Timed Petri Nets Research Articles

Scheduling of Resource Allocation Systems with Timed Petri Nets: A Survey

Resource allocation systems (RASs) belong to a kind of discrete event system commonly seen in the industry. In such systems, available resources are allocated to concurrently running processes to optimize some performance criteria. Search strategies in the reachability graph (RG) of a timed Petri net (PN) attracted much attention in the past decades to cope with RAS scheduling problems (RSPs), since PNs are very suitable to model and analyze RASs and their RGs fully reflect systems’ behavior. However, there has been no existing related survey and review paper till now. In this work, we present a tutorial and comprehensive literature survey of RG-based RSP methods. Many state-of-the-art RG-based RAS scheduling strategies are reviewed and summarized. First, we present a framework of RSPs and classify RSPs and their PNs in terms of resource usage and net structures. The differences and relations among the PNs are also given. Then, we introduce timed PN construction methods for RSPs and scheduling objectives and search strategies for RG-based RSPs. Next, we summarize different heuristic functions adopted in a frequently used A * search to solve RG-based RSPs. Finally, we discuss some important future research directions and open issues.

On tick automata for distributed timed DESs with synchronisations and minimal time constraints

This paper is about the representations of distributed timed discrete event systems (DESs) with synchronisation events and minimal time constraints. A subclass of Timed Petri nets with specific time semantics is first recalled as a reference model for the considered systems. Such a model is reformulated according to modular tick automata with minimal times that behave like logical automata. A synchronous composition of such automata is then defined and the result of this composition is a new tick automaton that has the same timed language as the original Petri net, which is not the case for the earlier model of tick automata with constant (exact) times.

Design and verification of pipelined circuits with Timed Petri Nets

Design and verification of pipelined circuits with Timed Petri Nets

Situation Representation and Strategic Reasoning Method of Hybrid Game System Based on Modified Hybrid Stochastic Timed Petri Net

The hybrid game system suffers from the constantly time-varying environment and the heterogeneous behaviors of decision-making subjects (players) when the hybrid situation emerges, bringing significant challenges to situation representation and strategic reasoning. With the help of the powerful modeling ability of hybrid systems for situation representation and reasoning in network topology, we propose the six-tuple hybrid game model based on modified hybrid stochastic timed Petri nets (M-HSTPN) to reveal the internal cross-level operating mechanisms. Then, we carry out the nonlinear mapping relationship between the hybrid game elements and the symbols of M-HSTPN. Taking the intelligent drones combat problem as an example, we compare differential game, event game, and proposed hybrid game scheme in pursuit–evasion trajectories, action timing, bombing sequence, and formation. The proposed M-HSTPN-based hybrid game scheme can effectively describe the constantly changing pursuit–evasion trajectories and select the action timing to reduce the energy consumption of interception and the target-missing quantity. Furthermore, the proposed hybrid game scheme can reason the optimal bombing sequence and narrow the scope of optimal formation through enriching payoff evaluation indexes with the current hybrid situation compared with the event game scheme and differential game scheme.

An Efficient Method to Assess Resilience and Robustness Properties of a Class of Cyber-Physical Production Systems

Widely available real-time data from the sensors of IoT infrastructure enables and increases the adoption and use of cyber-physical production systems (CPPS) to provide enterprise-wide status information to promptly respond to business opportunities through real-time monitoring, supervision and control of resources and activities in production systems. In CPPS, the failures of resources are uncertainties that are inevitable and unexpected. The failures of resources usually lead to chaos on the shop floor, delayed production activities and overdue orders. This calls for the development of an effective method to deal with failures in CPPS. An effective method to assess the impacts of failures on performance and create an alternative plan to mitigate the impacts is important. Robustness, which refers to the ability to tolerate perturbations, and resilience, which refers to the capability to recover from perturbations, are two concepts to evaluate the influence of resource failures on CPPS. In this study, we developed a method to evaluate the influence of resource failures on CPPS based on the concepts of robustness and resilience. We modeled CPPS by a class of discrete timed Petri nets. A model of CPPS consists of asymmetrically decomposed models of tasks. The dynamics of tasks can be represented by spatial-temporal networks (STN) with a similar but asymmetrical structure. A joint spatial-temporal networks (JSTN) model constructed based on the fusion of the asymmetrical STNs is used to develop an efficient algorithm to optimize performance. We characterized robustness and resilience as properties of CPPS with respect to the failures of resources. We analyzed the complexity of the proposed method and conducted experiments to illustrate the scalability and efficiency of the proposed method.

