Theory of two-parameter Markov chain with an application in warranty study

TL;DR

This paper extends classical Markov process theory to two-parameter models, providing a numerical method using double Laplace transforms and applying it to a warranty system with states of working and failure.

Contribution

It introduces an alternative numerical approach for two-parameter Markov processes using double Laplace transforms and demonstrates its application in a warranty study.

Findings

Derived transition density matrix for the warranty model

Calculated warranty costs using the proposed method

Validated the approach with an illustrative example

Abstract

In this paper we present the classical results of Kolmogorov's backward and forward equations to the case of a two-parameter Markov process. These equations relates the infinitesimal transition matrix of the two-parameter Markov process. However, solving these equations is not possible and we require a numerical procedure. In this paper, we give an alternative method by use of double Laplace transform of the transition probability matrix and of the infinitesimal transition matrix of the process. An illustrative example is presented for the method proposed. In this example, we consider a two-parameter warranty model, in which a system can be any of these states: working, failure. We calculate the transition density matrix of these states and also the cost of the warranty for the proposed model.