Fractional Brownian Heavy Traffic Approximations of Multiclass Feedforward Queueing Networks

We consider multiclass feedforward queueing networks under first in first out and priority service disciplines driven by long-range dependent arrival and service time processes. We show that in critical loading the normalized workload, queue length and sojourn time processes can converge to a multi-dimensional reflected fractional Brownian motion. This weak heavy traffic approximation is deduced from a deterministic pathwise approximation of the network behavior close to constant critical load in terms of the solution of a Skorokhod problem. Since we model the doubly infinite time interval, our results directly cover the stationary case.AMS subject classification: primary 90B15, secondary 60K25, 68M20     

Stability of Parallel Fluid Loaded Plates: A Nonlocal Approach

We consider the motion of a collection of fluid loaded elastic plates, situated horizontally in an infinitely long channel. We use a new, unified approach to boundary value problems, introduced by A.S. Fokas in the late 1990s, and show the problem is equivalent to a system of one‐parameter integral equations. We give a detailed study of the linear problem, providing explicit solutions and well‐posedness results in terms of standard Sobolev spaces. We show that the associated Cauchy problem is completely determined by a matrix, which depends solely on the mean separation of the plates and the horizontal velocity of each of the driving fluids. This matrix corresponds to the infinitesimal generator of the C₀ ‐semigroup for the evolution equations in Fourier space. By analyzing the properties of this matrix, we classify necessary and sufficient conditions for which the problem is asymptotically stable.     

A sufficient condition and a necessary condition for the diffusion approximations of multiclass queueing networks under priority service disciplines

We establish a sufficient condition for the existence of the (conventional) diffusion approximation for multiclass queueing networks under priority service disciplines. The sufficient condition relates to a sufficient condition for the weak stability of the fluid networks that correspond to the queueing networks under consideration. In addition, we establish a necessary condition for the network to have a continuous diffusion limit; the necessary condition is to require a reflection matrix (of dimension equal to the number of stations) to be completely-S. When applied to some examples, including generalized Jackson networks, single station multiclass queues, first-buffer-first-served re-entrant lines, a two-station Dai–Wang network and a three-station Dumas network, the sufficient condition coincides with the necessary condition.     

A Multiclass Feedback Queueing Network with a Regular Skorokhod Problem

We consider a four-class two-station network with feedback, with fluid inputs and a head-of-the-line generalized processor sharing discipline at each station. We derive the Skorokhod Problem associated with the network and obtain algebraic sufficient conditions for Lipschitz continuity of the associated Skorokhod Map. This provides the first example of a multiclass network with feedback for which the associated Skorokhod Problem has been proved to be regular. As an elementary application, we show that under the conditions which guarantee Lipschitz continuity the network is stable if and only if the usual load conditions apply.     

Convex duality and the Skorokhod Problem. I

The solution to the Skorokhod Problem defines a deterministic mapping, referred to as the Skorokhod Map, that takes unconstrained paths to paths that are confined to live within a given domain G⊂ℝ ⁿ . Given a set of allowed constraint directions for each point of ∂G and a path ψ, the solution to the Skorokhod Problem defines the constrained version φ of ψ, where the constraining force acts along one of the given boundary directions using the “least effort” required to keep φ in G. The Skorokhod Map is one of the main tools used in the analysis and construction of constrained deterministic and stochastic processes. When the Skorokhod Map is sufficiently regular, and in particular when it is Lipschitz continuous on path space, the study of many problems involving these constrained processes is greatly simplified. We focus on the case where the domain G is a convex polyhedron, with a constant and possibly oblique constraint direction specified on each face of G, and with a corresponding cone of constraint directions at the intersection of faces. The main results to date for problems of this type were obtained by Harrison and Reiman [22] using contraction mapping techniques. In this paper we discuss why such techniques are limited to a class of Skorokhod Problems that is a slight generalization of the class originally considered in [22]. We then consider an alternative approach to proving regularity of the Skorokhod Map developed in [13]. In this approach, Lipschitz continuity of the map is proved by showing the existence of a convex set that satisfies a set of conditions defined in terms of the data of the Skorokhod Problem. We first show how the geometric condition of [13] can be reformulated using convex duality. The reformulated condition is much easier to verify and, moreover, allows one to develop a general qualitative theory of the Skorokhod Map. An additional contribution of the paper is a new set of methods for the construction of solutions to the Skorokhod Problem. These methods are applied in the second part of this paper [17] to particular classes of Skorokhod Problems. Received: 17 April 1998 / Revised version: 8 January 1999     

A heavy traffic limit theorem for a class of open queueing networks with finite buffers

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 fd-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. This revised version was published online in June 2006 with corrections to the Cover Date.     

