Page 447 - A First Course In Stochastic Models

442 APPENDICES

F(t) of a gamma (α, λ) distributed random variable X is given by
1 λt −u α−1
F(t) = e u du, t ≥ 0.
Ŵ(α) 0
The latter integral is known as the incomplete gamma function. The class of gamma
distributions is closed in the following sense. If X 1 and X 2 are two independent ran-
dom variables that are gamma (α 1 , λ) and gamma (α 2 , λ) distributed, then X 1 +X 2
has a gamma (α 1 + α 2 , λ) distribution (an easy way to prove this is to use Laplace
transforms; see Appendix E). In particular, the sum of n independent random vari-
ables each having the same gamma (α, λ) distribution is gamma (nα, λ) distributed.
In queueing applications the gamma distribution is often used to model service-
time distributions and in inventory applications to model demand distributions. The
numerical evaluation of the gamma distribution function is hardly more difficult
than that of the standard normal distribution function. Fast numerical procedures
for the computation of the incomplete gamma function are widely available; see
for example Press et al. (1992).
The mean and the squared coefficient of variation of a gamma (α, λ) distributed
random variable X are given by
α 2 1
E(X) = and c = .
X
λ α
This result shows that a unique gamma distribution can be fitted to each positive
random variable with given first two moments. To characterize the shape and the
2
failure rate of the gamma density, we distinguish between the cases c < 1 (α > 1)
X
2
and c ≥ 1 (α ≤ 1). The gamma density is always unimodal; that is, the density
X
has only one maximum. For the case c 2 < 1 the density first increases to the
X
maximum at t = (α − 1)/λ > 0 and next decreases to zero as t → ∞, whereas
2
for the case c ≥ 1 the density has its maximum at t = 0 and thus decreases from
X
2
t = 0 onwards. The failure rate function is increasing from zero to λ if c < 1 and
X
2
2
is decreasing from infinity to zero if c > 1. The exponential distribution (c = 1)
X X
2
has a constant failure rate λ and is a natural boundary between the cases c < 1
X
2
and c > 1.
X
The Erlang distribution
The Erlang (E k ) distribution is a special case of the gamma distribution. For a pos-
itive integer k, the Erlang (k, λ) distribution is nothing else than the gamma (α, λ)
distribution with α = k. The probability density and the probability distribution
function of an Erlang (k, λ) distributed random variable X are
k k−1
λ t −λt k−1 −λt (λt) j

f (t) = e and F(t) = 1 − e , t ≥ 0.
(k − 1)! j!
j=0
The Erlang (k, λ) distribution has a very useful interpretation. An Erlang (k, λ)
distributed random variable X can be decomposed as the sum of k independent