Page 454 - A First Course In Stochastic Models

C. GENERATING FUNCTIONS 449

A two-stage process with negative probabilities
2
For c < 1 it is not possible to fit a Coxian-2 distribution to the first two moments
X 2
of the positive random variable X. A fit using an E k,k−1 distribution requires many
2
stages when c is close to zero and thus might be unattractive in (queueing) appli-
X
cations. A remarkable alternative involving two exponential stages was proposed
in Nojo and Watanabe (1987). The positive random variable X is approximated
through a two-stage process. The process starts in stage 1. It stays in stage 1 for
an exponentially distributed time with mean 1/γ . Upon completion of the sojourn
time in stage 1, the process expires with probability p 1 and moves to stage 2 with
probability 1 − p 1 . The sojourn time in stage 2 is also exponentially distributed
with the same mean 1/γ . Upon completion of the sojourn time in stage 2, the
process expires with probability p 2 and returns to stage 1 with probability 1 − p 2 .
In stage 1 the process starts anew. The idea is to approximate the random variable
X by the time until the process expires. Using results from Appendix E, it is not
difficult to verify that the Laplace transform of this lifetime is given by
2
γp 1 s + γ (p 1 + p 2 − p 1 p 2 )
∗
f (s) = .
2
2
s + 2γ s + γ (p 1 + p 2 − p 1 p 2 )
The moments of the lifetime are directly obtained from the Laplace transform
∗
f (s); see (E.2) in AppendixE. If c 2 X < 1 2 and the first three moments m 1 , m 2
3
2
and m 3 satisfy m 1 m 3 < m , it is nearly always possible to match the first three
2 2
∗
moments of f (s) with the first three moments of X by allowing for negative values
2
of p 1 and p 2 but requiring that γ > 0. This is particularly true when c = 0. A
X
surprising finding is that in many (queueing) applications excellent approximations
are obtained by replacing the random variable X through the two-stage process and
treating p 1 and p 2 as if they were probabilities.
APPENDIX C. GENERATING FUNCTIONS
The generating function (or z-transform) of a discrete probability distribution {p k ,
k = 0, 1, . . . } is defined by
∞
k
P (z) = p k z , |z| ≤ 1.
k=0
The variable z is usually taken as a real-valued variable, but in certain applications
it may be convenient to treat z as a complex-valued variable. It is easily verified
that the probability distribution {p k , k = 0, 1, . . . } can be recovered analytically
from the compressed function P (z) by
k !
1 d P (z) !
p k = ! , k = 0, 1, . . . . (C.1)
k! dz k ! z=0
The result (C.1) shows that a discrete probability distribution is uniquely determined
by its generating function. Also, the moments of the probability distribution {p k }