Search results for: high-level petri nets

Authors: Yurii Bloshko, Oksana Olar

Abstract:

This paper presents the analysis of six different classes of Petri nets: fuzzy Petri nets (FPN), generalized fuzzy Petri nets (GFPN), parameterized fuzzy Petri nets (PFPN), T2GFPN, flexible generalized fuzzy Petri nets (FGFPN), binary Petri nets (BPN). These classes were simulated in the special software PNeS® for the analysis of its pros and cons on the example of models which are dedicated to the decision-making process of passenger transport logistics. The paper includes the analysis of two approaches: when input values are filled with the experts’ knowledge; when fuzzy expectations represented by output values are added to the point. These approaches fulfill the possibilities of triples of functions which are replaced with different combinations of t-/s-norms.

Keywords: Fuzzy petri net, intelligent computational techniques, knowledge representation, triangular norms.

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Authors: M.H.Aziz, Erik L.J.Bohez, Manukid Parnichkun, Chanchal Saha

Abstract:

Petri Net (PN) has proven to be effective graphical, mathematical, simulation, and control tool for Discrete Event Systems (DES). But, with the growth in the complexity of modern industrial, and communication systems, PN found themselves inadequate to address the problems of uncertainty, and imprecision in data. This gave rise to amalgamation of Fuzzy logic with Petri nets and a new tool emerged with the name of Fuzzy Petri Nets (FPN). Although there had been a lot of research done on FPN and a number of their applications have been anticipated, but their basic types and structure are still ambiguous. Therefore, in this research, an effort is made to categorize FPN according to their structure and algorithms Further, literature review of the applications of FPN in the light of their classifications has been done.

Keywords: Discrete event systems, Fuzzy logic, Fuzzy Petri nets, and Petri nets.

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Authors: K. Shoylekova, K. Grigorova

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The Petri nets are the first standard for business process modeling. Most probably, it is one of the core reasons why all new standards created afterwards have to be so reformed as to reach the stage of mapping the new standard onto Petri nets. The paper presents a business process repository based on a universal database. The repository provides the possibility the data about a given process to be stored in three different ways. Business process repository is developed with regard to the reformation of a given model to a Petri net in order to be easily simulated. Two different techniques for business process simulation based on Petri nets - Yasper and Woflan are discussed. Their advantages and drawbacks are outlined. The way of simulating business process models, stored in the Business process repository is shown.

Keywords: Business process repository, Petri nets, Simulation, Woflan, Yasper.

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Authors: Aryanejad Mir Bahador Goli, Zahra Honarmand Shah Zileh

Abstract:

In most study fields, a phenomenon may not be studied directly but it will be examined indirectly by phenomenon model. Making an accurate model of system, there is attained new information from modeled phenomenon without any charge, danger, etc... there have been developed more solutions for describing and analyzing the recent complicated systems but few of them have analyzed the performance in the range of system description. Petri nets are of limited solutions which may make such union. Petri nets are being applied in problems related to modeling and designing the systems. Theory of Petri nets allow a system to model mathematically by a Petri net and analyzing the Petri net can then determine main information of modeled system-s structure and dynamic. This information can be used for assessing the performance of systems and suggesting corrections in the system. In this paper, beside the introduction of Petri nets, a real case study will be studied in order to show the application of generalized stochastic Petri nets in modeling a resource sharing production system and evaluating the efficiency of its machines and robots. The modeling tool used here is SHARP software which calculates specific indicators helping to make decision.

Keywords: Flexible manufacturing system, generalizedstochastic Petri nets, Markov chain, performance evaluation.

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Authors: Noura Boudiaf, Abdelhamid Djebbar

Abstract:

Colored Petri Nets (CPN) are very known kind of high level Petri nets. With sound and complete semantics, rewriting logic is one of very powerful logics in description and verification of non-deterministic concurrent systems. Recently, CPN semantics are defined in terms of rewriting logic, allowing us to built models by formal reasoning. In this paper, we propose an automatic translation of CPN to the rewriting logic language Maude. This tool allows graphical editing and simulating CPN. The tool allows the user drawing a CPN graphically and automatic translating the graphical representation of the drawn CPN to Maude specification. Then, Maude language is used to perform the simulation of the resulted Maude specification. It is the first rewriting logic based environment for this category of Petri Nets.

