On Optimizing the Initial Token Distribution for a Periodic Petri Net Firing Sequence with Prescribed Firing Numbers

Abstract

This paper proposes a technique of designing a periodic firing sequence σ=δδ… and its enabling initial marking M_I for a bounded live Petri Net PN=(P, T, E, M_I) with an aim of minimizing the number of tokens in M_I, |M_I|_??_ΣM_I(p).
Let A=A⁺-A^- be the place-transition incidence matrix of the PN. It is well known that if the initial token distribution M_I is sufficiently abundant (namely M_I=A^-X₀) then it is possible to construct a periodic firing sequence σ=δδ… with δ=X₀ satisfying AX₀=0. This paper presents a method of constructing multi-round firing of the period, δ=δ₁δ₂…δ_k, and its enabling initial marking M_I with an aim of minimizing the token number |M_I| through use of a partial ordering of the transition, induced by a given feed-back edge set F, and a decomposition X₀=Y₁+Y₂+…+Y_k such that in the i-th round δ_i transition t fires Y_i(t) times in succession in accordance with the ordering J.