MATLAB tools for the analysis of Petri net models∗

Jorge JulvezDept. of Computer Science and

Systems EngineeringUniv. of Zaragoza, Spain

{julvez}@unizar.es

Mihaela H. Matcovschi, Octavian PastravanuDept. of Automatic Control and

Applied Informatics, Technical University“Gh. Asachi” of Iasi, Romania

{mhanako,opastrav}@ac.tuiasi.ro

Abstract

The paradigm of Petri nets offers a rich modelingpower that has been successfully exploited in manydifferent application domains as manufacturing, lo-gistic, traffic, computer and biological domains. Inorder to develop and analyze Petri models in an effi-cient and systematic way, appropriate software toolsare required. Among the existing software platformsfor scientific software development, MATLAB placesat the user’s disposal a large set of toolboxes withan inutitive graphical interface. This paper describessome of the existing Petri net toolboxes that havebeen developed on MATLAB, and that cover theneeds for the modeling and analysis of discrete, con-tinuous and hybrid systems.

1 Introduction

The theory of Petri nets (abbreviated PNs) emergedfrom the computer-science-oriented research of theearly sixties [20] and two decades later became an ef-fective instrument for the exploration of a large classof systems, generically referred to as discrete-eventand hybrid systems. The original idea of a bipar-tite graph able to model properties such as processsynchronization, asynchronous events, concurrent op-erations, and conflicts or resource sharing, proved to

∗Submitted to the 19th IEEE International Conference onEmerging Technologies and Factory Automation; ETFA’2014as invited paper to track 4: “Automated Manufacturing Sys-tems”.

be seminal for further developments, in the sense ofthe theoretical extension, concomitantly with the ac-commodation of new types of practical models. Thus,the conceptual evolution allowed the incorporation oftime information into transitions [17, 21], places [24]or arcs [31], broadening the usage of PNs from qual-itative analysis of discrete-event systems to quanti-tative analysis and performance modeling. Differ-ent types of time constraints were allowed: deter-ministic values [21], time intervals [17], stochastic [1]or possibilistic forms [19]. Dealing with analysis orsynthesis of complex discrete-event systems based onPN models often leads to computational infeasibil-ity due to the so called state explosion when the setof reachable markings increases exponentially withrespect to the initial marking. To overcome thisdrawback in 1987 two different relaxation techniqueswere independently introduced in order to study dis-crete event systems through a continuous approxi-mated model [7] and [26]. If only a few transitionsin a PN model are fluidized, i.e., some transitionsremain discrete while some become continuous, theobtained models form a class of hybrid Petri nets [3].An overview of the development of the field of Petrinets in the half a century from the first introductionin Carl Adam Petris Ph.D. thesis is presented in therecent works [25, 22].

Besides computer science, manufacturing engineer-ing was one of the most important beneficiaries of thescientific potential offered by PNs at the beginningof the nineties. This is the period when robust in-struments supported by the PN theory were devisedfor flexible manufacturing systems analysis and de-

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2014 IEEE Emerging Technology and Factory Automation (ETFA)978-1-4799-4845-1/14/$31.00 c©2014 IEEE

sign, which yielded the publication of the first mono-graphs [9, 32]. The results of the mentioned periodstimulated a considerable increase of the effort in-vested during the next two decades, as reflected bythe state of the art presented in the book [6].The modeling power of the PNs combined with

the convenient handling of their graphical represen-tations motivated many academic or research groupsto create a wide variety of software tools, as sum-marized by section Tools and Software of Petri NetWorld (http://www.informatik.uni-hamburg.de/TGI/PetriNets/tools/db.html). On the otherhand, most of these tools were not meant as general-purpose packages, dealing with particular types ofPNs and focusing on sets of specialized problems inaccordance with the key interests of the developers.Therefore, the selection of such a package may be-come rather difficult for any user who cannot finda comfortable match between the available softwarefacilities and his/her investigation objectives.Users preferences in choosing PN tools also depend

