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67
Stochastic automata network for modeling parallel systems
 IEEE Trans. Software Eng
, 1991
"... AbstractThis paper is motivated by the study of the performance of parallel systems. The performance models of such systems are often complex to describe and hard to solve. The method presented here uses a modular representation of the system as a network of statetransition graphs. The state spac ..."
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Cited by 128 (10 self)
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AbstractThis paper is motivated by the study of the performance of parallel systems. The performance models of such systems are often complex to describe and hard to solve. The method presented here uses a modular representation of the system as a network of statetransition graphs. The state space explosion is handled by a decomposition technique. The dynamic behavior of the algorithm is analyzed under Markovian assumptions. The transition matrix of the chain is automatically derived using tensor algebra operators, under a format which involves a very limited storage cost. Index TermsPerformance evaluation, Markov chain, tensor product, parallel systems, discrete time scale.
From Boolean to Probabilistic Boolean Networks as Models of Genetic Regulatory Networks
 Proc. IEEE
, 2002
"... Mathematical and computational modeling of genetic regulatory networks promises to uncover the fundamental principles governing biological systems in an integrarive and holistic manner. It also paves the way toward the development of systematic approaches for effective therapeutic intervention in di ..."
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Cited by 124 (23 self)
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Mathematical and computational modeling of genetic regulatory networks promises to uncover the fundamental principles governing biological systems in an integrarive and holistic manner. It also paves the way toward the development of systematic approaches for effective therapeutic intervention in disease. The central theme in this paper is the Boolean formalism as a building block for modeling complex, largescale, and dynamical networks of genetic interactions. We discuss the goals of modeling genetic networks as well as the data requirements. The Boolean formalism is justified from several points of view. We then introduce Boolean networks and discuss their relationships to nonlinear digital filters. The role of Boolean networks in understanding cell differentiation and cellular functional states is discussed. The inference of Boolean networks from real gene expression data is considered from the viewpoints of computational learning theory and nonlinear signal processing, touching on computational complexity of learning and robustness. Then, a discussion of the need to handle uncertainty in a probabilistic framework is presented, leading to an introduction of probabilistic Boolean networks and their relationships to Markov chains. Methods for quantifying the influence of genes on other genes are presented. The general question of the potential effect of individual genes on the global dynamical network behavior is considered using stochastic perturbation analysis. This discussion then leads into the problem of target identification for therapeutic intervention via the development of several computational tools based on firstpassage times in Markov chains. Examples from biology are presented throughout the paper. 1
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.
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 113 (21 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.
The Möbius Modeling Tool
 IN PROCEEDINGS OF THE 9TH INTERNATIONAL WORKSHOP ON PETRI NETS AND PERFORMANCE MODELS
"... Despite the development of many modeling formalisms and model solution methods, most tool implementations support only a single formalism. Furthermore, models expressed in the chosen formalism cannot be combined with models expressed in other formalisms. This monolithic approach both limits the usef ..."
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Cited by 74 (12 self)
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Despite the development of many modeling formalisms and model solution methods, most tool implementations support only a single formalism. Furthermore, models expressed in the chosen formalism cannot be combined with models expressed in other formalisms. This monolithic approach both limits the usefulness of such tools to practitioners, and hampers new and existing formalisms and solvers. This paper describes the method that a new modeling tool, cal led Mobius, uses to eliminate these limitations. Mobius provides an infrastructure to support multiple interacting formalisms and solvers, and is extensible in that new formalisms and solvers can be added to the tool without changing those already implemented. Mobius provides this capability through the use of an abstract functional interface, which provides a formalismindependent interface to models. This allows models expressed in multiple formalisms to interact with each other, and with multiple solvers.
Numerical Analysis of Superposed GSPNs
 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING
, 1996
"... The numerical analysis of various modeling formalisms profits from a structured representation for the generator matrix Q of the underlying continuous time Markov chain, where Q is described by a sum of tensor (Kronecker) products of much smaller matrices. In this paper we describe such a representa ..."
