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Journal of Applied Mathematics
Volume 2012, Article ID 831909, 35 pages
Research Article

Stability and Probability 1 Convergence for Queueing Networks via Lyapunov Optimization

Electrical Engineering Department, University of Southern California, 3740 McClintock Avenue, Room 500, Los Angeles, CA 90089-2565, USA

Received 15 August 2011; Revised 17 March 2012; Accepted 11 April 2012

Academic Editor: P. G. L. Leach

Copyright © 2012 Michael J. Neely. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.