Subclass Of Petri Nets Research Articles

Structural Liveness of Immediate Observation Petri Nets

We look in detail at the structural liveness problem (SLP) for subclasses of Petri nets, namely immediate observation nets (IO nets) and their generalized variant called branching immediate multi-observation nets (BIMO nets), that were recently introduced by Esparza, Raskin, and Weil-Kennedy. We show that SLP is PSPACE-hard for IO nets and in PSPACE for BIMO nets. In particular, we discuss the (small) bounds on the token numbers in net places that are decisive for a marking to be (non)live.

Robust liveness enforcement of Petri nets with uncontrollable and unobservable transitions based on structural analysis

This paper concentrates on robust deadlock control problems for an automated manufacturing system with uncontrollable, unobservable events and resource failures, which is modeled by a subclass of Petri nets. A recovery subnet is added to the holder of each unreliable resource to model resource failure and recovery. In order to prevent a strict minimal siphon in a Petri net model from being emptied, an extended constraint set is constructed based on the complementary set of a strict minimal siphon to obtain a robust liveness constraint. Due to the uncontrollability and unobservability of the transitions, an automated manufacturing system cannot enforce inadmissible robust liveness constraints. It is necessary to decide the admissibility of robust liveness constraints and transform the inadmissible constraints into admissible ones. To this end, a monitor is designed for each admissible robust liveness constraint such that it can be enforced. Finally, an iterative algorithm is developed to perform the above steps and a robust liveness controlled system is obtained. The feasibility of the control strategy is demonstrated by examples.

A State-Equation-Based Method to Non-Reachability Analysis of Ordinary Petri Nets With Token-Free Circuit-Based Subnets

Petri nets (PNs) can model various event-driven systems well and detect a fault via reachability analysis. Some state-equation-based approaches are designed for only subclasses of PNs with specific structures. A key issue for PNs’ reachability analysis approach based on state equation is that a non-negative integer solution (NIS) to a state equation is necessary but not sufficient for a marking’s reachability. In this paper, we propose a circuit-based subnet. For ordinary Petri nets with token-free circuit-based subnets, we present an algorithm to determine that the state equation has NISs but the marking is non-reachable. We adopt a divide-and-conquer strategy to consider each circuit-based subnet. The proposed method is tested on a PN-based flexible manufacturing system with unreliable resources. When unreliable resources are lost, the non-reachable markings can be determined effectively. Besides, simulations are done, and the results verify the efficiency of the proposed algorithm compared with the reachability-tree-based method.

Decision on Maximal Permissiveness of Linear Constraints via Structural Analysis of a Subclass of Petri Nets

A maximally permissive (or optimal) supervisory control of an automated manufacturing system (AMS) modeled by Petri nets (PNs) can be usually implemented by imposing constraints in the form of a set of linear inequalities. To find such a set of linear constraints, in the existing work, an integer linear programming (ILP) problem is generally formulated and solved for some dominant markings obtained by reachability analysis and vector covering theory, which is computationally inefficient due to the combinatorial nature of solving ILPs to decide the coefficients of optimal linear constraints. This paper addresses the deadlock prevention problem for AMSs by developing efficient methods to reduce the computational overhead through the establishment of conditions on deciding the maximal permissiveness of linear constraints imposed on a system. By taking the advantage of structural properties of a PN model under consideration, we identify the most part of minimal covered illegal markings that can be optimally controlled via policies obtained by specific linear inequality constraints. Also, algorithms are developed to implement the proposed policies. The proposed approach can verify the optimality of a linear constraint efficiently without solving ILP problems. A linear programming method is developed to deal with the markings that cannot be processed by the proposed structural analysis. It is shown that for the considered class of PNs, called system of simple sequential processes with resources (S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> PR), no mixed ILP problem needs to solve and the computational burden is dramatically reduced. Two examples are employed to demonstrate the efficiency of the developed method.

Abstract processes and conflicts in place/transition systems

For one-safe Petri nets or condition/event-systems, a process as defined by Carl Adam Petri provides a notion of a run of a system where causal dependencies are reflected in terms of a partial order. Goltz and Reisig have generalised this concept for nets where places carry multiple tokens, by distinguishing tokens according to their causal history. However, this so-called individual token interpretation is often considered too detailed. Here we identify a subclass of Petri nets, called structural conflict nets, where no interplay between conflict and concurrency due to token multiplicity occurs. For this subclass, we define abstract processes as equivalence classes of Goltz-Reisig processes. We justify this approach by showing that there is a largest abstract process if and only if the underlying net is conflict-free with respect to a canonical notion of conflict.

