State space collapse with application to heavy traffic limits for multiclass queueing networks

Heavy traffic limits for multiclass queueing networks are a topic of continuing interest. Presently, the class of networks for which these limits have been rigorously derived is restricted. An important ingredient in such work is the demonstration of state space collapse. Here, we demonstrate state space collapse for two families of networks, first-in first-out (FIFO) queueing networks of Kelly type and head-of-the-line proportional processor sharing (HLPPS) queueing networks. We then apply our techniques to more general networks. To demonstrate state space collapse for FIFO networks of Kelly type and HLPPS networks, we employ law of large number estimates to show a form of compactness for appropriately scaled solutions. The limits of these solutions are next shown to satisfy fluid model equations corresponding to the above queueing networks. Results from Bramson [4,5] on the asymptotic behavior of these limits then imply state space collapse. The desired heavy traffic limits for FIFO networks of Kelly type and HLPPS networks follow from this and the general criteria set forth in the companion paper Williams [41]. State space collapse and the ensuing heavy traffic limits also hold for more general queueing networks, provided the solutions of their fluid model equations converge. Partial results are given for such networks, which include the static priority disciplines. This revised version was published online in June 2006 with corrections to the Cover Date.     

Convergence to equilibria for fluid models of head-of-the-line proportional processor sharing queueing networks

Fluid models have recently become an important tool for the study of open multiclass queueing networks. We are interested in a family of such models, which we refer to as head-of-the-line proportional processor sharing (HLPPS) fluid models. Here, the fraction of time spent serving a class present at a station is proportional to the quantity of the class there, with all of the service going into the first customer of each class. To study such models, we employ an entropy function associated with the state of the system. The corresponding estimates show that if the traffic intensity function is at most 1, then such fluid models converge exponentially fast to equilibria. When the traffic intensity function is strictly less than 1, the limit is always the empty state and occurs after a finite time. A consequence is that generalized HLPPS networks with traffic intensity strictly less than 1 are positive Harris recurrent. Related results for FIFO fluid models of Kelly type were obtained in Bramson [4].Partially supported by NSF Grants DMS-93-00612 and DMS-93-04580. The paper was written while the author was in residence at the Institute for Advanced Study.     

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.     

抢占型优先服务机制下多类排队网络的扩散逼近

证明一个满负荷交通极限定理以证实在抢占型优先服务机制下多类排队网络的扩散逼近,进而为该系统提供有效的随机动力学模型．所研究的排队网络典型地出现在现代通讯系统中高速集成服务分组数据网络,其中包含分组数据包的若干交通类型,每个类型涉及若干工作处理类（步骤）,并且属于同一交通类型的工作在可能接受服务的每一个网站被赋予相同的优先权等级,更进一步地,在整个网络中,属于不同交通类型的分组数据包之间无交互路由．     

Convergence to equilibria for fluid models of FIFO queueing networks

The qualitative behavior of open multiclass queueing networks is currently a topic of considerable activity. An important goal is to formulate general criteria for when such networks possess equilibria, and to characterize these equilibria when possible. Fluid models have recently become an important tool for such purposes. We are interested here in a family of such models, FIFO fluid models of Kelly type. That is, the discipline is first-in, first-out, and the service rate depends only on the station. To study such models, we introduce an entropy function associated with the state of the system. The corresponding estimates show that if the traffic intensity function is at most 1, then such fluid models converge exponentially fast to equilibria with fixed concentrations of customer types throughout each queue. When the traffic intensity function is strictly less than 1, the limit is always the empty state and occurs after a finite time. A consequence is that generalized Kelly networks with traffic intensity strictly less than 1 are positive Harris recurrent, and hence possess unique equilibria.1991Mathematics Subject Classification, 60K25, 68M20, 90B10. Partially supported by NSF Grant DMS-93-00612.     

A Simple proof for the stability of global FIFO queueing networks

We study the stability of multiclass queueing networks under the global FIFO （first in first out） service discipline, which was established by Bramson in 2001. For these networks, the service priority of a customer is determined by his entrance time. Using fluid models, we describe the entrance time of the most senior customer in the networks at time t, which is the key to simplify the proof for the stability of the global FIFO queueing networks.     

