On deciding stability of scheduling policies in queueing systems

We investigate stability of certain scheduling POlicies in a queueing system. To the day no algorithmic characterization exists for checking stability of a given policy in a given queueing system. In this paper we propose a certain generalized priority policy and prove that the stability of this policy is algorithmically undecidable. To the best of our knowledge this is the first undecidability result in the area of stability of queueing systems. We conjecture that stability of other common policies like First-In-First-Out and priority policy is also an undecidable problem. We also prove that stability of a homogeneous random walk in Z_~ is undecidable. 1 I n t r o d u c t i o n . We consider in this paper two types ofqueueing 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 part 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. Common scheduling policies include First-InFirst-Out (FIFO), Last-In-First-Out (LIFO), LongestIn-System (LIS), Shortest-In-System (SIS), priority policy, etc. The priority policy is an example of state dependent policy the scheduling decision depends only on the current configuration of the queueing system and is independent of the past past configurations and past decision rules. FIFO, LIS and SIS on the other hand are not entirely state dependent. 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 Watson Research Center, IBM Yorktown Heights, NY 10598 gamaxnik ~watson.ibm.com. any 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 ([19],[16],[5],[6],[9]), deterministic fluid networks ([6],[8],[7],[2]), deterministic adversarial networks ([4],[1],[13],[11]). One of the earliest result in the area were obtained by Rybko and Stolyar [19] and Lu and Kumar [16]. They showed that a simple priority policy can lead to instability in some queueing networks even if the traffic intensity in each station is smaller than one. Bramson [5] and Seidman [20] showed that even FIFO policy can be instable 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 [11], 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 Berts'imas, Gamarnik and Tsitsiklis [2] and Dai and Vande Vate [7]. Goel [13] constructed a complete characterization of adversarial queueing networks which are stable under the usual load condition. The result is extended by Gamarnik [12]. Motivated by a queueing network model a stability of homogeneous random walks in nonnegative orthant was considered by Malyshev [17], Menshikov [18], Fayolle [10]. Such random walks have Z~_ as a state space (Z+ is the set of nonnegative integers). The transition vectors have deterministically bounded length in max norm and the transition probabilities depend only on the face that the random walk is currently on (that is 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. A complete characterization of stable homogeneous random walks in Z~ and Z~ was obtained by Malyshev [17] and Menshikov [18]. But no extension of this classification to higher dimensions has been obtained, to the best of our knowledge. Likewise, no explicit or algorithmic characterization is