Embedded processes in stochastic Petri nets

Embedded discrete time processes are used to study a class of SPNs (stochastic Petri nets) which have a closed-form equilibrium distribution. These SPNs have probabilistic output bags, colored tokens, and alternating periods of arbitrarily distributing enabling and firing times (periods of time between transitions becoming enabled and absorption of tokens and between transitions absorbing tokens and depositing them in output places, respectively). In addition, an aggregation procedure is proposed which, in certain nets, not only reduces a complex SPN to a much simpler skeleton SPN but also obtains results for the skeleton SPN with are exact marginal distributions for the original SPN     

Perturbation analysis of stochastic Petri nets

This paper addresses the sensitivity analysis of stochastic Petri nets (SPNs) using simulations. The goal is to evaluate the derivatives of performance measures with respect to timing parameters. To characterize the underlying stochastic processes of SPNs, we use a generalized semi-Markov process (GSMP) representation and propose a new representation, called GSMP*, which differs from GSMP in the routing mechanism. By using existing results on perturbation analysis of GSMPs and by extending them to GSMP*, unbiased sensitivity estimators are obtained for SPNs simulated under a GSMP or GSMP* framework. Most importantly, we prove that only one simulation run is needed for evaluating both the performance measures and their derivatives for a class of free-choice nets simulated under a GSMP framework and for any free-choice net simulated under a GSMP* framework     

Implementing compositionality for stochastic Petri nets

An implementation of compositionality for stochastic well-formed nets (SWN) and, consequently, for generalized stochastic Petri nets (GSPN) has been recently included in the GreatSPN tool. Given two SWNs and a labelling function for places and transitions, it is possible to produce a third one as a superposition of places and transitions of equal label. Colour domains and arc functions of SWNs have to be treated appropriately. The main motivation for this extension was the need to evaluate a library of fault-tolerant “mechanisms” that have been recently defined, and are now under implementation, in a European project called TIRAN. The goal of the TIRAN project is to devise a portable software solution to the problem of fault tolerance in embedded systems, while the goal of the evaluation is to provide evidence of the efficacy of the proposed solution. Modularity being a natural “must” for the project, we have tried to reflect it in our modelling effort. In this paper, we discuss the implementation of compositionality in the GreatSPN tool, and we show its use for the modelling of one of the TIRAN mechanisms, the so-called local voter. Published online: 24 August 2001     

Estimation methods for nonregenerative stochastic Petri nets

When a computer, manufacturing, telecommunication, or transportation system is modeled as a stochastic Petri net (SPN), many long-run performance characteristics of interest can be expressed as time-average limits of the associated marking process. For nets with generally-distributed firing times, such limits often cannot be computed analytically or numerically, but must be estimated using simulation. Previous work on estimation methods for SPNs has focused on the case in which there exists a sequence of regeneration points for the marking process of the net, so that point estimates and confidence intervals for time-average limits can be obtained using the regenerative method for analysis of simulation output. This paper is concerned with SPNs for which the regenerative method is not applicable. We provide conditions on the clock-setting distributions and new-marking probabilities of an SPN under which time-average limits are well defined and the output process of the simulation obeys a multivariate functional central limit theorem. It then follows from results of Glynn and Iglehart (1990) that methods based on standardized time series can be used to obtain strongly consistent point estimates and asymptotic confidence intervals for time-average limits. In particular, the method of batch means is applicable. Moreover, the methods of Munoz and Glynn can be used to obtain point estimates and confidence intervals for ratios of time-average limits. We illustrate our results using an SPN model of an interactive video-on-demand system     

Discrete-event simulation of fluid stochastic Petri nets

The purpose of this paper is to describe a method for the simulation of the recently introduced fluid stochastic Petri nets. Since such nets result in rather complex system of partial differential equations, numerical solution becomes a formidable task. Because of a mixed (discrete and continuous) state space, simulative solution also poses some interesting challenges, which are addressed in the paper     

一种随机着色Petri网及模型的性能分析

针对随机Petri网（SPN）在系统性能分析时,其状态空间随着系统规模增大而指数性增长,造成求解稳定状态概率的复杂性的不足,提出了一种随机着色Petri网（SCPN）。分析了它的有界性和可达性,证明了它同构于一个一维连续时间的马尔可夫链;同时,也分析了随机着色Petri网用于建模和系统性能定量分析的方法。     

