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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 67 (21 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
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 40 (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
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 38 (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...
"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 37 (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 31 (12 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.
An efficient diskbased tool for solving very large Markov models
 Proc. 9th Int. Conf. on Modelling Techniques and Tools for Computer Performance Evaluation
, 1997
"... Very large Markov models often result when modeling realistic computer systems and networks. We describe a new tool for solving large Markov models on a typical engineering workstation. This tool does not require any special properties or a particular structure in the model, and it requires only s ..."
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Cited by 28 (3 self)
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Very large Markov models often result when modeling realistic computer systems and networks. We describe a new tool for solving large Markov models on a typical engineering workstation. This tool does not require any special properties or a particular structure in the model, and it requires only slightly more memory than what is necessary to hold the solution vector itself. It uses a disk to hold the statetransitionrate matrix, a variant of block GaussSeidel as the iterative solution method, and an innovative implementation that involves two parallel processes: the first process retrieves portions of the iteration matrix from disk, and the second process does repeated computation on small portions of the matrix. We demonstrate its use on two realistic models: a Kanban manufacturing system and the Courier protocol stack, which have up to 10 million states and about 100 million nonzero entries. The tool can solve the models efficiently on a workstation with 128 Mbytes of memory and 4Gbytes of disk.
Sequential and Distributed Model Checking of Petri Net Specifications
 STTT
, 2002
"... In this paper we present algorithms for model checking CTL over systems specified as Petri nets. We present sequential as well as distributed model checking algorithms. The algorithms rely on an explicit representation of the system state space, but do not require the transition relation to be expli ..."
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Cited by 26 (2 self)
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In this paper we present algorithms for model checking CTL over systems specified as Petri nets. We present sequential as well as distributed model checking algorithms. The algorithms rely on an explicit representation of the system state space, but do not require the transition relation to be explicitly available; it is recomputed whenever required. This approach allows us to model check very large systems, with hundreds of millions of states, in a fast and efficient way. Furthermore, our distributed algorithms scale very well, as they show efficiencies in the range of 80 to 100%.
Complexity of Kronecker Operations on Sparse Matrices with Applications to the Solution of Markov Models
, 1997
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A Kronecker product approximate preconditioner for SANs
 NUMER. LINEAR ALGEBRA APPL. 2004; 11:723–752
, 2004
"... Many very large Markov chains can be modelled eciently as stochastic automata networks (SANs). A SAN is composed of individual automata which, for the most part, act independently, requiring only infrequent interaction. SANs represent the generator matrix Q of the underlying Markov chain compactly a ..."
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Cited by 20 (1 self)
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Many very large Markov chains can be modelled eciently as stochastic automata networks (SANs). A SAN is composed of individual automata which, for the most part, act independently, requiring only infrequent interaction. SANs represent the generator matrix Q of the underlying Markov chain compactly as the sum of Kronecker products of smaller matrices. Thus, storage savings are immediate. The benet of a SAN’s compact representation, known as the descriptor, is often outweighed by its tendency to make analysis of the underlying Markov chain tough. While iterative or projections methods have been used to solve the system Q=0, the time until these methods converge to the stationary solution is still unsatisfactory. SAN’s compact representation has made the next logical research step of preconditioning thorny. Several preconditioners for SANs have been proposed and tested, yet each has enjoyed little or no success. Encouraged by the recent success of approximate inverses as preconditioners, we have explored their potential as SAN preconditioners. One particularly relevant nding on approximate inverse preconditioning is the nearest Kronecker product approximation discovered by Pitsianis and Van Loan. In this paper, we extend the nearest Kronecker product technique to approximate the Q matrix for an SAN with a Kronecker product, A1 ⊗A2 ⊗ · · · ⊗AN. Then, we take M =A−11 ⊗A−12 ⊗ · · · ⊗A−1N as our SAN NKP preconditioner.