on their familiarity with the software environmentshosting those tools, because many applications re-quire supplementary studies preceding or followingthe PN-oriented tasks. Subsequently, for studentsand specialists working in different areas of engineer-ing (particularly skilled in the exploitation of MAT-LAB and its specialized toolboxes), bridging the PN-formalism with the generous computational resourcesof MATLAB appeared as an extremely profitable ap-proach to discrete-event and hybrid systems.Within the above commented context, our work

aims to reinforce the dissemination of the capa-bilities offered for PN applicative research by twosoftware products endorsed by the MathWoks Con-nections Program, namely Petri Net Toolbox de-veloped at “Gheorghe Asachi” Technical Universityof Iasi (http://www.ac.tuiasi.ro/pntool/) andSimHPN developed at Zaragoza University (http://webdiis.unizar.es/GISED/?q=tool/simhpn). Itis worth saying that in 2001, besides the release ofPetri Net Toolbox version 1 presented in [13], somePN analysis instruments for MATLAB were also de-scribed in paper [29], but their use was drasticallylimited by the lack of a graphical user interface.To ensure a comprehensive picture of the

MATLAB-embedded (MATLAB-compatible) PNtools, our work refers additionally to another two

packages, namely HYPENS developed at theUniversity of Cagliari (http://www.diee.unica.it/automatica/hypens/) and PNSB - developedat “Gheorghe Asachi” Technical University ofIasi (http://www.ac.tuiasi.ro/~pnsb/); thesepackages allow the simulation of hybrid systemsdescribed via PN models.Our exposition is structured as follows. The theo-

retical background needed for the presentation of thetwo software products is ensured by Section 2. Themain features of Petri Net Toolbox and SimHPN arediscussed in Sections 3 and 4, respectively. Section 5gives brief overviews of HYPENS and PNSB. In Sec-tion 6, a relevant example illustrates the software usein addressing problems typical to systems engineer-ing.

2 Discrete, Continuous and

Hybrid Petri nets

In the classical formalism of discrete Petri nets, tran-sitions are fired in natural amounts, that leads todiscrete event dynamic systems. In contrast to this,the firing amounts of transitions in continuous Petrinets is allowed to take real numbers, that leads toa continuous state space. On the other hand hybridPetri nets [8, 5] integrate both discrete and continu-ous transitions in a single net model. In fact, bothdiscrete and continuous Petri nets can be seen as par-ticular cases of hybrid Petri nets in which all transi-tions are considered as discrete and continuous re-spectively.This section defines the main concepts related to

the hybrid nets that are supported by the MATLABtoolboxes under consideration. In the following, thereader is assumed to be familiar with Petri nets (see[18, 10] for a gentle introduction).

2.1 Untimed Hybrid Petri nets

Definition 2.1 A Hybrid Petri Net (HPN) systemis a pair 〈N ,m0〉, where: N = 〈P, T,Pre,Post〉 isa net structure, with set of places P , set of transitionsT , pre and post incidence matrices Pre,Post ∈

R|P |×|T |≥0

, and m0 ∈ R|P |≥0

is the initial marking.

The token load of the place pi at marking m is

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denoted bym(pi) and the preset and postset of a nodex ∈ P ∪ T are denoted by •x and x•, respectively.For a given incidence matrix, e.g., Pre, Pre(pi, tj)denotes the element of Pre in row i and column j.In a HPN, the set of transitions T is partitioned

in two sets T = T c ∪ T d, where T c contains the setof continuous transitions and T d the set of discretetransitions. In contrast to other works, the set ofplaces P is not explicitly partitioned, i.e., the mark-ing of a place is a natural or real number depend-ing on the firings of its input and output transitions.Nevertheless, in order to make net models easier tounderstand, those places whose marking can be a realnon-integer number will be depicted as double circles,and the rest of places will be depicted as simple circles(such places will have integer markings). Continuoustransitions are graphically depicted as two bars, whilediscrete transitions are represented as empty bars.Right and left non negative annullers of the token

flow matrix C, where C = Post−Pre, are called T-and P-semiflows, respectively. A semiflow v is mini-mal when its support, ‖v‖ = {i | v(i) 6= 0}, is not aproper superset of the support of any other semiflow,and the greatest common divisor of its elements isone. If there exists y > 0 such that y ·C = 0, the netis said to be conservative, and if there exists x > 0satisfying C · x = 0, the net is said to be consistent.The enabling degree of a transition tj ∈ T is:

enab(tj,m) =

minpi∈•tj

⌊

m(pi)

