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Proxel-Based Simulation of Stochastic Petri Nets.
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This paper discusses the analysis of stochastic Petri nets using the proxel-based simulation method. The paradigm of the proxel ("probability element") was recently introduced in order to provide a new algorithmic approach to analysing discrete-state stochastic models such as are represented by stochastic Petri nets (SPNs) or queueing systems. Proxel-based simulation is not related to either of the standard simulation approaches: it is in no way analogous to discrete-event simulation, and, although it is based on the model's underlying stochastic process and makes use of supplementary variables, it does not require the use of differential equations. Instead, the proxels trace the movement of probability from one state of the model to another using discretized time steps. Since stochastic Petri nets are a powerful and widespread tool for modelling stochastic processes, we are interested in finding out how the proxel-based method applies to them. The formal foundations of the analysis of SPNs with the use of the proxel-based method are the subject of this paper.
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... The method has been successfully applied to fault-tree analysis [3], stochastic Petri nets (SPN) [4] and the analysis of project schedules [5]. Besides an extension for immediate transitions in SPN's [6], recently a new model description framework was developed [1] that exploits the power of the proxel-based method beyond SPN's or fault trees. ...
... By leaving out phase-type transitions when specifying the model, the algorithm behaves exactly like an ordinary proxel algorithm. The necessary modifications can be adapted to the more advanced versions described in [6] or [1]. ...
The analysis of discrete stochastic models such as generally dis- tributed stochastic Petri nets can be done by using state space- based methods. They describe the behavior of a model by a Markov chain that can be solved mathematically. Formerly this ap- proach often had to be discarded as unfeasible due to high memory and runtime costs. The recently developed Proxel-based algorithm is a state space-based simulation method, and has already performed well in several application fields. Experiments suggest, that the selec- tive use of discrete phase approximations could further improve the method, because they can often represent infinite support dis- tribution functions with considerably fewer Markov chain states than proxels. By replacing certain on-the-fly proxel approxima- tions by predetermined phase-type approximations, the total run- time and memory requirement of the simulation method could be drastically reduced for some test models. An efficient algorithm for the approximation of discrete phase-type distributions based on common optimization methods was recently introduced. The formal inclusion of discrete phases into the proxel paradigm is another step toward a practically usable state space- based simulation method. Our hope is that such a combination of both approaches will lead to a competitive simulation algorithm.
... The Proxel-based simulation algorithm is a state space-based simulation method and has been introduced by Horton [26]. It was then further improved [48,50], specified and analyzed [47]. Proxels are based on the method of supplementary variables, and turn a user model (e.g. ...
... Assuming a time step of C Some Experiment Details ∆t = 0.1, which was also taken for producing the training sequences, one can already see that the trained transition probabilities representing the Markovian distributions are quite accurate. 50 Baum-Welch iterations. The leftmost shape was the result of the training process using five traces of length T = 300, 000, the middle shape resulted from traces of length T = 600, 000, and the rightmost one from traces of the maximum length T = 1, 410, 100. ...
... Dabei gibt es verschiedene Herangehensweisen. Die drei meist genutzten sind: 1 [6,7,8,9] ...
... [5,6] Das Evaluierungsproblem versucht die Frage zu beantworten, wie wahrscheinlich bestimme Ereignisse sind -z. B [8,9,7] Bei Expanded-Hidden-Markov-Modellen existieren hingegen bereits Algorithmen zum Berechnen der Forward-und Backward-Wahrscheinlichkeiten. ...
... These reliability and repair distribution functions of components associated with basic events, along with the fault tree structure, are inputs to the proxel-based simulation, which is then used to calculate system's reliability measures, in form of transient complete solutions. The proxel-based simulation is well-known for its ability to cope with stiff models, as fault models are typically (Lazarova-Molnar and Horton 2003;Lazarova-Molnar 2005). Namely, the proxel-based simulation has proved more accurate and less sensitive to rare events than Monte Carlo, and also efficient once the improvement of extrapolation of solutions has been implemented. ...
... In this section we present the algorithm for the proxel-based analysis of general models. The proxel simulation algorithm was introduced in [1] and developed further in [12] [5]. Essentially, the proxel-based method builds a discrete-time Markov chain [6] on the fly, which approximates the behaviour of the model at discrete time steps, using the method of supplementary variables [7]. ...
The motivation for this paper is to raise the recently introduced proxel-based simulation method to a higher level by defining its own model description framework, which at the same time would allow us to enhance the description and analysis of discrete sto- chastic models beyond the potential of stochastic Petri nets. The Proxel-based method is designed for transient analysis of discrete stochastic models, which are commonly described using stochastic Petri nets (SPNs). The approach is based on the method of supplementary variables, meaning it performs the analysis in a deterministic manner. It, however, works in a purely algorithmic style, without employing partial differential equations. Experi- ments and applications have so far shown the method to be promis- ing, especially in analysing classes of models which are known to be difficult or problematic to be deterministically analysed using standard methods (such as Markov chains and partial differential equations). Infinite state space models, such as queuing systems with unbounded queues (with generally distributed arrivals and processing times) where servers can fail, and the number of fail- ures and servers' age determine the parameters of the distribution function of the processing time, is one of those problematic cases. As mentioned, up to now, a starting point for the proxel-based analysis was the Petri net model of the system to be simulated. The models, however, had to be adapted, using hard coding, to trans- form them into an appropriate input for the proxel-based simulator, mainly for efficiency reasons, but also for allowing properties that were not supported by SPNs. The transformation was implicitly the model description approach that we present and formalise in this paper. We believe that the modelling framework that we describe here will be able to exploit all or most of the beneficial properties of the proxel-based method and describe the model in a way that is di- rectly analysable by the proxel-based simulator. The framework is based upon Petri net features and modifies and extends them ac- cording to the properties and needs of the proxel-based method. Our approach is supported and demonstrated by experiments and characteristic examples, as well as comparison to the formal- ism of SPNs.
