Diffusion approximations for controlled stochastic networks: An asymptotic bound for the value function

Budhiraja, Amarjit; Ghosh, Arka P.

doi:10.1214/105051606000000457

Cited by 65 publications

(73 citation statements)

References 20 publications

(32 reference statements)

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“…From standard estimates for renewal processes (see e.g., [2] Lemma 3.5), we see that under assumptions of Section 2, (A8.p) holds for p = 2. We refer the reader to [13] for sufficient conditions on the primitives of renewal processes under which (A8.p) is satisfied for p ∈ (2, ∞).…”

Section: Convergence Of Invariant Measuresmentioning

confidence: 85%

Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic

2009

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In a recent paper [5] it was shown that under suitable conditions stationary distributions of the (scaled) queue lengths process for a generalized Jackson network converge to the stationary distribution of the associated reflected Brownian motion in the heavy traffic limit. The proof relied on certain exponential integrability assumptions on the primitives of the network. In this note we show that the above result holds under much weaker integrability conditions. We provide an alternative proof of this result making (in addition to natural heavy traffic and stability assumptions) only standard independence and square integrability assumptions on the network primitives that are commonly used in heavy traffic analysis. Furthermore, under additional integrability conditions we establish convergence of moments of stationary distributions.

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Section: Convergence Of Invariant Measuresmentioning

confidence: 85%

Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic

2009

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“…A solution to the BCP provides useful insight into the queueing network control problem. For a broad class of queueing problems, it has been shown that the value function of the BCP is a lower bound for the minimum cost in the queueing network control problem (see [7]). In some cases, the solution to the BCP can be utilized to obtain optimal strategies for the queueing network control problem (cf.…”

Section: Introductionmentioning

confidence: 99%

Optimal buffer size for a stochastic processing network in heavy traffic

Ghosh

Weerasinghe

2007

Queueing Syst

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We consider a one-dimensional stochastic control problem that arises from queueing network applications. The state process corresponding to the queue-length process is given by a stochastic differential equation which reflects at the origin. The controller can choose the drift coefficient which represents the service rate and the buffer size b > 0. When the queue length reaches b, the new customers are rejected and this incurs a penalty. There are three types of costs involved: A "control cost" related to the dynamically controlled service rate, a "congestion cost" which depends on the queue length and a "rejection penalty" for the rejection of the customers. We consider the problem of minimizing long-term average cost, which is also known as the ergodic cost criterion. We obtain an optimal drift rate (i.e. an optimal service rate) as well as the optimal buffer size b * > 0. When the buffer size b > 0 is fixed and where there is no congestion cost, this problem is similar to the work in Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145-1160, 2005). Our method is quite different from that of (Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145-1160, 2005).To obtain a solution to the corresponding Hamilton-JacobiBellman (HJB) equation, we analyze a family of ordinary A.P. Ghoshdifferential equations. We make use of some specific characteristics of this family of solutions to obtain the optimal buffer size b * > 0.

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“…A solution of the limiting control problem provides useful insights into the queueing network control problem (see, for instance, [13]). For a broad class of queueing problems, it has been shown that the value function of the Brownian control problem (BCP) is a lower bound for the minimum cost in the queueing network control problem (see [6]). In many situations, the solution to the BCP can be utilized to obtain optimal strategies for the queueing network control problem (cf.…”

Section: Motivating Examplementioning

confidence: 99%

Optimal Control of a Stochastic Processing System Driven by a Fractional Brownian Motion Input

Ghosh¹,

Roitershtein²,

Weerasinghe³

2010

Adv. Appl. Probab.

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We consider a stochastic control model driven by a fractional Brownian motion. This model is a formal approximation to a queueing network with an ON-OFF input process. We study stochastic control problems associated with the long-run average cost, the infinite-horizon discounted cost, and the finite-horizon cost. In addition, we find a solution to a constrained minimization problem as an application of our solution to the long-run average cost problem. We also establish Abelian limit relationships among the value functions of the above control problems.

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Diffusion approximations for controlled stochastic networks: An asymptotic bound for the value function

Cited by 65 publications

References 20 publications

Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic

Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic

Optimal buffer size for a stochastic processing network in heavy traffic

Optimal Control of a Stochastic Processing System Driven by a Fractional Brownian Motion Input

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