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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 157 (31 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.
Symbolic Representations and Analysis of Large Probabilistic Systems
 In Validation of Stochastic Systems
, 2004
"... Abstract. This paper describes symbolic techniques for the construction, representation and analysis of large, probabilistic systems. Symbolic approaches derive their efficiency by exploiting highlevel structure and regularity in the models to which they are applied, increasing the size of the stat ..."
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Cited by 16 (2 self)
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Abstract. This paper describes symbolic techniques for the construction, representation and analysis of large, probabilistic systems. Symbolic approaches derive their efficiency by exploiting highlevel structure and regularity in the models to which they are applied, increasing the size of the state spaces which can be tackled. In general, this is done by using data structures which provide compact storage but which are still efficient to manipulate, usually based on binary decision diagrams (BDDs) or their extensions. In this paper we focus on BDDs, multivalued decision diagrams (MDDs), multiterminal binary decision diagrams (MTBDDs) and matrix diagrams. 1
Efficient state space generation of gspns using decision diagrams
 In Proc. DSN
, 2002
"... Implicit techniques for representing and generating the reachability set of a highlevel model have become quite efficient. However, such techniques are usually restricted to models whose events have equal priority. Models containing events with differing classes of priority or complex priority stru ..."
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Cited by 6 (2 self)
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Implicit techniques for representing and generating the reachability set of a highlevel model have become quite efficient. However, such techniques are usually restricted to models whose events have equal priority. Models containing events with differing classes of priority or complex priority structure, in particular models with immediate events, have thus been required to use explicit reachability set generation techniques. In this paper, we present an efficient implicit technique, based on multivalued decision diagram representations for sets of states and matrix diagram representations for nextstate functions, that can handle models with complex priority structure. If the model contains immediate events, the vanishing states can be eliminated either during generation, by manipulating the matrix diagram, or after generation, by manipulating the multivalued decision diagram. We apply both techniques to several models and give detailed results. 1.
Solution of Large Markov Models using Lumping Techniques and Symbolic Data Structures
, 2005
"... Continuous time Markov chains (CTMCs) are among the most fundamental mathematical structures used for performance and dependability modeling of communication and computer systems. They are often constructed from models described in one of the various highlevel formalisms. Since the size of a CTMC u ..."
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Cited by 3 (0 self)
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Continuous time Markov chains (CTMCs) are among the most fundamental mathematical structures used for performance and dependability modeling of communication and computer systems. They are often constructed from models described in one of the various highlevel formalisms. Since the size of a CTMC usually grows exponentially with the size of the corresponding highlevel model, one often encounters the infamous statespace explosion problem, which often makes solution of the CTMCs intractable and sometimes makes it impossible. In statebased numerical analysis, which is the solution technique we have chosen to use to solve for measures defined on a CTMC, the statespace explosion problem is manifested in two ways: 1) large state transition rate matrices, and 2) large iteration vectors. The goal of this dissertation is to extend, improve, and combine existing solutions of the statespace explosion problem in order to make possible the construction and solution of very large CTMCs generated from highlevel models. Our new techniques follow largeness avoidance and largeness tolerance approaches. In the former approach, we reduce the size of the CTMC that needs to be solved in order to compute the measures of interest. That
What a Structural World
 Proceedings of the 9th International Workshop on Petri Nets and Performance Models
, 2001
"... Petri nets and stochastic Petri nets have been widely adopted as one of the best tools to model the logical and timing behavior of discretestate systems. However, their practical applicability is limited by the statespace explosion problem. We survey some of the techniques that have been used to c ..."
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Petri nets and stochastic Petri nets have been widely adopted as one of the best tools to model the logical and timing behavior of discretestate systems. However, their practical applicability is limited by the statespace explosion problem. We survey some of the techniques that have been used to cope with large state spaces, starting from early explicit methods, which require data structures of size proportional to the number of states or statetostate transitions, then moving to implicit methods, which borrow ideas from symbolic model checking (binary decision diagrams) and numerical linear algebra (Kronecker operators) to drastically reduce the computational requirements. Next, we describe the structural decomposition approach which has been the topic of our research in the last few years. This method only requires to specify a partition of the places in the net and, combining decision diagrams and Kronecker operators with the new concepts of event locality and node saturation, achieves fundamental gains in both memory and time efficiency. At the same, the approach is applicable to a wide range of models. We conclude by considering several research directions that could further push the range of solvable models, eventually leading to an even greater industrial acceptance of this simple yet powerful modeling formalism.
Numerical Solution Of Large Structured Markov Models
"... Abstract — Continuous time markov chains (CTMC) are methods for analysis of dynamic systems from a wide range of applications, especially wireless communications. The solution of a markov chain involves evaluating the steady state probabilities of each of the states associated in the chain. For very ..."
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Abstract — Continuous time markov chains (CTMC) are methods for analysis of dynamic systems from a wide range of applications, especially wireless communications. The solution of a markov chain involves evaluating the steady state probabilities of each of the states associated in the chain. For very large markov models, the transition rate matrix (of order nxn, where n is the total number of states) becomes very large and hence storing and solving them becomes a difficult task. Also the transition rate matrix (Q) happens to be sparse in large models. So we need to exploit the sparsity of this matrix for getting the solution. We will first explore the different methods of solving large markov models and then discuss about Kronecker algebra. The idea is to express Q as a Kronecker product of smaller matrices which can be built sequentially from the original model. The final solution can be obtained by using either GaussJacobi or GaussSeidel approaches. Sparse storage schemes like those of linked lists can be used for storing Q and the smaller matrices that are built up. I.
unknown title
"... Petri nets and stochastic Petri nets have been widely adopted as one of the best tools to model the logical and timing behavior of discretestate systems. However, their practical applicability is limited by the statespace explosion problem. We survey some of the techniques that have been used to c ..."
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Petri nets and stochastic Petri nets have been widely adopted as one of the best tools to model the logical and timing behavior of discretestate systems. However, their practical applicability is limited by the statespace explosion problem. We survey some of the techniques that have been used to cope with large state spaces, starting from early explicit methods, which require data structures of size proportional to the number of states or statetostate transitions, then moving to implicit methods, which borrow ideas from symbolic model checking (binary decision diagrams) and numerical linear algebra (Kronecker operators) to drastically reduce the computational requirements. Next, we describe the structural decomposition approach which has been the topic of our research in the last few years. This method only requires to specify a partition of the places in the net and, combining decision diagrams and Kronecker operators with the new concepts of event locality and node saturation, achieves fundamental gains in both memory and time efficiency. At the same, the approach is applicable to a wide range of models. We conclude by considering several research directions that could further push the range of solvable models, eventually leading to an even greater industrial acceptance of this simple yet powerful modeling formalism. 1.
unknown title
"... Computing response time distributions using stochastic Petri nets and matrix diagrams In this paper, we consider random variables expressed in terms of the time required for the state of a stochastic Petri net to pass from a set of starting markings to a set of stopping markings. These random variab ..."
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Computing response time distributions using stochastic Petri nets and matrix diagrams In this paper, we consider random variables expressed in terms of the time required for the state of a stochastic Petri net to pass from a set of starting markings to a set of stopping markings. These random variables have continuous phasetype distributions when the all transitions have exponentiallydistributed firing delays. We demonstrate how to numerically compute the distribution of the random variable using both explicit techniques and an implicit approach based on multiway decision diagrams and matrix diagrams. We present an efficient matrixvector multiplication algorithm for matrix diagrams that is necessary for numerical solution. We demonstrate the efficiency of our approaches using several models. The lower storage requirements of the implicit approach effectively increases the size of models that can be analyzed by about an order of magnitude. 1.