Page 453 - A First Course In Stochastic Models

448 APPENDICES

b
exp(m 1 ) exp(m 2 )

1 − b

Figure B.2 The Coxian distribution with two phases

A Coxian-2 distribution having parameters (b, µ 1 , µ 2 ) with µ 1 < µ 2 can be shown
to have the same probability density as the Coxian-2 distribution having parameters
(b , µ , µ ) with µ = µ 2 , µ = µ 1 and b = 1 − (1 − b)µ 1 /µ 2 . Assuming that
∗
∗
∗
∗
∗
∗
1 2 1 2
µ 1 ≥ µ 2 , the Coxian-2 distributed random variable X has the density
−µ 1 t −µ 2 t
p 1 µ 1 e + (1 − p 1 )µ 2 e if µ 1 = µ 2 ,
f (t) = −µ 1 t 2 −µ 1 t
p 1 µ 1 e + (1 − p 1 )µ te if µ 1 = µ 2 ,
1
where p 1 = 1 − bµ 1 /(µ 1 − µ 2 ) if µ 1 = µ 2 and p 1 = 1 − b if µ 1 = µ 2 . Thus the
class of H 2 densities is contained in the class of Coxian-2 densities. Note that the
H 2 distribution allows for two different but equivalent probabilistic interpretations.
The H 2 distribution can be interpreted in terms of exponential phases in parallel
and in terms of exponential phases in series.
The density of a Coxian-2 distributed random variable X always has a unimodal
shape. Moreover, it holds that
1
2
c ≥ ,
X
2
2
where c ≥ 1 only if the density has the form p 1 µ 1 exp(−µ 1 t) + p 2 µ 2 exp(−µ 2 t)
X
for non-negative p 1 and p 2 . The Coxian-2 density has three parameters (b, µ 1 , µ 2 ).
Hence an infinite number of Coxian-2 densities can in principle be used for a two-
2
moment fit to a random variable X with c > 1 (the E 2 density is the only possible
X 2
2
1
choice when c = ). A particularly useful choice for a two-moment match is the
X 2
Coxian-2 density with parameters

 
2 2 1 2 4 µ 2
c −
X
µ 1 =  1 + 2  , µ 2 = − µ 1 , b = {µ 1 E(X) − 1}.
E(X) c + 1 E(X) µ 1
X
This particular Coxian-2 density has the remarkable property that its third moment
is the same as that of the gamma density with mean E(X) and squared coefficient
2
of variation c . The unique Coxian-2 density having this property will therefore
X
be called the Coxian-2 density with gamma normalization. This normalization is a
natural one in many applications.