Segment theory to compute elementary siphons in Petri nets for deadlock control

Abstract

Unlike other techniques, Li and Zhou add control nodes and arcs for only elementary siphons greatly reducing the number of control nodes and arcs (implemented by costly hardware of I/O devices and memory) required for deadlock control in Petri net supervisors. Li and Zhou propose that the number of elementary siphons is linear to the size of the net. An elementary siphon can be synthesized from a resource circuit consisting of a set of connected segments. We show that the total number of elementary siphons, |O E |, is upper bounded by the total number of resource places |P R | lower than that min(|P|, |T|) by Li and Zhou where |P| (|T|) is the number of places (transitions) in the net. Also, we claim that the number of elementary siphons |O E | equals that of independent segments (simple paths) in the resource subnet of an S 3 PR (systems of simple sequential processes with resources). Resource circuits for the elementary siphons can be traced out based on a graph-traversal algorithm. During the traversal process, we can also identify independent segments (i.e. their characteristic T-vectors are independent) along with those segments for elementary siphons. This offers us an alternative and yet deeper understanding of the computation of elementary siphons. Also, it allows us to adapt the algorithm to compute elementary siphons in [2] for a subclass of S 3 PR (called S 4 PR) to more complicated S 3 PR that contains weakly dependent siphons.

An elementary siphon-based deadlock control algorithm with maximally reachable number to cope with deadlock problems in ordinary Petri nets

“…Motivated by the advantages of ESs 14,22,23 and CPs and CTs, 40 a two-stage DCA is proposed in this article. Unlike these DPPs with NMPB in the existing literature, it can obtain a live controlled system (N Ã , M Ã ) with MRN.…”

“…40 However, the difference of controllability for ESs and DSs is not considered in Chao, 40 which may lead to a live controlled PNM (N Ã , M Ã ) with a relatively complicated structure. Partially enlightened by the advantages of using ESs 14,22,23 and the addition of CPs and CTs, 40 the purpose of this work is to design a deadlock control algorithm (DCA) to obtain not only a liveness-enforcing supervisor with MRN but also a relatively simple structure in order to eliminate deadlocks in the typical classes of ordinary Petri nets (OPNs), namely, S 3 PR, 29 linear S 3 PR (L À S 3 PR), 31 and extended S 3 PR (ES 3 PR) 8,31 comparing with the method in Chao. 40 The proposed DCA focuses on solving ESs and DSs, executing controllability test for all DSs, and adding CPs and CTs for all ESs and a few of DSs properly, which is executed in two stages.…”

“…However, this method of adding a CP and CT to each solved SMS 40 without considering different controllability between ESs and DSs may result in a relatively complex structural (N Ã , M Ã ) with MRN. Partially motivated by the advantages of ESs to design deadlock control policies, 14,[22][23][24] an IPP of P-invariants of an extended net system (N 0 , M 0 ) is constructed to test controllability of all DSs in (N 0 , M 0 ) formed by all ESs which are controlled by the addition of CPs and CTs. By this IPP test, a few CPs and CTs are added to those0 DSs that cannot meet controllability, and the remaining DSs meeting controllability are implicitly controlled.…”

“…Li and Zhou-policy only adds the CPs to all ESs and some DSs that cannot meet the controllability and then achieves a liveness-enforcing supervisor, whose SC is enormously depressed compared with that of E-policy. Subsequently, Li and Zhou 9 and Chao and Chen 14 and Chao 40 propose the further research results including the novel methods to compute ESs, controllability of DSs (or compound siphons), and the liveness conditions associated with the controlled ESs and DSs for S 3 PR and S 3 PGR (systems of simple sequential processes with general resources requirement), which enrich the theory of elementary siphon and its applications. But BP and CC are not ideal for the livenessenforcing supervisors synthesized from the controlled ESs and DSs.…”

“…[7][8][9][10][11][12][13] In the past two decades, the research of resolving deadlocks has become the hot issues and many scholars and researchers have achieved a lot of remarkable results in theory and practical applications. 6,[14][15][16][17][18][19][20][21][22][23][24][25][26] Because of their own characteristics to easily and concisely describe the concurrent execution of processes and the reasonable distribution of shared resources in an FMS, 12,27 Petri nets have been widely used to model, analyze, and simulate the static and dynamic behaviors of an FMS, especially in deadlock problems. Using Petri nets, three policies, called deadlock detection and recovery (DDR), 1,28 deadlock avoidance (DA), 1,4 and deadlock prevention (DP) 6,10,[23][24][25][28][29][30][31][32] , respectively, are developed to cope with deadlock problems in FMSs.…”