On monotonicity of finite capacity queueing networks and their perfect simulation

Traditional product form property of Markovian queueing networks usually is vanished when capacity of queues are finite and clients are blocked or rejected. A new efficient simulation method, derived from Propp & Wilson (Propp 1996) perfect simulation, is applied to the finite capacity queues context. We present an algorithm that samples directly states of the network according to stationary distribution. This method has been applied to queueing networks with various monotonous routing policies. It is shown that, according to an adequate uniformization techniques usual Markovian queuing networks are monotonous. Such monotonous networks include networks with overflow or blocking, join the shortest queue routing policy, fork of customers... Consequently, perfect simulation could be improved by drawing trajectories from minimum and maximum states which reduces computation time. Moreover, for the estimation of a monotonous reward function, the simulation time could be reduced drastically as in (Vincent & Marchand 2004). Some examples are given : loss estimation on Erlang models, usage of the last queue in a line of queues with blocking, saturation estimation for a multi-stage interconnection switch...