Invited talk

Stability analysis of regenerative queues: recent results

Prof. Evsey Morozov

Prof. Evsey Morozov

Institute of Applied Mathematical Research, Karelian
Research Centre, Russian Academy of Sciences
http://mathem.krc.karelia.ru/member.php?id=19&plang=e

Abstract

In this work, we present recent achievements in stability analysis of regenerative queueing systems. This analysis is based on renewal theory and a characterization of the limiting behavior of the forward regeneration time in the renewal process generated by regenerations of the basic (queueing) process. This characterization gives a straightforward way to establish positive recurrence (finiteness of mean regeneration cycle length). This approach (described in detail in [3]) turns out to be effective in the stabil- ity analysis of many queues including multiserver system with non-identical servers [1], a general retrial queue [2], the systems with optical buffers [4, 5], multi-server non-reliable discrete queues [6], cascade network [7]. An important feature of the approach is that, instead of the direct proof of positive recurrence, we show that the forward regeneration time does not go to infinity (in probability). The latter condition is typically much easier to verify. Then the mentioned characterization implies positive recurrence of the basic process. The verification consists of i) using a negative drift assumption to show that the basic process does not go to infinity; ii) using a regeneration assumption to show that the process, starting into a compact set, hits a regeneration point with a positive probability during a finite time. The approach can be applied to non-Markovian queues as well.

Refernces:
[1] E. Morozov, Weak regeneration in modeling of queueing processes. Queueing Systems, 46, 2004, 295-315.
[2] E. Morozov, A multiserver retrial queue: regenerative stability analysis. Queueing Systems, 56, 2007, 157-168.
[3] E. Morozov and R. Delgado. Stability analysis of regenerative queues, Automation and Remote control, Vol. 70, No. 12, 2009, 1977-1991.
[4] W. Rogiest, E. Morozov, D. Fiems, K. Laevens, and H. Bruneel. Stability of Single-Wavelength Optical Buffers. European Transactions on Telecommunications Vol. 21, No. 3, 2010, 202-212.
[5] E. Morozov, W. Rogiest, K. De Turk, D. Fiems, and H. Bruneel. Stability of multi-wavelength optical buffers with delay-oriented scheduling, Transactions on Emerging Telecommunication Technologies, Vol. 23, No. 3, 2012, 217 - 226.
[6] Evsey Morozov, Dieter Fiems, Herwig Bruneel. Stability analysis of multiserver discrete-time queueing systems with renewal-type server interruptions, Performance Evaluation, 68, 1261-1275, 2011.
[7] E. Morozov, B. Steyaert. Stability analysis of a two-station cascade queueing network, Annals of Operations Research, 202, 1, 135-160, 2013.

Brief biography of the Speaker

Evsey V. Morozov obtained the Ph.D. from Kiev Institute of Cybernetics in 1979, and the Dr. Scientist Degree in Mathematics in Moscow Institute of Control Sciences in 1996. He is the leading researcher at the Institute for Applied Mathematical Research of Karelian Research Centre, Russian Academy of Sciences and also full Professor in the Faculty of Mathematics of the Petrozavodsk State University (Russia). His research interests include stability analysis of queues and queueing networks, regenerative and rare event simulation, Gaussian queueing systems, fluid analysis of queues. He has published more than 120 papers on these and related topics. He had a numerous invitation research visits and gave intensive lecture courses on the stochastic modeling of telecommunication systems, in particular, at the Universities of Helsinki, Oulu, Kuopio (Finland), Pisa (Italy), Zaragoza, Navarra (Spain), Aizu (Japan).

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