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Petri Nets: Properties, Analysis and Applications Petri Nets: Properties, Analysis and Appl kations TADAO MURATA, FELLOW, IEEE Invited Paper This is an invited tutorial-review paper on Petri nets-a graphical to the faculty of Mathematics and Physics at the Technical and mathematical modeling tool. Petri nets are a promising tool University of Darmstadt, West Germany. The dissertation for describing and studying information processing systems that was prepared while C. A. Petri worked as a scientist at the are characterized as being concurrent, asynchronous, distributed, parallel, nondeterministic, and/or stochastic. Universityof Bonn. Petri’swork[l], [2]came totheattention The paper starts with a brief review of the history and the appli- of A. W. Holt, who later led the Information System Theory cation areas considered in the literature. It then proceeds with Project of Applied Data Research, Inc., in the United States. introductory modeling examples, behavioral and structural prop- The early developments and applications of Petri nets (or erties, three methods of analysis, subclasses of Petri nets and their analysis. In particular, one section is devoted to marked graphs- their predecessor)arefound in the reports [3]-[8] associated the concurrent system model most amenable to analysis. In addi- with this project, and in the Record [9] of the 1970 Project tion, the paper presents introductory discussions on stochastic nets MAC Conference on Concurrent Systems and Parallel with their application to performance modeling, and on high-level Computation. From 1970 to 1975, the Computation Struc- nets with their application to logic programming. Also included are recent results on reachability criteria. Suggestions are provided ture Group at MIT was most active in conducting Petri-net for further reading on many subject areas of Petri nets. related research, and produced many reports and theses on Petri nets. In July 1975, there was a conference on Petri I. INTRODUCTION Nets and Related Methods at MIT, but no conference pro- ceedings were published. Most of the Petri-net related Petri netsareagraphical andmathematical modeling tool papers written in English before 1980 are listed in the anno- applicable to many systems. They are a promising tool for tated bibliography of the first book [IO] on Petri nets. More describing and studying information processing systems recent papers up until 1984 and those works done in Ger- that are characterized as being concurrent, asynchronous, many and other European countries are annotated in the distributed, parallel, nondeterministic, and/or stochastic. appendix of another book [Ill. Three tutorial articles [12]- As a graphical tool, Petri nets can be used as a visual-com- [I41 provide a complemental, easy-to-read introduction to munication aid similar to flow charts, block diagrams, and Petri nets. networks. In addition, tokens are used in these nets to sim- Sincethe late-I970‘s, the Europeans have been veryactive ulate the dynamic and concurrent activities of systems. As in organizing workshops and publishing conference pro- a mathematical tool, it is possible to set up state equations, ceedings on Petri nets. In October 1979, about 135 research- algebraic equations, and other mathematical models gov- ers mostly from European countries assembled in Ham- erning the behavior of systems. Petri nets can be used by burg, West Germany, for a two-week advanced course on both practitioners and theoreticians. Thus, they provide a General Net Theory of Processes and Systems. The 17 lec- powerful medium of communication between them: prac- turesgiven in thiscoursewere published in its proceedings titioners can learn from theoreticians how to make their [15], which is currently out of print. The second advanced models more methodical, and theoreticians can learn from course was held in Bad Honnef, West Germany, in Sep- practitioners how to make their models more realistic. tember 1986. The proceedings [16], [I7 of this course con- Historically speaking, the concept of the Petri net has its tain 34 articles, including two recent articles by C. A. Petri; origin in Carl Adam Petri’s dissertation [I],submitted in 1962 one[l8] isconcerned with hisaxiomsof concurrencytheory and the other [I91with his suggestions for further research. The first European Workshop on Applications and Theory Manuscript received May 20, 1988; revised November 4, 1988. of Petri Nets was held in 1980 at Strasbourg, France. Since This work was supported by the National Science Foundation under then, this series of workshops has been held every year at Grant DMC-8510208. different locations in Europe: 1981, Bad Honnef, West Ger- The author is with the Department of Electrical Engineering and Computer Science, University of Illinois, Chicago, IL 