Computer protocol and torus maps

We investigate the dynamics of maps and flows which arise from a class of models of closed queueing networks in computer science theory. The network consists of n+1 servers, one of which is a central server with a queue of size n-1. A protocol or scheduling discipline must be specified in this server to define the queueing network. The standard model gives rise to a flow on an n-torus. We consider the service protocols first in-first out (FIFO) and last in-first out (LIFO) in dimension three, for which the state spaces are modifications of a 3-torus. We present a sufficient condition on the time it takes each call to complete one cycle for the FIFO protocol which guarantees that the set of periodic orbits which involve no waiting in the queue is a global attractor for the associated semi-flow. We also investigate the dynamics for the LIFO service protocol via a return map derived from the associated area preserving flow.