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.     

Performance Analysis of Retrial Queueing Systems Using Generalized Stochastic Petri Nets

We consider retrial queueing systems, in which an arriving customer finding the server busy, may repeat his call after a random duration. The consideration of repeated calls introduces great analytical difficulties. In fact, detailed analytical results exist for some special retrial queueing systems, while for many others, the performance evaluation is limited to numerical algorithms, approximation methods and simulation. Retrial queues have been widely used to model many problems in telephone switching systems, telecommunication and computer networks. In this paper, we show a method of modelling and analysing retrial queueing systems, using the Generalized Stochastic Petri nets (GSPNs).     

非马尔可夫随机Petri网模型的混合状态分析法

讨论具有发射时间任意分布之变迁的随机Petri网的解析问题.定义了随机Petri网的混合状态和混合状态密度,提出混合状态分析法,并给出具有一步转移关系标识下的混合状态密度的递推公式.使非马尔可夫型随机Petri网的分析成为可能.通过算例说明了混合状态分析法在系统性能评估中的应用.     

Energy consumption and execution time estimation of embedded system applications

Embedded systems often have conflicting constraints such as energy and time which considerably harden the design of those systems. In this context, this work proposes a mechanism for supporting design decisions on energy consumption and performance of embedded system applications. In order to depict the practical usability of the proposed methodology, a real case study as well as customized examples are presented. The estimates obtained through the conceived model are 93% close to the respective measures obtained from the real hardware platform.     

Queueing Petri nets with product form solution

The product form results recently published for stochastic Petri nets are combined with the well-known product form results for queueing networks in the model formalism of queueing Petri nets yielding the class of product form queueing Petri nets. This model class includes stochastic Petri nets with product form solution and BCMP queueing networks as special cases. We introduce an arrival theorem for the model class and present an exact aggregation approach extending known approaches from queueing networks.     

On Markov reward modelling with FSPNs

In this paper, fluid stochastic Petri nets (FSPNs) will be used for modelling reward in a performability model. Two variations of a known performability model are presented in order to demonstrate the ability of FSPNs in modelling accumulated rate reward as well as accumulated impulse reward. In the first model two fluid places are used, one of which represents the profit (reward) obtained by operating the system and the other one the buffer that is approximated continuously. In the second model only one fluid place is used, representing the costs (negative reward) arising due to repair of system components. The costs increase continuously at deterministic rate while the system is in state of repair (which is a rate reward in the model). Additional costs incur each time the buffer fails (which is an impulse reward in the model). With a numerical solution algorithm the distribution of the reward and its mean are computed. The accuracy of the numerical algorithm is studied by showing for the first model the impact of the choice of the discretization stepsizes on the obtained solution. Different boundary conditions are discussed for the second model.     

基于服务组合模型结构特征的性能瓶颈定位方案

在满足功能需求的前提下,组合服务的性能是赢得用户的关键,如何发现和消除性能瓶颈则是服务组合面临的重大挑战。面对这个挑战,提出一种基于服务组合模型结构特征的性能瓶颈分析方案。为了确保方案的可行性和有效性,除了对方案技术路线的可行性进行论证,还指出了实现方案需要解决的关键基本问题,即找出服务组合模型的最小结构完备集。提出一种基于随机Petri网的Web服务组合性能分析模型,并给出此模型最小结构完备集的求解和证明过程。最后,通过 一个应用实例说明了方案的有效性。     

准自相似业务流的SPN模型及其评价

该文研究了近似自相似业务的马尔可夫模型,提出了该模型的相位型表达式并据此将模型从离散时间域转换到连续时间域,笔者对一个处理自相似业务到达的简单队列系统进行了研究,比较了系统在实际跟踪的仿真情况下和模型数值分析情况下的性能,模型数值分析用无穷状态随机Petri网实现,研究表明:马尔可夫模型虽能较好地近似自相似性,但仍需进行改进。     

Robust manufacturing system design using multi objective genetic algorithms,Petri nets and Bayesian uncertainty representation

Decisions involving robust manufacturing system configuration design are often costly and involve long term allocation of resources. These decisions typically remain fixed for future planning horizons and failure to design a robust manufacturing system configuration can lead to high production and inventory costs, and lost sales costs. The designers need to find optimal design configurations by evaluating multiple decision variables (such as makespan and WIP) and considering different forms of manufacturing uncertainties (such as uncertainties in processing times and product demand). This paper presents a novel approach using multi objective genetic algorithms (GA), Petri nets and Bayesian model averaging (BMA) for robust design of manufacturing systems. The proposed approach is demonstrated on a manufacturing system configuration design problem to find optimal number of machines in different manufacturing cells for a manufacturing system producing multiple products. The objective function aims at minimizing makespan, mean WIP and number of machines, while considering uncertainties in processing times, equipment failure and repairs, and product demand. The integrated multi objective GA and Petri net based modeling framework coupled with Bayesian methods of uncertainty representation provides a single tool to design, analyze and simulate candidate models while considering distribution model and parameter uncertainties.     

Steady-state performance evaluation of continuous mono-T-semiflow Petri nets

The number of states in discrete event systems can increase exponentially with respect to the size of the system. A way to face this state explosion problem consists of relaxing the system model, for example by converting it to a continuous one. In the scope of Petri nets, the firing of a transition in a continuous Petri net system is done in a real amount. Hence, the marking (state) of the net system becomes a vector of non-negative real numbers. The main contribution of the paper lies in the computation of throughput bounds for continuous Petri net systems with a single T-semiflow. For that purpose, a branch and bound algorithm is designed. Moreover, it can be relaxed and converted into a linear programming problem. Some conditions, under which the system always reaches the computed bounds, are extracted. The results related to the computation of the bounds can be directly applied to a larger class of nets called mono T-semiflow reducible.     

