On the Relation between Phase-Type Distributions and Positive Systems

Kyungsup Kim

doi:10.1155/2015/731261

Abstract

The relation between phase-type representation and positive system realization in both the discrete and continuous time is discussed. Using the Perron-Frobenius theorem of nonnegative matrix theory, a transformation from positive realization to phase-type realization is derived under the excitability condition. In order to explain the connection, some useful properties and characteristics such as irreducibility, excitability, transparency, and order reduction for positive realization and phase-type representation are discussed. In addition, the connection between the phase-type renewal process and the feedback positive system is discussed in the stabilization concept.

Highlights

Positive system problems have been developed in applications areas such as biological models, production systems, and economic applications
We note that the positive realization of the integration of a positive system is closely related to the representation of the phase-type distribution
A discrete phase-type (DPT) distribution is the distribution of the time until one absorbing state in a discrete-state discrete-time Markov chain (DTMC) with n transients states and one absorbing state [8, 21]

Summary

Introduction

Positive system problems have been developed in applications areas such as biological models, production systems, and economic applications. We will discuss the relationship between phase-type representation and positive realization by using the Perron-Frobenius theorem introduced in [10,11,12,13]. We will discuss the properties and characteristics, such as irreducibility, excitability, transparency, and stabilization introduced in [3, 8, 15,16,17,18]. The relation and the common characteristics between discrete phase-type distributions and discrete-time positive systems are discussed in a similar manner to that applied to the continuous case.

Preliminaries

Phase-Type Representation and the Positive Realization

Discrete Phase-Type Distributions and Discrete-Time Positive Systems

Conclusions