Pre(pi, tj)

⌋

if tj ∈ T d

minpi∈•tj

m(pi)

Pre(pi, tj)if tj ∈ T c

(1)where ⌊x⌋ denotes the largest integer not greater thanx.Transition tj ∈ T is enabled at m iff enab(tj,m) >

0. An enabled transition tj ∈ T can fire in anyamount α such that 0 ≤ α ≤ enab(tj,m), where α ∈N if tj ∈ T d and α ∈ R if tj ∈ T c. Such a fir-ing leads to a new marking m′ = m+ α ·C(·, tj),where C = Post−Pre is the token-flow matrix andC(·, tj) is its jth column. If m is reachable from m0

through a finite sequence σ, the state (or fundamen-tal) equation, m = m0 + C · σ is satisfied, where

σ ∈ R|T |≥0

is the firing count vector. According tothis firing rule the class of nets defined in Def 2.1 isequivalent to the class of nets defined in [8, 5].

Two enabled transitions ti and tj are in con-flict at m if there exists p ∈ •ti ∩ •tj such thatPre(p, ti) + Pre(p, ti) > m(p). For this, it is nec-essary that •ti ∩ •tj 6= ∅, and in that case it is saidthat ti and tj are in structural conflict relation. Inorder to establish a conflict resolution policy, a natu-ral number ηj can be associated to each transition tj .This number can represent either priority or proba-bility. If it is a priority, the transition with lowestvalue in a set of conflicting transition will fire first.If it is a probability, the probability to fire tj firstis equal to ηj divided by the sum of the ηi of theconflicting transitions.

2.2 Timed Hybrid Petri nets

Different time interpretations can be associated tothe firing of transitions. Once an interpretation ischosen, the state equation can be used to show thedependency of the marking on time, i.e., m(τ) =m0 + C · σ(τ). The term σ(τ) is the firing countvector at time τ . Depending on the chosen time in-terpretation, the firing count vector σj(τ) of a tran-sition tj ∈ T c is differentiable with respect to time,and its derivative fj(τ) = σj(τ) represents the con-tinuous flow of tj . As for the timing of discrete tran-sitions, several definitions exist for the flow of contin-uous transitions.

Definition 2.2 A Timed Hybrid Petri Net (THPN)system is a 4-tuple 〈N ,m0, T ype,λ〉 where 〈N ,m0〉is a HPN, Type : T → {id, pd, dd, ic, pc} establishesthe time semantics of transitions and λ : T → R≥0

associates a real parameter to each transition relatedto its semantics.

Any of the following semantics is allowed for a dis-crete transition ti ∈ T d:

• Infinite server semantics (Type(ti) = id): Un-der infinite server semantics, the time delay of atransition ti, at a given marking m, is an expo-nentially distributed random variable with pa-rameter λi · enab(ti,m), where the integer en-abling enab(ti,m) represents the number of ac-tive servers of ti at marking m.

• Product server semantics (Type(ti) = pd): Un-der product server semantics, the time delay of

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a transition ti at m is an exponentially dis-tributed random variable with parameter λi ·∏

pj∈•ti

⌊

m(pj)

Pre(pj , ti)

⌋

, where∏

pj∈•ti

⌊

m(pj)

Pre(pj , ti)

⌋

is the number of active servers.

• Deterministic delay (Type(ti) = dd): A transi-tion ti with deterministic delay is scheduled tofire 1/λi time units after it became enabled.