Weakly Complete Event Logs in Process Mining

Many information systems have a possibility to record their execution, and, in this way, to generate a trace about events describing the real system behaviour. From behaviour example records in traces of the event log, the α-algorithm automatically generates a process model that belongs to a subclass of Petri nets, known as workflow nets. One of the basic limiting assumptions of α-algorithm is that the event log needs to be complete. As a result of attempting to overcome the problem of completeness of the event log, we introduced the notion of weakly complete event logs, from which our modified technique and algorithm can produce the same result as the α-algorithm from complete logs on parallel processes. Thereby weakly complete logs can be significantly smaller than complete logs, considering the number of traces they consist of. Weakly complete logs were used for the realization of our idea of interactive parallel business process model generation.

Computation of an emptiable minimal siphon in a subclass of Petri nets using mixed-integer programming

Deadlock resolution strategies based on siphon control are widely investigated. Their computational efficiency largely depends on siphon computation. Mixed-integer programming &#x0028 MIP &#x0029 can be utilized for the computation of an emptiable siphon in a Petri net &#x0028 PN &#x0029 . Based on it, deadlock resolution strategies can be designed without requiring complete siphon enumeration that has exponential complexity. Due to this reason, various MIP methods are proposed for various subclasses of PNs. This work proposes an innovative MIP method to compute an emptiable minimal siphon &#x0028 EMS &#x0029 for a subclass of PNs named S4PR. In particular, many particular structural characteristics of EMS in S4PR are formalized as constraints, which greatly reduces the solution space. Experimental results show that the proposed MIP method has higher computational efficiency. Furthermore, the proposed method allows one to determine the liveness of an ordinary S4PR.

Target-oriented Petri Net Synthesis

When a Petri net is synthesised from a labelled transition system, it is frequently desirable that certain additional constraints are fulfilled. For example, in circuit design, one is often interested in constructing safe Petri nets. Targeting such subclasses of Petri nets is not necessarily computationally more efficient than targeting the whole class. For example, targeting safe nets is known to be NP-complete while targeting the full class of place/transition nets is polynomial, in the size of the transition system. In this paper, several classes of Petri nets are examined, and their suitability for being targeted through efficient synthesis from labelled transition systems is studied and assessed. The focus is on choice-free Petri nets and some of their subclasses. It is described how they can be synthesised efficiently from persistent transition systems, summarising and streamlining in tutorial style some of the authors’ and their groups’ work over the past few years.

Just-in-Time Balancing of Feedback Assembly Lines Via Constrained-Linear Tropical Model

Abstract Discrete Events Systems are systems that are subject to synchronization and delay phenomena. They can be modeled by a subclass of Petri nets named Timed Event Graphs whose dynamics can described by Tropical algebra (Max or Min plus Algebra). The aim of this article is to present a methodology to project controllers that respect the restrictions imposed to the system operation ensuring the just-in-time policy. This is a control policy that proposes the use of the biggest possible delay of the system input dates without changing production rates, keeping the system demands. To this end, the controller synthesis methodology is developed based on Residuation Theory and on the concept of super-eigenvector. Sufficient conditions to get the controller is presented with an example to illustrate the applicability of the results.

Stability Analysis of Discrete Event Systems Modeled by Petri Nets Using Unfoldings

The stability of discrete event systems (DESs) is a property related to its robustness; a stable DES is guaranteed to reach a set of desired states in a finite number of steps. Algorithms proposed in the literature are useful for analyzing the stability of DES modeled either by finite automata or a constrained subclass of Petri nets (PNs). In this paper, a novel method to decide the stability of safe PNs is presented. It is based on the analysis of the finite complete prefix of the net unfolding rather than the reachability graph. For concurrent systems, the prefix is usually smaller than the reachability graph and then the analysis using the proposed algorithm is more efficient than analyzing the reachability set of the PN. Note to Practitioners —The stability of a discrete event system (DES) is a property that guarantees that the DES will return to the normal operation behavior after a disturbance deviates the system from such a behavior. This is a key property that a controlled DES must fulfill. The algorithms proposed in this paper allow determining efficiently the stability of concurrent DES by analyzing the Petri net structure.

