State-Space Construction of Hybrid Petri Nets with Multiple Stochastic Firings

Abstract

Hybrid Petri nets have been extended to include general transitions that fire after a randomly distributed amount of time. With a single general one-shot transition the state space and evolution over time can be represented either as a Parametric Location Tree or as a Stochastic Time Diagram. Recent work has shown that both representations can be combined and then allow multiple stochastic firings. This work presents an algorithm for building the Parametric Location Tree with multiple general transition firings and shows how its transient probability distribution can be computed using multi-dimensional integration. We discuss the (dis-)advantages of an interval arithmetic and a geometric approach to compute the areas of integration. Furthermore, we provide details on how to perform a Monte Carlo integration either directly on these intervals or convex polytopes, or after transformation to standard simplices. A case study on a battery-backup system shows the feasibility of the approach and discusses the performance of the different integration approaches.

State-space Construction of Hybrid Petri Nets with Multiple Stochastic Firings

“…The current article extends the previously published version [32] by a thorough discussion on how the integration can be performed either on the restricted domains or on the equivalent geometric representation as regions that have to be intersected with a time constraint. The resulting arbitrary convex polytopes and their representation need to be adapted for the computation of the corresponding probabilities using multi-dimensional integration.…”

“…This results in O(2•n) half-spaces, i.e, upper and lower bounds for each potential domain. However, this does not change the worst-case complexity, denoted in Equation (32).…”

Computation of the normalising constant for product-form models of distributed systems with synchronisation

“…The exact or approximate analysis of timed Petri nets is still a very active research area both for non-Markovian [29] and Markovian [30,31] models. While we leave the deep review of existing results on product-form SPNs to Section 3.2, we mention here the approach proposed in [30] which refines the basic fluidisation approaches previously introduced in the literature to tackle the state space explosion problem.…”

ARCH-COMP20 Category Report: Stochastic Models

“…Each general transition firing results in a random variable which follows a continuous probability distribution. It efficiently implements and combines algorithms for a symbolic state-space creation [37], transformation to a geometric representation as convex polytopes [39], model checking a potentially nested STL formula and integrating over the resulting satisfaction set to yield the probability that the specification holds at a specific time.…”

“…modes and prohver, the HPnG has been transformed into a hybrid automaton with stochastic resets, according to the transformation proposed in [52]. This transformation constructs a hybrid automaton building on the symbolic state space, which is represented as a finite parametric location tree [37]. The parametric location tree, which also forms the basis for the analytical model checker hpnmg, is constructed up to a certain time t. Note that, for this model depending on the parameter setting, the state space is finite as after a certain time, no state changes occur.…”