρ) N+c
P rej = (∞) . (9.8.8)
1 − ρ
N+c
(∞)
Practitioners often use the tail probability from the infinite-capacity model as
N+c
an approximation to the rejection probability in the finite-capacity model. The for-
mula (9.8.8) shows that this is a poor approximation when ρ is not very small. The
(∞) −1
approximation differs by a factor (1−ρ) from the right-hand side of (9.8.8)
N+c
when N gets large. The improved approximation (9.8.8) is just as easy to use as
(∞)
the approximation . In queueing systems in which the proportionality relation
N+c
(∞) (∞)
(9.8.3) does not necessarily holds, the structural form (1 − ρ) /(1 − ρ )
N+c N+c
can nevertheless be used as an approximation to P rej . In Exercise 9.14 this will
be illustrated for the single-server queue with a Markov modulated arrival process.
(∞) (∞)
Here we illustrate the performance of the approximation (1−ρ) /(1−ρ )
N+c N+c
to the rejection probability in the D/M/c/c + N queue with deterministic arrivals.
Table 9.8.2 gives the approximate and exact values of P rej for several examples.
The numerical result shows an excellent performance of the approximation. In
all examples the approximate value of P rej is of the same order of magnitude
as the exact value. This is what is typically needed when a heuristic is used for
dimensioning purposes.
X
9.8.3 The M /G/c/c + N Queue with Batch Arrivals
X
Theorem 9.8.2 can be extended to the batch-arrival M /G/c/c +N queue. In this
model batches of customers arrive according to a Poisson process with rate λ and