Page 451 - A First Course In Stochastic Models

446 APPENDICES

the associated E k−1,k distribution fits the first two moments of a positive random
variable X provided that
1 2 1
≤ c ≤ .
X
k k − 1
We note that only coefficients of variation between 0 and 1 can be achieved by
mixtures of the E k−1,k type. Also, it is noteworthy that the E k−1,k density can be
shown to have an increasing failure rate.
Next we consider the E 1,k distribution, which is defined as a mixture of E 1 and
E k distributions with the same scale parameters. The density of the E 1,k distribution
has the form
t k−1
−µt k −µt
f (t) = pµe + (1 − p)µ e , t ≥ 0,
(k − 1)!
where 0 ≤ p ≤ 1. By choosing

2 2 2
2kc + k − 2 − k + 4 − 4kc
X X p + k(1 − p)
p = 2 and µ = ,
2(k − 1)(1 + c ) E(X)
X
the associated E 1,k distribution fits the first two moments of a positive random
variable X provided that
2
1 2 k + 4
≤ c ≤ .
X
k 4k
2
2
Hence the E 1,k distribution can also achieve values of c with c > 1.
X X
For use in applications the E k−1,k density is generally better suited than the E 1,k
density since the E k−1,k density is always unimodal and has a shape similar to the
frequently occurring gamma density. The E 1,k density may be useful in sensitivity
analysis. For both theoretical and practical purposes it is often easier to work with
mixtures of Erlangian distributions than with gamma distributions, since mixtures
of Erlangian distributions with the same scale parameters allow for the probabilistic
interpretation that they represent a random sum of independent exponentials with
the same means.

Hyperexponential distribution
A commonly used representation of a positive random variable with a coefficient of
variation greater than 1 is a mixture of two exponentials with different means. The
distribution of such a mixture is called a hyperexponential distribution of order 2,
an H 2 distribution. The density of the H 2 distribution has the form

−µ 1 t −µ 2 t
f (t) = p 1 µ 1 e + p 2 µ 2 e , t ≥ 0,
where 0 ≤ p 1 , p 2 ≤ 1. Note that always p 1 + p 2 = 1, since the density f (t)
represents a probability mass of 1. In words, a random variable having the H 2 den-
sity is distributed with probability p 1 (p 2 ) as an exponential variable with mean