Starts with a brief review of the history and the application areas considered in the literature. The author then proceeds with introductory modeling examples, behavioral and structural properties, three methods of analysis, subclasses of Petri nets and their analysis. In particular, one section is devoted to marked graphs, the concurrent system model most amenable to analysis. Introductory discussions on stochastic nets with their application to performance modeling, and on high-level nets with their application to logic programming, are provided. Also included are recent results on reachability criteria. Suggestions are provided for further reading on many subject areas of Petri nets. >

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Petri nets: Properties, analysis and applications

This paper develops a new method for synthesizing deadlock avoidance controllers (DACs) that realize job and machine dispatching policies of a flexible manufacturing system (FMS) into deadlock free control actions. Such controllers not only keep the FMS capable of repeating any of its operations, but also achieve a high resource utilization under any given dispatching policy. Our methodology is based on an untimed Petri net formalism. It consists of four ingredients: 1) a bottom-up approach for synthesizing a controlled production Petri net (CPPN) model of a FMS; 2) a necessary and sufficient liveness condition based on decomposition of the CPPN into controlled production subnets and the concept of minimal resource requirements; 3) a sufficient procedure to test whether the liveness condition is kept after a control action is executed; and 4) an algorithm that combines the test procedure with the given dispatching policy to generate valid and utilization maximizing control actions. We assess that this method is of polynomial time complexity and show that it results in a much larger class of controls than that of an existing deadlock avoidance scheme. >

Dispatching-driven deadlock avoidance controller synthesis for flexible manufacturing systems

Modern manufacturing systems are highly parallel and distributed. They need to be analyzed from qualitative and quantitative points of view. Qualitative analysis looks for properties like the absence of deadlocks, the absence of (store) overflows, or the presence of certain mutual exclusions in the use of shared resources (e.g. a robot). Its ultimate goal is to prove the correctness of the modeled system. Quantitative analysis looks for performance properties (e.g. throughput), responsiveness properties (e.g. average completion times) or utilization properties (e.g. average queue lengths or utilization rates). In other words, the quantitative analysis concerns the evaluation of the efficiency of the modeled system.

Introducing Petri nets

A class of synchronized queueing networks with deterministic routing is identified to be equivalent to a subclass of timed Petri nets called marked graphs. Some structural and behavioral properties of marked graphs are used to show interesting properties of this class of performance models. In particular, ergodicity is derived from the boundedness and liveness of the underlying Petri net representation. In the case of unbounded (i.e., nonstrongly connected) marked graphs, ergodicity is computed as a function of the average transition firing delays. For steady-state performance, linear programming problems defined on the incidence matrix of the underlying Petri nets are used to compute tight (i.e., attainable) bounds for the throughput of transitions for marked graphs with deterministic or stochastic time associated with transitions. These bounds depend on the initial marking and the mean values of the delays but not on the probability distortion functions. The benefits of interleaving qualitative and quantitative analysis of marked graph models are shown. >

Properties and performance bounds for timed marked graphs

: We present a wide range of problems concerning minimum cost network flows, and  give an overview of the classic linear single-commodity Minimum Cost Network Flow Problem (MCNFP) and some other closely related problems, either tractable or intractable. We also discuss state-of-the-art algorithmic approaches and recent advances in the solution methods for the MCNFP. Finally, optimization software packages for the MCNFP are presented. Keywords : Mathematical Programming, Combinatorial Optimization, Optimization Software. MSC: 65K05, 90C27, 90B10.

https://scindeks-clanci.ceon.rs/data/pdf/0354-0243/2013/0354-02431301003S.pdf

Minimum cost network flows: problems, algorithms, and software

Using a linear programming formulation, a unified treatment of the submarking-reachability problem for both capacitated and uncapacitated marked graphs is presented. In both cases, the problem reduces to that of testing feasibility of the dual transshipment problem of operations research. An algorithm called REACH is presented for the feasibility testing problem; its worst-case time complexity is O(mn), where m and n are, respectively, the number of edges and the number of nodes in the marked graph. The place of this work in the context of general network programming problems is indicated. >

Structure of the submarking-reachability problem and network programming

The problem of determining a maximum-weight marking in a marked graph is mathematically dual to the transshipment problem of operations research. The special structure of the transshipment problem facilitates efficient implementation of the simplex method of linear programming, for solving such problems. In this paper, we first show that the maximum-weight marking problem possesses as much structure as its dual, and then present an implementation of simplex for this problem in terms of marked graph concepts and operations. The pivoting operation in the simplex method is shown to correspond to the subgraph firing operation in marked graphs. A diakoptic reachability theorem is also proved. The formulations presented in this paper cover both live- and nonlive-marked graphs with or without capacity constraints.

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Maximum-weight markings in marked graphs: Algorithms and interpretations based on the simplex method

We present a parallel algorithm for solving the dual transshipment problem. The traditional dual simplex method does not offer much scope for parallelization, because it moves from one basic feasible solution to another, performing one pivot operation at a time. We present a new method called modified network dual simplex method which uses concurrent pivots. This departure from the traditional LP approach raises several issues such as the need to convert a non-basic feasible solution to a basic feasible solution. We present our strategies to handle these issues as well as the corresponding parallel algorithms. We also present results of testing this algorithm on large graphs to solve the integrated layout compaction and wire balancing problem. >

Parallel network dual simplex method on a shared memory multiprocessor

We present U uniJied approuch to the layout compaction und wire balancingproblem. We show that the layout compaction problem can be solved by an algorithm which also solves the primal-dual initialization problem. We formulate the wire balancing problem as a transshipment problem and show that the results of the compaction problem can also used to initialize the primal-dual method for solving this transshipment problem. In fact wire balancing reduces to applying the primal-dual method on a graph much smaller than the original constraint graph. DistributedlParallel algorithms for these problems have been implemented on BBN Butterfly machine and are now being tested on benchmark problems.

Parallel Network Primal-Dual Method on a Shared Memory Multiprocessor and A Unified Approach to VLSI Layout Compaction and Wire Balancing

Marked directed graphs are a special case of Petri nets introduced by Carl Petri as a model for information flow in systems exhibiting asynchronism and parallelism. Commoner, Holt, Even and Pnueli (1971) have studied several structural and algorithmic aspects of marked graphs using graph theory and network flow algorithms. Subsequently, Murata (1977) has studied these graphs using a circuit-theoretic approach. In this paper, the authors combine the ideas of both these works and present algorithms for certain problems on marked graphs. In particular, they introduce the concept of scatter in firing sequences and present algorithms for determining minimum scatter firing sequences for different classes of graphs. An approach for the general case and a lowerbound on the scatter of any firing sequence-leading from an initial marking to a reachable final marking on a given marked graph are presented.

M.A. Comeau

Papers

Structure of the submarking-reachability problem and network programming

Maximum-weight markings in marked graphs: Algorithms and interpretations based on the simplex method

Parallel network dual simplex method on a shared memory multiprocessor

Parallel Network Primal-Dual Method on a Shared Memory Multiprocessor and A Unified Approach to VLSI Layout Compaction and Wire Balancing

Algorithms on marked directed graphs