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55
Probabilistic Symbolic Model Checking with PRISM: A Hybrid Approach
 International Journal on Software Tools for Technology Transfer (STTT
, 2002
"... In this paper we introduce PRISM, a probabilistic model checker, and describe the ecient symbolic techniques we have developed during its implementation. PRISM is a tool for analysing probabilistic systems. It supports three models: discretetime Markov chains, continuoustime Markov chains and ..."
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Cited by 203 (32 self)
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In this paper we introduce PRISM, a probabilistic model checker, and describe the ecient symbolic techniques we have developed during its implementation. PRISM is a tool for analysing probabilistic systems. It supports three models: discretetime Markov chains, continuoustime Markov chains and Markov decision processes. Analysis is performed through model checking speci cations in the probabilistic temporal logics PCTL and CSL. Motivated by the success of model checkers such as SMV, which use BDDs (binary decision diagrams), we have developed an implementation of PCTL and CSL model checking based on MTBDDs (multiterminal BDDs) and BDDs. Existing work in this direction has been hindered by the generally poor performance of MTBDDbased numerical computation, which is often substantially slower than explicit methods using sparse matrices. We present a novel hybrid technique which combines aspects of symbolic and explicit approaches to overcome these performance problems. For typical examples, we achieve orders of magnitude speedup compared to MTBDDs and are able to almost match the speed of sparse matrices whilst maintaining considerable space savings.
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 125 (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.
Compositional Markovian modelling using a process algebra
 Numerical Solution of Markov Chains
, 1995
"... We introduce a stochastic process algebra, PEPA, as a highlevel modelling paradigm for continuous time Markov chains (CTMC). Process algebras are mathematical theories which model concurrent systems by their algebra and provide apparatus for reasoning about the structure and behaviour of the model ..."
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Cited by 56 (15 self)
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We introduce a stochastic process algebra, PEPA, as a highlevel modelling paradigm for continuous time Markov chains (CTMC). Process algebras are mathematical theories which model concurrent systems by their algebra and provide apparatus for reasoning about the structure and behaviour of the model. Recent extensions of these algebras, associating random variables with actions, make the models also amenable to Markovian analysis. A compositional structure is inherent in the PEPA language. As well as the clear advantages that this offers for model construction, we demonstrate how this compositionality may be exploited to reduce the state space of the CTMC. This leads to an exact aggregation based on lumpability. Moreover this technique, taking advantage of symmetries within the system, may be formally defined in terms of the PEPA description of the model. An equivalence relation, strong equivalence, developed as a process algebra bisimulation relation, is used to partition the derivation graph. 1
SMART: Simulation and Markovian Analyzer for Reliability and Timing
, 1996
"... SMART is a new tool designed to allow various highlevel stochastic modeling formalisms (such as stochastic Petri nets and queueing networks) to be described in a uniform environment and solved using a variety of solution techniques, including numerical methods and simulation. Since SMART is intende ..."
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Cited by 39 (13 self)
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SMART is a new tool designed to allow various highlevel stochastic modeling formalisms (such as stochastic Petri nets and queueing networks) to be described in a uniform environment and solved using a variety of solution techniques, including numerical methods and simulation. Since SMART is intended as a research tool, it is written in a modular way that permits the easy integration of new solution algorithms. I. SMART Language Models are described to SMART using a stronglytyped, declarative language. The three basic predefined types for the objects defined in SMART are: ffl bool: true or false. ffl int: integer values. ffl real: real values (machinedependent precision). Composite types can be defined using the concepts of: ffl sets: collection of homogeneous objects. ffl arrays: multidimensional data structures of homogeneous objects indexed by the elements of a set. ffl aggregates: analogous to the Pascal "record". A type can be further modified by the following natures, w...
Symbolic model checking for probabilistic processes using MTBDDs and the Kronecker representation
 In Tools and Algorithms for the Analysis and Construction of Systems, LNCS 1785
, 2000
"... Abstract. This paper reports on experimental results with symbolic model checking of probabilistic processes based on MultiTerminal Binary Decision Diagrams (MTBDDs). We consider concurrent probabilistic systems as models; these allow nondeterministic choice between probability distributions and ar ..."
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Cited by 35 (2 self)
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Abstract. This paper reports on experimental results with symbolic model checking of probabilistic processes based on MultiTerminal Binary Decision Diagrams (MTBDDs). We consider concurrent probabilistic systems as models; these allow nondeterministic choice between probability distributions and are particularly well suited to modelling distributed systems with probabilistic behaviour, e.g. randomized consensus algorithms and probabilistic failures. As a specification formalism we use the probabilistic branchingtime temporal logic PBTL which allows one to express properties such as “under any scheduling of nondeterministic choices, the probability of φ holding until ψ is true is at least 0.78/at most 0.04 ”. We adapt the Kronecker representation of (Plateau 1985), which yields a very compact MTBDD encoding of the system. We implement an experimental model checker using the CUDD package and demonstrate that model construction and reachabilitybased model checking is possible in a matter of seconds for certain classes of systems consisting of up to 10 30 states. 1
Syntax, Semantics, Equivalences, and Axioms for MTIPP
 in Proc. of the 2nd Workshop on Process Algebras and Performance Modelling (PAPM '94
, 1994
"... The stochastic process algebra MTIPP has emerged from research in the field of process descriptions for random behaviour through time. This calculus has recently been shown to allow the calculation of performance measures (e.g. response times), purely functional statements (e.g. occurrences of deadl ..."
