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.