Page 404 - A First Course In Stochastic Models

THE GI/G/c QUEUE 399

holds that
p j ∼ στ −j as j → ∞ (9.7.1)

and
σδ −δx
1 − W q (x) ∼ e as x → ∞. (9.7.2)
2 c−1
λ(τ − 1) τ
Assuming that the interarrival time and the service time have probability densities
a(x) and b(x), the constant δ is the unique solution to the characteristic equation
∞ ∞

−δx δy/c
e a(x) dx e b(y) dy = 1 (9.7.3)
0 0

∞ st
on the interval (0, B) with B = sup{s | 0 e {1 − B(ct)} dt < ∞}. The constant
τ (> 1) is given by
∞ −1

τ = e −δx a(x) dx . (9.7.4)
0
An explicit expression for the constant σ cannot be given in general. A proof of
the above asymptotic expansions is beyond the scope of this book. The asymp-
totic expansions were established by Takahashi (1981) for the case of a phase-
type interarrival-time distribution and a phase-type service-time distribution. How-
ever, the class of phase-type distributions is dense in the class of all probabil-
ity distributions on the non-negative axis. Thus, one might conjecture that the
asymptotic expansions hold for a general interarrival-time distribution and a gen-
eral service-time distribution provided that the service-time distribution is not
heavy-tailed.

Two-moment approximations
In this section we restrict ourselves to the particular models of the GI/M/c queue
with exponential services and the GI/D/c queue with deterministic services. These
models allow for a relatively simple algorithmic analysis. The results for these
models may serve as a basis for approximations to the complex GI/G/c queue.
Several performance measures P , such as the average queue length, the average
waiting time per customer and the (conditional) waiting-time percentiles, can be
approximated by using the familiar interpolation formula
2 2
P app = (1 − c )P GI/D/c + c P GI/M/c (9.7.5)
S S
2
provided c is not too large and the traffic load on the system is not very light.
S
In this formula P GI/D/c and P GI/M/c denote the exact values of the specific per-
formance measure for the special cases of the GI/D/c queue and the GI/M/c
queue with the same mean service time E(S). Table 9.7.1 gives for several values