Page 421 - A First Course In Stochastic Models

416 ALGORITHMIC ANALYSIS OF QUEUEING MODELS

X
Table 9.8.3 The M /G/1/N + 1 queue with complete rejection
Geometric Two-point
c 2 N = 0 N = 50 N = 250 N = 0 N = 50 N = 250
S
0.1 app 8.99E-1 1.40E-2 1.62E-7 8.86E-1 8.88E-3 1.29E-8
exa 9.40E-1 1.59E-2 1.82E-7 8.86E-1 9.01E-3 1.31E-8

10 app 8.99E-1 6.09E-2 2.64E-4 8.86E-1 5.58E-2 1.79E-4
exa 9.40E-1 6.21E-2 2.68E-4 8.86E-1 5.55E-2 1.79E-4

X
Table 9.8.3 gives some numerical results for P rej in the M /G/1/N + 1 queue
with complete rejection. For the batch size we consider both the two-point distri-
bution P {X = 1} = P {X = 7} = 0.5 and the geometric distribution P {X = j} =
(1/4)(3/4) j−1 for j ≥ 1. In both cases the mean batch size β = 4. The service-
2
time distributions are the E 10 distribution (c = 0.1) and the H 2 distribution with
S
2
the gamma normalization (c = 10). The offered load ρ is taken equal to 0.8. The
S
results in Table 9.8.3 indicate that the approximation (9.8.13) performs quite well
for practical purposes.
Asymptotic expansion for P rej
For larger values of the buffer capacity N, the calculation of P rej can further be
simplified when an asymptotic expansion for the tail probabilities in the infinite-
(∞) (∞)

buffer model is known. If P rej = (1−ρ) ∞ π /[1−ρ ∞ π ] and
j=N+c j j=N+c j
(∞) j
an asymptotic expansion π ∼ ση as j → ∞ is known, then
j
(1 − ρ)ση N+c /(1 − η) N+c
P rej ≈ N+c ≈ (1 − ρ)ση /(1 − η) for large N.
1 − ρση /(1 − η)
X
To illustrate this, consider the single-server M /G/1/N + 1 queue with partial
j
X
rejection. For the M /G/1 queue the asymptotic expansion π (∞) ∼ ση as j →
j
∞ holds when the service time is not heavy-tailed, where the constants σ and
η = 1/τ are determined by the relations (9.3.5) and (9.3.6). When using the
asymptotic expansion one needs only to calculate the root of a non-linear equation
in a single variable.
Two-moment approximation
The practical applicability of the formulas for P rej stands or falls with the computa-
(∞)
tion of the state probabilities π . In some queueing models it is computationally
j
feasible to calculate these probabilities using embedded Markov chain analysis or
continuous-time Markov chain analysis. However, in many queueing models the
(∞)
exact computation of the state probabilities π is not practically feasible. This
j