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...

This paper studies the instability of multiclass queueing networks. We prove that if a fluid limit model of the queueing network is weakly unstable, then the queueing network is unstable in the sense that the total number of customers in the queueing network diverges to infinity with probability one as time t ! 1. Our result provides a converse to a recent result of Dai [2] which states that a queueing network is positive Harris recurrent if a corresponding fluid limit model is stable. Examples are provided to illustrate the usage of the result. AMS 1991 subject classification: Primary 60K25, 90B22; Secondary 60K20, 90B35. Key words and phrases: multiclass queueing networks, instability, transience, Harris positive recurrent, fluid approximation, fluid model. Running title: Instability of queueing networks First draft in June 1995 Revised in July 1995 Revised in February 1996 Annals of Applied Probability, Vol. 6, pp 751--757 (1996) 1 Introduction This paper studies the transience ...

This paper studies the uid models of two-station multiclass queueing networks with deterministic routing. A uid model is globally stable if the uid network eventually empties under each nonidling dispatch policy. We explicitly characterize the global stability region in terms of the arrival and service rates. We show that the global stability region is dened by the nominal workload conditions and the \virtual workload conditions" and we introduce two intuitively appealing phenomena: virtual stations and push starts, that explain the virtual workload conditions. When any of the workload conditions is violated, we construct a uid solution that cycles to innity, showing that the uid network is unstable. When all of the workload conditions are satised, we solve a network ow problem to nd the coecients of a piecewise linear Lyapunov function. The Lyapunov function decreases to zero proving that the uid level eventually reaches zero under any non-idling dispatch policy. Under certain assumptions on the interarrival and service time distributions, a queueing network is stable or positive Harris recurrent if the corresponding uid network is stable. Thus, the workload conditions are sucient to ensure the global stability of two-station multiclass queueing networks with deterministic routing. To appear in Operations Research

This paper studies the stability of a three-station fluid network. We prove that the global stability region of our three-station network is not monotone in the service times and so, we may move a service time vector out of the global stability region by reducing the service time for a class. We introduce the monotone global stability region and show that a linear program (LP) related to a piecewise linear Lyapunov function characterizes this largest monotone subset of the global stability region for our three-station network. The linear program proposed by Bertsimas, Gamarnik and Tsitsiklis [1] does not characterize either the global stability region or even the monotone global stability region of our three-station network.

We present the Multi-user EXP rule (M-EXP) for multiple-queue multiple-server scheduling motivated by a model for transmission to multiple users over a wireless broadcast channel. M-EXP is shown to be throughput optimal (i.e. achieves the largest possible stable input rate region) and pathwise optimal (i.e. minimizes system workload and maximum queue length) in the heavy traffic limit. This work generalizes the EXP rule proposed in the context of a single-server queueing system over a wireless channel to a multi-server scenario. While the service rate region for single-server EXP rule [1] is limited to a convex combination of simplexes, the rate region for the multi-server case is a convex combination of general convex polytopes. In the context of the M-EXP rule, this geometry does not permit traditional stability/pathwise-optimality analysis in the fluid and diffusion scales. Hence, we utilize a timescale separation argument that leads to an appropriate limiting process on a finer timescale, and show that the M-EXP rule reduces to a modified MaxWeight [2] scheduling rule over this finer timescale