The input-output map of a monotone discrete-time quasireversible node [queueing theory]

A class of discrete-time quasi-reversible nodes called monotone, which includes discrete-time analogs of the ./M/∞ and ./M/1 nodes, is considered. For stationary ergodic nonnegative integer valued arrival processes, the existence and uniqueness of stationary regimes are proven when a natural rate condition is met. Coupling is used to prove the contractiveness of the input-output map relative to a natural distance on the space of stationary arrival processes that is analogous to Ornstein´s d¯ distance. A consequence is that the only stationary ergodic fixed points of the input-output map are the processes of independent and identically distributed Poisson random variables meeting the rate condition