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

Loading next page...
 
/lp/springer_journal/weak-convergence-and-fluid-limits-in-optimal-time-to-empty-queueing-pMCQ0gSyAm
Publisher
Springer-Verlag
Copyright
Copyright © 2011 by Springer Science+Business Media, LLC
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

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off