Sensitivity analysis and optimization of stochastic Petri nets

In this paper we define the notion of controlled stochastic Petri net (CSPN), which is a stochastic Petri net with controlled parameters and performance indices. Specifically, transition times and/or conflict resolution rules can depend on controlled parameters and transition times can have arbitrary distribution functions. A method for computing statistical estimates of performance indices and their gradients (sensitivities) with respect to controlled parameters is described. This method, which needs only one simulation of a CSPN, is considerably superior to conventional finite differences both in terms of precision and required amount of simulation and is based on likelihood ratio/score function approach, other possibilities based on extensions of infinitesimal perturbation analysis are outlined. These gradient estimates are used in stochastic optimization algorithms to obtain the optimal value of the aggregated performance function of the CSPN. A combined optimization and simulation tool is developed which includes approaches to the gradient estimation mentioned above. The numerical experiments presented in this paper confirm the efficiency of the proposed techniques.     

Comparison properties of stochastic decision free Petri nets

Stochastic comparison problems arising in the analysis of stochastic decision free Petri nets are treated. The analysis is based on the evolution equations satisfied by firing times. Various structural properties are obtained, including the association of the firing times and their stochastic and convex monotonicity with respect to the holding time sequences, the initial marking, and the topology. The association and the stochastic monotonicity properties are extended to the counters using an inversion relation. It is also proven that the counters and the throughput are stochastically concave in the initial marking, provided that the holding times are independent, identically distributed, and belong to a subclass of the log-concave distribution functions that are introduced. Various bounds for the asymptotic cycle time and the throughput are then derived from these stochastic ordering results. Comparative results are presented for the marking and response times     

Product form solution for generalized stochastic Petri nets

In this paper, we show the structural characteristics that a particular class of generalized stochastic Petri nets must exhibit in order for their stationary probabilities to have a product-form. Sufficient conditions for identifying such a class are derived and proven with the development of a series of transformations that can also be used to construct, for any GSPN of the class, an equivalent SPN. These resulting SPNs represent the structures that can be analyzed with standard methods for product-form SPNs to establish whether the original GSPNs have product-form solutions and to compute their performance indices with effective approaches based on computationally efficient algorithms that avoid the generation of their underlying state spaces.     

生化网络的随机Petri网建模与分析

细胞的行为是随机性的,学习细胞中的随机性有助于理解细胞的组织,设计和进化。建立、确认和分析随机的生化网络模型是当前计算系统生物学领域的一个重要研究主题。当前,标准的Petri网模型已经成为生化网络模拟和定性分析的有力工具。尝试使用随机Petri网对生化网络进行建模与分析,简单描述了随机Petri网理论对标准Petri网的扩充,通过对二聚作用和肌动蛋白这两个典型例子的建模与演化模拟,介绍、论证了随机Petri网理论的新应用。     

Efficient steady-state analysis of second-order fluid stochastic Petri nets

This paper presents an efficient solution technique for the steady-state analysis of the second-order stochastic fluid model underlying a second-order fluid stochastic Petri net (FSPN) with constant flow and transition rates, and a single bounded fluid place. The new solution technique is an extension of existing solution techniques developed for (first-order) Fluid Models; the solution algorithm uses upwind semi-discretization and Matrix Geometric techniques to efficiently compute the steady-state probabilities of the mixed discrete and continuous state space of the model. The effectiveness of our technique is proven first on a simple producer–consumer second-order FSPN model and then on a complex example taken from the literature where the analysis of the completion time distribution and the packet loss probability of short-lived TCP connections is investigated through the decomposition of a model for the whole system into several (simpler) sub-models; the interaction between the different sub-models is handled by iterating their solution until the complete model solution converges according to a fixed point algorithm. The introduction of a second-order FSPN in a fixed point iteration scheme has been made possible thanks to the efficiency of the proposed solution technique.     

