STABILITY ANALYSIS OF REGENERATIVE QUEUES

One of the topics which were intensively studied in the last decade is the stability of non-Markovian queueing systems. It is well-known that stability is one of the hard and actual problems and requires refinement and laborious mathematical technique especially outside the limits of Markovian queues. Stability analysis establishes the region of predefined parameters where the stability of the basic process holds. Various notions of stability are applied. We mention weak and strong stability, Chen [4], global weak stability, global pathwise stability, Dai and Vande Vate [10], and so on. An effective and developed approach to stability analysis of a wide class queueing systems and networks is the fluid approximation. Among many works which treat this topic we mention Chen and Mandelbaum [6], Chen [4], Chen and Yao [5], Dai [7], Dai [8], Dai and Weiss [11], Dai and Vande Vate [10]. At the same time, the fluid approach is not direct in the sense that we study originally the stability/instability of the associated fluid limit model (and deal with deterministic fluid processes instead of original stochastic ones) to establish the similar property of the corresponding queueing process. The most recent overview on stability analysis methods (with focus on networks) is the paper [12]. Unlike the mentioned above approaches, our approach to the stability is based on the regeneration property of the basic queueing process [1, 32]. We focus on the regenerative queues since they have numerous applications (for instance, [30, 31]). Also the regeneration of Harris recurrent Markov chains extends an area of this approach, [1]. The notable monograph [16] contains detailed description of stability analysis of Markov chains. For Markovian setting, this approach has paralellism with the one described in