On deciding stability of constrained random walks and queueing systems

We consider in this paper two types of queueing systems which operate under a specific and fixed scheduling policy. The first system consists of a single server and several buffers in which arriving jobs are stored. We assume that arriving parts may require several stages of processing in which case each stage corresponds to a different buffer. The second system is a communication type queueing network given by a graph. The arriving jobs (packets) request a simple path along which they need to be processed. In both models the jobs arrive in a completely deterministic fashion: the interarrival times are fixed and known. All the processing times are also deterministic. A scheduling policy specifies a rule using which arriving parts are processed in the queueing system.A scheduling policy is defined to be stable if there is a finite uniform upper bound on the total number of parts in the system at all times. A necessary condition for stability of any work conserving policy is that the traffic intensity of the station (of each link in the graph in the communication model) is not bigger than one. Many results have demonstrated that this condition is not sufficient for stability. The results were obtained primarily in the context of stochastic networks ([24],[20],[5],[6],[10]), deterministic fluid networks ([6],[9],[7],[8],[2]), deterministic adversarial networks ([4],[1],[14],[12]).One of the earliest result in the area were obtained by Rybko and Stolyar [24] and Lu and Kumar [20]. They showed that a simple priority policy can lead to instability in some queueing networks. Bramson [5] and Seidman [25] showed that even FIFO policy can be unstable in stochastic networks. Instability of FIFO was later demonstrated in an adversarial queueing setting by Andrews et. al. [1]. Dai [6] established that stability of a fluid deterministic queueing network implies stability of a stochastic queueing network. A similar result was established by Gamarnik [12], which connects stability of fluid and adversarial queueing networks. A complete characterization of two-station fluid networks which are stable under any work conserving policy was established by Bertsimas, Gamarnik and Tsitsiklis [2] and Dai and Vande Vate [7]. Goel [14] constructed a complete characterization of adversarial queueing networks which are stable under the usual load condition. The result is extended by Gamarnik [13].Motivated by a queueing network model stability of homogeneous random walks in a nonnegative orthant was considered in several papers: Malyshev [21], Menshikov [23], Fayolle [11], Ignatyuk and Malyshev [18], Malyshev [22]. Such random walks Q(t),t = 0, 1, 2,i­ have Z+d as a state space (Z+ is the set of nonnegative integers). The transition vectors ¦¤ have deterministically bounded length in max norm and the transition probability p(¦«, ¦¤) along the direction ¦¤ depends only on the face ¦« that the random walk is currently on (the transition probabilities depend only on which coordinates of the current state are positive and which are zero). Such a random walk is defined to be stable if it is positive recurrent. We will also consider deterministic walks, for which p(¦«, ¦¤) is always zero or one (the transition deterministically depends on the face that the walk is currently on).A complete characterization of stable homogeneous random walks in Z+d for d iU 4 was obtained in Malyshev [21], Menshikov [23] and Ignatyuk and Malyshev [18], but no extension of this classification to higher dimensions has been obtained. Malyshev in [22] establishes a connection between homogeneous random walks and general dynamical systems on compact manifolds and shows that the difficulty of classifying stable random walks is of the same nature as the difficulty of understanding the dynamics of these dynamical systems. Specifically, the complicated dynamics precludes obtaining classification of stable random walks for d = 5.Thus despite many efforts the classification of stable random walks for general dimensions is not known. Likewise classification of stable policies in queueing systems is an open problem.