## Section: New Results

### Perfect Simulation

Participants : Bruno Gaujal, Brigitte Plateau, Florence Perronnin, Jean-Marc Vincent.

Perfect simulation enables one to compute samples distributed according to the stationary distribution of the Markov process with no bias. The following sections summarize the various new results obtained using this technique, or on this technique.

#### Different Monotonicity Definitions in Stochastic Modelling

In [32] , we discuss different monotonicity definitions applied in stochastic modelling. Obviously, the relationships between the monotonicity concepts depends on the relation order that we consider on the state space. In the case of total ordering, the stochastic monotonicity used to build bounding models and the realizable monotonicity used in perfect simulation are equivalent to each other while in the case of partial order there is only implication between them. Indeed, there are cases of partial order, where we cannot move from the stochastic monotonicity to the realizable monotonicity. This is why we will try to find the conditions for which there are equivalences between these two notions.

#### Perfect simulation and non-monotone Markovian systems

Perfect simulation, or coupling from the past, is an efficient technique for sampling the steady state of non-monotone discrete time Markov chains over lattices. Indeed, one only needs to consider two trajectories corresponding to inf. and sup. trajectories. We have shown that these envelopes can be efficiently computed for piece-wise space homogenenous Markov chains. In particular for Almost Sapce Homogeneous Events (ASHEs), closed form formulas for the envelopes are known. This can be extended to events with polytopic piece-wise zones where linear programming can be used to compute envelopes. Linear progamming can be replaced by a fast computations of Minkowsky sums of polytopes and cubes up to a slight loss in the tightness of the bounds. These approaches are useful for example, to simulate queuing networks with “ join the shortest queue” routing schemes.