Conflict resolution: When several discrete exponen-tial transitions, under either infinite or product serversemantics, are in conflict, a racing policy is adopted,i.e., the one with smaller time delay will fire first.If a discrete transition with deterministic delay is

not in conflict with other transitions, it is fired asscheduled, if it is in conflict then it is fired only ifits schedule firing time is less than the firing time ofthe conflicting transition. In the case of several con-flicting deterministic transitions with same scheduledfiring instance, the transition to fire is chosen prob-abilistically assigning the same probability to eachconflicting transition. Furthermore after the firing ofa deterministic transition, the timers of all the tran-sitions in the same conflict are discarded.For a continuous transition ti ∈ T c the following

semantics are allowed:

• Infinite server semantics (Type(ti) = ic): Underinfinite server the flow of a transition ti is:

fi = λi · enab(ti,m) = λi · minpj∈•ti

{

m(pj)

Pre(pj , ti)

}

(2)Such an expression for the flow is obtained from afirst order approximation of the discrete case [28]and corresponds to the variable speed of [3].

• Product server semantics (Type(ti) = pc): In asimilar way to discrete transitions, the continu-ous flow under product server semantics is givenby:

fi = λi ·∏

pj ∈ •ti

{

m(pj)

Pre(pj , ti)

}

The described supported semantics cover the mod-eling of a large variety of actions usually associated

to transitions. For instance, infinite server seman-tics, which is more general than finite server seman-tics, is well suited for modeling actions in manufac-turing, transportation and logistic systems [8]; prod-uct server semantics is especially useful to model-ing population dynamics [27] and biochemical reac-tions [11]; and deterministic delays allow one to repre-sent pure delays and clocks that appear, for instance,when modeling traffic lights in automotive traffic sys-tems [30].

3 Discrete nets in Petri Net

Toolbox

The Petri Net Toolbox [15, 14, 13] (abbreviated asPN Toolbox ) for MATLAB bridges the Petri net for-malism with the widely spread usage of MATLAB. Itaccepts three types of PN models, namely: untimed,transition-timed and place-timed. The timed netscan be deterministic or stochastic, and the stochasticcase allows using the appropriate probability distri-bution function with positive support. The capac-ity of a place can be finite or infinite. The usercan set priorities / probabilities for conflicting tran-sitions. Both single and infinite server semantics canbe used for transition firings. The PN Toolbox canalso operate with stochastic and generalized stochas-tic PNs [2].An easy to exploit Graphical User Interface GUI

(see Fig. 1) allows the user to get hold of the facilitiesoffered by the PN Toolbox. The GUI may be utilizedin two modes: first, the Draw Mode allows the userto draw a new PN model or modify the properties ofan existing one in a natural fashion; second, the Ex-plore Mode enables access to simulation, analysis anddesign tools, exploiting the computational resourcesof the MATLAB environment.The GUI exhibits eight control panels (see Fig. 1):

Menu Bar (1), Quick Access Toolbar (2), DrawingArea (3), Drawing Panel (4), Draw/Explore Switch(5), Simulation Panel (6), Status Panel (7) and aMessage Box (8).The user can access all the facilities available in

the PN Toolbox from the Menu Bar placed hori-zontally, on top of the main window of the GUI.It displays a set of nine drop-down menus, namelyFile, Modeling, View, Properties, Simulation, Per-

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Figure 1: GUI of the Petri net Toolbox.

formance, Max-Plus, Design and Help, that are en-abled depending on the exploitation mode of the PNToolbox.

The File menu offers facilities for file-handling op-erations such as creating a new model, loading apreviously saved model, saving, printing or closinga model, and exiting the GUI. The Modeling menuprovides tools for graphical editing a model in theDrawing Area. After adding places or transitions anddrawing arcs between two different nodes of the cur-rent PN model, the user can add tokens in a specificplace or edit the properties of different net objects.The user may also set the policy for the resolutionof conflicting transitions assigning priorities or prob-abilities to conflicting transitions.

The View menu allows choosing specific conditionsfor visualization of the current model by zooming inor out, by displaying or hiding the current values ofarc weights. The desired representation of a transi-tion as a square, rectangle or bar can also be selectedby the user.

