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HeavyTraffic Limits for Nearly Deterministic Queues
"... We establish heavytraffic limits for “nearly deterministic” queues, such as the G/D/n manyserver queue. Waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated Gn/D/1 model, where the Gn denotes “cyclic thinning ” of order n, indicating that the or ..."
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We establish heavytraffic limits for “nearly deterministic” queues, such as the G/D/n manyserver queue. Waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated Gn/D/1 model, where the Gn denotes “cyclic thinning ” of order n, indicating that the original (possibly general) point process of arrivals is thinned to contain only every n th point. We thus focus on the Gn/D/1 model and the generalization to Gn/Gn/1, where “cyclic thinning ” is applied to both the arrival and service processes. As n → ∞, the Gn/Gn/1 models approach the deterministic D/D/1 model. The classical example is the Erlang En/En/1 queue, where cyclic thinning of order n is applied to both the interarrival times and the service times, starting from a “base ” M/M/1 model. We establish different limits in two cases: (i) when (1−ρn) √ n → β as n → ∞ and (ii) (1 − ρn)n → β as n → ∞, where ρn is the traffic intensity in model n, and 0 < β < ∞. The nearly deterministic feature leads to interesting nonstandard scaling. We also establish revealing heavytraffic limits for the stationary waiting times and other performance measures in the Gn/Gn/1 queues, by letting ρn ↑ 1 as n → ∞.
2008b. Continuity of a Queueing Integral Representation in the M1 Topology. Submitted to Annals of Applied Probability
"... We establish continuity of the integral representation y(t) = x(t) + ∫ t0 h(y(s)) ds, t ≥ 0, mapping a function x into a function y when the underlying function space D is endowed with the Skorohod M1 topology. We apply this integral representation with the continuous mapping theorem to establish h ..."
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Cited by 3 (1 self)
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We establish continuity of the integral representation y(t) = x(t) + ∫ t0 h(y(s)) ds, t ≥ 0, mapping a function x into a function y when the underlying function space D is endowed with the Skorohod M1 topology. We apply this integral representation with the continuous mapping theorem to establish heavytraffic stochasticprocess limits for manyserver queueing models when the limit process has jumps unmatched in the converging processes as can occur with bursty arrival processes or service interruptions. The proof of M1continuity is based on a new characterization of the M1 convergence, in which the time portions of the parametric representations are absolutely continuous with respect to Lebesgue measure, and the derivatives are uniformly bounded and converge in L1. (1.1) 1. Introduction. The
Applied Probability Trust (1 December 2010) A PIECEWISE LINEAR SDE DRIVEN BY A LÉVY PROCESSES
"... We consider an SDE with piecewise linear drift driven by a spectrally onesided Lévy process. We show this SDE has some connections with queueing and storage models and apply this to obtain the invariant distribution. 1. ..."
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We consider an SDE with piecewise linear drift driven by a spectrally onesided Lévy process. We show this SDE has some connections with queueing and storage models and apply this to obtain the invariant distribution. 1.
HeavyTraffic Limits via an Averaging Principle for Service Systems Responding to Unexpected Overloads
, 2010
"... This dissertation considers how two networked largescale service systems, such as call centers, that normally operate separately, can help each other in face of an unexpected overload, caused by a sudden shift in the arrival rates. We assume that the time of the shift and the values of the new arri ..."
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This dissertation considers how two networked largescale service systems, such as call centers, that normally operate separately, can help each other in face of an unexpected overload, caused by a sudden shift in the arrival rates. We assume that the time of the shift and the values of the new arrival rates are not known apriori, and are hard to detect in real time. We also assume that staffing cannot be increased immediately. We propose the fixedqueue ratio with thresholds (FQRT) control, and show that it is optimal in a deterministic fluid approximation. The FQRT control activates serving some customers from the other system when a ratio of the two queue lengths (numbers of waiting customers) exceeds a threshold. Two thresholds, one for each direction of sharing, automatically detect the overload condition and prevent undesired sharing under normal loads. After a threshold has been exceeded, the control aims to keep the ratio of the two queue lengths at a specified value. To gain insight, we introduce an idealized X model, i.e., a stochastic model with two customer classes, each with its own dedicated service pool, containing a large number of agents. The agents in both pools are assumed to be crossedtrained, so that they are able
Critically loaded multiserver queues with abandonments, retrials, and timevarying parameters
, 2009
"... In this paper, we consider modeling timedependent multiserver 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 ..."
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In this paper, we consider modeling timedependent multiserver 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. 1