60680, USA. many; 1982, Varenna, Italy; 1983, Toulouse, France; 1984, IEEE Log Number 8926700. Aarhus, Denmark; 1985, Espoo, Finland; 1986, Oxford, Great 0018-9219/89/0400-0541$01.000 1989 IEEE PROCEEDINGS OF THE IEEE, VOL. 77, NO. 4, APRIL 1989 541 Britain; 1987, Zaragoza, Spain; 1988, Venice, Italy; and 1989, The rest of this paper consists of the following topics. Bad Honnef, West Germany (planned). The distribution of Section I I discusses informally the transition enabling and the proceedings of theseworkshops is limited to mostly the firing rule with and without capacity constraints. Several workshop participants. However, selected papers from introductory modeling examples are given in Section Ill to these workshops and other articles have been published illustrate modeling capabilities and concepts such as con- by Springer-Verlag as Advances in Petri Nets [20]-[25]. The flict (choice or decision), concurrency, synchronization, etc. 1987 volume [24] contains the most comprehensive bibli- Section IV describes behavioral or marking-dependent ographyof Petri nets[26] listing2074entries published from properties that can be studied using Petri nets. Section V 1962 to early1987. The”recent publications” section of Petri presents three methods of analysis: the coverability tree, Net Newsletter [27l lists short abstracts of recent publica- matrix equations, and reduction techniques. Section VI is tions three times ayear, and is agood sourceof information concerned with subclasses of Petri nets and their analysis. about the most recent Petri net literature. In-depth analysis and synthesis methods are given in Sec- In July 1985, another series of international workshops tion VI I for one of the subclasses known as marked graphs. was initiated. This series places emphasis on timed and sto- Structural or marking-independent properties are dis- chastic nets and their applications to performance evalu- cussed in Section VIII. Section IX presents an introduction ation. The first internationl workshop on timed Petri nets to timed nets, stochastic nets, and high-level nets, together was held in Torino, Italy, in July 1985; the second was held with their applications. Concluding remarks are given in in Madison, Wisconsin, in August 1987; the third is to be Section X. held in Kyoto, Japan, in December 1989; and the fourth is planned in Australia in 1991. The proceedings of the first II. TRANSITIONENABLING AND FIRING two workshops [28], [29] are available from the IEEE Com- In this section,wegivetheonly ruleone hastolearn about puter Society Press. Petri-net theory: the rule for transition enabling and firing. The above is a brief history of Petri nets. Now, we look Although this rule appears very simple, its implication in at some application areas considered in the literature. Petri Petri-net theory is very deep and complex. nets have been proposed for a very wide variety of appli- A Petrinet is a particular kind of directed graph, together cations. This is due to the generality and permissiveness with an initial state called the initialmarking, MO.The under- inherent in Petri nets. They can beapplied informallyto any lying graph N of a Petri net is a directed, weighted, bipartite area or system that can be described graphically like flow graph consisting of two kinds of nodes, called places and charts and that needs some means of representing parallel transitions, where arcs are either from a place toa transition or concurrent activities. However, careful attention must or from a transition to a place. In graphical representation, be paid to a tradeoff between modeling generalityand anal- places are drawn as circles, transitions as bars or boxes. Arcs ysis capability. That is, the more general the model, the less are labeled with their weights (positive integers), where a amenable it is to analysis. In fact, a major weakness of Petri k-weighted arc can be interpreted as the set of k parallel nets is the complexity problem, i.e., Petri-net-based models arcs. Labels for unity weight are usually omitted. A marking tend to become too large for analysis even for a modest-size (state)assignstoeach placeanonnegative integer. If amark- system. In applying Petri nets, it is often necessary to add ing assigns to place p a nonnegative integer k, we say that special modifications or restrictions suited to the particular p is marked with k tokens. Pictorially, we place k black dots application. Two successful application areas are perfor- (tokens) in placep. A marking is denoted by M, an m-vector, mance evaluation [28]-[50] and communication protocols where m is the total number of places. Thepth component [51]-[62]. Promising areas of applications include modeling of M, denoted by M(p),is the number of tokens in placep. and analysis of distributed-software
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