Heavy traffic limit for the workload plateau process in a tandem queue with identical service times

We consider a two-node tandem queueing network in which the upstream queue is GI/GI/1 and each job reuses its upstream service requirement when moving to the downstream queue. Both servers employ the first-in-first-out policy. To investigate the evolution of workload in the second queue, we introduce and study a process M, called the plateau process, which encodes most of the information in the workload process. We focus on the case of infinite-variance service times and show that under appropriate scaling, workload in the first queue converges, and although the workload in the second queue does not converge, the plateau process does converges to a limit that is a certain function of two independent Levy processes. Using excursion theory, we compare a time changed version of the limit to a limit process derived in previous work

Fluid Models of Many-server Queues with Abandonment

We study many-server queues with abandonment in which customers have general service and patience time distributions. The dynamics of the system are modeled using measure- valued processes, to keep track of the residual service and patience times of each customer. Deterministic fluid models are established to provide first-order approximation for this model. The fluid model solution, which is proved to uniquely exists, serves as the fluid limit of the many-server queue, as the number of servers becomes large. Based on the fluid model solution, first-order approximations for various performance quantities are proposed

Fluid limits for processor-sharing queues with impatience

Abstract. We investigate a processor sharing queue with renewal arrivals and generally distributed service times. Impatient jobs may abandon the queue, or renege, before completing service. The random time representing a job’s patience has a general distribution and may be dependent on its initial service time requirement. A scaling procedure that gives rise to a fluid model with nontrivial yet tractable steady state behavior is presented. This fluid model model captures many essential features of the underlying stochastic model

Diffusion approximation for a processor sharing queue in heavy traffic

Consider a single server queue with renewal arrivals and i.i.d. service times in which the server operates under a processor sharing service discipline. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. From this measure valued process, one can recover the traditional performance processes, including queue length and workload. We show that under mild assumptions, including standard heavy traffic assumptions, the (suitably rescaled) measure valued processes corresponding to a sequence of processor sharing queues converge in distribution to a measure valued diffusion process. The limiting process is characterized as the image under an appropriate lifting map, of a one-dimensional reflected Brownian motion. As an immediate consequence, one obtains a diffusion approximation for the queue length process of a processor sharing queue

Fluid limits for networks with bandwidth sharing and general document size distributions

We consider a stochastic model of Internet congestion control, introduced by Massoulie and Roberts, that represents the randomly varying number of flows in a network where bandwidth is shared amongst document transfers. In contrast to an earlier work by Kelly and Williams, the present paper allows inter arrival times and document sizes to be generally distributed, rather than exponentially distributed. Furthermore, we allow a fairly general class of bandwidth sharing policies that includes the weighted a-fair policies of Mo and Walrand, as well as certain other utility based scheduling policies. To describe the evolution of the system, measure valued processes are used to keep track of the residual document sizes of all flows through the network. We propose a fluid model (or formal functional law of large numbers approximation) associated with the stochastic flow level model. Under mild conditions, we show that the appropriately rescaled measure valued processes corresponding to a sequence of such models (with fixed network structure) are tight, and that any weak limit point of the sequence is almost surely a fluid model solution. For the special case of weighted affair policies, we also characterize the invariant states of the fluid model

Fluid limits for networks with bandwidth sharing and general document size distributions

We consider a stochastic model of Internet congestion control, introduced by Massoulie and Roberts, that represents the randomly varying number of flows in a network where bandwidth is shared amongst document transfers. In contrast to an earlier work by Kelly and Williams, the present paper allows inter arrival times and document sizes to be generally distributed, rather than exponentially distributed. Furthermore, we allow a fairly general class of bandwidth sharing policies that includes the weighted a-fair policies of Mo and Walrand, as well as certain other utility based scheduling policies. To describe the evolution of the system, measure valued processes are used to keep track of the residual document sizes of all flows through the network. We propose a fluid model (or formal functional law of large numbers approximation) associated with the stochastic flow level model. Under mild conditions, we show that the appropriately rescaled measure valued processes corresponding to a sequence of such models (with fixed network structure) are tight, and that any weak limit point of the sequence is almost surely a fluid model solution. For the special case of weighted affair policies, we also characterize the invariant states of the fluid model

Fluid model for a data network with alpha-fair bandwidth sharing and general document size distributions : two examples of stability

The design and analysis of congestion control mechanisms for modern data networks such as the Internet is a challenging problem. Mathematical models at various levels have been introduced in an effort to provide insight to some aspects of this problem. A model introduced and studied by Roberts and Massoulie [13] aims to capture the dynamics of document arrivals and departures in a network where bandwidth is shared fairly amongst flows that correspond to continuous transfers of individual elastic documents. With gener- ally distributed interarrival times and document sizes, except for a few special cases, it is an open problem to establish stability of this stochastic flow level model under the nominal condition that the average load on each resource is less than its capacity. As a step towards the study of this model, in a separate work [8], we introduced a measure valued process to describe the dynamic evolution of the residual document sizes and proved a fluid limit result: under mild assumptions, rescaled measure valued processes corresponding to a sequence of connection level models (with fixed network structure) are tight, and any weak limit point of the sequence is almost surely a solution of a certain fluid model. The invariant states for the fluid model were also characterized in [8]. In this paper, we review the structure of the stochastic flow level model, describe our fluid model approximation and then give two interesting examples of network topologies for which stability of the fluid model can be established under a nominal condition. The two types of networks are linear networks and tree networks. The result for tree networks is particularly interesting as there the distribution of the number of documents process in steady state is expected to be sensitive to the (non-exponential) document size distribution [2]. Future work will be aimed at further analysis of the fluid model and at using it for studying stability and heavy traffic behavior of the stochastic flow level model