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 shuffle-based 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 Jacobi-style and Gauss-Seidel-style 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 high-level modeling formalism, then a software tool is employed to generate the state space and generator matrix of the underlying CTMC and compute the stationary

We consider the problem of storing and searching a large state space obtained from a high-level 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

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SMART is a new tool designed to allow various high-level 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 strongly-typed, 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 (machine-dependent 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...

Use of a high-level 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 transition-ratematrix on-the-fly, eliminating the need to explicitly store the matrix at all. Next, we introduce a new iterative solution method, called modified adaptive Gauss-Seidel, 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 computationally-efficient technique for Gauss-Seidel-based 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.

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.

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 state-transition-rate matrix, a variant of block Gauss-Seidel 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.

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Measure-adaptive state-space construction is the process of exploiting symmetry in high-level model and performance measure specifications to automatically construct reduced state-space Markov models that support the evaluation of the performance measure. This paper describes a new reward variable specification technique, which, combined with recently developed state-space construction techniques, will allow us to build tools capable of measure-adaptive state-space construction. That is, these tools will automatically adapt the size of the state space to constraints derived from the system model and the user-specified reward variables. The work described in this paper extends previous work in two directions. First, standard reward variable definitions are extended to allow symmetry in the reward variable to be identified and exploited. Then, symmetric reward variables are further extended to include the set of path-based reward variables described in earlier work. In addition to the theory, several examples are introduced to demonstrate these new techniques.

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%.