Security Analysis for Distributed IoT-Based Industrial Automation

Internet of Things (IoT) technologies enable development of reconfigurable manufacturing systems—a new generation of modularized industrial equipment suitable for highly customized manufacturing. Sequential control in these systems is largely based on discrete events, whereas their formal execution semantics is specified as control interpreted Petri nets (CIPN). Despite industry-wide use of programming languages based on the CIPN formalism, formal verification of such control applications in the presence of adversarial activity is not supported. Consequently, in this article, we introduce security-aware modeling and verification techniques for CIPN-based sequential control applications. Specifically, we show how CIPN models of networked industrial IoT controllers can be transformed into time Petri net (TPN)-based models and composed with plant and security-aware channel models in order to enable system-level verification of safety properties in the presence of network-based attacks. Additionally, we introduce realistic channel-specific attack models that capture adversarial behavior using nondeterminism. Moreover, we show how verification results can be utilized to introduce security patches and facilitate design of attack detectors that improve system resiliency and enable satisfaction of critical safety properties. Finally, we evaluate our framework on an industrial case study. Note to Practitioners—Our main goal is to provide formal security guarantees for distributed sequential controllers. Specifically, we target smart automation controllers geared toward Industrial IoT applications that are typically programed in C/C++ and are running applications originally designed in, for example, GRAFCET (IEC 60848)/SFC (IEC 61131-3) automation programming languages. Since existing tools for the design of distributed automation do not support system-level verification of relevant safety properties, we show how security-aware transceiver and communication models can be developed and composed with distributed controller models. Then, we show how existing tools for verification of time Petri nets can be used to verify relevant properties including safety and liveness of the distributed automation system in the presence of network-based attacks. To provide an end-to-end analysis as well as security patching, results of our analysis can be used to deploy suitable firmware updates during the stage when executable code for target controllers (e.g., in C/C++) is generated based on GRAFCET/SFC control models. We also show that security guarantees can be improved as the relevant safety/liveness properties can be verified after corresponding security patches are deployed. Finally, we show applicability of our framework on a realistic distributed pneumatic manipulator.

Symbolic Scheduling of Robotic Cellular Manufacturing Systems With Timed Petri Nets

To reduce the computational burden in the scheduling of robotic cellular manufacturing (RCM) systems based on Petri nets’ (PNs) reachability graphs, existing methods mainly focus on the relaxation of a graph search algorithm at the cost of schedule optimality. Different from that, this article presents a method that accelerates the search process by using symbolic representations and manipulations. The proposed method uses binary decision diagrams (BDDs) to functionally represent and evolve discrete and place-timed PNs of RCM systems and then schedule them with a symbolic A* search to find an optimal solution with minimal makespan. It uses compact BDD structures to represent sets of states, instead of individual states, and then performs efficient Boolean manipulations to evolve the net and search for a system schedule. A heuristic function and its Boolean implementations for the symbolic A* search are developed. The admissibility of the proposed heuristic function is proven, which guarantees the optimality of the obtained schedule. The computational time is thus reduced without sacrificing the schedule optimality. Finally, experiments are carried out to show the effectiveness and efficiency of the presented method.

A Clustering Approach to Approximate the Timed Reachability Graph for a Class of Time Petri Nets

Timed extended reachability graphs (TERG) of time Petri nets abstract the temporal specifications and represent the feasible trajectories under the earliest firing policy. One drawback of such graphs is the rapid increase in the number of states with respect to time specifications. For this reason, approximations of TERG that remove some states have been studied in recent works. In this article, we improve the approximation of a TERG. New objects—called vertices—are defined to manipulate the time constraints and algorithms are proposed to cluster nearby vertices. A metric based on the time constraints is defined for this purpose and a cluster TERG of reduced size is obtained.

Fluid Net Models: From Behavioral Properties to Structural Objects

Increasing the production in manufacturing systems is one of the main demands in modern systems. The naive approach that this goal can be achieved when more or faster resources are used is not always valid. In fact, the complex interactions among system’s elements may lead to paradoxical behaviors; for example, using faster machines could reduce the equilibrium throughput (number of part fabricated per unit time in steady state) of the system, or even worse, block all system activities, reducing it to zero. This work leverages the concepts about fluidization and analysis techniques used in Timed Continuous Petri nets (TCPN) presented in earlier works to study the behavior of the equilibrium throughput when more/faster machines are used. Herein, we illustrate how discontinuities induced bifurcations of the equilibrium throughput are due to the existence of paths that can increase/decrease the marking of certain subnets. In particular, if paths gaining/losing tokens are fired without a particular balance, then the equilibrium throughput exhibits discontinuities since the equilibrium marking loses hyperbolicity. Moreover, these discontinuities imply other undesired throughput behaviors; for example, the existence of non-monotonicities of the equilibrium throughput (when more/faster resources are used in the system, its equilibrium throughput is reduced). The discontinuities together with a homothecy property are used to explain non-monotonicities in the equilibrium throughput. A relevant aspect is that these undesired system behaviors appear when the net has structural objects named problematic configurations that are associated with certain subnets in which there are no P-semiflows. Although the number of these configurations increase exponentially in the size of the net, some reduction rules are introduced to remove configurations, while the problematic ones are kept (or can be recovered) in the reduced net. This saves computation time in the analysis and, more importantly, provides useful insights about the root of undesired behaviors. This work focus on systems that can be modeled with fluid (or continuous) mono T-semiflow Timed Continuous Petri nets. Even if under certain constraints, they are capable of capturing many characteristics of modern systems, such as interleaving of cooperation and competition.

