Correctness verification and performance analysis of real-time systems using stochastic preemptive time Petri nets

Time Petri nets describe the state of a timed system through a marking and a set of clocks. If clocks take values in a dense domain, state space analysis must rely on equivalence classes. These support verification of logical sequencing and quantitative timing of events, but they are hard to be enriched with a stochastic characterization of nondeterminism necessary for performance and dependability evaluation. Casting clocks into a discrete domain overcomes the limitation, but raises a number of problems deriving from the intertwined effects of concurrency and timing. We present a discrete-time variant of time Petri nets, called stochastic preemptive time Petri nets, which provides a unified solution for the above problems through the adoption of a maximal step semantics in which the logical location evolves through the concurrent firing of transition sets. We propose an analysis technique, which integrates the enumeration of a succession relation among sets of timed states with the calculus of their probability distribution. This enables a joint approach to the evaluation of performance and dependability indexes as well as to the verification of sequencing and timeliness correctness. Expressive and analysis capabilities of the model are demonstrated with reference to a real-time digital control system.     

Classification and evaluation of timed running schemas for workflow based on process mining

The system running logs of a workflow contain much information about the behavior and logical structure between activities. In this paper, a mining approach is proposed to discover the structural and temporal model for a workflow from its timed running logs. The mining results are represented in the formalized form of Petri nets extended with two timing factors that allows validation or verification the actual behaviors, especially the temporal constraints between activities. According to the reachability graph of the extended Petri net model mined, all running schemas of a workflow can be generated, which defines the temporal constraints between running activities. By calculating the earliest and latest start time of each activity, the earliest starting and latest existing time of each state in the running schema can be determined. Based on the temporal relations between the timing factors of each running state, the running schemas can be classified into six classes. The effects of the six classes of running schemas on the implementation of the whole workflow are evaluated so as to obtain the best one that can ensure the workflow is finished in the shortest time. The standards for the ideal, reliable and favorable running schemas and their existence conditions are discussed, which can be used to evaluate the running logs and control the future running of a workflow.     

Compositional semantics of a real-time prototyping language

The formal semantics of a prototyping language for hard real-time systems, PSDL, is given. PSDL provides a data flow notation augmented by application-orientation timing and control constraints to describe a system as a hierarchy of networks of processing units communicating via data streams. The semantics of PSDL are defined in terms of algebraic high-level Petri nets. This formalism combines algebraic specifications of abstract data types with process and concurrency concepts of Petri nets. Its data abstraction facilities are used to define the meaning of PSDL data types, while high-level Petri nets serve to model the casual and timing behavior of a system. The net model exposes potential concurrency of computation and makes all synchronization needs implied by timing and control constraints explicit and precise. Time is treated as state of clocks, and clocks are modeled as ordinary system components. The net semantics provides the basis for applying analysis techniques and semantic tools available for high-level Petri nets     

On the Regularity of Petri Net Languages

Petri nets are known to be useful for modeling concurrent systems. Once modeled by a Petri net, the behavior of a concurrent system can be characterized by the set of all executable transition sequences, which in turn can be viewed as a language over an alphabet of symbols corresponding to the transitions of the underlying Petri net. In this paper, we study the language issue of Petri nets from a computational complexity viewpoint. We analyze the complexity of theregularity problem(i.e., the problem of determining whether a given Petri net defines an irregular language or not) for a variety of classes of Petri nets, includingconflict-free,trap-circuit,normal,sinkless,extended trap-circuit,BPP, andgeneralPetri nets. (Extended trap-circuit Petri nets are trap-circuit Petri nets augmented with a specific type ofcircuits.) As it turns out, the complexities for these Petri net classes range from NL (nondeterministic logspace), PTIME (polynomial time), and NP (nondeterministic polynomial time), to EXPSPACE (exponential space). In the process of deriving the complexity results, we develop adecomposition approachwhich, we feel, is interesting in its own right, and might have other applications to the analysis of Petri nets as well. As a by-product, an NP upper bound of the reachability problem for the class of extended trap-circuit Petri nets (which properly contains that of trap-circuit (and hence, conflict-free) and BPP-nets, and is incomparable with that of normal and sinkless Petri nets) is derived.     

