Journal of Applied Mathematics

Stability and Probability 1 Convergence for Queueing Networks via Lyapunov Optimization

Michael J. Neely

Full-text: Open access

Abstract

Lyapunov drift is a powerful tool for optimizing stochastic queueing networks subject to stability. However, the most convenient drift conditions often provide results in terms of a time average expectation, rather than a pure time average. This paper provides an extended drift-plus-penalty result that ensures stability with desired time averages with probability 1. The analysis uses the law of large numbers for martingale differences. This is applied to quadratic and subquadratic Lyapunov methods for minimizing the time average of a network penalty function subject to stability and to additional time average constraints. Similar to known results for time average expectations, this paper shows that pure time average penalties can be pushed arbitrarily close to optimality, with a corresponding tradeoff in average queue size. Further, in the special case of quadratic Lyapunov functions, the basic drift condition is shown to imply all major forms of queue stability.

Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 831909, 35 pages.

Dates
First available in Project Euclid: 17 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.jam/1350479387

Digital Object Identifier
doi:10.1155/2012/831909

Mathematical Reviews number (MathSciNet)
MR2948168

Zentralblatt MATH identifier
1251.90096

Citation

Neely, Michael J. Stability and Probability 1 Convergence for Queueing Networks via Lyapunov Optimization. J. Appl. Math. 2012 (2012), Article ID 831909, 35 pages. doi:10.1155/2012/831909. https://projecteuclid.org/euclid.jam/1350479387


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