Stability Analysis of a MAP/M/s Cluster Model by Matrix-Analytic Method

  • Evsey Morozov
  • Alexander RumyantsevEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9951)


In this paper, we study the stability conditions of the multiserver system in which each customer requires a random number of servers simultaneously and a random service time, identical at all occupied servers. We call it cluster model since it describes the dynamics of the modern multicore high performance clusters (HPC). Stability criterion of an M/M/s cluster model has been proved by the authors earlier. In this work we, again using the matrix-analytic approach, prove that the stability criterion of a more general MAP/M/s cluster model (with Markov Arrival Process) has the same form as for M/M/s system. We verify by simulation that this criterion (in an appropriate form) allows to delimit stability region of a MAP/PH/s cluster model with phase-type (PH) service time distribution. Finally, we discuss asymptotic results related to accelerated stability verification, as well as to the new method of accelerated regenerative estimation of the performance metrics.


Stability condition High performance cluster Map arrivals Simultaneous service multiserver system 



This research is partially supported by Russian Foundation for Basic Research, grants 15-07-02341, 15-07-02354, 15-07-02360, 15-29-07974, 16-07-00622 and the Program of Strategic Development of Petrozavodsk State University. The authors thank Udo Krieger for a few useful comments.


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© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Institute of Applied Mathematical ResearchKarelian Research Centre of RASPetrozavodskRussia
  2. 2.Petrozavodsk State UniversityPetrozavodskRussia

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