Networks and dynamical systems

A new approach to the problem of classification of (deflected) random walks in or Markovian models for queueing networks with identical customers is introduced. It is based on the analysis of the intrinsic dynamical system associated with the random walk. Earlier results for small dimensions are presented from this novel point of view. We give proofs of new results for higher dimensions related to the existence of a continuous invariant measure for the underlying dynamical system. Two constants are shown to be important: the free energy M < 0 corresponds to ergodicity, the Lyapounov exponent L < 0 defines recurrence. General conjectures, examples, unsolved problems and surprising connections with ergodic theory, classical dynamical systems and their random perturbations are largely presented. A useful notion naturally arises, the so-called scaled random perturbation of a dynamical system.