... The method is deterministic and easy to implement because it can be thought of as an easy iteration loop that distributes the continuous flow of probability between the states for discrete steps in time. The method has shown to be applicable for the numerical simulation of Stochastic Petri Nets (SPN) [14]. It has dramatically reduced the analysis time of warranty models of automotive industry and, for this reason, was implemented in an industrial software tool for the analyses of reliability, safety, and costs [15]. ...
Proxel-based simulation is a state-space based method for the transient so- lution of discrete stochastic models with general distributions. The method is very promising for the simulation and evaluation of rare events, even in sti models. It computes the probability of model states and events determinis- tically. Contrary to rare event simulation techniques like, e.g., Importance Sampling or Splitting, it requires no further knowledge from the user and is free from sample variances, allowing for an easy handling even by simulation beginners. Further on, proxel-based simulation can be accelerated by using dierent time steps sizes. We show how to implement dierent time steps and how to choose their sizes. Two small examples are given to illustrate the application.
... Rare events and the thereby reached system states are guaranteed to be considered. The method has shown to be applicable for the numerical simulation of Stochastic Petri Nets (SPN) [4]. The method has dramatically reduced the analysis time of warranty models of automotive industry and, for this reason, was implemented in an industrial software tool for the analyses of reliability, safety, and costs [5]. ...
State space-based simulation method of general stochastic continuous-time models suffers from its sensitivity to state space-explosion. In this paper, we address this problem by the introduction of variable time steps for the proxel-based simulation method. We show how the complexity of the simulation is reduced by expressing rare event proba-bility using fewer computational units. It is shown by means of experi-mental results that for models with rare events the modified proxel-based simulation algorithm computes solutions with comparable accuracy in less time.
KEYWORDS Proxel, supplementary variables, stochastic Petri net, state space-based simulation. ABSTRACT The Proxel method is a state space-based approach to the simulation of discrete stochastic models. It implements the method of supplementary variables in an algorithmic, rather than mathematical way, without using partial differential equations and is able to solve a wide class of stochastic models. The approach has proven to be very competitive with discrete-event simulation for a class of models that is used in reliability modelling. In this paper we report on our experience with the application of the proxel-based method to a problem provided by the DaimlerChrysler Corporation. The model that portrays the problem describes the correlation between system reliability and warranty costs for automobiles. Until recently, this model was analysed using discrete-event simulation, which resulted in very time-consuming computations. By contrast, the proxel-based method achieved comparable accuracy within a few seconds of computation time.
At DaimlerChrysler, the reliability and safety of automobiles are analysed using sim-ulation. While discrete-event simulation can be used for these models, this approach may not deliver accurate results in acceptable time. Markov chain modelling for a state-space-based analysis is too complex for the user. Proxel-based simulation has shown to combine the best of both worlds and even to be more efficient for small models. In this paper we present a general-purpose proxel-based simulator that has been implemented into an analysis tool of DaimlerChrysler. Furthermore, a caching strategy is presented which reduces the computational effort of proxel-based simulation.
This paper is concerned with the simulation analysis of discrete-state stochastic models such as queueing systems or stochastic Petri nets, in which arbitrary probability distributions may be assigned to the activities. The analysis is performed on the state space using a numerical approach, rather than the usual discrete-event simulation at the model level. A new computational paradigm, the so-called Proxel (probability element) is introduced, which allows an approximation to the continuous stochastic process of the Petri net to be developed which does not require the use of differential equations. This proxel-based computational model directly yields a simulation algorithm which is readily understood and implemented. Simulation experiments are used to illustrate the behaviour of the method and to discuss the advantages and disadvantages of the method compared to the alternatives.
The paradigm of the proxel ("probability element") was recently introduced in order to provide a new algorithmic approach to analysing discrete-state stochastic models such as are represented by stochastic Petri nets or queueing systems. Proxel-based simulation is not related to either of the standard simulation approaches: it is in no way analogous to discrete-event simulation, and, although it is based on the model's underlying stochastic process, it does not require the use of differential equations. Instead, the proxels dynamically trace the movement of probability from one state of the model to another using discretized time steps. Since the method is new, it still needs to be studied and experiments must be performed in order to fully understand its properties and behaviour. Results of such experiments will be presented and analysed in this paper. 1 Goals of the Paper Proxels are a new method for the simulation of discrete-state stochastic models. The goal of this paper is to present the method, and to complement theoretical research into the proxel-based method by experiments which study its behaviour. When a new algorithm is proposed, its usefulness must be determined, in particular in comparison with existing (and competing) methods. The properties of interest in this paper are the complexity of the algorithm (in terms of both memory and computation time) and the accuracy of the simulation results. Our experiments are designed to determine the effects of the principal algorithmic and model parameters, such as the size of the time step, the method of proxel storage, and the number of discrete states and competing activities in the model. A comparison with the standard approach, discrete- event simulation, will also be made. This initial study is limited to small, simple models.