It is now known that the usual traffic condition (the nominal load being less than one at each station) is not sufficient for stability for a multiclass open queueing network. Although there has been some progress in establishing the stability conditions for a multiclass network, there is no unified approach to this problem. In this paper, we prove that a queueing network is positive Harris recurrent if the corresponding fluid limit model eventually reaches zero and stays there regardless of the initial system configuration. As an application of the result, we prove that single class networks, multiclass feedforward networks and first-buffer-first-served preemptive resume discipline in a re-entrant line are positive Harris recurrent under the usual traffic condition. AMS 1991 subject classification: Primary 60K25, 90B22; Secondary 60K20, 90B35. Key words and phrases: multiclass queueing networks, Harris positive recurrent, stability, fluid approximation Running title: Stability of mu...

Abstract. Part II continues the development of policy synthesis techniques for multiclass queueing networks based upon a linear fluid model. The following are shown: (i) A relaxation of the fluid model based on workload leads to an optimization problem of lower dimension. An analogous workload-relaxation is introduced for the stochastic model. These relaxed control problems admit pointwise optimal solutions in many instances. (ii) A translation to the original fluid model is almost optimal, with vanishing relative error as the networkload ρ approaches one. It is pointwise optimal after a short transient period, provided a pointwise optimal solution exists for the relaxed control problem. (iii) A translation of the optimal policy for the fluid model provides a policy for the stochastic networkmodel that is almost optimal in heavy traffic, over all solutions to the relaxed stochastic model, again with vanishing relative error. The regret is of order | log(1 − ρ)|.

Algorithms for perfect or exact simulation of random samples from the invariant measure of a Markov chain have received considerable recent attention following the introduction of the "coupling-from-the-past" (CFTP) technique of Propp and Wilson. Here we place such algorithms in the context of backward coupling of stochastically recursive sequences. We show that although general backward couplings can be constructed for chains with finite mean forward coupling times, and can even be thought of as extending the classical "Loynes schemes" from queueing theory, successful "vertical" CFTP algorithms such as those of Propp and Wilson can be constructed if and only if the chain is uniformly geometric ergodic. We also relate the convergence moments for backward coupling methods to those of forward coupling times: the former typically lose at most one moment compared to the latter. Work supported in part by NSF Grant DMS-9504561 and by CRDF Grant RM1-226 y Postal Address: Institute of Math...

Using stochastic dominance, in this paper we provide a new characterization of point processes. This characterization leads to a unified proof for various stability results of open Jackson networks where service times are i.i.d. with a general distribution, external interarrival times are i.i.d. with a general distribution and the routing is Bernoulli. We show that if the traffic condition is satisfied, i.e., the input rate is smaller than the service rate at each queue, then the queue length process (the number of customers at each queue) is tight. Under the traffic condition, the p th moment of the queue length process is bounded for all t if the p+1 th moment of the service times at all queues are nite. If, furthermore, the moment generating functions of the service times at all queues exist, then all the moments of the queue length process are bounded for all t. When the interarrival times are unbounded and non-lattice (resp. spread-out), the queue lengths and the remaining service times converge in distribution (resp. in total variation) to a steady state. Also, the moments converge if the corresponding moment conditions are satisfied.

We consider the problem of determining the optimal policy for staffing a queueing system over multiple periods, using a model that takes into account transient queueing effects. Formulating the problem in a dynamic programming setting, we show that the optimal policy follows a monotone optimal control by establishing the submodularity of the objective function with respect to the staffing level and initial queue size in a period. In particular, this requires proving that the system occupancy in a G=M=s queue is submodular in the number of servers and initial system occupancy. Keywords: monotone optimal control, submodularity, transient queues, G=M=s queue, dynamic programming, staffing, service operations Monotone Optimal Policies for a Transient Queueing Staffing Problem In Yoo (1996), a dynamic programming (DP) model is formulated to address the problem of setting staffing levels at a post office's service windows over multiple time periods in a day. A major component of the model...

This paper gives a survey of recent results on generalized Jackson networks, where classical exponential or i.i.d. assumptions on services and routings are replaced by stationary and ergodic assumptions. We first show that the most basic features of the network may exhibit unexpected behavior. Several probabilistic properties are then discussed, including a strong law of large numbers for the number of events in the stations, the existence, uniqueness and representation of stationary regimes for queue size and workload.

This paper establishes new criteria for stability and for instability of multiclass network models under a given stationary policy. It also extends previous results on the approximation of the solution to the average cost optimality equations through an associated uid model: It is shown that an optimized network possesses a uid limit model which is itself optimal with respect to a total cost criterion. A general framework for constructing control algorithms for multiclass queueing networks is proposed based on these general results. Network sequencing and routing problems are considered as special cases. The following aspects of the resulting feedback regulation policies are developed in the paper: (i) The policies are stabilizing, and are in fact geometrically ergodic for a Markovian model. (ii) Numerical examples are given. In each case it is shown that the feedback regulation policy closely resembles the average-cost optimal policy. (iii) A method is proposed for reducing va...

We derive a lower bound for the staffing levels required to meet a projected load in a retail service facility. We model the queueing system as a Markovian process with non-homogeneous Poisson arrivals. Motivated by an application from the postal services, we assume that the arrival rate is piecewise constant over the time horizon and retain such transient effects as build-up in the system. The optimal staffing decision is formulated as a multiperiod dynamic programming problem where staff is allocated to each time period to minimize the total costs over the horizon. The main result is the derivation of a lower bound on the staffing requirements that is computed by decoupling successive time periods. Keywords: dynamic programming, staffing, service operations 1 M.C. Fu is supported in part by the National Science Foundation under Grant No. NSF EEC 94-02384. 1 Introduction Queueing theory is frequently used to determine the staffing required to meet a desired level of service. Stan...

This paper gives a pathwise construction of Jackson-type queueing networks allowing the derivation of stability and convergence theorems under general probabilistic assumptions on the driving sequences; namely, it is only assumed that the input process, the service sequences and the routing mechanism are jointly stationary and ergodic in a sense that is made precise in the paper. The main tools for these results are the subadditive ergodic theorem, which is used to derive a strong law of large numbers, and basic theorems on monotone stochastic recursive sequences. The techniques which are proposed here apply to other and more general classes of discrete event systems, like Petri nets or GSMP’s. The paper also provides new results on the Jackson-type networks with i.i.d. driving sequences which were studied in the past.

Abstract not found