Stability and Capacity Regions or Discrete Time Queueing Networks

We consider stability and network capacity in discrete time queueing systems. Relationships between four common notions of stability are described. Specifically, we consider rate stability, mean rate stability, steady state stability, and strong stability. We then consider networks of queues with random events and control actions that can be implemented over time to affect arrivals and service at the queues. The control actions also generate a vector of additional network attributes. We characterize the network capacity region, being the closure of the set of all rate vectors that can be supported subject to network stability and to additional time average attribute constraints. We show that (under mild technical assumptions) the capacity region is the same under all four stability definitions. Our capacity achievability proof uses the drift-plus-penalty method of Lyapunov optimization, and provides full details for the case when network states obey a decaying memory property, which holds for finite state ergodic systems and more general systems.