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Cited by 30 (1 self)
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The stochastic process algebra MTIPP has emerged from research in the field of process descriptions for random behaviour through time. This calculus has recently been shown to allow the calculation of performance measures (e.g. response times), purely functional statements (e.g. occurrences of deadlocks), as well as combined statements (e.g. optimal timeout values) [9, 11]. In contrast to classical process calculi each atomic action is supposed to happen after a delay that is characterised by a certain exponentially distributed random variable. In this report we present the language together with its operational semantics, that defines Markovian labelled transition systems as a combination of classical actionoriented transition systems and Markovian processes, especially continuous time Markov chains. In order to reflect different behavioural aspects we define a hierarchy of bisimulation equivalences and show that two of them are congruences. Finally we present equational laws for our...
QPNTool for the Specification and Analysis of Hierarchically Combined Queueing Petri Nets
 BAUSE (EDS.) QUANTITATIVE EVALUATION OF COMPUTING AND COMMUNICATION SYSTEMS, LECTURE NOTES IN COMPUTER SCIENCE
, 1995
"... This article describes a new version of the QPNTool now supporting specification and analysis of hierarchically combined Queueing Petri nets (HQPNs). HQPNs are an extension of QPNs allowing the refinement of places by QPN subnets and/or queues. HQPNs can be analysed with respect to qualitative ..."
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Cited by 26 (4 self)
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This article describes a new version of the QPNTool now supporting specification and analysis of hierarchically combined Queueing Petri nets (HQPNs). HQPNs are an extension of QPNs allowing the refinement of places by QPN subnets and/or queues. HQPNs can be analysed with respect to qualitative and quantitative aspects. Quantitative analysis is based on numerical Markov chain analysis. In contrast to conventional techniques the Markov chain underlying a HQPN is analysed by an approach exploiting the hierarchical structure of the model which results in a tensor representation of the generator matrix. This technique extends the size of solvable state spaces by one order of magnitude. Qualitative analysis of HQPNs relies on efficient analysis techniques based on Petri net theory. The new version of QPNTool implements the above analysis approaches supported by a graphical interface for a convenient specification of complex models.
Specification Techniques for Markov Reward Models
 in Discrete Event Dynamic Systems: Theory and Applications 3:219–247
, 1993
"... Abstract. Markov reward models (MRMs) are commonly used for the performance, dependability, and performability analysis of computer and communication systems. Many papers have addressed solution techniques for MRMs. Far less attention has been paid to the specification of MRMs and the subsequent de ..."
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Cited by 25 (1 self)
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Abstract. Markov reward models (MRMs) are commonly used for the performance, dependability, and performability analysis of computer and communication systems. Many papers have addressed solution techniques for MRMs. Far less attention has been paid to the specification of MRMs and the subsequent derivation of the underlying MRM. In this paper we only briefly address the mathematical aspects of MRMs. Instead, emphasis is put on specification techniques. In an application independent way, we distinguish seven classes of specification techniques: stochastic Petri nets, queuing networks, fault trees, production rule systems, communicating processes, specialized languages, and hybrid techniques. For these seven classes, we discuss the main principles, give examples and discuss software tools that support the use of these techniques. An overview like this has not been presented in the literature before. Finally, the paper addresses the generation of the underiying MRM from the highlevel specification, and indicates important future research areas.
Symbolic Performance Modeling of Parallel Systems
 In Proc. of IEEE Transactions on Parallel and Distributed Systems
, 2003
"... Abstract—Performance prediction is an important engineering tool that provides valuable feedback on design choices in program synthesis and machine architecture development. We present an analytic performance modeling approach aimed to minimize prediction cost, while providing a prediction accuracy ..."
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Cited by 24 (0 self)
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Abstract—Performance prediction is an important engineering tool that provides valuable feedback on design choices in program synthesis and machine architecture development. We present an analytic performance modeling approach aimed to minimize prediction cost, while providing a prediction accuracy that is sufficient to enable major code and data mapping decisions. Our approach is based on a performance simulation language called PAMELA. Apart from simulation, PAMELA features a symbolic analysis technique that enables PAMELA models to be compiled into symbolic performance models that trade prediction accuracy for the lowest possible solution cost. We demonstrate our approach through a large number of theoretical and practical modeling case studies, including six parallel programs and two distributedmemory machines. The average prediction error of our approach is less than 10 percent, while the average worstcase error is limited to 50 percent. It is shown that this accuracy is sufficient to correctly select the best coding or partitioning strategy. For programs expressed in a highlevel, structured programming model, such as dataparallel programs, symbolic performance modeling can be entirely automated. We report on experiments with a PAMELA model generator built within a dataparallel compiler for distributedmemory machines. Our results show that with negligible program annotation, symbolic performance models are automatically compiled in seconds, while their solution cost is in the order of milliseconds. Index Terms—Performance prediction, parallel processing, analytic performance modeling. æ 1
The Kronecker Product and Stochastic Automata Networks
 In Journal of Computational and Applied Mathematics
, 2003
"... This paper can be thought of as a companion paper to Van Loan's The Ubiquitous Kronecker Product paper [23]. We collect and catalog the most useful properties of the Kronecker product and present them in one place. We prove several new properties that we discovered in our search for a Stochasti ..."
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Cited by 24 (0 self)
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This paper can be thought of as a companion paper to Van Loan's The Ubiquitous Kronecker Product paper [23]. We collect and catalog the most useful properties of the Kronecker product and present them in one place. We prove several new properties that we discovered in our search for a Stochastic Automata Network preconditioner. We conclude by describing one application of the Kronecker product, omitted from Van Loan's list of applications, namely Stochastic Automata Networks. Key words: Stochastic automata networks, Kronecker products, Kronecker product properties, Preconditioning.