State Equation for Interval-Timed Petri Nets

Timed Petri Nets (also called Duration Petri Nets, DPNs) [7, 8] are a commonly used class for modeling formally complex technical systems as embedded real time systems, for instance in aeronautics. Engineers stated that intervals allowing some variable duration of events would be a more realistic view, so Interval-Timed Petri Nets (ITPNs) have been proposed as generalizations in [2]. Hierarchical extensions and behavioral analysis of those ITPNs under partial order aspects has been studied in [1, 4]. Timed Petri Nets, as considered in [6] are a proper subclass of ITPNs, when both classes have their duration limits as natural resp. rational, numbers. Interval-Timed Petri Nets can be simulated by or translated into other time dependent net classes, as in causal time nets [3,4] or Time Petri Nets [5]. The state space explosion makes direct reachability analysis of Interval-Timed Petri Nets non efficient, in particular to prove non-reachability, which implies to build the whole state graph. The approach to prove the nonreachability on the timeless skeleton of the ITPN only yields a sufficient but not necessary condition, not taking in account the time constraints of the net, thus this is not very satisfactory, too. Translation in causal or Time Petri Nets does not bring more efficiency. For other, simpler net classes, an algebraic way exists to calculate non-reachability, using the state equation of the net. Such state equations can be calculated for usual timeless Petri nets and DPNs, cf. [6]. In this paper, we propose a generalization to ITPNs of the time dependent state equation for DPNs from [6] which needs to be more sophisticated, taking into account each possible actual duration of a transition. Our result may look more complicated as it actually is, and is practically computable, because the used matrices are sparse matrices, i.e. their elements are mainly zeros.