The ORIS tool

ORIS is a tool for quantitative modeling and evaluation of concurrent systems with non-Markovian durations. It provides a Graphical User Interface (GUI) for model specification as Stochastic Time Petri Nets (STPNs), validation by interactive simulation, and evaluation by several techniques, computing instantaneous and cumulative rewards. It also provides an open-source Java Application Programming Interface (API) to automate the workflow, and it can be used as a toolkit for derivation and evaluation of STPNs in model driven engineering. As distinguishing features, ORIS implements transient and steady-state analysis of STPNs with underlying Markov Regenerative Process (MRP), and transient analysis of STPNs with underlying Generalized Semi- Markov Process (GSMP). It also implements nondeterministic analysis of Time Petri Nets (TPNs), simulation of STPNs, and solution methods for Continuous-Time Markov Chains (CTMCs) and MRPs with at most one non-exponential timer in each state. The well-engineered software architecture of ORIS supports agile implementation of new STPN features, new modeling formalisms, and new analysis methods.

Supervisory Control of Timed Discrete-Event Systems With Logical and Temporal Specifications

A novel framework is introduced for the supervisory control (SC) of timed discrete event systems based on Time Petri nets. The method encompasses both logical (markings to reach or avoid) and temporal specifications (arrival and departure times in specific markings). It relies on the construction of a partial forward reachability graph of the modified state class graph type and the formulation of integer linear programming problems to establish suitable firing time intervals (FTIs) for the controllable transitions. For each enabled controllable transition, the SC algorithm provides the largest FTI that that the specifications are met, irrespectively of the firing times of the uncontrollable transitions.

A Graph Transformation Approach for Modeling UML Diagrams

This article proposes an approach for transforming UML Statecharts, Sequence Diagrams and activity Diagrams to Time Petri Nets with Action Duration (DTPN) using graph transformation. By this transformation the author aims to bridge the gap between semi-formal models generated by UML and formal models DTPN. UML is considered to be the standardized language for modelling and describing systems behaviors for analysis. In other hand, DTPN models are tools for the specification and performance analysis of distributed and concurrent systems. However, the proposed approach allows to generate automatically a visual modeling tool for DTPN. The cost of building a visual modeling tool from scratch is prohibitive. Meta-modeling approach is useful to deal with this problem since it allows the modeling of the formalisms themselves, by means of graph grammars. The meta-modeling tool AToM3 is used.

A Theoretical Foundation for Context-Aware Cyber-Physical Production Systems

The complex workflows and interactions between heterogeneous entities in Cyber-Physical Production Systems (CPPS) call for the use of context-aware computing technology to operate effectively and meet the order requirements in a timely manner. In addition to the objective to meet the order due date, due to resource contention between production processes, CPPS may enter undesirable states. In undesirable states, all or part of the production activities are in waiting states or blocked situation due to improper allocation of resources. The capability to meet the order due date and prevent the system from entering an undesirable state poses challenges in the development of context-aware computing applications for CPPS. In this study, we formulate two situation awareness problems, including a Deadline Awareness Problem and a Future States Awareness Problem to address the above issues. In our previous study, we found that Discrete Timed Petri Nets provide an effective tool to model and analyze CPPS. In this paper, we present a relevant theory to support the operation of CPPS by extending the Discrete Timed Petri Nets to lay a foundation for developing context-aware applications of CPPS with deadline awareness and future states awareness capabilities. We illustrate the theory developed in this study by an example and conduct experiments to verify the computational feasibility of the proposed method.

Scenic Tourist Intelligent Shunt Based on Timed Petri Nets

As the scale of tourism continues to expand, the huge passenger flow during peak tourism periods has brought tremendous pressure and challenges to the management of popular scenic spots. During peak periods of scenic spots, tourists are overloaded in popular attractions and the distribution of tourists is unbalanced. Based on the dynamic tour of tourists in scenic spots, a tourist triage model and four triage strategies are proposed for timed Petri nets. Tourist satisfaction and variance of scenic area load rate are taken as evaluation criteria, the advantages and disadvantages of different strategies are analyzed through simulation experiments, and appropriate strategies are proposed for different evaluation criteria, it has practical reference value for tourist diversion management of scenic spots.

