Weak Convergence and Fluid Limits in Optimal Time-to-Empty Queueing Control Problems

Weak Convergence and Fluid Limits in Optimal Time-to-Empty Queueing Control Problems We consider a class of controlled queue length processes, in which the control allocates each server’s effort among the several classes of customers requiring its service. Served customers are routed through the network according to (prescribed) routing probabilities. In the fluid rescaling, $X^{n}(t)=\frac{1}{n} X(nt)$ , we consider the optimal control problem of minimizing the integral of an undiscounted positive running cost until the first time that X n =0. Our main result uses weak convergence ideas to show that the optimal value functions V n of the stochastic control problems for X n ( t ) converge (as n →∞) to the optimal value V of a control problem for the limiting fluid process. This requires certain equicontinuity and boundedness hypotheses on { V n }. We observe that these are essentially the same hypotheses that would be needed for the Barles-Perthame approach in terms of semicontinuous viscosity solutions. Sufficient conditions for these equicontinuity and boundedness properties are briefly discussed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Weak Convergence and Fluid Limits in Optimal Time-to-Empty Queueing Control Problems

, Volume 64 (3) – Dec 1, 2011
24 pages

/lp/springer_journal/weak-convergence-and-fluid-limits-in-optimal-time-to-empty-queueing-pMCQ0gSyAm
Publisher
Springer-Verlag
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics; Numerical and Computational Physics; Mathematical Methods in Physics; Systems Theory, Control
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-011-9144-y
Publisher site
See Article on Publisher Site

Abstract

We consider a class of controlled queue length processes, in which the control allocates each server’s effort among the several classes of customers requiring its service. Served customers are routed through the network according to (prescribed) routing probabilities. In the fluid rescaling, $X^{n}(t)=\frac{1}{n} X(nt)$ , we consider the optimal control problem of minimizing the integral of an undiscounted positive running cost until the first time that X n =0. Our main result uses weak convergence ideas to show that the optimal value functions V n of the stochastic control problems for X n ( t ) converge (as n →∞) to the optimal value V of a control problem for the limiting fluid process. This requires certain equicontinuity and boundedness hypotheses on { V n }. We observe that these are essentially the same hypotheses that would be needed for the Barles-Perthame approach in terms of semicontinuous viscosity solutions. Sufficient conditions for these equicontinuity and boundedness properties are briefly discussed.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Dec 1, 2011

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