The input-output map of a monotone discrete time quasireversible node

The author considers a class of discrete-time quasi-reversible nodes called monotone which includes discrete-time analogs of the ./M/∞ and ./M/1 nodes. For stationary ergodic nonnegative integer-valued arrival processes, the existence and uniqueness of stationary regimes are proved 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 the distance of D. Ornstein (1973). 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. The problem is of interest in connection with the construction of product form network models