Subexponential asymptotics for steady state tail probabilities in a single-server queue with regenerative input flow
Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 3, pp. 423-445
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The present paper is devoted to queueing systems with regenerative input flow in the presence of heavy tails. Our goal is to develop an asymptotics for the probability of the waiting time process in a stationary regime to exceed a high level. In this paper, we consider the total service time of customers arriving during the time-interval $[0,t]$ as an input flow $X(t)$. This allows us to consider a case when the service times $\{\eta_n\}_{n=1}^\infty$ are dependent random variables that, besides, may be dependent on a number of customers arriving in $[0,t]$. We obtain conditions for the virtual waiting time process in steady state to have a subexponential distribution function. We apply this result to a system with a Markov modulated semi-Markov input flow. We also consider a queue with a doubly stochastic Poisson flow in the case when the random intensity is a regenerative process. We show that these results could be transferred to corresponding systems with an unreliable server.
Keywords:
large deviations, regenerative flow, subexponential distributions, waiting timeю.
@article{TVP_2017_62_3_a0,
author = {S. Zh. Aibatov and L. G. Afanasyeva},
title = {Subexponential asymptotics for steady state tail probabilities in a single-server queue with regenerative input flow},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {423--445},
publisher = {mathdoc},
volume = {62},
number = {3},
year = {2017},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2017_62_3_a0/}
}
TY - JOUR AU - S. Zh. Aibatov AU - L. G. Afanasyeva TI - Subexponential asymptotics for steady state tail probabilities in a single-server queue with regenerative input flow JO - Teoriâ veroâtnostej i ee primeneniâ PY - 2017 SP - 423 EP - 445 VL - 62 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TVP_2017_62_3_a0/ LA - ru ID - TVP_2017_62_3_a0 ER -
%0 Journal Article %A S. Zh. Aibatov %A L. G. Afanasyeva %T Subexponential asymptotics for steady state tail probabilities in a single-server queue with regenerative input flow %J Teoriâ veroâtnostej i ee primeneniâ %D 2017 %P 423-445 %V 62 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/item/TVP_2017_62_3_a0/ %G ru %F TVP_2017_62_3_a0
S. Zh. Aibatov; L. G. Afanasyeva. Subexponential asymptotics for steady state tail probabilities in a single-server queue with regenerative input flow. Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 3, pp. 423-445. http://geodesic.mathdoc.fr/item/TVP_2017_62_3_a0/