  Lyapunov drift and Lyapunov optimization are powerful techniques for
optimizing time averages in stochastic queueing networks subject to stability.
However, there are various definitions of queue stability in the literature,
and the most convenient Lyapunov drift conditions often provide stability and
performance bounds only in terms of a time average expectation, rather than a
pure time average. We extend the theory to show that for quadratic Lyapunov
functions, the basic drift condition, together with a mild bounded fourth
moment condition, implies all major forms of stability. Further, we show that
the basic drift-plus-penalty condition implies that the same bounds for queue
backlog and penalty expenditure that are known to hold for time average
expectations also hold for pure time averages with probability 1. Our analysis
combines Lyapunov drift theory with the Kolmogorov law of large numbers for
martingale differences with finite variance.

Optimization and Control