Optimal Control of Timed Petri Nets Under Temporal Logic Constraints with Generalized Mutual Exclusion

Optimal Control of Timed Petri Nets Under Temporal Logic Constraints with Generalized Mutual Exclusion

Crucial States Estimation in Radio Block Center Handover Using Petri Nets With Unobservable Transitions

The radio block center (RBC) is one of the most essential ground systems in a high-speed train control system both in Europe and in China. The RBC handover procedure is an important function of RBC, which affects the transport efficiency, reliability and safety of railways. Analysis of crucial states in the RBC handover procedure is helpful to determine whether there are potential risks in the procedure, and to locate the fault in time when a fault occurs. In this paper, we study a property, called C-detectability, of the RBC handover. This property has been defined in discrete event systems and requires that the crucial states can be determined uniquely by observing the system output. Taking the RBC handover procedure in the Chinese train control system level 3 (CTCS-3) as an example, we first model the RBC handover procedure using Petri nets, which are a graphical and mathematical modeling tool to formalize the behavior of discrete event systems. Then, based on the notion of basis reachability graph, an efficient approach is used to check C-detectability of the Petri net modeling the handover procedure. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The railway system is a safety-critical system. System safety is essential to the railway system. It is necessary to provide formal methods to model the system and analyze its properties. The motivation of this work is to present a general modeling framework of railway systems for the crucial states estimation. The crucial states estimation of the railway system has not been discussed yet in the literature despite its importance due to its relationship with the safety of the railway. In particular, estimating the crucial states in the railway system helps to determine whether there are potential risks in the system. In this paper, we investigate a new state estimation property (C-detectability), which is related to fault diagnosis, marking estimation and fault identification. Future research will consider the use of timed Petri nets to increase the modeling power and to allow the performance analysis of the railway system.

Cost Problems for Parametric Time Petri Nets*

We investigate the problem of parameter synthesis for time Petri nets with a cost variable that evolves both continuously with time, and discretely when firing transitions. More precisely, parameters are rational symbolic constants used for time constraints on the firing of transitions and we want to synthesise all their values such that some marking is reachable, with a cost that is either minimal or simply less than a given bound. We first prove that the mere existence of values for the parameters such that the latter property holds is undecidable. We nonetheless provide symbolic semi-algorithms for the two synthesis problems and we prove them both sound and complete when they terminate. We also show how to modify them for the case when parameter values are integers. Finally, we prove that these modified versions terminate if parameters are bounded. While this is to be expected since there are now only a finite number of possible parameter values, our algorithms are symbolic and thus avoid an explicit enumeration of all those values. Furthermore, the results are symbolic constraints representing finite unions of convex polyhedra that are easily amenable to further analysis through linear programming. We finally report on the implementation of the approach in Romeo, a software tool for the analysis of time Petri nets.

Piecewise Affine Dynamical Models of Petri Nets – Application to Emergency Call Centers*

We study timed Petri nets, with preselection and priority routing. We represent the behavior of these systems by piecewise affine dynamical systems. We use tools from the theory of nonexpansive mappings to analyze these systems. We establish an equivalence theorem between priority-free fluid timed Petri nets and semi-Markov decision processes, from which we derive the convergence to a periodic regime and the polynomial-time computability of the throughput. More generally, we develop an approach inspired by tropical geometry, characterizing the congestion phases as the cells of a polyhedral complex. We illustrate these results by a current application to the performance evaluation of emergency call centers in the Paris area. We show that priorities can lead to a paradoxical behavior: in certain regimes, the throughput of the most prioritary task may not be an increasing function of the resources.

Dynamic Slicing of Time Petri Net Based on MTL Property

The time Petri net (TPN) is a powerful tool for modeling, simulating, and verifying real-time systems. Unfortunately, the state spaces of the time Petri net grow exponentially due to the complexity of real-time systems. The enormous size of the state spaces could also cause state explosion problems in model checking. This paper proposes an alternative dynamic slicing algorithm written as a metric temporal logic (MTL) formula to reduce the size of the time Petri net model by considering specific criteria. Our algorithm proposes an alternative dependency graph representing the global firing time interval of the transition to remove the transitions that are never fired and do not affect the state space. The dynamic slicing algorithm involves the initial marking and the properties of the target metric temporal logic formula. Unlike the unsliced time Petri net, the result preserves all necessary execution paths for the model checking of a particular metric temporal logic formula. Therefore, model checking can generate sufficient state space to conduct verification equivalent to the unsliced time Petri net but may take less time, including if the unsliced time Petri net causes state space explosion.

Critical pairs based diagnosability analysis of timed fault in Time Petri Nets

Time Petri Nets are a suitable tool for the modeling of timed Discrete Event Systems. This paper is about the diagnosability of single timed fault in Time Petri Nets. To check the existence of critical pairs, an abstraction of the systems infinite behaviours, called path, is presented as a set of observable constraints associated with a particular sequence of transitions. Properties on the set of solutions of a partition of this abstraction are then provided to check the diagnosability of the timed fault.