TY - JOUR

T1 - The correlation function of a queue with Lévy and Markov additive input

AU - Berkelmans, Wouter

AU - Cichocka, Agata

AU - Mandjes, Michel

PY - 2020/3/1

Y1 - 2020/3/1

N2 - Let (Qt)t∈R be a stationary workload process, and r(t) the correlation coefficient of Q0 and Qt. In a series of previous papers (i) the transform of r(⋅) has been derived for the case that the driving process is spectrally-positive (SP) or spectrally-negative (SN) Lévy, (ii) it has been shown that for SP-Lévy and SN-Lévy input r(⋅) is positive, decreasing, and convex, (iii) in case the driving Lévy 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 Lévy processes; we do so for the case the Lévy 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(⋅) in the light-tailed MAP case, thus showing that the tail behavior of r(⋅) 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.

AB - Let (Qt)t∈R be a stationary workload process, and r(t) the correlation coefficient of Q0 and Qt. In a series of previous papers (i) the transform of r(⋅) has been derived for the case that the driving process is spectrally-positive (SP) or spectrally-negative (SN) Lévy, (ii) it has been shown that for SP-Lévy and SN-Lévy input r(⋅) is positive, decreasing, and convex, (iii) in case the driving Lévy 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 Lévy processes; we do so for the case the Lévy 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(⋅) in the light-tailed MAP case, thus showing that the tail behavior of r(⋅) 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.

KW - Lévy processes

KW - Markov additive processes

KW - Reflection

KW - Workload

UR - http://www.scopus.com/inward/record.url?scp=85066468606&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066468606&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2019.05.015

DO - 10.1016/j.spa.2019.05.015

M3 - Article

AN - SCOPUS:85066468606

VL - 130

SP - 1713

EP - 1734

JO - Stochastic Processes and Their Applications

JF - Stochastic Processes and Their Applications

SN - 0304-4149

IS - 3

ER -