On Deadlockability, Liveness and Reversibility in Subclasses of Weighted Petri Nets

Liveness, (non-)deadlockability and reversibility are behavioral properties of Petri nets that are fundamental for many real-world systems. Such properties are often required to be mono-tonic, meaning preserved upon any increase of the marking. However, their checking is intractable in general and their monotonicity is not always satisfied. To simplify the analysis of these features, structural approaches have been fruitfully exploited in particular subclasses of Petri nets, deriving the behavior from the underlying graph and the initial marking only, often in polynomial time. In this paper, we further develop these efficient structural methods to analyze deadlockability, live-ness, reversibility and their monotonicity in weighted Petri nets. We focus on the join-free subclass, which forbids synchronizations, and on the homogeneous asymmetric-choice subclass, which allows conflicts and synchronizations in a restricted fashion. For the join-free nets, we provide several structural conditions for checking liveness, (non-)deadlock-ability, reversibility and their monotonicity. Some of these methods operate in polynomial time. Furthermore , in this class, we show that liveness, non-deadlockability and reversibility, taken together or separately, are not always monotonic, even under the assumptions of structural boundedness and structural liveness. These facts delineate more sharply the frontier between monotonicity and non-monotonicity of the behavior in weighted Petri nets, present already in the join-free subclass. In addition, we use part of this new material to correct a flaw in the proof of a previous characterization of monotonic liveness and boundedness for homogeneous asymmetric-choice nets, published in 2004 and left unnoticed.

Liveness characteristic analysis of a class of Petri nets

Petri nets are an effective tool for analyzing and modeling the dynamic behavior of flexible manufacturing systems. Finite capacity systems of simple sequential processes with resources are an important subclass of Petri nets, for which this article gives a liveness characteristic analysis. First, an effective algorithm for deciding the liveness of finite capacity systems of simple sequential processes with resources is developed by analyzing the relation between the structural properties of resource subnets and the strict minimal siphons. Then, a liveness condition of finite capacity systems of simple sequential processes with resources is accordingly established. Based on the proposed liveness condition, an algorithm for configuring an initial marking for a finite capacity systems of simple sequential processes with resources is given, and therefore, a live finite capacity systems of simple sequential processes with resources net with a configured initial marking can be obtained, which avoids the siphon enumerations and the addition of any control actions. It is shown that the computational complexity of both the developed liveness deciding and the initial marking configuration algorithms is polynomial. Examples are finally provided to illustrate the mentioned results.

Performance Evaluation of Discrete Event Systems with GPenSIM

Petri nets are a useful tool for the modeling and performance evaluation of discrete event systems. Literature reveals that the Petri Net models of real-world discrete event systems are most frequently event graphs (a subclass of Petri nets). Literature also reveals that there are some simple methods for the performance evaluation of event graphs. The general-purpose Petri Net simulator (GPenSIM) is a new simulator that runs on the MATLAB platform. GPenSIM provides a Petri net language, with which Petri net classes and extensions can be developed. GPenSIM also provides functions for performance analysis. Since real-world discrete event systems usually possess a large number of resources, the Petri net models of these systems tend to become huge. Activity-Oriented Petri Nets (AOPN) is an approach that reduces the size of the Petri nets. In addition to the simulator functions, GPenSIM also realizes the AOPN approach on the MATLAB platform. Thus, AOPN is an integral part of GPenSIM. As a running example, a flexible manufacturing system is firstly modeled as an event graph, and then the size of the model is reduced with the AOPN approach. The advantages of GPenSIM and AOPN are discussed in this paper.

Firing Rate Optimization of Deterministic Timed Event Graphs by Server Performance Improvement

Timed event graphs are a subclass of Petri nets that are commonly applied for modeling and controlling manufacturing systems. In this paper, we investigate the problem of improving the performance of machines (servers) in timed event graphs in order to maximize the firing rate of a system. We show that the logic constraints on the cost of machines and the operation time of machines (firing delay of transitions) can be transformed into linear algebraic constraints. Furthermore, we formulate a mixed integer linear programming problem to maximize the firing rate of the net under a given budget for improving the performance of servers (decreasing the firing delay of transitions) that can provide an optimal solution. Finally, application to an assembly line is presented to show the effectiveness of the developed methodology.

