Bilateral phase‐type distributions

Abstract

In this article we define a class of distributions called bilateral phase type (BPH), and study its closure and computational properties. The class of BPH distributions is closed under convolution, negative convolution, and mixtures. The one-sided version of BPH, called generalized phase type (GPH), is also defined. The class of GPH distributions is strictly larger than the class of phase-type distributions introduced by Neuts, and is closed under convolution, negative convolution with nonnegativity condition, mixtures, and formation of cohere'nt systems. We give computational schemes to compute the resulting distributions from the above operations and extend them to analyze queueing processes. In particular, we present efficient algorithms to compute the steady-state and transient waiting times in GPH/GPH/ 1 queues and a simple algorithm to compute the steady-state waiting time in M/GPH/ 1 queues. INTRODUCTIONNeuts [ 171, using the properties of exponential, generalized Erlang, and hyperexponential distributions as a base, developed a phase-type probability distribution function. Through a sequence of several articles, he has established its closure and computational properties, which make its use very attractive in applied probability modeling (see [20] for a recent account of these results). The purpose of this article is to present two classes of distributions, called bilateral phase type and generalized phase type, and to establish their closure and computational properties. The class of generalized phase-type distributions is strictly larger than the class of phase-type distributions.The phase-type distribution of Neuts can be interpreted as a random sum of independent exponential random variables via the uniformization procedure [9; 18,p.78;23;24]. The stopping time for this sum has an associated discrete phase-type distribution. A natural extension of the phase-type distribution is to consider the sum of independent and identically distributed (iid) exponential random variables (rv) over an arbitrary random number. We will call such a sum a generalized phase-type random variable. Let (g(n))," be the probability distribution function of a random number L , and define where (En): are iid exponential rv's with mean l / A and Eo = 0 with probability (wp) 1. The survival function Fx of X is then *Present address:

First failure time of dependent parallel systems with safety periods

“…When F 0 , F, G and H are all absolutely continuous with support (0, o) one may use either the Laguerre transform (Keilson andNunn (1979), andSumita (1981)) or the generalised phase type (Shanthikumar (1985)) method to compute…”

A note on order statistics in the mixed Erlang case

“…). The reader is referred to Arnold and Balakrishnan (1989), 8 Cao and West (1997), Barakat and Abdelkader (2004), as well as references therein for a more complete discussion on the 9 topic. More specifically, an extensive body of literature exists on the moments of the order statistics; see, e.g., Barakat 10 and Abdelkader (2004), Nadarajah (2007) and references therein.…”

Modelling heavy-tails with phase-type scale mixture distributions

“…exponentially distributed random variables. This above considered BPH distributions form a subclass of the one defined by Shanthikumar (1983). However, the later generalization lost the nature of Markov jump process thus not admitted in this thesis.…”

“…Note that Shanthikumar (1983) also defined a BPH random variable as the one whose positive and negative parts can be represented as the sums of i.i.d. exponentially distributed random variables.…”