The Properties menu provides computational tools

for the analysis of the behavioral and structural prop-erties of the current PN model. After drawing a PNmodel, the user can: a) visualize the Incidence Ma-trix, which is automatically built from the net topol-ogy; b) explore the Behavioral Properties (such asliveness, boundedness, reversibility etc.) by consult-ing the Coverability Tree, which is automatically builtfrom the net topology and initial marking; c) explorethe Structural Properties (such as structural bound-edness, repetitiveness, conservativeness and consis-tency): d) calculate T- and P-semiflows.

Using the Simulation menu the user can simulatethe PN model in three ways: step by step and runslow, which are accompanied by animation, and runfast. This menu also allows the user to set the con-ditions of the simulation and to store or retrieve thesimulation log file.

At the end of a simulation experiment, the Per-formance menu allows the visualization of the globalperformance indices that are separately recorded fortransitions (such as total number of firings, averagefrequency of firings, etc.) and for places (such as to-

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tal number of arrived/ departed tokens, average fre-quency of token arrival/ departure, average numberof tokens in place, average waiting time per token,etc.). These indices may be saved in HTML formatin a file placed in the working directory.The Max-Plus menu allows performing the simula-

tion and analysis of a place-timed event graph basedon its max-plus state-space model derived directlyfrom its topology and initial marking in an implicitform [4].For the synthesis, via automated iterative simula-

tions, of timed or (generalized) stochastic PN models,the PN Toolbox offers the menu (command) Design,which allows exploring the dependence of a DesignIndex on one or two Design Parameters that varywithin intervals defined by the user. A Design Pa-rameter may be selected as (i) the initial markingof a place, (ii) a parameter of the distribution func-tion defining the duration associated with a place ora transition in timed PNs, (iii) the mean value ofthe exponential distribution function associated witha transition in (generalized) stochastic PNs. The De-sign Index may be selected as a global performanceindex associated with any node of the net.The Help menu provides complete information

about the PN Toolbox and allows visualization of fourmovies initiating the user in its exploitation.The Quick Access Toolbar is placed horizontally,

below the Menu Bar, and presents six image buttonsthat give access to the highly used commands NewModel, Open Model, Save, Zoom In, Zoom Out andShow Grid.The Drawing Area is located in the central and

right side of the main window. It is provided witha grid, where the nodes of the PN graph are to beplaced, and with two scrollbars (on the right and bot-tom sides) for moving the desired parts of the graphinto view.The Draw/Explore Switch allows switching be-

tween the Draw Mode (in which the user can drawa new model or modify an existing one) and the Ex-plore Mode (in which the user can access all functionsavailable for simulation and analysis). When switch-ing from one exploitation mode to the other, differentmenus become available in the Menu Bar.In Draw Mode, for a quick access to the main

commands used for drawing a model, the PN Tool-box is equipped with a Drawing Panel placed verti-

cally, in the left side of the main window, above theDraw/Explore Switch. In Explore Mode, the DrawingPanel is hidden and the user gains access to the Sim-ulation Panel placed vertically, in the left side of themain window, just below the Draw/Explore Switch.The Status Panel is a message board (placed in

the bottom left-hand corner of the main window),where the PN Toolbox displays the current simulationtime and the total number of events. In addition, theStatus Panel displays the file name corresponding tothe current model.The Message Box is a MATLAB text object used

by the PN Toolbox to display different messages tothe user, depending on the exploitation mode.

