Informatics and Applications
2018, Volume 12, Issue 2, pp 24-28
SUFFICIENT ERGODICITY CONDITIONS FOR PRIORITY QUEUES
- A. V. Mistryukov
- V. G. Ushakov
Abstract
Known results in ergodicity of priority queues are based on the assumption that interarrival times in each queue have exponential distribution. The aim of this paper is to relax this assumption, namely, to establish sufficient conditions of ergodicity for queues with two priority classes GI\GI 11 under assumption that interarrival times only in high priority class queue have exponentional distribution. Queues with nonpreemptive and different kinds of preemptive priority are considered. To formulate desired conditions, the authors use Lindley's recursion for waiting times of each priority class queue. Using Lyapunov-Foster criteria, the authors obtain sufficient conditions for a given recursion to be a Harris-ergodic Markov chain, meaning to have a unique invariant measure, to which its transition probabilities converge in total variation.
[+] References (8)
- Walters, P. 1982. An introduction to ergodic theory. New York, NY: Springer-Verlag. 259 p.
- Borovkov, A. A. 1999. Ergodichnost' i ustoichivost' sluchaynykh protsessov [Ergodicity and stability of stochastic processes]. Moscow: EditorialURSS. 450 p.
- Gnedenko, B. V., E. A. Danielyan, B. N. Dimitrov, G. P. Klimov, and V. F. Matveev. 1973. Prioritetnye siste- my obsluzhivaniya [Priority queueing systems]. Moscow: Izd-vo Moskovskogo un-ta. 448 p.
- Meyn, S.,and R. Tweedie. 1993. Markov chains and stochas-tic stability. New York, NY: Springer Verlag. 562 p.
- Asmussen, S. 1998. Subexponential asymptotics for stochastic processes: Extremal behavior, stationary distri-butions and first passage probabilities. Ann. Appl. Probab. 8:354-374.
- Cohen, J. W 1982. The single server queue. Amsterdam: North-Holland Publishing Co. 694 p.
- Asmussen, S. 2003. Applied probability and queues. New York, NY: Springer-Verlag. 439 p.
- Matveev, V. F, and V. G. Ushakov. 1984. Sistemy massovogo obsluzhivaniya [Queueing systems]. Moscow: Izd-vo Moskovskogo un-ta. 240 p.
[+] About this article
Title
SUFFICIENT ERGODICITY CONDITIONS FOR PRIORITY QUEUES
Journal
Informatics and Applications
2018, Volume 12, Issue 2, pp 24-28
Cover Date
2018-05-30
DOI
10.14357/19922264180204
Print ISSN
1992-2264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
head of the line priority; preemptive priority; ergodicity; the method of test functions; waiting time; Lindley recursion
Authors
A. V. Mistryukov  and V. G. Ushakov , 
Author Affiliations
Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, M.V. Lomonosov Moscow State University, 1-52 Leninskiye Gory, GSP-1, Moscow 119991, Russian Federation
Institute of Informatics Problems, Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
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