时间约束Petri网的可达性分析研究

时间约束Ｐｅｔｒｉ风是具有广义时间约束的一类Ｐｅｔｒｉ网。目前有关ＴＣＰＮ状态可达性的研究仅局限于一些较简单的网,本文通过对ＴＣＰＮ的进一步研究,给出了更一般的状态可达性分析方法,并讨论了一般拓扑结构ＴＣＰＮ的可调度分析。     

Oris: a tool for modeling, verification and evaluation of real-time systems

Oris is a tool for qualitative verification and quantitative evaluation of reactive timed systems, which supports modeling and analysis of various classes of timed extensions of Petri Nets. As most characterizing features, Oris implements symbolic state space analysis of preemptive Time Petri Nets, which enable schedulability analysis of real-time systems running under priority preemptive scheduling; and stochastic Time Petri Nets, which enable an integrated approach to qualitative verification and quantitative evaluation. In this paper, we present the current version of the tool and we illustrate its application to two different case studies in the areas of qualitative verification and quantitative evaluation, respectively.     

Compositional schedulability analysis of real-time systems using time Petri nets

This paper presents an approach to the schedulability analysis of real-time systems modeled in time Petri nets by separating timing properties from other behavioral properties. The analysis of behavioral properties is conducted based on the reachability graph of the underlying Petri net, whereas timing constraints are checked in terms of absolute and relative firing domains. If a specific task execution is schedulable, we calculate the time span of the task execution, and pinpoint nonschedulable transitions to help adjust timing constraints. A technique for compositional timing analysis is also proposed to deal with complex task sequences, which not only improves efficiency but also facilitates the discussion of the reachability issue with regard to schedulability. We identified a class of well-structured time Petri nets such that their reachability can be easily analyzed.     

Timed high-level nets

Petri nets have been widely used for modeling and analyzing concurrent systems. Several reasons contribute to their success: the simplicity of the model, the immediate graphical representation, the easy modeling of asynchronous aspects, the possibility of reasoning about important properties such as reachability, liveness, boundedness. However, the original model fails in representing two important features: complex functional aspects, such as conditions which rule the flow of control, and time. Due to that, two different classes of extensions of Petri nets have been proposed: high-level nets and timed Petri nets. High-level nets allow the representation of functional aspects in full details, but do not provide a means for representing time; on the other hand, timed Petri nets have been thought for time representation, but they do not provide a means for representing detailed functinal aspects. Thus, these two important aspects cannot be mastered together. In particular, it is difficult to express relationships between time and functional aspects.This paper investigates the relationships between high-level nets and timed Petri nets, thus extending a first set of results published in a previous paper, where a unifying Petri net based model for time representation has been proposed. It first recalls how time can be represented in a Petri net extension called ER nets, and assesses its generality. It then investigates the relationships of ER nets with the best known high-level nets. In particular it shows the overall equivalence of ER nets, Colored Petri nets and Predicate/Transition nets, and extends the mechanism for time representation introduced in ER nets to both Colored Petri nets and Predicate/Transition nets. It also shows that these models cannot be simplified without significantly constraining the timing aspects that can be modeled.     

延迟时间Petri网的验证分析

延迟时间Petri网（Delay Time Petri Nets，DTPN）是一类重要的时间扩展Petri网系统，解决了其他时间扩展Petri网（如时间Petri网）在保存时间约束时所面临的困难。可调度验证的目的是验证工作流模型时间约束的合理性，对流程实例的时间可达性进行仿真。提出一种基于DTPN的时间约束工作流验证分析方法。给出了DTPN的相关定义，并结合工作流控制结构描述了变迁可触发的时间条件；提出了DTPN触发点的概念以及基于此的验证分析算法；简要分析了DTPN的特性。DTPN的研究丰富完善了现有时间Petri网体系，具有积极的意义。     

