Page 378 - A First Course In Stochastic Models

THE GI /G/1 QUEUE 373

where η 1 and η 2 with 0 < η 1 < min(µ 1 , µ 2 ) ≤ η 2 are the roots of

∞
2 −xt
x − (µ 1 + µ 2 )x + µ 1 µ 2 − {µ 1 µ 2 − (1 − b)µ 1 x} e a(t) dt = 0. (9.5.6)
0
The function a(t) denotes the interarrival-time density and
2
a 1 = [−η η 2 + η 1 η 2 (µ 1 + µ 2 ) − η 2 µ 1 µ 2 ]/ [µ 1 µ 2 (η 1 − η 2 )]
1
2
a 2 = [η 1 η − η 1 η 2 (µ 1 + µ 2 ) + η 1 µ 1 µ 2 ]/ [µ 1 µ 2 (η 1 − η 2 )] .
2
A derivation of this explicit result can be found in Cohen (1982). In particular,
P delay and W q are given by

η 1 η 2 (µ 1 + µ 2 ) 1 1
P delay = 1 − and W q = − + + . (9.5.7)
µ 1 µ 2 µ 1 µ 2 η 1 η 2
Since the computation of the roots of a function of a single variable is standard fare
in numerical analysis, the above results are very easy to use for practical purposes.
Bisection is a safe and fast method to compute the roots.

9.5.3 The GI /Ph/1 Queue
The results in Section 9.5.2 can be extended to the GI/P h/1 queue with phase-

∞ −st
∗
type services. Let b (s) = e b(t) dt denote the Laplace transform of the
0
service-time density b(t). For phase-type service b (s) can be written as
∗
b 1 (s)
∗
b (s) =
b 2 (s)
for polynomials b 1 (s) and b 2 (s), where the degree of b 1 (s) is smaller than the
degree of b 2 (s). Let m be the degree of b 2 (s). It is no restriction to assume that
b 1 (s) and b 2 (s) have no common zeros and that the coefficient of s m in b 2 (s) is

∞ −st
∗
equal to 1. Also, let a (s) = e a(t) dt denote the Laplace transform of the
0
interarrival-time density a(t). It is assumed that a (s) and b 2 (s) have no common
∗
zero. In Cohen (1982) it has been proved that

m
∞ 1 b 2 (s)
−sx η i
e {1 − W q (x)} dx = 1 − , (9.5.8)
0 s b 2 (0) η i + s
i=1
where η 1 , . . . , η m are the roots of
b 2 (−s) − a (s)b 1 (−s) = 0 (9.5.9)
∗
in the right half-plane {s|Re(s) > 0}. Moreover,
m
1
P delay = 1 − η i (9.5.10)
b 2 (0)
i=1