Vhs Case Study 5: Semantics of Timed Petri Nets, Equivalence and Model Reduction

1 Semantics for timed Petri nets used in case study 5 model An (untimed) Petri net 6] is deened by the triple (P; T; F) with P a nite set of places, T a nite set of transitions, P \T = ;, and F (P T)(T P) a set of relations. The state of the untimed Petri net is the marking m : P ! Z + , assigning to each place p the number of tokens m(p) present in the place p: The set t := fp 2 P j (p; t) 2 Fg (t := fp 2 P j (t; p) 2 Fg) contains the input (output) places of transition t 2 T: The sets of input, resp. output transitions p := ft 2 T j (t; p) 2 Fg, resp. p := ft 2 T j (t; p) 2 Fg for place p 2 P are deened analogously. A transition t is called state-enabled under marking m if all its input places are marked, i.e. 8p 2 t : m(p) > 0. When t is state-enabled then t can be (but does not have to be) executed or red. When t is executed, then one token is removed from each of the places in t and one token is added to each of the places in t : A timed Petri net (TPN) N is a ve-tuple (P; T; F; L; U) in which (P; T; F) is a Petri net. The timed Petri net model associates to each transition t 2 T a lower bound L(t) and an upper bound U(t). The functions L and U : T ! 0; 1] (8t 2 T : 0 L(t) U(t)) associate to each transition t 2 T a time interval L(t); U(t)] which is deenes the ring times (t) of t : transition t must re as soon as a clock, which is started as soon as t becomes enabled, reaches a value in L(t); U(t)]: We assume that U(t) > 0 and L(t) < 1: A path in the TPN(P, T, F, L, U) is an ordered sequence r n of places and transitions, such that 8m : r m 2 r m+1 : In words: a path is an ordered sequence of elements which are alternatingly places and transitions, and such that each place is an input place to the transition following it in the sequence, …