Page 396 - A First Course In Stochastic Models

MULTI-SERVER QUEUES WITH POISSON INPUT 391

where τ is the unique solution to the equation
∞ 1

e −λ(1−τ)t {1 − B(ct)} dt = (9.6.26)
0 λ
on the interval (1, 1 + B/λ). The constant σ app is given by
app c−1
∞ −λ(1−τ)t c−1
p τ e {1 − B e (t)} {1 − B(t)}dt
c−1 0
σ app =
. (9.6.27)
λ ∞ te −λ(1−τ)t {1 − B(ct)} dt
0
In Section 9.7 we give asymptotic expansions for the state probabilities and the
waiting-time probabilities in the general GI/G/c queue. Using equation (9.6.26)
app app
and equation (9.7.4), it is not difficult to verify that p /p is asymptoti-
j j−1
cally exact as j → ∞. Also, an approximation to the asymptotic expansion of
the waiting-time probabilities can be given. Using (9.6.25) and (9.7.1) to (9.7.4),
we find
1 − W q (x) ∼ γ e −λ(τ−1)x as x → ∞, (9.6.28)

where an approximation to γ is given by
σ app
γ app = c−1 . (9.6.29)
(τ − 1)τ
Two-moment approximations for the waiting-time percentiles
It is convenient to work with the percentiles η(p) of the waiting-time distribution
of the delayed customers. The percentiles η(p) are defined for all 0 ≤ p < 1; see
Section 9.2.2. Just as in the M/G/1 case, we suggest the first-order approximation
1 2
η app1 (p) = (1 + c )η exp (p) (9.6.30)
S
2
and the second-order approximation
2
2
η app2 (p) = (1 − c )η det (p) + c η exp (p), (9.6.31)
S S
where η exp (p) and η det (p) are the corresponding percentiles for the M/M/c queue
and the M/D/c queue. Both approximations require that the squared coefficient
2
of variation of the service time is not too large (say, 0 ≤ c ≤ 2) and the traffic
S
load on the system is not very small. In the multi-server case the fraction of
time that all servers are busy is an appropriate measure for the traffic load on the
system. This fraction is given by P delay . The second-order approximation (9.6.31)
performs quite satisfactorily for all parameter values. The simple approximation
(9.6.30) is only useful for quick engineering calculations when P delay is not small
and p is sufficiently close to 1 (say, p > 1 − P delay ). Table 9.6.2 gives for several
examples the exact value and the approximate values (9.6.30) and (9.6.31) for the
conditional waiting-time percentiles. It also includes the asymptotic value based on
2
the approximation (9.6.28). We consider the cases of E 2 services (c = 0.5) and
S
2
H 2 services with gamma normalization (c = 2).
S