Communicating Piecewise Deterministic Markov Processes

Abstract

In this chapter we introduce the automata framework CPDP, which stands for Communicating Piecewise Deterministic Markov Processes. CPDP is developed for compositional modelling and analysis for a class of stochastic hybrid systems. We define a parallel composition operator, denoted as $\left\vert {P \atop A} \right\vert$, for CPDPs, which can be used to interconnect component-CPDPs, to form the composite system (which consists of all components, interacting with each other). We show that the result of composing CPDPs with $\left\vert {P \atop A} \right\vert$ is again a CPDP (i.e., the class of CPDPs is closed under $\left\vert {P \atop A} \right\vert$). Under certain conditions, the evolution of the state of a CPDP can be modelled as a stochastic process. We show that for these CPDPs, this stochastic process can always be modelled as a PDP (Piecewise Deterministic Markov Process) and we present an algorithm that finds the corresponding PDP of a CPDP. After that, we present an extended CPDP framework called value-passing CPDP. This framework provides richer interaction possibilities, where components can communicate information about their continuous states to each other. We give an Air Tra.c Management example, modelled as a value-passing CPDP and we show that according to the algorithm, this CPDP behavior can be modelled as a PDP. Finally, we define bisimulation relations for CPDPs. We prove that bisimilar CPDPs exhibit equal stochastic behavior. Bisimulation can be used as a state reduction technique by substituting a CPDP (or a CPDP component) by a bisimulation-equivalent CPDP (or CPDP component) with a smaller state space. This can be done because we know that such a substitution will not change the stochastic behavior.