Page 452 - A First Course In Stochastic Models

B. USEFUL PROBABILITY DISTRIBUTIONS 447

1/µ 1 (1/µ 2 ). The hyperexponential density always has a coefficient of variation
of at least 1 and is unimodal with a maximum at t = 0. The failure rate func-
tion of the hyperexponential distribution is decreasing. The H 2 density has three
parameters and is therefore not uniquely determined by its first two moments. For
a two-moment fit, the H 2 density with balanced means is often used; that is, the
normalization p 1 /µ 1 = p 2 /µ 2 is used. The parameters of the H 2 density having
balanced means and fitting the first two moments of a positive random variable X
2
with c ≥ 1 are
X

2
1 c − 1 2p 1 2p 2
X
p 1 = 1 + 2 , p 2 = 1 − p 1 , µ 1 = , µ 2 = .
2 c + 1 E(X) E(X)
X
In the context of a Coxian-2 distribution we give below another normalization we
believe to be a more natural one. A three-moment fit by an H 2 density is not
always possible, but it is unique whenever it exists. An H 2 density can only be
fitted to the first three moments m 1 , m 2 and m 3 of a positive random variable X
2
3
2
with c > 1 when the requirement m 1 m 3 ≥ m is satisfied; see Whitt (1982). If
X 2 2
3
2
m 1 m 3 = m then the H 2 fit is the exponential density, otherwise the parameters
2 2
of the three-moment fit are given by
1 2 µ 1 (1 − µ 2 m 1 )
µ 1,2 = a 1 + a − 4a 2 , p 1 = , p 2 = 1 − p 1 ,
1
2 µ 1 − µ 2
2
1
2
3
where a 2 = (6m − 3m 2 )/( m − m 1 m 3 ) and a 1 = (1 + m 2 a 2 )/m 1 . The require-
1
2
2
2
3
2
ment m 1 m 3 ≥ m holds for both a gamma distributed and a lognormal distributed
2 2
2
random variable X with c > 1.
X
Coxian-2 distribution
The hyperexponential density requires that the weights p 1 and p 2 are non-negative.
However, in order that p 1 µ 1 exp(−µ 1 t) + p 2 µ 2 exp(−µ 2 t) represents a probability
density, it is not necessary to require that p 1 and p 2 are both non-negative. The
class of H 2 distributions can be shown to be a subclass of the class of so-called
Coxian-2 (C 2 ) distributions. A random variable X is said to be Coxian-2 distributed
if X can be represented as

X 1 + X 2 with probability b,
X =
X 1 with probability 1 − b,
where X 1 and X 2 are independent random variables having exponential distribu-
tions with respective means 1/µ 1 and 1/µ 2 . In words, the lifetime X first goes
through an exponential phase X 1 and then through a second exponential phase X 2
with probability b or it goes out with probability 1 − b; see Figure B.2. It can be
assumed without loss of generality that µ 1 ≥ µ 2 .