Keywords: Colored Petri Nets, Rewriting Logic, Maude, Graphical Edition, Automatic Translation, Simulation.

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Authors: M. Zareiee, A. Dideban

Abstract:

This paper addresses the problem of forbidden states in non safe Petri Nets. In the system, for preventing it from entering the forbidden states, some linear constraints can be assigned to them. Then these constraints can be enforced on the system using control places. But when the number of constraints in the system is large, a large number of control places must be added to the model of system. This concept complicates the model of system. There are some methods for reducing the number of constraints in safe Petri Nets. But there is no a systematic method for non safe Petri Nets. In this paper we propose a method for reducing the number of constraints in non safe Petri Nets which is based on solving an integer linear programming problem.

Keywords: discrete event system, Supervisory control, Petri Net, Constraint

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Authors: W. Louhichi, M.Berrima, N. Ben Rajeb Robbana

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Elasticity is the essential property of cloud computing. As the name suggests, it constitutes the ability of a cloud system to adjust resource provisioning in relation to fluctuating workloads. There are two types of elasticity operations, vertical and horizontal. In this work, we are interested in horizontal scaling, which is ensured by two mechanisms; scaling in and scaling out. Following the sizing of the system, we can adopt scaling in the event of over-supply and scaling out in the event of under-supply. In this paper, we propose a formal model, based on temporized and colored Petri nets (TdCPNs), for the modeling of the duplication and the removal of a virtual machine from a server. This model is based on formal Petri Nets (PNs) modeling language. The proposed models are edited, verified, and simulated with two examples implemented in colored Petri nets (CPNs)tools, which is a modeling tool for colored and timed PNs.

Keywords: Cloud computing, elasticity, elasticity controller, petri nets, scaling in, scaling out.

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Authors: M. Zareiee, A. Dideban, P. Nazemzadeh

Abstract:

This paper deals with the problem of constructing constraints in non safe Petri Nets and then reducing the number of the constructed constraints. In a system, assigning some linear constraints to forbidden states is possible. Enforcing these constraints on the system prevents it from entering these states. But there is no a systematic method for assigning constraints to forbidden states in non safe Petri Nets. In this paper a useful method is proposed for constructing constraints in non safe Petri Nets. But when the number of these constraints is large enforcing them on the system may complicate the Petri Net model. So, another method is proposed for reducing the number of constructed constraints.

Keywords: discrete event system, Supervisory control, Petri Net, Constraint

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Authors: Noura Boudiaf, Allaoua Chaoui

Abstract:

One major difficulty that faces developers of concurrent and distributed software is analysis for concurrency based faults like deadlocks. Petri nets are used extensively in the verification of correctness of concurrent programs. ECATNets [2] are a category of algebraic Petri nets based on a sound combination of algebraic abstract types and high-level Petri nets. ECATNets have 'sound' and 'complete' semantics because of their integration in rewriting logic [12] and its programming language Maude [13]. Rewriting logic is considered as one of very powerful logics in terms of description, verification and programming of concurrent systems. We proposed in [4] a method for translating Ada-95 tasking programs to ECATNets formalism (Ada-ECATNet). In this paper, we show that ECATNets formalism provides a more compact translation for Ada programs compared to the other approaches based on simple Petri nets or Colored Petri nets (CPNs). Such translation doesn-t reduce only the size of program, but reduces also the number of program states. We show also, how this compact Ada-ECATNet may be reduced again by applying reduction rules on it. This double reduction of Ada-ECATNet permits a considerable minimization of the memory space and run time of corresponding Maude program.

Keywords: Ada tasking, ECATNets, Algebraic Petri Nets, Compact Representation, Analysis, Rewriting Logic, Maude.

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Authors: N. Boudiaf

Abstract:

One major difficulty that faces developers of concurrent and distributed software is analysis for concurrency based faults like deadlocks. Petri nets are used extensively in the verification of correctness of concurrent programs. ECATNets are a category of algebraic Petri nets based on a sound combination of algebraic abstract types and high-level Petri nets. ECATNets have 'sound' and 'complete' semantics because of their integration in rewriting logic and its programming language Maude. Rewriting logic is considered as one of very powerful logics in terms of description, verification and programming of concurrent systems We proposed previously a method for translating Ada-95 tasking programs to ECATNets formalism (Ada-ECATNet) and we showed that ECATNets formalism provides a more compact translation for Ada programs compared to the other approaches based on simple Petri nets or Colored Petri nets. We showed also previously how the ECATNet formalism offers to Ada many validation and verification tools like simulation, Model Checking, accessibility analysis and static analysis. In this paper, we describe the implementation of our translation of the Ada programs into ECATNets.

