We introduce a new approach to the study of dynamic (or continuous) packet routing, where packets are being continuously injected into a network. Our objective is to study what happens to packet routing under continuous injection

Usually, the stability of queueing networks is established by explicitly determining the invariant distribution. However, outside of the narrow class of queueing networks possessing a product form solution, such explicit solutions are rare, and consequently little is known concerning stability too. We develop here a programmatic procedure for establishing the stability of queueing networks and scheduling policies. The method uses linear or nonlinear programming to determine what is an appropriate quadratic functional to use as a Lyapunov function. If the underlying system is Markovian, our method establishes not only positive recurrence and the existence of a steady--state probability distribution, but also the geometric convergence of an exponential moment. We illustrate this method on several example problems. For an example of an open re-entrant line, we show that all stationary non-idling policies are stable for all load factors less than one. This includes the well known First Com...

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...

Several proposals exist for the introduction of synchronization constraints into Queueing Networks (QN). We show that many monoclass QN with synchronizations can naturally be modelled with a subclass of Petri Nets (PN) called Free Choice nets (FC), for which a wide gamut of qualitative behavioural and structural results have been derived. We use some of these net theoretic results to characterize the ergodicity, boundedness and liveness of closed Free Choice Synchronized Queueing Networks (FCSQN). Moreover we define upper and lower throughput bounds based on the mean value of the service times, without any assumption on the probability distributions (thus including both the deterministic and the stochastic cases). We show that monotonicity properties exist between the throughput bounds and the parameters of the model in terms of population and service times. We propose (theoretically polynomial and practically linear complexity) algorithms for the computation of these bounds, based on ...

In this paper, we establish a sufficient condition for the stability of a multiclass fluid network and queueing network under priority service disciplines. The sufficient condition is based on the existence of a linear Lyapunov function, and it is stated in terms of the feasibility of a set of inequalities that are defined by network parameters. In all the networks we have tested, this sufficient condition actually gives a necessary and sufficient condition for their stability.

We analyze the performance of greedy routing for array networks by providing bounds on the average delay and the average number of packets in the system for the dynamic routing problem. In this model packets are generated at each node according to a Poisson process, and each packet is sent to a destination chosen uniformly at random. Our bounds are based on comparisons with computationally simpler queueing networks, and the methods used are generally applicable to other network systems. A primary contribution we provide is a new lower bound technique that also improves on the previous lower bounds by Stamoulis and Tsitsiklis for heavily loaded hypercube networks. On heavily loaded array networks, our lower and upper bounds differ by only a small constant factor. We further examine extensions of the problem where our methods prove useful. For example, we consider variations where edges can have different transmission rates or the destination distribution is non-uniform. In particular, we study to what extent optimally configured array networks outperform the standard array network.] 1996 Academic Press, Inc. 1.

In many large systems, such as manufacturing systems and communication networks, whenever a resource becomes available, one has to decide which of several tasks contending for its attention should be performed next. Such problems are called "scheduling " problems. They are control problems of immense economic importance, and have often been formulated and addressed in an open-loop setting. To both illustrate the types of problems encountered and to serve as focus, in this article we will address scheduling problems in a technological area of much topical interest- semiconductor manufacturing. Being of relatively recent origin, and organized differently from more traditional manufacturing systems such as flow shops and job shops, they are relatively less explored. They are also of significant economic interest, and much in the public limelight, and thus a fertile area for systems and control researchers. We provide an account of some problems in the area, as well as some suggested solutions.

Product-form solutions in Markovian process algebra (MPA) are constructed using properties of reversed processes. The compositionality of MPAs is directly exploited, allowing a large class of hierarchically constructed systems to be solved for their state probabilities at equilibrium. The paper contains new results on both reversed stationary Markov processes as well as MPA itself and includes a mechanisable proof in MPA notation of Jackson’s theorem for product-form queueing networks. Several examples are used to illustrate the approach.

We present a survey of the main ergodic theory techniques which are used in the study of iterates of monotone and homogeneous stochastic operators. It is shown that ergodic theorems on discrete event networks (queueing networks and/or Petri nets) are a generalization of these stochastic operator theorems. Kingman's subadditive ergodic Theorem is the key tool for deriving what we call rst order ergodic results. We also show how to use backward constructions (also called Loynes schemes in network theory) in order to obtain second order ergodic results. We will propose a review of systems entering the framework insisting on two models, precedence constraints networks and Jackson type networks.

Since their introduction nearly ten years ago, compositionality has been reported as one of the major attractions of stochastic process algebras. The benefits that compositionality provides for model construction are readily apparent and have been demonstrated in numerous case studies. Early research on the compositionality of the languages focussed on how the inherent structure could be used, in conjunction with equivalence relations, for model simplification and aggregation. In this paper we consider how far we have been able to take advantage of compositionality when it comes to solving the Markov process underlying a stochastic process algebra model and outline directions for future work in order for current results to be fully exploited. 1 Introduction Stochastic process algebras (SPA) were first proposed as a tool for performance and dependability modelling in 1989 [24]. At that time there was already a plethora of techniques for constructing performance models so the introducti...