5,663 research outputs found
Critically loaded multi-server queues with abandonments, retrials, and time-varying parameters
In this paper, we consider modeling time-dependent multi-server queues that
include abandonments and retrials. For the performance analysis of those, fluid
and diffusion models called "strong approximations" have been widely used in
the literature. Although they are proven to be asymptotically exact, their
effectiveness as approximations in critically loaded regimes needs to be
investigated. To that end, we find that existing fluid and diffusion
approximations might be either inaccurate under simplifying assumptions or
computationally intractable. To address that concern, this paper focuses on
developing a methodology by adjusting the fluid and diffusion models so that
they significantly improve the estimation accuracy. We illustrate the accuracy
of our adjusted models by performing a number of numerical experiments
Diffusion models and steady-state approximations for exponentially ergodic Markovian queues
Motivated by queues with many servers, we study Brownian steady-state
approximations for continuous time Markov chains (CTMCs). Our approximations
are based on diffusion models (rather than a diffusion limit) whose
steady-state, we prove, approximates that of the Markov chain with notable
precision. Strong approximations provide such "limitless" approximations for
process dynamics. Our focus here is on steady-state distributions, and the
diffusion model that we propose is tractable relative to strong approximations.
Within an asymptotic framework, in which a scale parameter is taken large,
a uniform (in the scale parameter) Lyapunov condition imposed on the sequence
of diffusion models guarantees that the gap between the steady-state moments of
the diffusion and those of the properly centered and scaled CTMCs shrinks at a
rate of . Our proofs build on gradient estimates for solutions of the
Poisson equations associated with the (sequence of) diffusion models and on
elementary martingale arguments. As a by-product of our analysis, we explore
connections between Lyapunov functions for the fluid model, the diffusion model
and the CTMC.Comment: Published in at http://dx.doi.org/10.1214/13-AAP984 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
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