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45
Efficient DescriptorVector Multiplications in Stochastic Automata Networks
, 1996
"... This paper examines numerical issues in computing solutions to networks of stochastic automata. It is wellknown that when the matrices that represent the automata contain only constant values, the cost of performing the operation basic to all iterative solution methods, that of matrixvector multi ..."
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Cited by 119 (20 self)
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This paper examines numerical issues in computing solutions to networks of stochastic automata. It is wellknown that when the matrices that represent the automata contain only constant values, the cost of performing the operation basic to all iterative solution methods, that of matrixvector multiply, is given by ae N = N Y i=1 n i \Theta N X i=1 n i ; where n i is the number of states in the i th automaton and N is the number of automata in the network. We introduce the concept of a generalized tensor product and prove a number of lemmas concerning this product. The result of these lemmas allows us to show that this relatively small number of operations is sufficient in many practical cases of interest in which the automata contain functional and not simply constant transitions. Furthermore, we show how the automata should be ordered to achieve this.
Process Algebras for Quantitative Analysis
, 2005
"... In the 1980s process algebras became widely accepted formalisms for describing and analysing concurrency. Extensions of the formalisms, incorporating some aspects of systems which had previously been abstracted, were developed for a number of different purposes. In the area of performance analysis m ..."
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Cited by 47 (6 self)
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In the 1980s process algebras became widely accepted formalisms for describing and analysing concurrency. Extensions of the formalisms, incorporating some aspects of systems which had previously been abstracted, were developed for a number of different purposes. In the area of performance analysis models must quantify both timing and probability. Addressing this domain led to the formulation of stochastic process algebras. In this paper we give a brief overview of stochastic process algebras and the problems which motivated them, before focussing on their relationship with the underlying mathematical stochastic process. This is presented in the context of the PEPA formalism.
A fluid analysis framework for a Markovian process algebra
, 2010
"... Markovian process algebras, such as PEPA and stochastic πcalculus, bring a powerful compositional approach to the performance modelling of complex systems. However, the models generated by process algebras, as with other interleaving formalisms, are susceptible to the state space explosion problem. ..."
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Cited by 41 (27 self)
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Markovian process algebras, such as PEPA and stochastic πcalculus, bring a powerful compositional approach to the performance modelling of complex systems. However, the models generated by process algebras, as with other interleaving formalisms, are susceptible to the state space explosion problem. Models with only a modest number of process algebra terms can easily generate so many states that they are all but intractable to traditional solution techniques. Previous work aimed at addressing this problem has presented a fluidflow approximation allowing the analysis of systems which would otherwise be inaccessible. To achieve this, systems of ordinary differential equations describing the fluid flow of the stochastic process algebra model are generated informally. In this paper, we show formally that for a large class of models, this fluidflow analysis can be directly derived from the stochastic process algebra model as an approximation to the mean number of component types within the model. The nature of the fluid approximation is derived and characterised by direct comparison with the Chapman–Kolmogorov equations underlying the Markov model. Furthermore, we compare the fluid approximation with the exact solution using stochastic simulation and we are able to demonstrate that it is a very accurate approximation in many cases. For the first time, we also show how to extend these techniques naturally to generate systems of differential equations approximating higher order moments of model component counts. These are important performance characteristics for estimating, for instance, the variance of the component counts. This is very necessary if we are to understand how precise the fluidflow calculation is, in a given modelling situation.
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.
Performance Analysis of Stochastic Timed Petri Nets using Linear Programming Approach
 IEEE Transactions on Software Engineering
, 1995
"... Stochastic timed Petri nets are a useful tool in 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 problems of practical sizes. ..."
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Cited by 16 (0 self)
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Stochastic timed Petri nets are a useful tool in 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 problems of practical sizes. In this paper, we provide a method to compute efficiently upper and lower bounds for the throughputs and mean token numbers in general Markovian timed Petri nets. Our approach is based on uniformization technique and linear programming.
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...
Reversed processes, product forms and a nonproduct form
 LINEAR ALGEBRA AND ITS APPLICATIONS
"... ..."
A general result for deriving productform solutions of markovian models
 In Proc. of First Joint WOSP/SIPEW Int. Conf. on Perf. Eng
"... In this paper we provide a general method to derive productform solutions for stochastic models. We take inspiration from the Reversed Compound Agent Theorem [14] and we provide a different formulation using labeled automata, a generalization which encompasses a bigger class of productform solutions ..."
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Cited by 11 (8 self)
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In this paper we provide a general method to derive productform solutions for stochastic models. We take inspiration from the Reversed Compound Agent Theorem [14] and we provide a different formulation using labeled automata, a generalization which encompasses a bigger class of productform solutions, and a new proof based on the solution of the system of global balance equations. We show that our result may have practical applications in the performance evaluation of complex software and hardware architectures and can be the base for the development of new analysis tools or the extension of existing ones.