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The correlation function of a queue with Lévy and Markov additive input
- Source :
- Stochastic Processes and Their Applications, 130(3), 1713-1734. Elsevier, Stochastic Processes and their Applications, 130(3), 1713-1734. Elsevier, Berkelmans, W, Cichocka, A & Mandjes, M 2020, ' The correlation function of a queue with Lévy and Markov additive input ', Stochastic Processes and Their Applications, vol. 130, no. 3, pp. 1713-1734 . https://doi.org/10.1016/j.spa.2019.05.015
- Publication Year :
- 2020
- Publisher :
- Elsevier, 2020.
-
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.<br />20 pages
- Subjects :
- Statistics and Probability
Correlation coefficient
Reflection
Workload
Transition rate matrix
01 natural sciences
Lévy process
Combinatorics
010104 statistics & probability
Correlation function
60K25
FOS: Mathematics
Markov additive processes
0101 mathematics
Mathematics
Conjecture
Series (mathematics)
Markov chain
Applied Mathematics
Probability (math.PR)
010102 general mathematics
Distribution (mathematics)
Lévy processes
Modeling and Simulation
Mathematics - Probability
Subjects
Details
- Language :
- English
- ISSN :
- 03044149
- Volume :
- 130
- Issue :
- 3
- Database :
- OpenAIRE
- Journal :
- Stochastic Processes and their Applications
- Accession number :
- edsair.doi.dedup.....001bddcd7b8a6e23659235f0f72f90a1