# Stationary Markovian Arrival Processes, Results and Open Problems

## TL;DR

This paper investigates properties of Markovian arrival processes, especially MMPPs, exploring overdispersion, hazard rates, and stochastic orders, and highlights open problems and conjectures in the field.

## Contribution

It provides new results and counterexamples for properties of Markovian arrival processes, and formulates conjectures about their stochastic behavior.

## Key findings

- (I) Overdispersion holds for general MSPPs and two-state MMPPs.
- (II) Non-increasing hazard rate is false for general MMPPs due to counterexamples.
- (III) Variance-related property follows from stochastic order, but its validity is open for MMPPs.

## Abstract

We consider two classes of irreducible Markovian arrival processes specified by the matrices $C$ and $D$. The Markov Modulated Poison Process (MMPP) and the Markovian Switched Poison Process (MSPP). The former exhibits a diagonal $D$ while the latter exhibits a diagonal $C$. For these two classes, we consider the following statements: (I) Overdispersion of the counts process. (II) A non-increasing hazard rate of the stationary inter-event time. (III) The squared coefficient of variation of the event stationary process is greater or equal to unity. (IV) A stochastic order showing that the time stationary inter-arrival time dominates the event-stationary time. For general MSPPs and two-state MMPPs, we show that (I)-(IV) hold. Then for general MMPPs, it is easy to establish (I), while (II) is false due to a counter-example of Miklos Telek and Illes Horvath. For general simple point processes, (III) follows from (IV). For MMPPs we conjecture and numerically test that (IV) and thus (III) hold. Importantly, modeling folklore has often treated MMPPs as ``bursty'' and implicitly assumed that (III) holds. However, this is still an open question.