A Transformation of Hybrid Petri Nets with Stochastic Firings into a Subclass of Stochastic Hybrid Automata

Abstract

We present a transformation of Hybrid Petri nets extended with stochastic firings (HPnGs) into a subclass of Stochastic Hybrid Automata (SHA), thereby making HPnGs amenable to techniques from that domain. While (non-stochastic) Hybrid Petri nets have previously been transformed into Hybrid Automata, we consider also stochastic aspects and transform HPnGs into Singular Automata, which are Hybrid Automata restricted to piecewise constant derivatives for continuous variables, extended by random clocks. We implemented our transformation and show its usefulness by comparing results for time-bounded reachability for HPnGs extended with non-determinism on the one hand, and for the transformed SHAs using the ProHVer tool on the other hand.

Appendix

1.1 Parametric Location Tree

Fig. 4.

Parametric location tree for the feasibility study example

Figure 4 shows the Parametric Location Tree for the model of the feasibility study (r.t. Sect. 6). t is the entry time into each parametric location, x denotes the fluid level of the continuous place $P^c_\text {battery}$ and $\dot{x}$ its drift upon entry. Further $s_0$ denotes the random variable, which describes the firing of $T^G_\text {switch}$, and similarly $s_1$ denotes the one of $T^G_{\text {stop}\_\text {high}}$. Every node shows how many discrete places contain one token each and which non-continuous transitions are enabled. The restrictions on the random variables describe the values which the random variables need to take to get into the specific parametric location.

The PLT is split right at its root ($\varLambda _1$) into two sub-trees, depending on whether $T^G_\text {switch}$ fires before or after $P^c_\text {battery}$ reaches zero. If it fires within 9 time units, this leads us to $\varLambda _2$ and otherwise to $\varLambda _6$, via $\varLambda _3$. From there, in both sub-trees the non-deterministic choice is taken, further splitting the tree. If $T^I_{\text {choose}\_\text {high}}$ is chosen ($\varLambda _4$ and $\varLambda _{11}$), $T^G_{\text {stop}\_\text {high}}$ becomes enabled and either it fires ($\varLambda _7, \varLambda _{13}$) or $P^c_\text {battery}$ reaches its upper boundary ($\varLambda _8, \varLambda _{14}$). If instead $T^I_{\text {choose}\_\text {low}}$ is chosen ($\varLambda _5, \varLambda _{12}$), $T^D_{\text {stop}\_\text {low}}$ becomes enabled and is might fire ($\varLambda _9, \varLambda _{15}$). Only if $P^c_\text {battery}$ has not been empty before ($\varLambda _5$), it can reach its upper boundary before $T^D_{\text {stop}\_\text {low}}$ fires ($\varLambda _{10}$).