Penalization methods for reflecting stochastic differential equations with jumps

The problem of approximation of a solution to a reflecting stochastic differential equation (SDE) with jumps by a sequence of solutions to SDEs with penalization terms is considered. The approximating sequence is not relatively compact in the Skorokhod topology J ¹ and so the methods of approximation based on the J ¹-topology break down. In the paper, we prove our convergence results in the S-topology on the Skorokhod space D(R⁺,?R ^d ) introduced recently by Jakubowski. The S-topology is weaker than J ¹ but stronger than the Meyer-Zheng topology and shares many useful properties with J ¹.     

Stability of Earliest-Due-Date,First-Served Queueing Networks

We study multiclass queueing networks with the earliest-due-date, first-served (EDDFS) discipline. For these networks, the service priority of a customer is determined, upon its arrival in the network, by an assigned random due date. First-in-system, first-out queueing networks, where a customer's priority is given by its arrival time in the network, are a special case. Using fluid models, we show that EDDFS queueing networks, without preemption, are stable whenever the traffic intensity satisfies _j <1 for each station j.     

An invariance principle for semimartingale reflecting Brownian motions in an orthant

Semimartingale reflecting Brownian motions in an orthant (SRBMs) are of interest in applied probability because of their role as heavy traffic approximations for open queueing networks. It is shown in this paper that a process which satisfies the definition of an SRBM, except that small random perturbations in the defining conditions are allowed, is close in distribution to an SRBM. This perturbation result is called an invariance principle by analogy with the invariance principle of Stroock and Varadhan for diffusions with boundary conditions. A crucial ingredient in the proof of this result is an oscillation inequality for solutions of a perturbed Skorokhod problem. In a subsequent paper, the invariance principle is used to give general conditions under which a heavy traffic limit theorem holds for open multiclass queueing networks. This revised version was published online in June 2006 with corrections to the Cover Date.     

A new sufficient condition in the invariance principle for the partial sum processes of moving averages

We obtain a new sufficient condition for the C-convergence of distributions of the partial sum processes of moving averages to the distribution of a Wiener process in the metric space D[0,1] with the Skorokhod metric.     

A multiclass network with non-linear, non-convex, non-monotonic stability conditions

We consider a stochastic queueing network with fixed routes and class priorities. The vector of class sizes forms a homogeneous Markov process of countable state space Z⁶ ₊. The network is said “stable” (resp.“unstable”) if this Markov process is ergodic (resp. transient). The parameters are the traffic intensities of the different classes. An unusual condition of stability is obtained thanks to a new argument based on the characterization of the “essential states”. The exact stability conditions are then detected thanks to an associated fluid network: we identify a zone of the parameter space in which diverging, fluid paths appear. In order to show that this is a zone of instability (and that the network is stable outside this zone), we resort to the criteria of ergodicity and transience proved by Malyshev and Menshikov for reflected random walks in Z⁶ ₊ (Malyshev and Menshikov, 1981). Their approach allows us to neglect some “pathological” fluid paths that perturb the dynamics of the fluid model. The stability conditions thus determined have especially unusual characteristics: they have a quadratic part, the stability domain is not convex, and increasing all the service rates may provoke instability (Theorem 1.1 and section 7). This revised version was published online in June 2006 with corrections to the Cover Date.     

Heavy and light traffic in fluid models with burst arrivals

We consider the problem of finding a heavy and light traffic limits for the steady-state workload in a fluid model having a continuous burst arrival process. Such a model is useful for describing (among other things) the packetwise transmission of data in telecommunications, where each packet is approximated to be a continuous flow. Whereas in a queueing model, each arrival epoch,t _n , corresponds to a customer with a service timeS _n , the burst model is different: each arrival epoch,t _n , corresponds to a burst of work, that is, a continuous flow of work (fluid, information) to the system at rate 1 during the time interval [t _n ,t _n +S _n ]. In the present paper we show that the burst and queueing models share the same heavy-traffic limit for work, but that their behavior in light traffic is quite different.Research supported by the Japan Society for the Promotion of Science, during the author's fellowship in Tokyo.Research funded by C & C Information Technology Research Laboratories, NEC, and the International Science Foundation.     

