This paper studies the fluid approximation (also known as the functional strong law-of-large-numbers) and the stability (positive Harris recurrent) for a multiclass queueing network. Both of these are related to the stabilities of a linear fluid model, constructed from the first-order parameters (i.e., long-run average arrivals, services and routings) of the queueing network. It is proved that the fluid approximation for the queueing network exists if the corresponding linear fluid model is weakly stable, and that the queueing network is stable if the corresponding linear fluid model is (strongly) stable. Sufficient conditions are found for the stabilities of a linear fluid model. Keywords and phrases: Multiclass queueing networks, fluid models, fluid approximations, stability, positive Harris recurrent, and work-conserving service disciplines. Preliminary Versions: September 1993 Revisions: June 1994; September 1994; January 1995 To appear in Annals of Applied Probability AMS 1980 su...

Re-entrant lines can be used to model complex manufacturing systems such as wafer fabrication facilities. As the first step to the optimal or near-optimal scheduling of such lines, we investigate their stability. In light of a recent theorem of Dai (1995) which states that a scheduling policy is stable if the corresponding fluid model is stable, we study the stability and instability of fluid models. To do this we utilize piecewise linear Lyapunov functions. We establish stability of First-Buffer-First-Served (FBFS) and Last-Buffer-First-Served (LBFS) disciplines in all reentrant lines, and of all work-conserving disciplines in any three buffer re-entrant lines. For the four buffer network of Lu and Kumar we characterize the stability region of the Lu and Kumar policy, and show that it is also the global stability region for this network. We also study stability and instability of Kelly-type networks. In particular, we show that not all work-conserving policies are stable for such netw...

We develop the use of piecewise linear test functions for the analysis of stability of multiclass queueing networks and their associated fluid limit models. It is found that if an associated LP admits a positive solution, then a Lyapunov function exists. This implies that the fluid limit model is stable and hence that the network model is positive Harris recurrent with a finite polynomial moment. Also, it is found that if a particular LP admits a solution, then the network model is transient. Running head : Stability and Instability of Queueing Networks Keywords : Multiclass queueing networks, ergodicity, stability, performance analysis. 1 Introduction It has generally been taken for granted in queueing theory that stability of a network is guaranteed so long as the overall traffic intensity is less than unity and in recent years there has been much analysis which supports this belief for special classes of systems, such as single class queueing networks (see Borovkov [2], Sig...

We introduce a new method to investigate stability of work-conserving policies in multiclass queueing networks. The method decomposes feasible trajectories and uses linear programming to test stability. We show that this linear program is a necessary and sufficient condition for the stability of all work-conserving policies for multiclass fluid queueing networks with two stations. Furthermore, we find new sufficient conditions for the stability of multiclass queueing networks involving any number of stations and conjecture that these conditions are also necessary. Previous research had identified sufficient conditions through the use of a particular class (monotone piecewise linear convex) Lyapunov functions. Using linear programming duality, we show that for two-station systems the Lyapunov function approach is equivalent to ours and therefore characterizes stability exactly. 1 Introduction The problem of establishing conditions under which a multiclass queueing network is stable und...

In a live and bounded Free Choice Petri net, pick a non-conflicting transition. Then there exists a unique reachable marking in which no transition is enabled except the selected one. For a routed live and bounded Free Choice net, this property is true for any transition of the net. Consider now a live and bounded stochastic routed Free Choice net, and assume that the routings and the firing times are independent and identically distributed. Using the above results, we prove the existence of asymptotic firing throughputs for all transitions in the net. Furthermore, the vector of the throughputs at the different transitions is explicitly computable up to a multiplicative constant. 1.

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.

In heavy traffic analysis of open queueing networks, processes of interest such as queue lengths and workload levels are generally approximated by a multidimensional reflected Brownian motion (RBM). Decomposition approximations, on the other hand, typically analyze stations in the network separately, treating each as a single queue with adjusted interarrival time distribution. We present a hybrid method for analyzing generalized Jackson networks that employs both decomposition approximation and heavy traffic theory: stations in the network are partitioned into groups of "bottleneck subnetworks" that may have more than one station; the subnetworks then are analyzed "sequentially" with heavy traffic theory. Using the numerical method of Dai and Harrison for computing the stationary distribution of multidimensional RBM's, we compare the performance of this technique to other methods of approximation via some simulation studies. Our results suggest that this hybrid method generally perform...

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.

Abstract. The paper establishes necessary and sufficient conditions for the stability of different join-the-shortest-queue models including the load-balanced network with general input and output processes. It is shown that the necessary and sufficient condition for the stability of the load-balanced network is related to the solution of the linear programming problem precisely formulated in the paper. It is proved that if the minimum of the objective function of that linear programming problem is less than 1, then the associated load-balanced network is stable.

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.