Markovian Transition Counting Processes: An Alternative to Markov Modulated Poisson Processes

TL;DR

This paper introduces the Markovian transition counting process (MTCP) as an alternative to the widely used Markov Modulated Poisson Process (MMPP), demonstrating its advantages in moment matching and performance analysis.

Contribution

The paper establishes a duality between MTCPs and a class of MMPPs, showing MTCP's superiority in moment matching and applicability in queueing performance analysis.

Findings

MTCP can match the first and second moments of counts of MMPP.

Duality between MTCPs and slow MMPPs is established.

MTCP is a competitive model for queueing performance analysis.

Abstract

Stochastic models for performance analysis, optimization and control of queues hinge on a multitude of alternatives for input point processes. In case of bursty traffic, one very popular model is the \textit{Markov Modulated Poisson Process} (MMPP), however it is not the only option. Here, we introduce an alternative that we call \textit{Markovian transition counting process} (MTCP). The latter is a point process counting the number of transitions of a finite continuous-time Markov chain. For a given MTCP one can establish an MMPP with the same first and second moments of counts. In this paper, we show the other direction by establishing a duality in terms of first and second moments of counts between MTCPs and a rich class of MMPPs which we refer to as slow MMPPs (modulation is slower than the events). Such a duality confirms the applicability of the MTCP as an alternative to the MMPP which is superior when it comes to moment matching and finding the important measures of the inter-event process. We illustrate the use of such equivalence in a simple queueing example, showing that the MTCP is a comparable and competitive model for performance analysis.