Healthcare Operation Improvement Based on Simulation of Cooperative Resource Preservation Nets for None‐Consumable Resources

Healthcare systems are growing very fast, especially emergency departments (EDs) which constitute the major bottleneck of these complex concurrent systems. Emergency departments, where patients arrive without any prior notice, are considered real‐time complex dynamic systems. Enhancing these systems requires tailored modeling techniques and a process optimization approach. A new mathematical approach is proposed in order to help multiple emergency units cooperate and share none‐consumable resources to achieve the required flow. To achieve the cooperation, the process is modeled by a new subclass of Petri nets. The new Petri net model was proposed in a previous work and is used in this study in order to tackle the problem of modeling and managing these emergency units. The proposed Petri net is named Resource Preservation Net (RPN). Few theorems and lemmas are proposed to support the proposed Petri net model and to prove the correctness of cooperation and resource sharing. In this contribution, a model of cooperative healthcare units is proposed to achieve sound resource sharing and collaboration. The objective function of the proposed model is to improve the key performance indicators: patients length of stay (LoS), resource utilization rates, and patients waiting time. The cooperation among multiple EDs is then proposed through the study of merging two or more units. The cooperative and noncooperative behavior are also studied through theorems of soundness, separability and serializability, and a proof of scalability.

Qualitative Assessment of the Exceeding Behaviour of Discovered Petri Nets

In this paper, the precision evaluation of a subclass of Petri nets, called workflow nets (WFN), is addressed. A novel technique for qualitatively assessing the precision is proposed; it estimates the exceeding language of a sound WFN including cycles and silent transitions with respect to the actual language provided by the events log. The estimation, which does not consider the frequency of traces in the log, involves three sub-metrics dealing with a) activity completeness, b) finite behaviour of the model, and c) cyclic behaviour of the model. The compound metric is computed through a partial reachability graph called log-based marking graph (LMG); it is iteratively evaluated by simulating the WFN and adding new behaviour to such a LMG until the computed value converges asymptotically. The proposed technique provides a good approximation to the precision of sound WFNs modelling finite or infinite languages.

On the Decidability and Complexity of Diagnosability for Labeled Petri Nets

In this paper, we investigate the decidability and complexity of the fault diagnosis problem in unbounded labeled Petri nets. First, we show that checking diagnosability for unbounded Petri nets is decidable. We present a new necessary and sufficient condition for diagnosability, which can be reduced to a model checking problem for unbounded Petri nets. Then, we show that checking diagnosability for unbounded Petri nets is EXPSPACE-complete. This complexity result is further extended to various subclasses of Petri nets. To the best of our knowledge, this is the first paper that establishes decidability and complexity results for diagnosability of unbounded Petri nets.

Output feedback control of discrete processes under time constraint: application to cluster tools

Practical control of timed behaviours in industrial applications is difficult. One of the few efficient modelling approaches to deal with time constraints on discrete event systems is to use timed event graph (TEG), a subclass of Petri nets. The dynamic behaviour of these graphs is represented by a linear equations system over the max-plus algebra. Up to date, these models were assumed to be fully observable. In this paper, it is demonstrated that if a TEG is strongly connected, one can derive a realisable output feedback control. In other words, the control law depends solely on the control itself and the system’s outputs, considered to be the only observable events. The satisfaction of a set of constraints is addressed in the case of a single input model (i.e. having only one input transition) and the procedure to derive a realisable output feedback control law is provided. Finally, the present control method is applied to a dual-armed cluster tool, a well-known industrial practical application.

Virtual Control Policy for Binary Ordered Resources Petri Net Class.

Prevention and avoidance of deadlocks in sensor networks that use the wormhole routing algorithm is an active research domain. There are diverse control policies that will address this problem being our approach a new method. In this paper we present a virtual control policy for the new specialized Petri net subclass called Binary Ordered Resources Petri Net (BORPN). Essentially, it is an ordinary class constructed from various state machines that share unitary resources in a complex form, which allows branching and joining of processes. The reduced structure of this new class gives advantages that allow analysis of the entire system’s behavior, which is a prohibitive task for large systems because of the complexity and routing algorithms.

CTL Model Checking of Web Services Composition based on Open Workflow Nets Modeling

Web services composition (WSC) has an enormous potential for the organizations in the B2B area. In fact, different services collaborate through the exchange of messages to implement complex business processes. BPEL is one of the most used languages to develop such cooperation. However, it has been proved that its use is complex and can require some expertise in XML syntax. Even its graphical representation is not evident to handle. This is why the authors propose to model Web services using oWF-nets, a subclass of Petri nets, and then, to translate them to BPEL. Whilst, a WSC is with added value only if the involved services are compatible. So in this context, across the translation proposed the researchers develop a verification layer of the WSC compatibility. Hence, they propose a framework named D&amp;A4WSC which allows to model the WSC by oWF-nets, to check their compatibility with the model checker NuSMV and to translate them if they are compatible in BPEL processes using the oWFN2BPEL compiler. D&amp;A4WSC permits, furthermore, to formally analyze a BPEL process.