4 Continuous and hybrid nets

in SimHPN

The SimHPN simulator [12] supports infinite serverand product server semantics for both discrete andcontinuous transitions. Moreover, deterministic de-lays with single server semantics are also supportedfor discrete transitions. Both the data related to themodel description, i.e., net structure, initial markingand timing parameter, and the output results, i.e.,time trajectories, are MATLAB variables. At theend of the simulation, the user can export the datato the MATLAB workspace where it can be used forfurther analysis. The next subsection describes thefunctionality of the tool as shown in its graphical in-terface.The SimHPN toolbox provides a Graphical User

Interface (GUI) that enables the user to easily per-form simulations and carry out analysis methods.This GUI consists of a MATLAB figure window, ex-hibiting a Menu bar and three control panels: (i)Drawing Area, (ii) Options panel, and (iii) ModelManagement panel. Fig. 2 presents a hard-copyscreenshot of the main window opened by SimHPNtoolbox, where all the component parts of the GUIare visible.The Menu bar (placed horizontally, on the top of

the window in Fig. 2) displays a set of four drop-downmenus at the top of the window, where the user canselect different features available in the SimHPNtoolbox. These menus are: Model, Options, Simu-lation, and Optimal.

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Figure 2: Sketch of the main window of SimHPN

The Model menu contains the pop-up menus Im-port from Pmeditor, Import from TimeNet and Im-port from .mat file that implement several importingoptions for the matrices, Pre, Post, m0, etc, thatdescribe the net system: Such matrices can be in-troduced manually or through two Petri nets editors:PMEditeur and TimeNet [34]. Moreover, the ma-trices can be automatically loaded from a .mat file(MATLAB file format) or loaded from variables de-fined in the workspace, this is done just by writing thename of the variable to be used in the correspondingedit boxes.

The Options menu contains only the pop-up menuShow Figure Toolbar allows to show the characteristictoolbar of the MATAB figure object that permits,for example, the use of zoom tool on the displayed

graphic in the Drawing Area.

The Simulation menu contains the pop-up menusMarkings to plot, Flows to plot, and Save resultsto workspace. The pop-up menus Markings to plot,Flows to plot allow the user to select the componentsof marking vector and flow vector that will be plot-ted after simulation in the Drawing area. The pop-up menu Save results to workspace permits to export,after simulation, the marking and flow evolution tovariables in the MATLAB workspace.

The Optimal menu contains the pop-up menus Op-timal Observability and Optimal Control. Such pop-up menus perform calls to the algorithms for comput-ing optimal steady state and optimal sensor place-ment for continuous Petri nets with infinite serversemantics.

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The Drawing area (located in the left and centralside of the window in Fig. 2), is a MATLAB axesobject where the trajectories of the simulation resultsare plotted. The components of markings and flowsthat will be represented are selected from the menu.

The Options panel (placed, as a horizontal bar, onthe right part of the window Fig. 2) presents a num-ber of options related to the model. From top tobottom: (a) two radio buttons to select the firing se-mantics for continuous and discrete exponential tran-sitions; (b) three radio buttons allowing to select thevariables to be plotted in the Drawing Area, the sim-ulator allows one to plot the evolution of the markingof the places, the evolution of the flow of the transi-tions and the evolution of the marking of one place vs.the marking of other place; (c) three edit boxes to fixthe maximum absolute and relative errors allowed bythe simulated trajectory and the sampling time usedin simulations (see next subsection for more detailson the selection of the sampling time); (d) a Simu-late button to start a new simulation; (e) a ComputeBounds button that computes performance boundsfor continuous nets under infinite server semantics;(f) a P T semiflows button to compute the minimalP- and T-semiflows of the net, the results are dis-played on the MATLAB command window and canbe used for future analysis tasks; and (g) a Closebutton to close the SimHPN toolbox.

The Model Management Panel panel is composedof different edit boxes (placed in the bottom left cor-ner of the window in Fig. 2), where the SimHPNtoolbox displays the current values of the matricesdescribing the net system and permits to select thesimulation time and the number of simulations to beperformed (this last parameter is ignored if the netcontains no stochastic transitions). The required ma-trices for a system in order to be simulated are: Pre

and Post matrices, initial marking m0, the parame-ter λ of each transition, and the type of each transi-tion. This last parameter is equal to ’c’ for continuoustransitions, to ’d’ for stochastic discrete transitionsand to ’q’ for deterministic discrete transitions. No-tice that if the type of a transition is ’q’ then singleserver semantics is adopted for its firing and thereforethe selection of firing semantics in the Options panelwill be ignored for this transition.

