Brownian models of open queueing networks with homogeneous customer populations<sup>∗</sup>

Harrison, J. Michael; Williams, Ruth

doi:10.1080/17442508708833469

Cited by 302 publications

(301 citation statements)

References 20 publications

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“…Arratia in [5] (see also [4]) observes that instead of Brownian motion, if one considers the exclusion process associated with the nearest neighbor random walk on Z, then the corresponding stationary system of spacings between particles can be interpreted as a finite or infinite series of queues. We direct the reader to the seminal articles by Harrison and his coauthors for a background on systems of Brownian queues: [16], [17], [18], [19], and [20]. Baryshnikov [7] establishes the connections between Brownian queues and GUE random matrices.…”

Section: Introductionmentioning

confidence: 99%

A phase transition behavior for Brownian motions interacting through their ranks

Chatterjee

Pal

2009

Probab. Theory Relat. Fields

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Abstract. Consider a time-varying collection of n points on the positive real axis, modeled as Exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both central objects of study in models of equity markets introduced by Banner, Fernholz, and Karatzas. In this paper, we look at the behavior of this point process under the stationary measure as n tends to infinity. Under a certain 'continuity at the edge' condition on the drifts, we show that one of the following must happen: either (i) all points converge to 0, or (ii) the maximum goes to 1 and the rest go to 0, or (iii) the processes converge in law to a non-trivial Poisson-Dirichlet distribution. The underlying idea of the proof is inspired by Talagrand's analysis of the low temperature phase of Derrida's Random Energy Model of spin glasses. The main result establishes a universality property for the BFK models and aids in explicit asymptotic computations using known results about the Poisson-Dirichlet law.

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Section: Introductionmentioning

confidence: 99%

A phase transition behavior for Brownian motions interacting through their ranks

Chatterjee

Pal

2009

Probab. Theory Relat. Fields

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“…if b ∈ C then the corresponding constrained diffusion is transient [5]. In the context of the constant drift case, this necessary and sufficient condition for positive recurrence, for constrained diffusions which correspond to single class networks, was first proved in [14].…”

Section: Introductionmentioning

confidence: 99%

An Ergodic Control Problem for Constrained Diffusion Processes: Existence of Optimal Markov Control

Budhiraja¹

2003

SIAM J. Control Optim.

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An ergodic control problem for a class of constrained diffusion processes is considered. The goal is the almost sure minimization of long term cost per unit time. The main result of the paper is that there exists an optimal Markov control for the considered problem. It is shown that under the assumption of regularity of the Skorohod map and appropriate conditions on the drift coefficient the class of controlled diffusion processes considered have strong, uniform in control, stability properties. The stability results on the class of controlled constrained diffusion processes considered in this work are non-trivial in that the domains are unbounded and the corresponding unconstrained diffusions are typically transient. These stability properties are key in obtaining appropriate tightness estimates. Once these estimates are available the remaining work lies in identifying weak limits of a certain family of occupation measures. In this regard an extension to the Echeverria-Weiss-Kurtz characterization of invariant measures of Markov processes, to the case of constrained-controlled processes considered in this paper, is proved. This characterization result is also crucially used in proving the compactness of the family of invariant measures of Markov processes corresponding to all possible Markov controls.

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“…Static planning problems are formulated by constructing a system where the fluid-scaled processes are replaced by their long-run averages (or fluid limits) and solving a suitable optimization problem involving those averages (see [19,25,28,29]). In the fluid limit, we formulate a deterministic problem to choose ρ andμ that maximize the profit rate subject to stability conditions on both queues.…”

Section: Static Planning Problemmentioning

confidence: 99%

Heavy Traffic Analysis of a Simple Closed-Loop Supply Chain

et al. 2010

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We consider a closed-loop supply chain where new products are produced to order and returned products are refurbished for reselling. The solution to aprice-setting problem enforces the heavy traffic condition, under which we address the production rate control problem for two types of cost functions. Wesolve a drift-control problem for an approximate system driven by a correlated two-dimensional Brownian motion. The solutions to this system are then used to obtain asymptotically optimal control policies. We also conduct a numerical study to explore the effects of different parameters on the optimal production rates and the resulting costs. AbstractWe consider a closed loop supply chain where new products are produced to order and returned products are refurbished for reselling. The solution to a price-setting problem enforces the "heavy traffic" condition, under which we address the production rate control problem for two types of cost functions. We solve a drift-control problem for an approximate system driven by a correlated two-dimensional Brownian motion. The solutions to this system are then used to obtain asymptotically optimal control policies. We also conduct a numerical study to explore the effects of different parameters on the optimal production rates and the resulting costs.

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Brownian models of open queueing networks with homogeneous customer populations^∗

Cited by 302 publications

References 20 publications

A phase transition behavior for Brownian motions interacting through their ranks

A phase transition behavior for Brownian motions interacting through their ranks

An Ergodic Control Problem for Constrained Diffusion Processes: Existence of Optimal Markov Control

Heavy Traffic Analysis of a Simple Closed-Loop Supply Chain

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Brownian models of open queueing networks with homogeneous customer populations∗

Cited by 302 publications

References 20 publications

A phase transition behavior for Brownian motions interacting through their ranks

A phase transition behavior for Brownian motions interacting through their ranks

An Ergodic Control Problem for Constrained Diffusion Processes: Existence of Optimal Markov Control

Heavy Traffic Analysis of a Simple Closed-Loop Supply Chain

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Brownian models of open queueing networks with homogeneous customer populations^∗