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.
Highlights
This paper considers Lyapunov methods for analysis and control of queueing networks
Time average expectations can often be translated into pure time averages when the system has a Markov structure and some additional assumptions are satisfied, such as when the system regularly returns to a renewal state
Much prior work on queue stability uses quadratic Lyapunov functions, including 1, 2, 19– for ergodic systems defined on countably infinite Markov chains, and for more general systems but where stability is defined in terms of a time average expectation
Summary
This paper considers Lyapunov methods for analysis and control of queueing networks. Landmark work in [1, 2] uses the drift of a quadratic Lyapunov function to design general max-weight algorithms for network stability. In the special case of quadratic Lyapunov functions, it is shown that the basic drift condition implies all major forms of queue stability, including rate and mean rate stability, as well as four stronger versions These results require only mild bounded fourth moment conditions on queue difference processes and bounded second-moment conditions on penalty processes and do not require a Markov structure or renewal assumptions. Algorithms based on subquadratics are shown to provide desired results whenever the problem is feasible, without requiring the additional slackness assumption This analysis is useful for control of queueing networks, and for other stochastic problems that seek to optimize time averages subject to time average constraints
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
