The Analysis of Hypo-Exponential and Hyper-Exponential Distributions in Solving Performance Measures for General Phase Type Distributions

Abstract

The importance of phase-type distribution in modeling activities cannot be under emphasized when both a distribution's initial and second moments are accessible, or when the sequence of data points for computing moments is the information available. In continuous time process for an absorbing finite state Markov chain, the phase-type distribution can be thought of as the of the time until absorption, and it is widely used in queueing theories and other fields of applied probabilities. The common phase-type distributions are generalized Erlang, Coxian, Hypo-exponential, and Hyper-exponential distributions. In this study, performance measures of phase-type distribution using Hypo-exponential, and Hyper-exponential distributions have been looked into, in order to provide meaningful study into the probability function, mean, moment, variance, Laplace Stieltjes transform and squared coefficient of variation of phase type distribution. The study started by considering the tractability and memory less properties of exponential distribution, and since these properties are not enough, we examined the journey through a series of exponential phases to arrive at performance measures. Illustrative examples are demonstrated for various cases to arrive at various values for probability functions, Laplace Stieltjes transform, squared coefficient of variation, moment, mean and variance for the phase type distribution.