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Cited by 71 (10 self)
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The numerical analysis of various modeling formalisms profits from a structured representation for the generator matrix Q of the underlying continuous time Markov chain, where Q is described by a sum of tensor (Kronecker) products of much smaller matrices. In this paper we describe such a representation for the class of superposed generalized stochastic Petri nets (SGSPNs), which is less restrictive than in previous work. Furthermore a new iterative analysis algorithm is proposed. It pays special attention to a memory efficient representation of iteration vectors as well as to a memory efficient structured representation of Q. In consequence the new algorithm is able to solve models which have state spaces with several millions of states, where other exact numerical methods become impracticable on a common workstation.
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 68 (19 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
The Numerical Solution of Stochastic Automata Networks
, 1994
"... Stochastic Automata Networks (SAN's) have recently received attention in the literature as an efficient means of modelling parallel systems such as communicating processes, concurrent processors, shared memory, etc. The advantage that the SAN approach has over generalized stochastic Petri nets, ..."
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Cited by 48 (11 self)
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Stochastic Automata Networks (SAN's) have recently received attention in the literature as an efficient means of modelling parallel systems such as communicating processes, concurrent processors, shared memory, etc. The advantage that the SAN approach has over generalized stochastic Petri nets, and indeed over any Markovian analysis that requires the generation of a transition matrix, is that its representation remains compact even as the number of states in the underlying Markov chain begins to explode. Our concern in this paper is with the numerical issues that are involved in solving SAN networks. We introduce stochastic automata and consider the numerical difficulties that result from their interaction. We examine how the product of a vector with a compact SAN descriptor may be formed, for this operation is basis to all iterative solution methods. We describe possible solution methods, including the power method, the method of Arnoldi and GMRES, and show that the two latter methods...
"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 38 (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.
Systemlevel power/performance analysis for embedded systems design.
 In Proceedings of the annual conference on Design automation (DAC),
, 2001
"... Abstract Introduction and objectives Portable embedded multimedia systems have a few distinctive features that make them special within the general class of embedded systems. First, they are characterized by 'soft' realtime constraints and then may tolerate missed deadlines. In other wor ..."
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Cited by 33 (6 self)
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Abstract Introduction and objectives Portable embedded multimedia systems have a few distinctive features that make them special within the general class of embedded systems. First, they are characterized by 'soft' realtime constraints and then may tolerate missed deadlines. In other words, their behavior is not necessarily characterized by a single number (the hard realtime constraint), as is the case for reactive embedded systems used in safety critical applications, but by a probability (or distribution of probabilities) which captures some sort of variability in the performance metrics. Another important characteristic is the notion of Quality of Service (QoS) which embraces all the nonfunctional properties of a system (e.g. power consumption, latency, jitter, cost, etc.). In multimedia systems, QoS requirements vary considerably from one media type to another. For example, video connections require consistently high throughput, but can tolerate reasonable levels of jitter and bit or packet errors. In contrast, audio applications do not require such high bandwidth, but place tighter restrictions on jitter and error rates. The ability to explore several design alternatives while trying to satisfy QoS requirements is of crucial importance, especially early in the design cycle to avoid costly redesign steps Based on the distinctive features discussed above, the objective of this paper is to propose a new modeling and analysis methodology for multimedia systems communicating over wireless channels. More precisely, we provide an analytical technique for averagecase power/performance analysis that can be used early in the design cycle to identify the best match (in terms of performance and power consumption) among several possible applicationarchitecture combinations. Indeed, the computational requirements of such systems show such large statistical variations that designing them based on a single fixed number that represents the worstcase behavior (typically, one or two orders of magnitude larger than the actual execution time [3]) would result in completely inefficient systems. Despite the great potential for embedded system design, the area of averagecase analysis has received little attention While our formalism is completely general, to illustrate its technicalities, we consider the MPEG2 video application throughout the paper