Phase-Type Distributions

Abstract

The class of distributions on [0, ∞) having a rational Laplace transform (i.e., a Laplace transform that is the fraction between two polynomials) will, for reasons that will become apparent in the next chapter, be referred to as matrix-exponential distributions. Within the class of matrix-exponential distributions there is the subclass of phase-type distributions, which are defined in terms of an underlying Markov jump process (or Markov chain). As opposed to a general matrix-exponential distribution, we can for a phase-type distribution use the behavior of the underlying Markov jump process (or chain) in the deduction of its properties and in applications. Deduction in which we condition on an underlying Markov process is often referred to as probabilistic reasoning, as opposed to deduction in the general class of matrix-exponential distributions, where more analytic methods may be necessary. The probabilistic reasoning provides both elegance and power to the theory of matrix-exponential methods and to applications in stochastic modeling.