On deciding stability of constrained random walks and queueing systems

We consider in this paper two types of queueing systems whichoperate under a specific and fixed scheduling policy. The firstsystem consists of a single server and several buffers in whicharriving jobs are stored. We assume that arriving parts may requireseveral stages of processing in which case each stage correspondsto a different buffer. The second system is a communication typequeueing network given by a graph. The arriving jobs (packets)request a simple path along which they need to be processed. Inboth models the jobs arrive in a completely deterministic fashion:the interarrival times are fixed and known. All the processingtimes are also deterministic. A scheduling policy specifies a ruleusing which arriving parts are processed in the queueingsystem. A scheduling policy is defined to be stable if there is a finiteuniform upper bound on the total number of parts in the system atall times. A necessary condition for stability of any workconserving policy is that the traffic intensity of the station (ofeach link in the graph in the communication model) is not biggerthan one. Many results have demonstrated that this condition is notsufficient for stability. The results were obtained primarily inthe context of stochastic networks ([24],[20],[5],[6],[10]),deterministic fluid networks ([6],[9],[7],[8],[2]), deterministicadversarial networks ([4],[1],[14],[12]). One of the earliest result in the area were obtained by Rybkoand Stolyar [24] and Lu and Kumar [20]. They showed that a simplepriority policy can lead to instability in some queueing networks.Bramson [5] and Seidman [25] showed that even FIFO policy can beunstable in stochastic networks. Instability of FIFO was laterdemonstrated in an adversarial queueing setting by Andrews et. al.[1]. Dai [6] established that stability of a fluid deterministicqueueing network implies stability of a stochastic queueingnetwork. A similar result was established by Gamarnik [12], whichconnects stability of fluid and adversarial queueing networks. Acomplete characterization of two-station fluid networks which arestable under any work conserving policy was established byBertsimas, Gamarnik and Tsitsiklis [2] and Dai and Vande Vate [7]. Goel [14] constructed a complete characterization of adversarialqueueing networks which are stable under the usual load condition.The result is extended by Gamarnik [13]. Motivated by a queueing network model stability of homogeneousrandom walks in a nonnegative orthant was considered in severalpapers: Malyshev [21], Menshikov [23], Fayolle [11], Ignatyuk andMalyshev [18], Malyshev [22]. Such random walksQ(t),t = 0, 1, 2,¡­ haveZ+d as a state space(Z+ is the set of nonnegative integers). Thetransition vectors ¦¤ have deterministically boundedlength in max norm and the transition probabilityp(¦«, ¦¤) along the direction¦¤ depends only on the face ¦« that therandom walk is currently on (the transition probabilities dependonly on which coordinates of the current state are positive andwhich are zero). Such a random walk is defined to be stable if itis positive recurrent. We will also consider deterministic walks,for which p(¦«, ¦¤) is alwayszero or one (the transition deterministically depends on the facethat the walk is currently on). A complete characterization of stable homogeneous random walksin Z+d for d¡Ü 4 was obtained in Malyshev [21], Menshikov [23] andIgnatyuk and Malyshev [18], but no extension of this classificationto higher dimensions has been obtained. Malyshev in [22]establishes a connection between homogeneous random walks andgeneral dynamical systems on compact manifolds and shows that thedifficulty of classifying stable random walks is of the same natureas the difficulty of understanding the dynamics of these dynamicalsystems. Specifically, the complicated dynamics precludes obtainingclassification of stable random walks for d = 5. Thus despite many efforts the classification of stable randomwalks for general dimensions is not known. Likewise classificationof stable policies in queueing systems is an open problem.