First Passage Moments of Finite-State Semi-Markov Processes

TL;DR

This paper derives analytical formulas and estimators for the expected first passage times to universal accessible states in finite-state semi-Markov processes, extending existing irreducible SMP results.

Contribution

It introduces a matrix expression for first passage moments to UA states and provides consistent estimators, expanding semi-Markov process analysis.

Findings

Derived analytical matrix expression for first passage moments to UA states.

Provided consistent estimators for these moments.

Extended irreducible SMP results to UA states.

Abstract

In this paper, we discuss the computation of first passage moments of a time-homogeneous semi-Markov process (SMP) with finite state space to certain of its states that possess the property of universal accessibility (UA). A UA state is one which is accessible from any other state of the SMP, but which may or may not connect back to one or more other states. An important characteristic of UA is that it is the state-level version of the oft-invoked process-level property of irreducibility. We adapt existing results for irreducible SMPs to the derivation of an analytical matrix expression for the first passage moments to a single UA state of the SMP. In addition, consistent estimators for these first passage moments are given.