5 Hybrid models in HYPENS

and PNSB

The HYbrid PEtri Net Simulator (HYPENS) [23] isa tool that allows to simulate several classes of timeddiscrete, continuous and hybrid Petri nets. More pre-cisely, the class of First Order Hybrid Petri Nets isconsidered. In this class of nets, the flow of thecontinuous transitions is piecewise constant and isobtained by solving a linear programming problem.Both finite and infinite server semantics can be asso-ciated to the firing of discrete transitions. The toolcan be easily interfaced with other MATLAB pro-grams and can also be used for analysis and opti-mization via simulation. The large set of plot func-tions available in MATLAB allows one to representthe results of the simulation in a clear and intuitiveway.

The Petri Net Simulink Block (PNSB) [16, 14] en-sures the integration of the simulation facilities avail-able in the PN Toolbox with Simulink and facilitatesthe simulation of hybrid systems when the event-driven part of their dynamics is modeled using thePetri net formalism. The PN model stored in a PNSBcan be untimed, P-timed or T-timed. It containstransitions whose firing is synchronized (triggered)with external events generated by a set of input sig-nals. A synchronized transition is fired whenever (i)it is enabled by the net marking and (ii) one of itsassociated triggering events, defined at the PN level,occurs. Both finite and infinite server semantics canbe used for transition firings. A Simulink model cancontain any number of PNSBs needed to model acomplex system. The PNSB is equipped with a GUIthat allows a user to draw a PN model (PNSB Ed-itor), define the triggering events (PNSB Event Ex-plorer) and debug the Simulink model (PNSB Debug-ger). The functions of the PNSB Editor are similarto the editing facilities available in the Draw Mode ofthe PN Toolbox.

This approach to hybrid systems modeling andanalysis allows incorporating accurate nonlinearmodels for the continuous dynamics (built fromblocks available in the standard Simulink libraries)with PN models for discrete event dynamics.

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idleM1

M3B

M2A

M2B

finished

P AinM1emptyP

loadP

loadedP

sAGV1

agv1

AGV1

choice

partA

partB

M1A

AtoM23

sConv

ConvFree

conv

Conv

idleAGV

agv2

sAGV2AGV2

AwM23

BwM23

AsM3

AsM2

BsM2

BsM3

AinM3

BinM3

AinM2

BinM2

idleM2

idleM3

M3A

Figure 3: A GSPN model of a FMS taken from [33].

6 Case study

Let us consider the Generalized Stochastic Petri Net(GSPN) [2] model of a Flexible Manufacturing Sys-tem (FMS) adapted from [33] and presented in Fig. 3.The FMS consists of three machines called M1, M2

and M3, and handles two types of products named Aand B. Parts of type A are processed first by machineM1. Then, they are processed by M3, but if M2 isnot processing a product of type B, i.e., M2 is idle, itcan be processed on M2 as well. On the other hand,parts of type B are processed by M2 but if M3 is idle,it can be processed on M3 as well. The productionof a product of type A is modeled by place AinM1(production by M1) while the activity of manufactur-ing is modeled by transition M1A. The products arethen transported by a conveyor with capacity of threeproducts (place conv) and then in place AwM23 adecision is taken. Products are moved to M3 by fir-ing AsM3 or to M2 by firing AsM2. Notice thattransition AsM2 has an input inhibitor arc (an in-hibitor arc from pi to tj with weight w disables tj ifm(pi) > k) enabling the transition only if there areno products of part B in place BwM23. For prod-ucts of type B, in place BwM23 a decision is taken.The products are produced on M2 by firing BsM2 or

on M3 by firing BsM3 (if no products of type A arewaiting to be processed in place AwM23).

In place choice there is a decision on which type ofproduct should be produced. There is a conflict be-tween transitions partA and partB and a fixed per-centage is given: 30% to produce products of part A(transition partA) and 70% for partB. An AGV isavailable in order to unload the finished products andload the first machines.

