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The Möbius Framework and Its Implementation
"... The Möbius framework is an environment for supporting multiple modeling formalisms and solution techniques. Models expressed in formalisms that are compatible with the framework are translated into equivalent models using Mobius framework components. This translation preserves the structure of the m ..."
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Cited by 73 (19 self)
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The Möbius framework is an environment for supporting multiple modeling formalisms and solution techniques. Models expressed in formalisms that are compatible with the framework are translated into equivalent models using Mobius framework components. This translation preserves the structure of the models, allowing e#cient solutions. The framework is implemented in the tool by a welldefined abstract functional interface. Models and solution techniques interact with one another through the use of the standard interface, allowing them to interact with Mobius framework components, not formalism components. This permits novel combinations of modeling techniques, and will be a catalyst for new research in modeling techniques. This paper describes our approach, focusing on the "atomic model." We describe the formal description of the Mobius components as well as their implementations in our software tool.
Complexity of memoryefficient Kronecker operations with applications to the solution of Markov models
 INFORMS J. Comp
, 2000
"... We present new algorithms for the solution of large structured Markov models whose infinitesimal generator can be expressed as a Kronecker expression of sparse matrices. We then compare them with the shufflebased method commonly used in this context and show how our new algorithms can be advantageo ..."
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Cited by 65 (18 self)
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We present new algorithms for the solution of large structured Markov models whose infinitesimal generator can be expressed as a Kronecker expression of sparse matrices. We then compare them with the shufflebased method commonly used in this context and show how our new algorithms can be advantageous in dealing with very sparse matrices and in supporting both Jacobistyle and GaussSeidelstyle methods with appropriate multiplication algorithms. Our main contribution is to show how solution algorithms based on Kronecker expression can be modified to consider probability vectors of size equal to the "actual" state space instead of the "potential" state space, thus providing space and time savings. The complexity of our algorithms is compared under different sparsity assumptions. A nontrivial example is studied to illustrate the complexity of the implemented algorithms. Continuous time Markov chains (CTMCs) are an established technique to analyze the performance, reliability, or performability of dynamic systems from a wide range of application areas. CTMCs are usually specied in a highlevel modeling formalism, then a software tool is employed to generate the state space and generator matrix of the underlying CTMC and compute the stationary
Modelbased evaluation: From dependability to security
 IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING
, 2004
"... The development of techniques for quantitative, modelbased evaluation of computer system dependability has a long and rich history. A wide array of modelbased evaluation techniques are now available, ranging from combinatorial methods, which are useful for quick, roughcut analyses, to statebased ..."
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Cited by 56 (2 self)
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The development of techniques for quantitative, modelbased evaluation of computer system dependability has a long and rich history. A wide array of modelbased evaluation techniques are now available, ranging from combinatorial methods, which are useful for quick, roughcut analyses, to statebased methods, such as Markov reward models, and detailed, discreteevent simulation. The use of quantitative techniques for security evaluation is much less common, and has typically taken the form of formal analysis of small parts of an overall design, or experimental red teambased approaches. Alone, neither of these approaches is fully satisfactory, and we argue that there is much to be gained through the development of a sound modelbased methodology for quantifying the security one can expect from a particular design. In this work, we survey existing modelbased techniques for evaluating system dependability, and summarize how they are now being extended to evaluate system security. We find that many techniques from dependability evaluation can be applied in the security domain, but that significant challenges remain, largely due to fundamental differences between the accidental nature of the faults commonly assumed in dependability evaluation, and the intentional, human nature of cyber attacks.
On the use of Kronecker operators for the solution of generalized stochastic Petri nets
, 1996
"... We discuss how to describe the Markov chain underlying a generalized stochastic Petri net using Kronecker operators on smaller matrices. We extend previous approaches by allowing both an extensive type of markingdependent behavior for the transitions and the presence of immediate synchronizations. ..."
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Cited by 45 (13 self)
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We discuss how to describe the Markov chain underlying a generalized stochastic Petri net using Kronecker operators on smaller matrices. We extend previous approaches by allowing both an extensive type of markingdependent behavior for the transitions and the presence of immediate synchronizations. The derivation of the results is thoroughly formalized, including the use of Kronecker operators in the treatment of the vanishing markings and the computation of impulsebased reward measures. We use our techniques to analyze a model whose solution using conventional methods would fail because of the statespace explosion. In the conclusion, we point out ideas to parallelize our approach.
Efficient symbolic statespace construction for asynchronous systems
 Application and Theory of Petri Nets 2000 (Proc. 21th Int. Conf. on Applications and Theory of Petri Nets, Aarhus, Denmark), Lecture Notes in Computer Science 1825
, 2000
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Storage alternatives for large structured state spaces
 Proc. 9th Int. Conf. on Modelling Techniques and Tools for Computer Performance Evaluation, Lecture Notes in Computer Science 1245
, 1997
"... We consider the problem of storing and searching a large state space obtained from a highlevel model such as a queueing network or a Petri net. After reviewing the traditional technique based on a single search tree, we demonstrate how an approach based on multiple levels of search trees offers adv ..."
