32,148 research outputs found
Large deviations analysis for the queue in the Halfin-Whitt regime
We consider the FCFS queue in the Halfin-Whitt heavy traffic
regime. It is known that the normalized sequence of steady-state queue length
distributions is tight and converges weakly to a limiting random variable W.
However, those works only describe W implicitly as the invariant measure of a
complicated diffusion. Although it was proven by Gamarnik and Stolyar that the
tail of W is sub-Gaussian, the actual value of was left open. In subsequent work, Dai and He
conjectured an explicit form for this exponent, which was insensitive to the
higher moments of the service distribution.
We explicitly compute the true large deviations exponent for W when the
abandonment rate is less than the minimum service rate, the first such result
for non-Markovian queues with abandonments. Interestingly, our results resolve
the conjecture of Dai and He in the negative. Our main approach is to extend
the stochastic comparison framework of Gamarnik and Goldberg to the setting of
abandonments, requiring several novel and non-trivial contributions. Our
approach sheds light on several novel ways to think about multi-server queues
with abandonments in the Halfin-Whitt regime, which should hold in considerable
generality and provide new tools for analyzing these systems
Heavy traffic analysis of open processing networks with complete resource pooling: asymptotic optimality of discrete review policies
We consider a class of open stochastic processing networks, with feedback
routing and overlapping server capabilities, in heavy traffic. The networks we
consider satisfy the so-called complete resource pooling condition and
therefore have one-dimensional approximating Brownian control problems.
We propose a simple discrete review policy for controlling such networks.
Assuming 2+\epsilon moments on the interarrival times and processing times,
we provide a conceptually simple proof of asymptotic optimality of the proposed
policy.Comment: Published at http://dx.doi.org/10.1214/105051604000000495 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
The pseudo-self-similar traffic model: application and validation
Since the early 1990¿s, a variety of studies has shown that network traffic, both for local- and wide-area networks, has self-similar properties. This led to new approaches in network traffic modelling because most traditional traffic approaches result in the underestimation of performance measures of interest. Instead of developing completely new traffic models, a number of researchers have proposed to adapt traditional traffic modelling approaches to incorporate aspects of self-similarity. The motivation for doing so is the hope to be able to reuse techniques and tools that have been developed in the past and with which experience has been gained. One such approach for a traffic model that incorporates aspects of self-similarity is the so-called pseudo self-similar traffic model. This model is appealing, as it is easy to understand and easily embedded in Markovian performance evaluation studies. In applying this model in a number of cases, we have perceived various problems which we initially thought were particular to these specific cases. However, we recently have been able to show that these problems are fundamental to the pseudo self-similar traffic model. In this paper we review the pseudo self-similar traffic model and discuss its fundamental shortcomings. As far as we know, this is the first paper that discusses these shortcomings formally. We also report on ongoing work to overcome some of these problems
Return to the Poissonian City
Consider the following random spatial network: in a large disk, construct a
network using a stationary and isotropic Poisson line process of unit
intensity. Connect pairs of points using the network, with initial / final
segments of the connecting path formed by travelling off the network in the
opposite direction to that of the destination / source. Suppose further that
connections are established using "near-geodesics", constructed between pairs
of points using the perimeter of the cell containing these two points and
formed using only the Poisson lines not separating them. If each pair of points
generates an infinitesimal amount of traffic divided equally between the two
connecting near-geodesics, and if the Poisson line pattern is conditioned to
contain a line through the centre, then what can be said about the total flow
through the centre? In earlier work ("Geodesics and flows in a Poissonian
city", Annals of Applied Probability, 21(3), 801--842, 2011) it was shown that
a scaled version of this flow had asymptotic distribution given by the 4-volume
of a region in 4-space, constructed using an improper anisotropic Poisson line
process in an infinite planar strip. Here we construct a more amenable
representation in terms of two "seminal curves" defined by the improper Poisson
line process, and establish results which produce a framework for effective
simulation from this distribution up to an L1 error which tends to zero with
increasing computational effort.Comment: 11 pages, 2 figures Various minor edits, corrections to
multiplicative constants in Theorem 5.1. Version 2: minor stylistic
corrections, added acknowledgement of grant support. Version 3: three further
minor corrections. This paper is due to appear in Journal of Applied
Probability, Volume 51
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