Since the model contains inhibitor arcs, manystructural properties cannot be directly computed be-cause this information is not presented in the tokenflow matrix. If the net is bounded, the inhibitor arcscan be removed by using a complementary place. Toapply this construction the maximum number of to-kens in the input place of the inhibitor arcs must beknown. This can be obtained from PN Toolbox byexploiting the reachability graph. For a number ofpallets P = 5, the reachability graph has 6582 nodesand the maximum number of tokes in places AwM23and BwM23 is 5. Inhibitor arcs (AwM23, BsM3)and (BwM23, AsM2) can be replaced by using thecomplementary places p20 and p21 and connected tothe other nodes as in Fig. 1. Notice that transitionBsM3 can be fired only if place p20 has 5 tokes thatis equivalent to have zero tokens in AwM23.

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The PN Toolbox allows the check of different struc-tural properties. For example, the P and T semi-flows. In this case, there are 7 P-semiflows (associ-ated to the conservation of the following resources:main process stream, AGV, conveyor, M2, M3 and 2associated to the complementary places introduced:p20 and p21) and 4 T-semiflows (corresponding to thefour repetitive components). Each place is in the sup-port of at least one P-semiflow meaning that the netis conservative. Furthermore, all transitions are inthe support of at least one T-semiflow, hence the netis consistent. These structural propertes can be eas-ily checked by the PN Toolbox.

The main facility of the PN Toolbox is the simula-tion of the GSPN system. Assuming the same firingrate for all transition equal to 1, single server seman-tics for transition M1A and infinite server semanticsto the other transitions, and simulating for 10, 000time units, the number of products produced is equalto 4, 846 (Service Sum index of transition sAGV 2),from which 1, 454 products of part A (Service Sumof transitions M3A and M2A) and 3, 393 products ofpart B (Service Sum of transitions M3B and M2B).The throughput of the system is obtained by consult-ing the index Service rate of transition AGV 2. In thiscase, it is equal to 0.4845.

An important feature of the PN Toolbox is the pos-sibility of running a design experiment. Assume thatwe want to see how the throughput of the systemchanges when the firing rate of transition M1A de-creases from 1 to 0.2 (the average of the firing delayis varying from 1 to 5). The results are sketched inFig.4 where it is easy to see that the throughput isalmost constant when the firing rate of M1A is be-tween 0.33 and 1.

The GSPN can be automatically exported toSimHPN by using the corresponding pushdown but-ton from PN Toolbox. However, in the consideredcontinuous nets, all transitions must have the samefiring semantics, in particular infinite server seman-tics. For this reason, in the GSPN in Fig. 3, placeidleM1 is introduced to model the finite server se-mantics of M1A. Since the initial marking of idleM1is one, the enabling degree of M1A is upper boundedby 1, and the resulting net has the desired behav-ior. The throughput of the continuous system inthe steady state computed by SimHPN is 0.497512,which is very similar to the one of the discrete system.

7 Conclusions

Petri nets are widely used in industry and academyfor the modeling and analysis of dynamical systems.Although Petri nets were originally oriented (andsomehow constrained) to discrete event systems, theadvent of continuous and hybrid Petri nets has fur-ther widened the application scope of the paradigmand has simplified the study of large discrete systemsas they may admit a suitable continuous approxima-tion.This paper has presented some tools that offer

modeling, analysis and simulation functions for dis-crete, continuous and hybrid Petri nets. More pre-cisely, the main focus has been on the Petri NetToolbox which is mainly oriented to discrete Petrinets, and on SimHPN , which is a complementarytool mainly oriented to continuous and hybrid Petrinets. Thus, the choice of the appropriate softwaretool strongly depends on the nature (discrete, contin-uous or hybrid) of the model to be analyzed. Besidesthe existing algorithms for analysis and simulation,both tools offer a clear an intuitive graphical inter-face which makes the user easier to grasp the way thetools work. In addition to the mentioned tools, a briefdiscussion of two other MATLAB tools has been in-cluded, namely Hybrid Petri Net Siulator (HYPENS)and Petri Net Simulink Block (PNSB).

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