Stability Analysis of a Multi-server Model with Simultaneous Service and a Regenerative Input Flow
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Stability Analysis of a Multi-server Model with Simultaneous Service and a Regenerative Input Flow Larisa Afanaseva1 · Elena Bashtova1 · Svetlana Grishunina1,2 Received: 30 September 2017 / Revised: 17 February 2019 / Accepted: 28 April 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We study the stability conditions of a multi-server queueing system in which each customer requires a random number of servers simultaneously. The input flow is assumed to be a regenerative one and random service times are identical for all occupied servers. The service time has a hypoexponential distribution which belongs to the class of phase-type distributions. We introduce an auxiliary queueing system in which there are always customers in the queue and define an auxiliary service process as the number of served customers in this system. Then we construct the sequence of common regeneration points for the regenerative input flow and the auxiliary service process. Based on the relationship between the real and the auxiliary service processes we obtain upper and lower estimates for the mean of the number of actually served customers during the common regeneration period. It allows us to deduce the stability criterion of the model under consideration. It turns out that the stability condition does not depend on the structure of the input flow. It only depends on the rate of this process. Keywords Stability criterion · Cluster systems · Regeneration · Queueing systems Mathematics Subject Classification (2010) 60K25
Larisa Afanaseva
[email protected] Elena Bashtova [email protected] Svetlana Grishunina [email protected] 1
Department of Probability Theory, Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, Moscow, Russia
2
Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, Moscow, Russia
Methodology and Computing in Applied Probability
1 Introduction This paper is devoted to the stability analysis of a multi-server queueing system in which each new customer requires a random number of servers simultaneously and random service times are identical across all occupied servers. For this model which has a regenerative input flow we deduce the stability criterion using the synchronization method. The most important feature of the system is the fact that a customer cannot begin getting served until all required servers are available. Therefore, servers may be idle even when there are customers waiting to be served. Queueing systems belonging to this class are found in a variety of applications (see e.g. Rumyantsev and Morozov 2015). The increasing interest to multi-server systems with simultaneous service is motivated by the modeling of high performance clusters (HPC) and cloud/distributed computing containing a huge number of servers working in parallel. The class of systems with simultaneous service can be divided into two subclasses. The first one is the class of systems with independent service times (the s
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