On Occupation Time for On-Off Processes with Multiple Off-States

TL;DR

This paper derives exact distributions for the occupation times of specific off-states in Markov renewal on-off processes with multiple off-states, aiding understanding in various scientific fields.

Contribution

It provides the first explicit formulas for marginal and joint occupation time distributions in multi-off-state Markov renewal processes.

Findings

Exact marginal and joint distributions derived

Numerical confirmation through simulations

Special case with Levy-distributed holding times analyzed

Abstract

The need to model a Markov renewal on-off process with multiple off-states arise in many applications such as economics, physics, and engineering. Characterization of the occupation time of one specific off-state marginally or two off-states jointly is crucial to understanding such processes. We derive the exact marginal and joint distributions of the off-state occupation times. The theoretical results are confirmed numerically in a simulation study. A special case when all holding times have Levy distribution is considered for the possibility of simplification of the formulas.