# The correlation function of a queue with Levy and Markov additive input

## TL;DR

This paper investigates the correlation function of queue workloads driven by Levy and Markov additive processes, proving conjectures, providing counterexamples, and analyzing decay rates and tail behaviors in different process settings.

## Contribution

It extends the analysis of correlation functions to spectrally two-sided Levy processes and Markov additive processes, including proofs, counterexamples, and decay rate identification.

## Key findings

- Correlation function properties extend to spectrally two-sided Levy processes with reflection.
- Counterexamples show correlation can be negative, decreasing, and concave for Markov additive processes.
- Decay rate analysis reveals tail behavior differences between Levy and Markov additive inputs.

## Abstract

Let $(Q_t)$ be a stationary workload process, and $r(t)$ the correlation coefficient of $Q_0$ and $Q_t$. In a series of previous papers (i) the transform of $r(\cdot)$ has been derived for the case that the driving process is spectrally-positive (sp) or spectrally-negative (sn) Levy, (ii) it has been shown that for sp-Levy and sn-Levy input $r(\cdot)$ is positive, decreasing, and convex, (iii) in case the driving Levy process is light-tailed (a condition that is automatically fulfilled in the sn case), the decay of the decay rate agrees with that of the tail of the busy period distribution. In the present paper we first prove the conjecture that property (ii) carries over to spectrally two-sided Levy processes; we do so for the case the Levy process is reflected at 0, and the case it is reflected at 0 and $K > 0$. Then we focus on queues fed by Markov additive processes (maps). We start by the establishing the counterpart of (i) for sp- and sn-maps. Then we refute property (ii) for maps: we construct examples in which the correlation coefficient can be (locally) negative, decreasing, and concave. Finally, in relation to (iii), we point out how to identify the decay rate of $r(\cdot)$ in the light-tailed map case, thus showing that the tail behavior of $r(\cdot)$ does not necessarily match that of the busy-period tail; singularities related to the transition rate matrix of the background Markov chain turn out to play a crucial role here.