Keywords: Ada tasking, Analysis, Automatic Translation, ECATNets, Maude, Rewriting Logic.

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Authors: Noura Boudiaf, Allaoua Chaoui

Abstract:

To analyze the behavior of Petri nets, the accessibility graph and Model Checking are widely used. However, if the analyzed Petri net is unbounded then the accessibility graph becomes infinite and Model Checking can not be used even for small Petri nets. ECATNets [2] are a category of algebraic Petri nets. The main feature of ECATNets is their sound and complete semantics based on rewriting logic [8] and its language Maude [9]. ECATNets analysis may be done by using techniques of accessibility analysis and Model Checking defined in Maude. But, these two techniques supported by Maude do not work also with infinite-states systems. As a category of Petri nets, ECATNets can be unbounded and so infinite systems. In order to know if we can apply accessibility analysis and Model Checking of Maude to an ECATNet, we propose in this paper an algorithm allowing the detection if the ECATNet is bounded or not. Moreover, we propose a rewriting logic based tool implementing this algorithm. We show that the development of this tool using the Maude system is facilitated thanks to the reflectivity of the rewriting logic. Indeed, the self-interpretation of this logic allows us both the modelling of an ECATNet and acting on it.

Keywords: ECATNets, Rewriting Logic, Maude, Finite-stateSystems, Infinite-state Systems, Boundness Property Checking.

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Authors: Nidhi Gaur, Padmaja Joshi, Vijay Jain, Rajeev Srivastava

Abstract:

Web application architecture is important to achieve the desired performance for the application. Performance analysis studies are conducted to evaluate existing or planned systems. Web applications are used by hundreds of thousands of users simultaneously, which sometimes increases the risk of server failure in real time operations. We use Coloured Petri Net (CPN), a very powerful tool for modelling dynamic behaviour of a web application system. CPNs extend the vocabulary of ordinary Petri nets and add features that make them suitable for modelling large systems. The major focus of this work is on server side of web applications. The presented work focuses on modelling restructuring aspects, with major focus on concurrency and architecture, using CPN. It also focuses on bringing out the appropriate architecture for web and database servers given the number of concurrent users.

Keywords: Coloured petri nets, concurrent users, performance modelling, web application architecture.

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Authors: Julija Asmuss, Gunars Lauks, Viktors Zagorskis

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We consider the problem of bandwidth allocation in a substrate network as an optimization problem for the aggregate utility of multiple applications with diverse requirements and describe a simulation scheme for dynamically adaptive bandwidth allocation protocols. The proposed simulation model based on Coloured Petri Nets (CPN) is realized using CPN Tools.

Keywords: Bandwidth Allocation Problem, Coloured Petri Nets, CPN Tools, Simulation

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Authors: Lotfi Nabli, Achraf Jabeur Telmoudi, Radhi M'hiri

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This paper proposes a modeling method of the laws controlling manufacturing systems with temporal and non temporal constraints. A methodology of robust control construction generating the margins of passive and active robustness is being elaborated. Indeed, two paramount models are presented in this paper. The first utilizes the P-time Petri Nets which is used to manage the flow type disturbances. The second, the quality model, exploits the Intervals Constrained Petri Nets (ICPN) tool which allows the system to preserve its quality specificities. The redundancy of the robustness of the elementary parameters between passive and active is also used. The final model built allows the correlation of temporal and non temporal criteria by putting two paramount models in interaction. To do so, a set of definitions and theorems are employed and affirmed by applicator examples.

Keywords: Manufacturing systems control, flow, quality, robustness, redundancy, Petri Nets.

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Authors: Husam Kaid, Abdulrahman Al-Ahmari, Zhiwu Li

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This paper develops a robust deadlock control technique for shared and unreliable resources in automated manufacturing systems (AMSs) based on structural analysis and colored Petri nets, which consists of three steps. The first step involves using strict minimal siphon control to create a live (deadlock-free) system that does not consider resource failure. The second step uses an approach based on colored Petri net, in which all monitors designed in the first step are merged into a single monitor. The third step addresses the deadlock control problems caused by resource failures. For all resource failures in the Petri net model a common recovery subnet based on colored petri net is proposed. The common recovery subnet is added to the obtained system at the second step to make the system reliable. The proposed approach is evaluated using an AMS from the literature. The results show that the proposed approach can be applied to an unreliable complex Petri net model, has a simpler structure and less computational complexity, and can obtain one common recovery subnet to model all resource failures.