高负荷下非强占优先排队网络系统中队长过程的扩散逼近

本文是在高负荷下非强占优先排除网络系统中给出了队长过程的扩散逼近 .证明了其队长过程的扩散极限是半鞅反射的布朗运动 .     

State space collapse and stability of queueing networks

We study the stability of subcritical multi-class queueing networks with feedback allowed and a work-conserving head-of-the-line service discipline. Assuming that the fluid limit model associated to the queueing network satisfies a state space collapse condition, we show that the queueing network is stable provided that any solution of an associated linear Skorokhod problem is attracted to the origin in finite time. We also give sufficient conditions ensuring this attraction in terms of the reflection matrix of the Skorokhod problem, by using an adequate Lyapunov function. State space collapse establishes that the fluid limit of the queue process can be expressed in terms of the fluid limit of the workload process by means of a lifting matrix.     

On the Analysis of the Virtual Waiting Time in Open Queueing Networks

The model of an open queueing network in heavy traffic has been developed. These models are mathematical models of computer networks in heavy traffic. A limit theorem has been presented for the virtual waiting time of a customer in heavy traffic in open queueing networks. Finally, we present an application of the theorem—a reliability model from computer network practice.     

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.     

Long time asymptotics for constrained diffusions in polyhedral domains

We study long time asymptotic properties of constrained diffusions that arise in the heavy traffic analysis of multiclass queueing networks. We first consider the classical diffusion model with constant coefficients, namely a semimartingale reflecting Brownian motion (SRBM) in a d$d$-dimensional positive orthant. Under a natural stability condition on a related deterministic dynamical system [P. Dupuis, R.J. Williams, Lyapunov functions for semimartingale reflecting brownian motions, Annals of Probability 22 (2) (1994) 680–702] showed that an SRBM is ergodic. We strengthen this result by establishing geometric ergodicity for the process. As consequences of geometric ergodicity we obtain finiteness of the moment generating function of the invariant measure in a neighborhood of zero, uniform time estimates for polynomial moments of all orders, and functional central limit results. Similar long time properties are obtained for a broad family of constrained diffusion models with state dependent coefficients under a natural condition on the drift vector field. Such models arise from heavy traffic analysis of queueing networks with state dependent arrival and service rates.     

Heavy traffic limit theorems for a queue with Poisson ON/OFF long-range dependent sources and general service time distribution

In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet sizes (hence service times) can be generally distributed. In this paper, we establish heavy traffic limit theorems to provide suitable approximations for the system under first-in first-out (FIFO) and work-conserving service discipline, which state that, when the lengths of both ON- and OFF-periods are lightly tailed, the sequences of the scaled queue length and workload processes converge weakly to short-range dependent reflecting Gaussian processes, and when the lengths of ON- and/or OFF-periods are heavily tailed with infinite variance, the sequences converge weakly to either reflecting fractional Brownian motions (FBMs) or certain type of longrange dependent reflecting Gaussian processes depending on the choice of scaling as the number of superposed sources tends to infinity. Moreover, the sequences exhibit a state space collapse-like property when the number of sources is large enough, which is a kind of extension of the well-known Little??s law for M/M/1 queueing system. Theory to justify the approximations is based on appropriate heavy traffic conditions which essentially mean that the service rate closely approaches the arrival rate when the number of input sources tends to infinity.     

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.     