随机Petri网理论在GSM-R通信系统中的仿真应用

列车和无线闭塞中心在多数情况下通信处于正常状态,但在实际通信过程中可能出现以下几种引发故障因素：传输数据出错、连接丢失、越区切换、端到端传输延时。应用仿真主要考虑高速列车在移动闭塞区间条件下GSM-R无线通信系统的可靠性,结合随机Petri网理论对GSM-R铁路无线通信系统故障恢复事件进行建模和分析,给出了随机Petri网表示的列车与无线闭塞中心通信的故障恢复模型,并采用TimeNET仿真工具对GSM-R通信系统的可靠性进行分析得出相应结论。     

Product-form and stochastic Petri nets: a structural approach

Stochastic Petri nets (SPNs) with product-form solution are nets for which there is an analytic expression of the steady-state probabilities with respect to place markings, as it is the case for product-form queueing networks with respect to queue lengths. The most general kind of SPNs with product-form solution introduced by Coleman et al. (and denoted here by -nets) suffers a serious drawback: the existence of such a solution depends on the values of the transition rates. Thus since their introduction, it is an open question to characterize -nets with product-form solution for any values of the rates. A partial characterization has been obtained by Henderson et al. However, this characterization does not hold for every initial marking and it is expressed in terms of the reachability graph. In this paper, we obtain a purely structural characterization of -nets for which a product-form solution exists for any value of probabilistic parameters of the SPN and for any initial marking. This structural characterization leads to the definition of -nets (Stochastic Parametric Product-form Petri nets). We also design a polynomial time (with respect to the size of the net structure) algorithm to check whether a SPN is a -net. Then, we study qualitative properties of -nets and -nets, the non-stochastic versions of -nets and -nets: we establish two results on the complexity bounds for the liveness and the reachability problems, which are central problems in Petri nets theory. This set of results complements previous studies on these classes of nets and improves the applicability of product-form solutions for SPNs.     

基于Gillespie算法的生化随机Petri网演化分析

Petri网既有严格的数学表达方式,又有直观的图形描述形式,是当前生物化学系统模拟和定性分析的一个有力工具。作为标准Petri网的一个重要扩展,随机Petri网使得可以定量地分析生物化学系统。建立了两个典型的生物化学系统的随机Petri网模型：Schlogl系统和转录调控系统,并使用Gillespie随机仿真算法模拟了这些生化随机Petri网模型的状态演化。     

随机Petri网可视化软件的设计与实现

针对随机Petri网建模设计与分析的复杂性问题,通过面向对象方法在VisualC++环境下利用MFC开发了可视化的随机Petri网建模和分析工具SPNSim。利用广度优先思想,设计了可达图的生成算法,给出了随机Petri网模型的编辑器,用户可以直观方便地进行随机Petri网编辑,同时还能进行模型分析,得到可达图、稳态概率等随机Petri网特性和系统性能指标。通过对一个实例的建模与分析,结果验证了软件工具SPNSim的正确性和可用性。     

Performance analysis of stochastic timed Petri nets using linearprogramming approach

Stochastic timed Petri nets are a useful tool in the performance analysis of concurrent systems such as parallel computers, communication networks and flexible manufacturing systems. In general, performance measures of stochastic timed Petri nets are difficult to obtain for practical problems due to their sizes. In this paper, we provide a method to efficiently compute upper and lower bounds for the throughputs and mean token numbers for a large class of stochastic timed Petri nets. Our approach is based on uniformization technique and linear programming     

Prediction of business process durations using non-Markovian stochastic Petri nets

Companies need to efficiently manage their business processes to deliver products and services in time. Therefore, they monitor the progress of individual cases to be able to timely detect undesired deviations and to react accordingly. For example, companies can decide to speed up process execution by raising alerts or by using additional resources, which increases the chance that a certain deadline or service level agreement can be met. Central to such process control is accurate prediction of the remaining time of a case and the estimation of the risk of missing a deadline.To achieve this goal, we use a specific kind of stochastic Petri nets that can capture arbitrary duration distributions. Thereby, we are able to achieve higher prediction accuracy than related approaches. Further, we evaluate the approach in comparison to state of the art approaches and show the potential of exploiting a so far untapped source of information: the elapsed time since the last observed event. Real-world case studies in the financial and logistics domain serve to illustrate and evaluate the approach presented.     

基于随机Petri网的机群系统可信赖性研究

随机Petri网(SPN)是一种灵活有力的建模工具,在可信赖性研究方面已形成较为成熟的理论体系.采用SPN网对机群系统前端分配器及后端服务器进行建模,通过在系统建模时对前端分配器给予必要的修复保证,给出整个机群系统的SPN模型,在模型分析的基础上给出了机群系统的可信赖性参数,并对模型进行了仿真分析.仿真结果表明,在对关键系统前端分配器给予修复保证后,系统模型的稳态可用性将大大提高.