Regenerative processes

TL;DR

This paper reviews the theory of regenerative processes, highlighting their structure, generalizations, and connections to queues and Markov chains, with a focus on conditions for the existence of limiting distributions.

Contribution

It introduces a more general definition of regenerative processes allowing dependence between cycles and discusses their applications to queues and Markov chains.

Findings

Connection to queues and Markov chains

Conditions for limiting distribution existence

Extensions of the classical regenerative process definition

Abstract

We review the theory of regenerative processes, which are processes that can be intuitively seen as comprising of i.i.d.\ cycles. Although we focus on the classical definition, we present a more general definition that allows for some form of dependence between two adjacent cycles, and mention two further extensions of the second definition. We mention the connection of regenerative processes to the single-server queue, to multi-server queues and more generally to Harris ergodic Markov chains and processes. In the main theorem, we pay some attention to the conditions under which a limiting distribution exists and provide references that should serve as a starting point for the interested reader.