Bounds Computation for Symmetric Nets

Monotonicity in Markov chains is the starting point for quantitative abstraction of complex probabilistic systems leading to (upper or lower) bounds for probabilities and mean values relevant to their analysis. While numerous case studies exist in the literature, there is no generic model for which monotonicity is directly derived from its structure. Here we propose such a model and formalize it as a subclass of Stochastic Symmetric (Petri) Nets (SSNs) called Stochastic Monotonic SNs (SMSNs). On this subclass the monotonicity is proven by coupling arguments that can be applied on an abstract description of the state (symbolic marking). Our class includes both process synchronizations and resource sharings and can be extended to model open or cyclic closed systems. Automatic methods for transforming a non monotonic system into a monotonic one matching the MSN pattern, or for transforming a monotonic system with large state space into one with reduced state space are presented. We illustrate the interest of the proposed method by expressing standard monotonic models and modelling a flexible manufacturing system case study

Stochastic bounds for loss rates

We propose to use stochastic comparison method in order to compute packet loss rates in IP routers for MPLS networks. We study an IP buffer with finite capacity B and we consider two kinds of packets. Those who are delay sensitive and those who are loss sensitive. We propose a new service discipline which we denoted by "LATIN" service discipline and for buffer management, we propose to use push-out. The state of the buffer can be represented by a discrete Markov chain defined on a large state space size O(B^3). We propose bounded models with HOL disciplines, which are easier models to solve and using stochastic comparisons, we prove that the bounded models provide really upper and lower bounds on the packet loss rates. We present some numerical results to show the accurate bounds on loss rates.ou