Markov regenerative stochastic Petri nets with general execution policies: supplementary variable analysis and a prototype tool

Stochastic Petri nets (SPNs) with general firing time distributions are considered. Generally timed transitions can have general execution policies: the preemption policy may be preemptive repeat different (prd) or preemptive resume (prs) and the firing time distribution can be marking-dependent. A stationary analysis method covering all possible combinations is presented by means of supplementary variables. The method is implemented in a prototype tool SPNica which is based on Mathematica. The use of the general execution policies is illustrated by a WWW server model.     

Iterative analysis of Markov regenerative models

Conventional algorithms for the steady-state analysis of Markov regenerative models suffer from high computational costs which are caused by densely populated matrices. In this paper, a new algorithm is suggested which avoids computing these matrices explicitly. Instead, a two-stage iteration scheme is used. An extended version of uniformization is applied as a subalgorithm to compute the required transient quantities “on-the fly”. The algorithm is formulated in terms of stochastic Petri nets. A detailed example illustrates the proposed concepts.     

Risk assessment and management for supply chain networks: A case study

The aim of this study is to show how a timed Petri nets framework can be used to model and analyze a supply chain (SC) network which is subject to various risks. The method is illustrated by an industrial case study. We first investigate the disruption factors of the SC network by a failure mode, effects and criticality analysis (FMECA) technique. We then integrate the risk management procedures into design, planning, and performance evaluation process of supply chain networks through Petri net (PN) based simulation. The developed PN model provides an efficient environment for defining uncertainties in the system and evaluating the added value of the risk mitigation actions. The findings of the case study shows that the system performance can be improved using risk management actions and the overall system costs can be reduced by mitigation scenarios.     

Reward model solution methods with impulse and rate rewards: an algorithm and numerical results

Reward models have become an important method for specifying performability models for many types of systems. Many methods have been proposed for solving reward models, but no method has proven itself to be applicable over all system classes and sizes. Furthermore, specification of reward models has usually been done at the state level, which can be extremely cumbersome for realistic models. We describe a method to specify reward models as stochastic activity networks (SANs) with impulse and rate rewards, and a method by which to solve these models via uniformization. The method is an extension of one proposed by de Souza e Silva and Gail in which impulse and rate rewards are specified at the SAN level, and solved in a single model. Furthermore, we propose a new technique for discarding paths in the uniformized process whose contribution to the reward variable is minimal, which greatly reduces the time and space required for a solution. A bound is calculated on the error introduced by this discarding, and its effectiveness is illustrated through the study of the performability and availability of a degradable multi-processor system.     

网格监控体系结构的一种可执行性模型

为网格监控体系结构建立可执行性模型有助于网格监控系统的服务质量提升.因为网格环境的动态性和不可靠性,所以对网格监控体系结构建模时要从可用性和性能两方面综合考虑.讨论系统可用性、响应时间分布、事件丢失概率、公平性等问题.用随机Petri网建立网格监控体系结构的可执行性模型并讨论了模型的应用.网格监控体系结构的系统模型有一个关键特性：事件发布信息和监控事件数据由不同通道传递,在模型中重点关注这一特性.     

An algorithmic approach for analysis of finite-source retrial systems with unreliable servers

This paper aims at presenting an approach for analyzing finite-source retrial systems with servers subject to breakdowns and repairs, using Generalized Stochastic Petri Nets (GSPNs). This high-level formalism allows a simple representation of such systems with different breakdown disciplines. From the GSPN model, a Continuous Time Markov Chain (CTMC) can be automatically derived. However, for multiserver retrial systems with unreliable servers, the models may have a huge state space. Using the GSPN model as a support, we propose an algorithm for directly computing the infinitesimal generator of the CTMC without generating the reachability graph. In addition, we develop the formulas of the main stationary performance and reliability indices, as a function of the number of servers, the size of the customer source and the stationary probabilities. Through numerical examples, we discuss the effect of the system parameters and the breakdown disciplines on performance.     

Integrating predicate transition nets with first order temporal logic in the specification and verification of concurrent systems

This paper presents some results of integrating predicate transition nets with first order temporal logic in the specification and verification of concurrent systems. The intention of this research is to use predicate transition nets as a specification method and to use first order temporal logic as a verification method so that their strengths — the easy comprehension of predicate transition nets and the reasoning power of first order temporal logic can be combined. In this paper, a theoretical relationship between the computation models of these two formalisms is presented; an algorithm for systematically translating a predicate transition net into a corresponding temporal logic system is outlined; and a special temporal refutation proof technique is proposed and illustrated in verifying various concurrent properties of the predicate transition net specification of the five dining philosophers problem.     

Modeling and formal verification of embedded systems based on a Petri net representation

In this paper we concentrate on aspects related to modeling and formal verification of embedded systems. First, we define a formal model of computation for embedded systems based on Petri nets that can capture important features of such systems and allows their representation at different levels of granularity. Our modeling formalism has a well-defined semantics so that it supports a precise representation of the system, the use of formal methods to verify its correctness, and the automation of different tasks along the design process. Second, we propose an approach to the problem of formal verification of embedded systems represented in our modeling formalism. We make use of model checking to prove whether certain properties, expressed as temporal logic formulas, hold with respect to the system model. We introduce a systematic procedure to translate our model into timed automata so that it is possible to use available model checking tools. We propose two strategies for improving the verification efficiency, the first by applying correctness-preserving transformations and the second by exploring the degree of parallelism characteristic to the system. Some examples, including a realistic industrial case, demonstrate the efficiency of our approach on practical applications.