模拟实时系统的点区间优先级时间Petri网与TCTL验证

时间Petri网为实时系统提供了一种形式化的建模方法, 时间计算树逻辑(TCTL)为描述实时系统与时间相关的设计需求提供了一种逻辑化的表达方式, 因此基于时间Petri网的TCTL模型检测广泛应用于实时系统的正确性验证.然而对于一些涉及优先级的实时系统, 例如多核多任务实时系统, 这里不仅需要考虑任务之间的时间约束还要考虑任务执行的优先级以及引入优先级带来的抢占式调度问题, 致使相应的建模和分析变得更加困难.为此, 本文提出了点区间优先级时间Petri网, 通过在时间Petri网上定义变迁发生的优先级以及变迁的可挂起性, 从而可以模拟实时系统的抢占式调度机制, 即首先高优先级的任务抢占低优先级的任务所占用的资源, 导致后者被中断, 然后前者执行完毕后释放资源, 最后后者再次获得资源从中断的地方恢复.本文通过点区间优先级时间Petri网来模拟多核多任务实时系统, 使用TCTL来描述它们的设计需求, 设计了相应的模型检测算法, 开发了相应的模型检测器以验证它们的正确性.我们通过一个实例来说明我们的模型和方法的有效性.     

Compositional validation of time-critical systems usingcommunicating time Petri nets

An extended Petri net model which considers modular partitioning along with timing restrictions and environment models is presented. Module constructs permit the specification of a complex system as a set of message passing modules with the timing semantics of time Petri nets. The state space of each individual module can be separately enumerated and assessed under the assumption of a partial specification of the intended module operation environment. State spaces of individual modules can be recursively integrated, to permit the assessment of module clusters and of the overall model, and to check the satisfaction of the assumptions made in the separate analysis of elementary component modules. In the intermediate stages between subsequent integration steps, the state spaces of module and module clusters can be projected onto reduced representations concealing local events that are not essential to the purposes of the analysis. The joint use of incremental enumeration and intermediate concealment of local events allows for a flexible management of state explosion, and permits a scalable approach to the validation of complex systems     

Coloured Petri Nets and CPN Tools for modelling and validation of concurrent systems

Coloured Petri Nets (CPNs) is a language for the modelling and validation of systems in which concurrency, communication, and synchronisation play a major role. Coloured Petri Nets is a discrete-event modelling language combining Petri nets with the functional programming language Standard ML. Petri nets provide the foundation of the graphical notation and the basic primitives for modelling concurrency, communication, and synchronisation. Standard ML provides the primitives for the definition of data types, describing data manipulation, and for creating compact and parameterisable models. A CPN model of a system is an executable model representing the states of the system and the events (transitions) that can cause the system to change state. The CPN language makes it possible to organise a model as a set of modules, and it includes a time concept for representing the time taken to execute events in the modelled system. CPN Tools is an industrial-strength computer tool for constructing and analysing CPN models. Using CPN Tools, it is possible to investigate the behaviour of the modelled system using simulation, to verify properties by means of state space methods and model checking, and to conduct simulation-based performance analysis. User interaction with CPN Tools is based on direct manipulation of the graphical representation of the CPN model using interaction techniques, such as tool palettes and marking menus. A license for CPN Tools can be obtained free of charge, also for commercial use.     

A unified high-level Petri net formalism for time-critical systems

The authors introduce a high-level Petri net formalism-environment/relationship (ER) nets-which can be used to specify control, function, and timing issues. In particular, they discuss how time can be modeled via ER nets by providing a suitable axiomatization. They use ER nets to define a time notation that is shown to generalize most time Petri-net-based formalisms which appeared in the literature. They discuss how ER nets can be used in a specification support environment for a time-critical system and, in particular, the kind of analysis supported     

Deciding a class of path formulas for conflict-free petri nets

The aim of this paper is to develop a unified approach for deriving complexity results for problems concerning conflict-free Petri nets. To do so, we first define a class of formulas for paths in Petri nets. We then show that answering the satisfiability problem for conflict-free Petri nets is tantamount to solving a system of linear inequalities (which is known to be in P). Since a wide spectrum of Petri net problems (including various fairness-related problems) can be reduced to the satisfiability problem in a straightforward manner, our approach offers an umbrella under which many Petri net problems for conflict-free Petri nets can be shown to be solvable in polynomial time. As a side-product, our analysis provides evidence as to why detecting unboundedness for conflict-free Petri nets is easier (provided P ≠ NP) than for normal and sinkless Petri nets (which are two classes that properly contain conflict-free Petri nets). A preliminary version was presented at the 14th International Conference on Application and Theory of Petri Nets, Chicago, IL, USA, June 1993.     

