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31
Recent Developments in NonMarkovian Stochastic Petri Nets
, 1998
"... Analytical modeling plays a crucial role in the analysis and design of computer systems. Stochastic Petri Nets represent a powerful paradigm, widely used for such modeling in the context of dependability, performance and performability. Many structural and stochastic extensions have been proposed in ..."
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Cited by 22 (4 self)
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Analytical modeling plays a crucial role in the analysis and design of computer systems. Stochastic Petri Nets represent a powerful paradigm, widely used for such modeling in the context of dependability, performance and performability. Many structural and stochastic extensions have been proposed in recent years to increase their modeling power, or their capability to handle large systems. This paper reviews recent developments by providing the theoretical background and the possible areas of application. Markovian Petri nets are first considered together with very well established extensions known as Generalized Stochastic Petri nets and Stochastic Reward Nets. Key ideas for coping with large state spaces are then discussed. The challenging area of nonMarkovian Petri nets is considered, and the related analysis techniques are surveyed together with the detailed elaboration of an example. Finally new models based on Continuous or Fluid Stochastic Petri Nets are briefly discussed.
A Syntactical Analysis of Reversible PEPA Models
 University of Verona
, 1998
"... Product form solutions have played an important role in the development of performance modelling and in particular in the popularity of queueing network models. The notions of reversibility and quasireversibility underpin many product form results in queueing theory. Stochastic process algebras (SPA ..."
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Cited by 16 (12 self)
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Product form solutions have played an important role in the development of performance modelling and in particular in the popularity of queueing network models. The notions of reversibility and quasireversibility underpin many product form results in queueing theory. Stochastic process algebras (SPAs) support a compositional approach to performance modelling without severe restriction on the form of the underlying Markov chain. The textual nature of SPAs makes them ideal for large scale tool support for both the structural analysis and numerical solution of models. Clearly, therefore, there is substantial potential and great advantage in developing product form solutions for classes of SPA models. In this paper we identify a class of PEPA components that exhibit reversibility and show how these components can be combined. Although developed here in the context of PEPA the results presented can be easily generalised to any of the other stochastic process algebra languages. keywords: Ma...
Exploiting Structure in Solution: Decomposing Composed Models
 Proceedings of 6th International Workshop on Process Algebra and Performance Modelling
, 1998
"... Since their introduction nearly ten years ago, compositionality has been reported as one of the major attractions of stochastic process algebras. The benefits that compositionality provides for model construction are readily apparent and have been demonstrated in numerous case studies. Early researc ..."
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Cited by 16 (3 self)
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Since their introduction nearly ten years ago, compositionality has been reported as one of the major attractions of stochastic process algebras. The benefits that compositionality provides for model construction are readily apparent and have been demonstrated in numerous case studies. Early research on the compositionality of the languages focussed on how the inherent structure could be used, in conjunction with equivalence relations, for model simplification and aggregation. In this paper we consider how far we have been able to take advantage of compositionality when it comes to solving the Markov process underlying a stochastic process algebra model and outline directions for future work in order for current results to be fully exploited. 1 Introduction Stochastic process algebras (SPA) were first proposed as a tool for performance and dependability modelling in 1989 [24]. At that time there was already a plethora of techniques for constructing performance models so the introducti...
Stochastic automata networks: product forms and iterative solutions
 INRIA Research Report
, 1996
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A Class of PEPA Models Exhibiting Product Form Solution over Submodels
, 1998
"... The advantages of the compositional structure within PEPA for model construction and simplification have already been demonstrated. In this paper we show that for some PEPA models this structure may also be used to advantage during the solution of the model. Several papers offering product form solu ..."
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Cited by 7 (5 self)
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The advantages of the compositional structure within PEPA for model construction and simplification have already been demonstrated. In this paper we show that for some PEPA models this structure may also be used to advantage during the solution of the model. Several papers offering product form solutions of stochastic Petri nets have been published during the last ten years. In a recent paper [1], Boucherie showed that some of these solutions were a special case of a simple exclusion mechanism for the product process of a collection of Markov chains. The results presented in this paper take advantage of his observation. Moreover, we show that PEPA models that generate such processes may be readily identified and show how the product form solution may be obtained. Although developed here in the context of PEPA the results presented can be easily generalised to any of the other stochastic process algebra languages. 1
Bulò. A general algorithm to compute the steadystate solution of productform cooperating Markov chains
 In Proc. of MASCOTS 2009
, 2009
"... Abstract—In the last few years several new results about productform solutions of stochastic models have been formulated. In particular, the Reversed Compound Agent Theorem (RCAT) and its extensions play a pivotal role in the characterization of cooperating stochastic models in productform. Alth ..."
