Discrete Time Networks with Product Form Steady States

Abstract

We consider networks of queues in discrete time, where the steady state distribution can be computed explicitly in closed form (product form networks): (i) Closed cycles and open tandems of single server FCFS Bernoulli nodes with state dependent service probabilities, where customers flow linearly, (ii) networks of doubly stochastic and geometrical queues (which are discrete time analogues of Kelly’s symmetric, resp. general, servers), where customers of different types move through the network governed by a general routing mechanism and request for service according to general, resp. geometrical, distributions, (iii) networks with batch movements of customers and batch service, where the service and routing mechanism is defined via an abstract transition scheme.

We describe recent developments of product form networks where nodes are unreliable, break down and are repaired. This opens the possibility to investigate performance and availability of networks in an integrated model.