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Cited by 39 (16 self)
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We consider the problem of storing and searching a large state space obtained from a highlevel model such as a queueing network or a Petri net. After reviewing the traditional technique based on a single search tree, we demonstrate how an approach based on multiple levels of search trees offers advantages in both memory and execution complexity. Further execution time improvements are obtained by exploiting the concept of “event locality”. We apply our technique to three large parametric models, and give detailed experimental results. 1
"OntheFly" Solution Techniques for Stochastic Petri Nets and Extensions
 IEEE Transactions on Software Engineering
, 1997
"... Use of a highlevel modeling representation, such as stochastic Petri nets, frequently results in a very large state space. In this paper, we propose new methods that can tolerate such large state spaces and that do not require any special structure in the model. First, we develop methods that gener ..."
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Cited by 32 (5 self)
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Use of a highlevel modeling representation, such as stochastic Petri nets, frequently results in a very large state space. In this paper, we propose new methods that can tolerate such large state spaces and that do not require any special structure in the model. First, we develop methods that generate rows and columns of the state transitionratematrix onthefly, eliminating the need to explicitly store the matrix at all. Next, we introduce a new iterative solution method, called modified adaptive GaussSeidel, that exhibits locality in its use of data from the state transitionrate matrix. This permits the caching of portions of the matrix, hence reducing the solution time. Finally, we develop a new memory and computationallyefficient technique for GaussSeidelbased solvers that avoids the need for generating rows of A in order to solve Ax = b. Taken together, these new results show that one can solve very large SPN, GSPN, SRN, and SANmodels without any special structure.
A toolbox for functional and quantitative analysis of DEDS
 Proc. 10th Int. Conf. on Modelling Techniques and Tools for Computer Performance Evaluation, Lecture Notes in Computer Science 1469
, 1998
"... Abstract This paper presents a toolbox for the construction of modular tools for functional and quantitative (performance) analysis of discrete event dynamic systems (DEDS). The intention is to simplify the usage of appropriate analysis algorithms, thus supporting the development of appropriate tool ..."
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Cited by 25 (9 self)
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Abstract This paper presents a toolbox for the construction of modular tools for functional and quantitative (performance) analysis of discrete event dynamic systems (DEDS). The intention is to simplify the usage of appropriate analysis algorithms, thus supporting the development of appropriate tools.
The Need for and the Advantages of Generalized Tensor Algebra for Kronecker Structured Representations
 International Journal of Simulation: Systems, Science & Technology
, 2004
"... Abstract. This paper presents the advantages in extending Classical Tensor Algebra (CTA), also known as Kronecker Algebra, to allow the definition of functions, i.e., functional dependencies among its operands. Such extended tensor algebra have been called Generalized Tensor Algebra (GTA). Stochasti ..."
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Cited by 24 (13 self)
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Abstract. This paper presents the advantages in extending Classical Tensor Algebra (CTA), also known as Kronecker Algebra, to allow the definition of functions, i.e., functional dependencies among its operands. Such extended tensor algebra have been called Generalized Tensor Algebra (GTA). Stochastic Automata Networks (SAN) and Superposed Generalized Stochastic Petri Nets (SGSPN) formalisms use such Kronecker representations. The advantages of GTA do not imply in a reduction or augmentation of application scope, since there is a representation equivalence between SAN, which uses GTA, and SGSPN, which uses only CTA. Two modeling examples are presented in order to draw comparisons between the memory needs and CPU time required for the generation and solution using both formalisms, showing the computational advantages in using GTA. 1
Structured Analysis Approaches for Large Markov Chains  A Tutorial
 Applied Numerical Mathematics
, 1996
"... The tutorial introduces structured analysis approaches for continuous time Markov chains (CTMCs) which are a means to extend the size of analyzable state spaces significantly compared with conventional techniques. It is shown how generator matrices of large CTMCs can be represented in a very compact ..."
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Cited by 19 (8 self)
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The tutorial introduces structured analysis approaches for continuous time Markov chains (CTMCs) which are a means to extend the size of analyzable state spaces significantly compared with conventional techniques. It is shown how generator matrices of large CTMCs can be represented in a very compact form, how this representation can be exploited in numerical solution techniques and how numerical analysis profits from this exploitation. Additionally, recent results covering implementation issues, tool support, and advanced analysis techniques are surveyed. 1 Introduction Analysis of continuous time Markov chains (CTMCs) is a well established approach to analyze the performance, dependability and performability of computer and communication systems. Systems are modeled using specification techniques like queueing networks (QNs), stochastic Petri nets (SPNs), formal specification techniques to mention only a few. Unfortunately, the size of CTMCs underlying most realistic examples can be ...