Keywords: Automated manufacturing system, colored Petri net, deadlock, siphon.

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Authors: Nabli L., Dhouibi H., Collart Dutilleul S., Craye E.

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The indistinctness of the manufacturing processes makes that a parts cannot be realized in an absolutely exact way towards the specifications on the dimensions. It is thus necessary to assume that the effectively realized product has to belong in a very strict way to compatible intervals with a correct functioning of the parts. In this paper we present an approach based on mixing tow different characteristics theories, the fuzzy system and Petri net system. This tool has been proposed to model and control the quality in an assembly system. A robust command of a mechanical assembly process is presented as an application. This command will then have to maintain the specifications interval of parts in front of the variations. It also illustrates how the technique reacts when the product quality is high, medium, or low.

Keywords: Petri nets, production rate, performance evaluation, tolerant system, fuzzy sets.

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Authors: P. Nazemzadeh, A. Dideban, M. Zareiee

Abstract:

In this paper the effect of faults in the elements and parts of discrete event systems is investigated. In the occurrence of faults, some states of the system must be changed and some of them must be forbidden. For this goal, different states of these elements are examined and a model for fail-safe behavior of each state is introduced. Replacing new models of the target elements in the preliminary model by a systematic method, leads to a fail-safe discrete event system.

Keywords: Discrete event systems, Fail-safe, Petri nets, Supervisory control.

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Authors: Amel Ltifi, Ahmed Zouinkhi, Med Salim Bouhlel

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Spontaneous networks such as VANET are in general deployed in an open and thus easily accessible environment. Therefore, they are vulnerable to attacks. Trust management is one of a set of security solutions dedicated to this type of networks. Moreover, the strong mobility of the nodes (in the case of VANET) makes the establishment of a trust management system complex. In this paper, we present a concept of ‘Active Vehicle’ which means an autonomous vehicle that is able to make decision about trustworthiness of alert messages transmitted about road accidents. The behavior of an “Active Vehicle” is modeled using Petri Nets.

Keywords: Component, active vehicle, cooperation, petri nets, trust management, VANET.

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Authors: Mourtou Efstratia, Abdel-Badeeh M. Salem, Pavlidis George

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Hierarchical high-level PNs (HHPNs) with time versions are a useful tool to model systems in a variety of application domains, ranging from logistics to complex workflows. This paper addresses an application domain which is receiving more and more attention: procedure that arranges the final inpatient charge in payment-s office and their management. We shall prove that Petri net based analysis is able to improve the delays during the procedure, in order that inpatient charges could be more reliable and on time.

Keywords: eHealth, Petri-Nets, Hospital Services, InpatientCharges, Workflow Modeling.

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Authors: Joselito Medina Marin, Norberto Hernández Romero, Juan Carlos Seck Tuoh Mora, Erick S. Martinez Gomez

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The Flow Shop Scheduling Problem (FSSP) is a typical problem that is faced by production planning managers in Flexible Manufacturing Systems (FMS). This problem consists in finding the optimal scheduling to carry out a set of jobs, which are processed in a set of machines or shared resources. Moreover, all the jobs are processed in the same machine sequence. As in all the scheduling problems, the makespan can be obtained by drawing the Gantt chart according to the operations order, among other alternatives. On this way, an FMS presenting the FSSP can be modeled by Petri nets (PNs), which are a powerful tool that has been used to model and analyze discrete event systems. Then, the makespan can be obtained by simulating the PN through the token game animation and incidence matrix. In this work, we present an adaptive PN to obtain the makespan of FSSP by applying PN analytical tools.

Keywords: Flow-shop scheduling problem, makespan, Petri nets, state equation.

