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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
Logical and stochastic modeling with SMART
, 2003
"... We describe the main features of SmArT, a software package providing a seamless environment for the logic and probabilistic analysis of complex systems. SmArT can combine dierent formalisms in the same modeling study. For the analysis of logical behavior, both explicit and symbolic statespace g ..."
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Cited by 23 (13 self)
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We describe the main features of SmArT, a software package providing a seamless environment for the logic and probabilistic analysis of complex systems. SmArT can combine dierent formalisms in the same modeling study. For the analysis of logical behavior, both explicit and symbolic statespace generation techniques, as well as symbolic CTL modelchecking algorithms, are available. For the study of stochastic and timing behavior, both sparsestorage and Kronecker numerical solution approaches are available when the underlying process is a Markov chain. In addition,
Aggregation Of Stochastic Automata Networks With Replicas
, 2004
"... We present techniques for computing the solution of large Markov chain models whose generators can be represented in the form of a generalized tensor algebra, such as Stochastic Automata Networks (SAN). Many large systems include a number of replications of identical components. This paper exploits ..."
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Cited by 17 (5 self)
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We present techniques for computing the solution of large Markov chain models whose generators can be represented in the form of a generalized tensor algebra, such as Stochastic Automata Networks (SAN). Many large systems include a number of replications of identical components. This paper exploits replication by aggregating similar components. This leads to a state space reduction, based on lumpability. We define SAN with replicas, and we show how such SAN models can be strongly aggregated, taking functional rates into account. A tensor representation of the matrix of the aggregated Markov chain is proposed, allowing to store this chain in a compact manner and to handle larger models with replicas more efficiently. Examples and numerical results are presented to illustrate the reduction in state space and, consequently, the memory and processing time gains.
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 15 (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
SMART  Stochastic Model Analyzer for Reliability and Timing
 In Tools of Aachen 2001 Int. Multiconference on Measurement, Modelling and Evaluation of Computer Communication Systems
, 2001
"... al collections of homogeneous objects indexed by set elements. a[3][0.2] aggregates : analogous to the Pascal \record". p:3 A type can be further modied by the following natures, which describe stochastic characteristics: const: (the default) a nonstochastic quantity. ph: a random variable w ..."
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Cited by 12 (2 self)
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al collections of homogeneous objects indexed by set elements. a[3][0.2] aggregates : analogous to the Pascal \record". p:3 A type can be further modied by the following natures, which describe stochastic characteristics: const: (the default) a nonstochastic quantity. ph: a random variable with discrete or continuous phasetype distribution. rand: a random variable with arbitrary distribution. ctmc, dtmc, spn, . . . : stochastic formalisms dening a stochastic process indexed by time. 1.1 Function declarations Syntactically, objects dened in SMART are functions, possibly recursive, and can be overloaded: real pi := 3.14; /* a parameterless function */ bool close(real a, real b) := abs(ab) < 0.00001; /* a twoparameter function */ int pow(int base, int e
Advances in Model Representations
 Proc. PAPM/PROBMIV 2001, Available as Volume 2165 of LNCS (2001
, 2001
"... We review highlevel specification formalisms for Markovian performability models, thereby emphasising the role of structuring concepts as realised par excellence by stochastic process algebras. Symbolic representations based on decision diagrams are presented, and it is shown that they quite id ..."
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Cited by 6 (2 self)
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We review highlevel specification formalisms for Markovian performability models, thereby emphasising the role of structuring concepts as realised par excellence by stochastic process algebras. Symbolic representations based on decision diagrams are presented, and it is shown that they quite ideally support compositional model construction and analysis.
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.
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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Cited by 3 (0 self)
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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.
Modeling Finite Capacity Queueing Networks with Stochastic Automata Networks
"... This paper presents a method to represent Finite Capacity Queueing Networks  FCQN  using an alternative formalism called Stochastic Automata Networks  SAN. This method can be performed automatically and the resulting SAN model can be solved by traditional solution methods. The solution of a SAN m ..."
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Cited by 3 (1 self)
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This paper presents a method to represent Finite Capacity Queueing Networks  FCQN  using an alternative formalism called Stochastic Automata Networks  SAN. This method can be performed automatically and the resulting SAN model can be solved by traditional solution methods. The solution of a SAN model can provide stationary results like throughput, servers utilization, response time and population of each queue. The FCQN models that can be handled by this method include, but are not limited to the following features: dierent classes of clients with or without priority; open and closed queueing systems with blocking due to restricted capacity; open systems with loss of clients due to restricted capacity or priority among classes; and variable routing patterns according to queues local states. The benets of the use of SAN are related to other similar approaches in the conclusion. keywords: performance evaluation, numerical solutions, nite capacity queueing networks, stochastic auto...