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Cited by 6 (3 self)
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Abstract—In the last few years several new results about productform solutions of stochastic models have been formulated. In particular, the Reversed Compound Agent Theorem (RCAT) and its extensions play a pivotal role in the characterization of cooperating stochastic models in productform. Although these results have been used to prove several wellknown theorems (e.g., Jackson queueing network and Gnetwork solutions) as well as novel ones, to the best of our knowledge, an automatic tool to derive the productform solution (if present) of a generic cooperation among a set of stochastic processes, is not yet developed. In this paper we address the problem of solving the nonlinear system of equations that arises from the application of RCAT. We present an iterative algorithm that is the base of a software tool currently under development. We illustrate the algorithm, discuss the convergence and the complexity, compare it with previous algorithms defined for the analysis of the Jackson networks and the Gnetworks. Several tests have been conducted involving the solutions of a (arbitrary) large number of cooperating processes in productform by RCAT. I.
Deficiency zero Petri nets and product form
 Petri Nets 2009, volume 5606 of LNCS
, 2009
"... Abstract. Consider a Markovian Petri net with race policy. The marking process has a “product form ” stationary distribution if the probability of viewing a given marking can be decomposed as the product over places of terms depending only on the local marking. First we observe that the Deficiency Z ..."
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Cited by 6 (1 self)
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Abstract. Consider a Markovian Petri net with race policy. The marking process has a “product form ” stationary distribution if the probability of viewing a given marking can be decomposed as the product over places of terms depending only on the local marking. First we observe that the Deficiency Zero Theorem of Feinberg, developped for chemical reaction networks, provides a structural and simple sufficient condition for the existence of a product form. In view of this, we study the classical subclass of freechoice nets. Roughly, we show that the only such Petri nets having a product form are the state machines which can alternatively be viewed as Jackson networks. 1
Graphs and Stochastic Automata Networks
, 1995
"... We show how some graph theoretical arguments may be used to reduce the complexity of the computation of the steadystate distribution of Markov chain. We consider the directed graph associated to a Markov chain derived from a Stochastic Automata Network (SAN). The structural properties of the automa ..."
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Cited by 6 (4 self)
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We show how some graph theoretical arguments may be used to reduce the complexity of the computation of the steadystate distribution of Markov chain. We consider the directed graph associated to a Markov chain derived from a Stochastic Automata Network (SAN). The structural properties of the automata are used to establish new various results. First, we establish the complexity of the resolution for Stochastic Automata Networks with a sparse matrix representation of the automata. This results are used to compare simple SAN (i.e. without functions) with methods which generates a sparse representation of Markov chains (i.e. Markovian Petri Nets for instance) on some examples. Then, we show how to apply state reduction techniques on a chain associated to a SAN. We present an algorithm to solve the steadystate equations and we prove its complexity. Finally, we extend our algorithm to allow the semi parametric analysis of Stochastic Automata Networks. 1 Introduction The complexity of the...
Embedded ProductForm Queueing Networks and the Improvement of Performance Bounds for Petri Net Systems
 Performance Evaluation
, 1993
"... This paper addresses the computation of upper bounds for the steadystate throughput of stochastic Petri net systems with immediate and generally distributed timed transitions. It is achieved through the use of a kind of decomposition of the whole net system. Results are obtained deeply bridging sto ..."
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Cited by 4 (3 self)
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This paper addresses the computation of upper bounds for the steadystate throughput of stochastic Petri net systems with immediate and generally distributed timed transitions. It is achieved through the use of a kind of decomposition of the whole net system. Results are obtained deeply bridging stochastic Petri net theory to untimed Petri net and queueing network theories. Previous results are improved by considering some embedded productform queueing networks (generated by the support of some left annullers of the incidence matrix of the net). The obtained results for the case of live and bounded free choice systems are of special interest. In this case, the subnets generated by the minimal left annullers of the incidence matrix always have a topology of productform closed monoclass queueing networks. Keywords: Stochastic Petri net systems, throughput bounds, embedded productform queueing networks. 1 Introduction