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

Heavy traffic limit for a processor sharing queue with soft deadlines

This paper considers a GI/GI/1 processor sharing queue in which jobs have soft deadlines. At each point in time, the collection of residual service times and deadlines is modeled using a random counting measure on the right half-plane. The limit of this measure valued process is obtained under diffusion scaling and heavy traffic conditions and is characterized as a deterministic function of the limiting queue length process. As special cases, one obtains diffusion approximations for the lead time profile and the profile of times in queue. One also obtains a snapshot principle for sojourn times.Comment: Published at http://dx.doi.org/10.1214/105051607000000014 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Invariance of fluid limits for the Shortest Remaining Processing Time and Shortest Job First policies

We consider a single-server queue with renewal arrivals and i.i.d. service times, in which the server employs either the preemptive Shortest Remaining Processing Time (SRPT) policy, or its non-preemptive variant, Shortest Job First (SJF). We show that for given stochastic primitives (initial condition, arrival and service processes), the model has the same fluid limit under either policy. In particular, we conclude that the well-known queue length optimality of preemptive SRPT is also achieved, asymptotically on fluid scale, by the simpler-to-implement SJF policy. We also conclude that on fluid scale, SJF and SRPT achieve the same performance with respect to response times of the longest-waiting jobs in the system.Comment: 24 page

Heavy Traffic Limit for a Tandem Queue with Identical Service Times

We consider a two-node tandem queueing network in which the upstream queue is M/G/1 and each job reuses its upstream service requirement when moving to the downstream queue. Both servers employ the first-in-first-out policy. We investigate the amount of work in the second queue at certain embedded arrival time points, namely when the upstream queue has just emptied. We focus on the case of infinite-variance service times and obtain a heavy traffic process limit for the embedded Markov chain

Diffusion limits for shortest remaining processing time queues

We present a heavy traffic analysis for a single server queue with renewal arrivals and generally distributed i.i.d. service times, in which the server employs the Shortest Remaining Processing Time (SRPT) policy. Under typical heavy traffic assumptions, we prove a diffusion limit theorem for a measure-valued state descriptor, from which we conclude a similar theorem for the queue length process. These results allow us to make some observations on the queue length optimality of SRPT. In particular, they provide the sharpest illustration of the well-known tension between queue length optimality and quality of service for this policy.Comment: 19 pages; revised, fixed typos. To appear in Stochastic System

Critical fluid limit of a gated processor sharing queue

We consider a sequence of single-server queueing models operating under a service policy that incorporates batches into processor sharing: arriving jobs build up behind a gate while waiting to begin service, while jobs in front of the gate are served according to processor sharing. When they have been completed, the waiting jobs move in front of the gate and the cycle repeats. We model this system with a pair of measure valued processes describing the jobs in front of and behind the gate. Under mild asymptotically critical conditions and a law-of-large-numbers scaling, we prove that the pair of measure-valued processes converges in distribution to an easily described limit, which has an interesting periodic dynamics

Clifford algebras and new singular Riemannian foliations in spheres

Using representations of Clifford algebras we construct indecomposable singular Riemannian foliations on round spheres, most of which are non-homogeneous. This generalizes the construction of non-homogeneous isoparametric hypersurfaces due to by Ferus, Karcher and Munzner.Comment: 21 pages. Construction of foliations in the Cayley plane added. Proofs simplified and presentation improved, according to referee's suggestions. To appear in Geom. Funct. Ana

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

We consider a stochastic model of Internet congestion control, introduced by Massouli\'{e} and Roberts [Telecommunication Systems 15 (2000) 185--201], that represents the randomly varying number of flows in a network where bandwidth is shared among document transfers. In contrast to an earlier work by Kelly and Williams [Ann. Appl. Probab. 14 (2004) 1055--1083], the present paper allows interarrival 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 $\alpha$-fair policies of Mo and Walrand [IEEE/ACM Transactions on Networking 8 (2000) 556--567], 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 $\alpha$-fair policies, we also characterize the invariant states of the fluid model.Comment: Published in at http://dx.doi.org/10.1214/08-AAP541 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org