基于时延Petri网的DEDS的代数模型

用一种新的算子建立了结构无竞争时延Petri网和资源分配型时延Petri网的动态方 程,为该类系统的分析和控制提供了研究工具.     

随机时间Petri网综述

随机时间Petri网作为一种灵活有力的建模机制,被广泛应用于计算机和通信系统的性能与可靠性评价。综述了随机时间Petri网各主要予类的发展历程与研究现状。首先介绍了随机时间Petrl网的一些基本概念,然后回顾了广义随机Petri网,简要总结了处理状态空间爆炸问题的主要途径,讨论了非马尔科夫随机Petri网及其分析技术,最后详细分析了流体随机Petri网。     

基于UML的CPN模型在软件测试中的应用

UML在被工业界广泛接受的同时也成为学术界遵循的一种标准建模语言。许多面向对象软件测试的研究都围绕UML模型开展。但UML模型属于半形式化模型,往往无法自动生成测试用例。将UML模型与Petri网相结合,能够弥补其数学支持的不足。文章给出了基于UML的CPN模型的测试框架和一个从UML模型构造CPN模型的算法,并根据该方法给出了自动化支持工具U2CPN的类图结构。     

Efficient Verification of Parallel Real–Time Systems

This paper presents an efficient model checking algorithm for one–safe time Petri nets and a timed temporal logic. The approach is based on the idea of (1) using only differences of timing variables to be able to construct a finite representation of the set of all reachable states and (2) further reducing the size of this representation by exploiting the concurrency in the net. This reduction of the state space is possible, because the considered linear–time temporal logic is stuttering invariant. The firings of transitions are only partially ordered by causality and a given formula; therefore the order of firings of independent transitions is irrelevant, and only one of several equivalent interleavings has to be generated for the evaluation of the given formula. In this paper the theory of timing verification with time Petri nets and temporal logic is presented, a concrete model checking algorithm is developed and proved to be correct, and some experimental results demonstrating the efficiency of the method are given.     

Reduction rules for time Petri nets

The goal of net reduction is to increase the effectiveness of Petri-netbased real-time program analysis. Petri-net-based analysis, like all reachabilitybased methods, suffers from the state explosion problem. Petri net reduction is one key method for combating this problem. In this paper, we extend several rules for the reduction of ordinary Petri nets to work with time Petri nets. We introduce a notion of equivalence among time Petri nets, and prove that our reduction rules yield equivalent nets. This notion of equivalence guarantees that crucial timing and concurrency properties are preserved.     

Timing constraint Petri nets and their application toschedulability analysis of real-time system specifications

We present timing constraint Petri nets (or TCPN's for short) and describe how to use them to model a real-time system specification and determine whether the specification is schedulable with respect to imposed timing constraints. The strength of TCPN's over other time-related Petri nets is in the modeling and analysis of conflict structures. Schedulability analysis is conducted in three steps: specification modeling, reachability simulation, and timing analysis. First, we model a real-time system by transforming its system specification along with its imposed timing constraints into a TCPN; we call this net N_s. Then we simulate the reachability of N_s to verify whether a marking, M_n, is reachable from an initial marking, M_o. It is important to note that a reachable marking in Petri nets is not necessarily reachable in TCPN's due to the imposed timing constraints, Therefore, in the timing analysis step, a reachable marking M_n, found in the reachability simulation step is analyzed to verify whether M_n, is reachable with the timing constraints. M_n is said to be reachable in the TCPN's if and only if we can find at least one firing sequence σ so that all transitions in σ are strongly schedulable with respect to M_o under the timing constraints. If such M_n can be found, then we can assert that the specification is schedulable under the imposed timing constraints, otherwise the system specification needs to be modified or the timing constraints need to be relaxed. We also present a synthesis method for determining the best approximation of the earliest fire beginning time (EFBT) and the latest fire ending time (LFET) of each strongly schedulable transition