Heavy traffic approximations of large deviations of feedforward queueing networks

We consider a multi-class feedforward queueing network with first come first serve queueing discipline and deterministic services and routing. The large deviation asymptotics of tail probabilities of the distribution of the workload vector can be characterized by convex path space minimization problems with non-linear constraints. So far there exists no numerical algorithm which could solve such minimization problems in a general way. When the queueing network is heavily loaded it can be approximated by a reflected Brownian motion. The large deviation asymptotics of tail probabilities of the distribution of this heavy traffic limit can again be characterized by convex path space minimization problems with non-linear constraints. However, due to their less complicated structure there exist algorithms to solve such minimization problems. In this paper we show that, as the network tends to a heavy traffic limit, the solution of the large deviation minimization problems of the network approaches the solution of the corresponding minimization problems of a reflected Brownian motion. Stated otherwise, we show that large deviation and heavy traffic asymptotics can be interchanged. We prove this result in the case when the network is initially empty. Without proof we state the corresponding result in the stationary case. We present an illuminating example with two queues. This revised version was published online in June 2006 with corrections to the Cover Date.     

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 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.     

Computing the Stationary Distribution of an SRBM in an Orthant with Applications to Queueing Networks

In [15], a BNAfm (Brownian network analyzer with finite element method) algorithm was developed for computing the stationary distribution of a semimartingale reflecting Brownian motion (SRBM) in a hypercube. In this companion paper, that BNAfm algorithm is extended to computing the stationary distribution of an SRBM in an orthant, which is achieved by constructing a converging sequence of SRBMs in hypercubes. The SRBM in the orthant serves as an approximation model of queueing networks with infinite buffers. We show that the constructed sequence of SRBMs in the hypercubes converges weakly to the SRBM in the orthant as the hypercubes approach the orthant. Under the conjecture that the set of the stationary distributions of the SRBMs in the hypercubes is relatively compact, we prove that the sequence of the stationary distributions of the SRBMs in the hypercubes converges weakly to the stationary distribution of the SRBM in the orthant. A three-machine job shop example is presented to illustrate the effectiveness of the SRBM approximation model and our BNAfm algorithm. The BNAfm algorithm is shown to produce good estimates for stationary probabilities of queueing networks.     

About the Sojourn Time Process in Multiphase Queueing Systems

Multiphase queueing systems (MQS) (tandem queues, queues in series) are of special interest both in theory and in practical applications (packet switch structures, cellular mobile networks, message switching systems, retransmission of video images, asembly lines, etc.). In this paper, we deal with approximations of MQS and present a heavy traffic limit theorems for the sojourn time of a customer in MQS. Functional limit theorems are proved for the customer sojourn time – an important probability characteristic of the queueing system under conditions of heavy traffic.      

有限多类别交通网络的扩展Wardrop均衡

研究有容量约束的交通网络中有限多类别用户的扩展Wardrop均衡.证明扩展Wardrop均衡定义的4种等价形式;并利用凸集分离定理得到在一定条件下.定义中A(v)的存在性.A(v)可以看作是一种收费,因而得到了有限多类别的交通网络中收费的存在性.     

Approximate analysis of a gordon-newell like non-product-form queueing network

This paper is tutorial in nature. It demonstrates how a particular heuristic extension of the arrival theorem, which was introduced earlier for very special network topologies, can be effectively applied (in an essentially unchanged manner) to obtain all mean performance measures for a rich class of Gordon-Newell like non-product-form queueing networks (QNs). All nodes in the class of queueing networks discussed are either FIFO or IS (pure delay), there is a single closed chain with probabilistic routing and each FIFO node also processes customers from a dedicated open chain. The number of FIFO nodesK, the number of IS nodesL and the closed chain populationN are finite but arbitrary and closed chain customers route probabilistically according to an arbitrary routing matrixQ. The think time distribution at an IS node is general, the service time distribution for both closed chain and open chain customers at the FIFO nodes is exponential with distinct service times for each, and both IS think times and FIFO service times are node dependent.The approximation technique is enhanced by an analytic study which demonstrates that it mirrors the expected behavior of the QN in many essential respects: monotonicity, bottleneck and asymptotic behavior. Moreover, in the case of balanced QNs, the approximation yields simple and explicit expressions for all quantities of interest. The analytic study and the numerical experiments presented complement one another and suggest that this approximation technique captures the essential structure of the QN, insofar as mean performance quantities are concerned.Currently on leave of absence from Bell Laboratories, at The Chinese University of Hong Kong, Department of Information Engineering, New Territories, Hong Kong.