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Authors: Sabina Baraković, Dragan Jevtić, Jasmina Baraković Husić

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Wireless mobile communications have experienced the phenomenal growth through last decades. The advances in wireless mobile technologies have brought about a demand for high quality multimedia applications and services. For such applications and services to work, signaling protocol is required for establishing, maintaining and tearing down multimedia sessions. The Session Initiation Protocol (SIP) is an application layer signaling protocols, based on request/response transaction model. This paper considers SIP INVITE transaction over an unreliable medium, since it has been recently modified in Request for Comments (RFC) 6026. In order to help in assuring that the functional correctness of this modification is achieved, the SIP INVITE transaction is modeled and analyzed using Colored Petri Nets (CPNs). Based on the model analysis, it is concluded that the SIP INVITE transaction is free of livelocks and dead codes, and in the same time it has both desirable and undesirable deadlocks. Therefore, SIP INVITE transaction should be subjected for additional updates in order to eliminate undesirable deadlocks. In order to reduce the cost of implementation and maintenance of SIP, additional remodeling of the SIP INVITE transaction is recommended.

Keywords: Colored Petri Nets, SIP INVITE, state space, dead marking

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Authors: Shuzhen Yao, Fengjing Zhao, Jianwei He

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The UML modeling of complex distributed systems often is a great challenge due to the large amount of parallel real-time operating components. In this paper the problems of verification of such systems are discussed. ECPN, an Extended Colored Petri Net is defined to formally describe state transitions of components and interactions among components. The relationship between sequence diagrams and Free Choice Petri Nets is investigated. Free Choice Petri Net theory helps verifying the liveness of sequence diagrams. By converting sequence diagrams to ECPNs and then comparing behaviors of sequence diagram ECPNs and statecharts, the consistency among models is analyzed. Finally, a verification process for an example model is demonstrated.

Keywords: Consistency, liveness, Petri Net, sequence diagram.

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Authors: Anthony Spiteri Staines

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This paper examines the modeling and analysis of a cruise control system using a Petri net based approach, task graphs, invariant analysis and behavioral properties. It shows how the structures used can be verified and optimized.

Keywords: Software Engineering, Real Time Analysis andDesign, Petri Nets, Task Graphs, Parallelism.

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Authors: J. Barilla, M. Lokajíček, H. Pisaková, P. Simr

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The final biological effect of ionizing particles may be influenced strongly by some chemical substances present in cells mainly in the case of low-LET radiation. The influence of oxygen may by particularly important because oxygen is always present in living cells. The corresponding processes are then running mainly in the chemical stage of radiobiological mechanism.

The radical clusters formed by densely ionizing ends of primary or secondary charged particles are mainly responsible for final biological effect. The damage effect depends then on radical concentration at a time when the cluster meets a DNA molecule. It may be strongly influenced by oxygen present in a cell as oxygen may act in different directions: at small concentration of it the interaction with hydrogen radicals prevails while at higher concentrations additional efficient oxygen radicals may be formed.

The basic radical concentration in individual clusters diminishes, which is influenced by two parallel processes: chemical reactions and diffusion of corresponding clusters. The given simultaneous evolution may be modeled and analyzed well with the help of Continuous Petri nets. The influence of other substances present in cells during irradiation may be studied, too. Some results concerning the impact of oxygen content will be presented.

Keywords: DSB formation, chemical stage, Petri nets, radiobiological mechanism.

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Authors: S. Ariffin, K. Hasnan, R.H. Weston

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A new generation of manufacturing machines so-called MIMCA (modular and integrated machine control architecture) capable of handling much increased complexity in manufacturing control-systems is presented. Requirement for more flexible and effective control systems for manufacturing machine systems is investigated and dimensioned-which highlights a need for improved means of coordinating and monitoring production machinery and equipment used to- transport material. The MIMCA supports simulation based on machine modeling, was conceived by the authors to address the issues. Essentially MIMCA comprises an organized unification of selected architectural frameworks and modeling methods, which include: NISTRCS, UMC and Colored Timed Petri nets (CTPN). The unification has been achieved; to support the design and construction of hierarchical and distributed machine control which realized the concurrent operation of reusable and distributed machine control components; ability to handle growing complexity; and support requirements for real- time control systems. Thus MIMCA enables mapping between 'what a machine should do' and 'how the machine does it' in a well-defined but flexible way designed to facilitate reconfiguration of machine systems.

Keywords: Machine control, architectures, Petri nets, modularity, modeling, simulation.

