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 Stochastic Systems > Vol. 6 (2016) open journal systems 


Construction of asymptotically optimal control for crisscross network from a free boundary problem

Amarjit Budhiraja, University of North Carolina at Chapel Hill
Xin Liu, Clemson University
Subhamay Saha, Indian Institute of Technology Guwahati


Abstract
An asymptotic framework for optimal control of multiclass stochastic processing networks, using formal diffusion approximations under suitable temporal and spatial scaling, by Brownian control problems (BCP) and their equivalent workload formulations (EWF), has been developed by Harrison (1988). This framework has been implemented in many works for constructing asymptotically optimal control policies for a broad range of stochastic network models. To date all asymptotic optimality results for such networks correspond to settings where the solution of the EWF is a reflected Brownian motion in \(\mathbb{R}_+\) or a wedge in \(\mathbb{R}_+^2\). In this work we consider a well studied stochastic network which is perhaps the simplest example of a model with more than one dimensional workload process. In the regime considered here, the singular control problem corresponding to the EWF does not have a simple form explicit solution. However, by considering an associated free boundary problem one can give a representation for an optimal controlled process as a two dimensional reflected Brownian motion in a Lipschitz domain whose boundary is determined by the solution of the free boundary problem. Using the form of the optimal solution we propose a sequence of control policies, given in terms of suitable thresholds, for the scaled stochastic network control problems and prove that this sequence of policies is asymptotically optimal. As suggested by the solution of the EWF, the policy we propose requires a server to idle under certain conditions which are specified in terms of thresholds determined from the free boundary.

AMS 2000 subject classifications: Primary 60K25, 68M20, 90B36; secondary 60J70.

Keywords: Stochastic networks, crisscross networks, dynamic control, heavy traffic, diffusion approximations, Brownian control problems, singular control problems, reflected Brownian motions, free boundary problems, threshold policies, large deviations.

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Budhiraja, Amarjit, Liu, Xin, Saha, Subhamay, Construction of asymptotically optimal control for crisscross network from a free boundary problem, Stochastic Systems, 6, (2016), 459-518 (electronic). DOI: 10.1214/15-SSY211.

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