ИСТИНА

We study the stability conditions of the multi-server system in which each customer requires a random number of servers simultaneously, a random service time is identical at all occupied servers. We call this system a cluster model since it may be employed in description of modern multicore high performance clusters. Stability criterion of an M|M|s cluster model has been proved by the authors earlier and for a MAP|M|r in the recent work by E. Morozov and A. Rumyantsev. The main contribution of this work is an extension of the stability criterion to the cluster model with a regenerative input flow. The class of these flows contains the most of fundamental flows that are exploited in the queueing theory. Thus, semi-Markov, Markovmodulated, Markov arrival processes and others belong to this class. So we consider the system Reg|M|r. Our analysis is based on synchronization of the input flow X(t) and auxiliary service process Y(t) that is the number of served customers under assumption that there are always customers for service. Since service time has an exponential distribution this process turns out to be Markov-modulated one. It is shown that the intensities of X(t) and Y(t) are equal to the means of the increments of these processes on common regeneration periods for X(t) and Y(t). Moreover, Y(t) dominates the real service process (the real number of served customers). Basing on these estimates we obtain the stability criterion.