We consider a model of Internet congestion control, that represents the randomly varying number of ows present in a network where bandwidth is shared fairly between document transfers. We study critical uid models, obtained as formal limits under law of large numbers scalings when the average load on at least one resource is equal to its capacity. We establish convergence to equilibria for uid models, and identify the invariant manifold. The form of the invariant manifold gives insight into the phenomenon of entrainment, whereby congestion at some resources may prevent other resources from working at their full capacity.

Using a slight modification of the framework in Bramson [7] and Williams [52], we prove heavy traffic limit theorems for six families of multiclass queueing networks. The first three families are single station systems operating under first-in first-out (FIFO), generalized head-of-the-line proportional processor sharing (GHLPPS) and static buffer priority (SBP) service disciplines. The next two families are re-entrant lines operating under first-buffer-first-serve (FBFS) and last-buffer-first-serve (LBFS) service disciplines; the last family consists of certain 2-station, 5-class networks operating under an SBP service discipline. Some of these heavy traffic limits have appeared earlier in the literature; our new proofs demonstrate the significant simplifications that can be achieved in the present setting.

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

We consider a queueing network of d single server stations. Each station has a finite capacity waiting buffer, and all customers served at a station are homogeneous in terms of service requirements and routing. The routing is assumed to be deterministic and hence feedforward. A server stops working when the downstream buffer is full. We show that a properly normalized d-dimensional queue length process converges in distribution to a d-dimensional semimartingale reflecting Brownian motion (RBM) in a d-dimensional box under a heavy traffic condition. The conventional continuous mapping approach does not apply here because the solution to our Skorohod problem may not be unique. Our proof relies heavily on a uniform oscillation result for solutions to a family of Skorohod problems. The oscillation result is proved in a general form that may be of independent interest. It has the potential to be used as an important ingredient in establishing heavy traffic limit theorems for general finite buffer networks. Key words and phrases: Finite capacity network, blocking probabilities, loss network, semimartingale reflecting Brownian motion, RBM, heavy traffic, limit theorems, oscillation estimates.

One of the primary tools in establishing the stability of a fluid network is to construct a Lyapunov function. In this paper, we establish the sufficiency in the use of a Lyapunov function. Specifically, we show that a necessary and sufficient condition for the stability of a generic fluid network (GFN) is the existence of a Lyapunov function for its fluid level process. Then by applying this result to various specific fluid networks, including a fluid network under all work-conserving service disciplines, a fluid network under a priority service discipline, and a fluid network under a FIFO service discipline, we establish the existence of a Lyapunov function for their fluid level processes is a necessary and sufficient condition for their stabilities. The result is also applied to various fluid limit models and a linear Skorohod problem.

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.

this paper we investigate the stability of a class of two-station multiclass uid networks with proportional routing. We obtain explicit necessary and sucient conditions for the global stability of such networks. By virtue of a stability theorem of Dai [14], these results also give sucient conditions for the stability of a class of related multiclass queueing networks. Our study extends the results of Dai and VandeVate [19], who provided a similar analysis for uid models without proportional routing, which arise from queueing networks with deterministic routing. The models we investigate include uid models which arise from a large class of two-station queueing networks with probabilistic routing. The stability conditions derived turn out to have an appealing intuitive interpretation in terms of virtual stations and push-starts which were introduced in earlier work on multiclass networks. Keywords: Multiclass queueing network, uid mod