State space collapse for asymptotically critical multi-class fluid networks

We consider a class of fluid queueing networks with multiple fluid classes and feedback allowed, which are fed by N heavy tailed ON/OFF sources. We study the asymptotic behavior when N→∞ of these queueing systems in a heavy traffic regime (that is, when they are asymptotically critical). As performance processes we consider the workload W ^N (the amount of time needed for each server to complete processing of all the fluid in queue), and the fluid queue Z ^N (the quantity of each fluid class in the system). We show the convergence of and (to and ) in heavy traffic if state space collapse (SSC) holds. (SSC) is a condition that establishes a relationship between those components of that correspond to fluid classes processed by the same server, which implies that for a deterministic lifting matrix Δ. Our main contribution is to prove that assuming that the other hypotheses are true, (SSC) is not only sufficient for this convergence, but necessary. Furthermore, we prove that processes and , conveniently scaled in time, converge to W (a reflected fractional Brownian motion) and Z (=ΔW). We illustrate the application of our results with some examples including a tandem queue. Supported by project MEC-FEDER ref. MTM2006-06427.     

Near optimal control of queueing networks over a finite time horizon

We propose a method for the control of multi-class queueing networks over a finite time horizon. We approximate the multi-class queueing network by a fluid network and formulate a fluid optimization problem which we solve as a separated continuous linear program. The optimal fluid solution partitions the time horizon to intervals in which constant fluid flow rates are maintained. We then use a policy by which the queueing network tracks the fluid solution. To that end we model the deviations between the queuing and the fluid network in each of the intervals by a multi-class queueing network with some infinite virtual queues. We then keep these deviations stable by an adaptation of a maximum pressure policy. We show that this method is asymptotically optimal when the number of items that is processed and the processing speed increase. We illustrate these results through a simple example of a three stage re-entrant line. Research supported in part by Israel Science Foundation Grant 249/02 and 454/05 and by European Network of Excellence Euro-NGI.     

Stability of Fluid Networks with Proportional Routing

In this paper we investigate the stability of a class of two-station multiclass fluid networks with proportional routing. We obtain explicit necessary and sufficient conditions for the global stability of such networks. By virtue of a stability theorem of Dai [14], these results also give sufficient 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 fluid models without proportional routing, which arise from queueing networks with deterministic routing. The models we investigate include fluid 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.     

Matrix-valued Schrödinger operators over local fields

We consider quantum systems that have as their configuration spaces finite dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally defined. For a wide class of potentials we prove that this Hamiltonian is well-defined as an unbounded self-adjoint operator. The free part of the operator gives rise to ameasure on the Skorokhod space of paths,D[0,∞), and with respect to this measure there is a path integral representation for the semigroup associated to the Hamiltonian. We prove this Feynman-Kac formula in the local field setting as a consequence of the Hille-Yosida theory of semi-groups. The text was submitted by the authors in English.     

Log-weight scheduling in switched networks

We consider switched queueing networks in which there are constraints on which queues may be served simultaneously. The scheduling policy for such a network specifies which queues to serve at any point in time. We introduce and study a variant of the popular maximum weight or backpressure policy which chooses the collection of queues to serve that has maximum weight. Unlike the maximum weight policies studied in the literature, the weight of a queue depends on logarithm of its queue-size in this paper. For any multihop switched network operating under such maximum log-weighted policy, we establish that the network Markov process is positive recurrent as long as it is underloaded. As the main result of this paper, a meaningful fluid model is established as the formal functional law of large numbers approximation. The fluid model is shown to be work-conserving. That is, work (or total queue-size) is nonincreasing as long as the network is underloaded or critically loaded. We identify invariant states or fixed points of the fluid model. When underloaded, null state is the unique invariant state. For a critically loaded fluid model, the space of invariant states is characterized as the solution space of an optimization problem whose objective is lexicographic ordering of total queue-size and the negative entropy of the queue state. An important contribution of this work is in overcoming the challenge presented by the log-weight function in establishing meaningful fluid model. Specifically, the known approaches in the literature primarily relied on the “scale invariance” property of the weight function that log-function does not possess.     

Piecewise linear test functions for stability and instability of queueing networks

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. This revised version was published online in June 2006 with corrections to the Cover Date.     

Mean queue size in a queue with discrete autoregressive arrivals of order <Emphasis Type="Italic">p</Emphasis>

We consider a discrete time single server queueing system where the arrival process is governed by a discrete autoregressive process of order p (DAR(p)), and the service time of a customer is one slot. For this queueing system, we give an expression for the mean queue size, which yields upper and lower bounds for the mean queue size. Further we propose two approximation methods for the mean queue size. One is based on the matrix analytic method and the other is based on simulation. We show, by illustrations, that the proposed approximations are very accurate and computationally efficient.