Augmented Marked Graphs by King Sing Cheung

By King Sing Cheung

Petri nets are a proper and theoretically wealthy version for the modelling and research of structures. A subclass of Petri nets, augmented marked graphs own a constitution that's in particular fascinating for the modelling and research of structures with concurrent strategies and shared resources.

This monograph involves 3 elements: half I offers the conceptual heritage for readers who've no previous wisdom on Petri nets; half II elaborates the speculation of augmented marked graphs; ultimately, half III discusses the appliance to approach integration. The ebook is appropriate as a primary self-contained quantity on augmented marked graphs, and may be priceless to either researchers and practitioners within the fields of Petri nets and procedure integration.

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Augmented Marked Graphs

Petri nets are a proper and theoretically wealthy version for the modelling and research of platforms. A subclass of Petri nets, augmented marked graphs own a constitution that's specially fascinating for the modelling and research of structures with concurrent approaches and shared assets. This monograph includes 3 elements: half I offers the conceptual historical past for readers who've no past wisdom on Petri nets; half II elaborates the idea of augmented marked graphs; ultimately, half III discusses the applying to process integration.

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36 3 Augmented Marked Graphs t4 t1 p5 p3 p1 t5 t6 p4 p6 p7 t3 t7 t8 t2 p2 p10 p8 p9 t9 t10 Fig. 3 shows a PT-net N ¼ h P, T, F i. There are five cycles in N. ΩN ¼ { γ1, γ2, γ3, γ4, γ5 }, where γ1 ¼ h p1, p3, p4 i, γ2 ¼ h p2, p3, p4 i, γ3 ¼ h p2, p5, p6 i, γ4 ¼ h p2, p5, p7, p8 i and γ5 ¼ h p7, p9, p10 i. ΩN[p1] ¼ { γ1 } is the set of cycles containing p1. ΩN[p2] ¼ { γ2, γ3, γ4 } is the set of cycles containing p2. Consider Y1 ¼ { γ1, γ2, γ3 }  ΩN. P[Y1] ¼ { p1, p2, p3, p4, p5, p6 } is the set of places contained in Y1, where •P[Y1] ¼ { t1, t2, t3, t4, t5, t7, t9 } and P[Y1]• ¼ { t1, t2, t3, t4, t5, t6, t7 }.

N0 , M00 ) is a marked graph. Every cycle in (N0 , M00 ) is marked. For example, the cycles γ1 ¼ h p1, p4, p7 i, γ2 ¼ h q1, p5, p8 i, γ3 ¼ h p5, p8, p10, q2, p6, p3 i, γ4 ¼ h q2, p6, p3, p5, p8, p10 i and γ5 ¼ h p2, p6, p9 i are marked. 20 Let (N, M0; R) be an augmented marked graph, and (N0 , M00 ) be the R-transform of (N, M0; R), where a place r ∈ R is replaced by a set of places Q ¼ { q1, q2, . , qk }. Then, for each qi in N0 , there exists a place invariant αi of N0 such that αi[qi] ¼ 1 and αi[q] ¼ 0 for any q ∈ (P0 \ {qi}), where P0 is the set of marked places in (N0 , M00 ).

Let S0 be a minimal siphon in S. 3, S0 is marked. Since r is the only one marked place in S, r is also the only one marked place in S0 . 6 shows an augmented marked graph (N, M0; R), where R ¼ { r1, r2 }. S1 ¼ { r1, p2, p4, p6, p7, p9 } is a minimal siphon. There exists Y1 ¼ { γ11, γ12 }  ΩN, where γ11 ¼ h r1, p4, p7 i and γ12 ¼ h r1, p2, p6, p9 i, such that S1 ¼ P[Y1] and •r1 ¼ { t9, t11 }  T(Y1). r1 ∈ R is the only one marked place in S1. S2 ¼ { r2, p3, p5, p6, p8, p10 } is another minimal siphon.

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