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Authors: Douglas A. Menezes, Isabel D. Nunes, Ulrich Schiel

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Most Virtual Learning Environments do not provide support mechanisms for the integrated planning, construction and follow-up of Instructional Design supported by Learning Analytic results. The present work aims to present an authoring tool that will be responsible for constructing the structure of an Instructional Design (ID), without the data being altered during the execution of the course. The visual interface aims to present the critical situations present in this ID, serving as a support tool for the course follow-up and possible improvements, which can be made during its execution or in the planning of a new edition of this course. The model for the ID is based on High-Level Petri Nets and the visualization forms are determined by the specific kind of the data generated by an e-course, a population of students generating sequentially dependent data.

Keywords: Educational data visualization, high-level petri nets, instructional design, learning analytics.

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Authors: Raida El Mansouri, Elhillali Kerkouche, Allaoua Chaoui

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The Petri net tool INA is a well known tool by the Petri net community. However, it lacks a graphical environment to cerate and analyse INA models. Building a modelling tool for the design and analysis from scratch (for INA tool for example) is generally a prohibitive task. Meta-Modelling approach is useful to deal with such problems since it allows the modelling of the formalisms themselves. In this paper, we propose an approach based on the combined use of Meta-modelling and Graph Grammars to automatically generate a visual modelling tool for INA for analysis purposes. In our approach, the UML Class diagram formalism is used to define a meta-model of INA models. The meta-modelling tool ATOM3 is used to generate a visual modelling tool according to the proposed INA meta-model. We have also proposed a graph grammar to automatically generate INA description of the graphically specified Petri net models. This allows the user to avoid the errors when this description is done manually. Then the INA tool is used to perform the simulation and the analysis of the resulted INA description. Our environment is illustrated through an example.

Keywords: INA, Meta-modelling, Graph Grammars, AToM3, Automatic Code Generation.

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Authors: Nabli L., Dhouibi H., Collart Dutilleul S., Craye E.

Abstract:

The existence of maximal durations drastically modifies the performance evaluation in Discrete Event Systems (DES). The same particularity may be found on systems where the associated constraints do not concern the time. For example weight measures, in chemical industry, are used in order to control the quantity of consumed raw materials. This parameter also takes a fundamental part in the product quality as the correct transformation process is based upon a given percentage of each essence. Weight regulation therefore increases the global productivity of the system by decreasing the quantity of rejected products. In this paper we present an approach based on mixing different characteristics theories, the fuzzy system and Petri net system to describe the behaviour. An industriel application on a tobacco manufacturing plant, where the critical parameter is the weight is presented as an illustration.

Keywords: Petri Net, Manufacturing systems, Performance evaluation, Fuzzy logic, Tolerant system.

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Authors: Prasun Hajra, Ranjan Dasgupta

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Petri Net being one of the most useful graphical tools for modelling complex asynchronous systems, we have used Petri Net to model multi-track railway level crossing system. The roadway has been augmented with four half-size barriers. For better control, a three stage control mechanism has been introduced to ensure that no road-vehicle is trapped on the level crossing. Timed Petri Net is used to include the temporal nature of the signalling system. Safeness analysis has also been included in the discussion section.

Keywords: Modelling, Timed Petri Net, Railway Level Crossing, Safeness Condition.

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Authors: S. P. Sharma, Monica Rani

Abstract:

In conventional reliability assessment, the reliability data of system components are treated as crisp values. The collected data have some uncertainties due to errors by human beings/machines or any other sources. These uncertainty factors will limit the understanding of system component failure due to the reason of incomplete data. In these situations, we need to generalize classical methods to fuzzy environment for studying and analyzing the systems of interest. Fuzzy set theory has been proposed to handle such vagueness by generalizing the notion of membership in a set. Essentially, in a Fuzzy Set (FS) each element is associated with a point-value selected from the unit interval [0, 1], which is termed as the grade of membership in the set. A Vague Set (VS), as well as an Intuitionistic Fuzzy Set (IFS), is a further generalization of an FS. Instead of using point-based membership as in FS, interval-based membership is used in VS. The interval-based membership in VS is more expressive in capturing vagueness of data. In the present paper, vague set theory coupled with conventional Lambda-Tau method is presented for reliability analysis of repairable systems. The methodology uses Petri nets (PN) to model the system instead of fault tree because it allows efficient simultaneous generation of minimal cuts and path sets. The presented method is illustrated with the press unit of the paper mill.

Keywords: Lambda -Tau methodology, Petri nets, repairable system, vague fuzzy set.

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