  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.
