The performance measures analysis of Erlang distribution in solving phase type distribution

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 distribution of the time until absorption and it is widely used in queueing theories and other fields of applied probabilities with the used of generalized Erlang, Coxian, Hypo-exponential, and Hyper-exponential distributions. In this study, performance measures of phase type distribution using Erlang - r distribution, and mix Erlang –(r − 1) with Erlang - r distributions have been looked into, in order to provide meaningful study into the probability function, mean, k^th moment, variance, Laplace Stieltjes transform and squared coefficient of variation of phase type distribution. We began from 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, k^th  moment, mean and variance for the phase type distribution. The result of its variation and mean value, as well as the likelihood that the waiting period will exceed 12 time units are obtained on the waiting time until the fourth arrival with Poisson arrival process. And we demonstrate that, using the Erlang-r distribution, the squared coefficient of variation might have a variety of values. Also, by increasing the number of phases (r) and setting the parameter at each phase to be rμ. The variance goes to zero and the expectation stays at 1/μ in the limit as r→∞