GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
The Effect of Execution Policies on the Semantics and Analysis of Stochastic Petri Nets
IEEE Transactions on Software Engineering
Numerical methods in Markov chain modeling
Operations Research
Structured analysis approaches for large Markov chains
Applied Numerical Mathematics
The Approximation of Maximum Subgraph Problems
ICALP '93 Proceedings of the 20th International Colloquium on Automata, Languages and Programming
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Queueing networks with blocking: a bibliography
ACM SIGMETRICS Performance Evaluation Review
Block SOR Preconditioned Projection Methods for Kronecker Structured Markovian Representations
SIAM Journal on Scientific Computing
Numerical Methods for Structured Markov Chains (Numerical Mathematics and Scientific Computation)
Numerical Methods for Structured Markov Chains (Numerical Mathematics and Scientific Computation)
A minimal representation of Markov arrival processes and a moments matching method
Performance Evaluation
Solving Real Linear Systems with the Complex Schur Decomposition
SIAM Journal on Matrix Analysis and Applications
Shifted Kronecker Product Systems
SIAM Journal on Matrix Analysis and Applications
Queueing Theory: A Linear Algebraic Approach
Queueing Theory: A Linear Algebraic Approach
Stochastic Petri nets with matrix exponentially distributed firing times
Performance Evaluation
Analysis of queues with rational arrival process components: a general approach
ACM SIGMETRICS Performance Evaluation Review
Rational Automata Networks: A Non-Markovian Modeling Approach
INFORMS Journal on Computing
A component-based solution for reducible Markov regenerative processes
Performance Evaluation
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Matrix exponential distributions and rational arrival processes have been proposed as an extension to pure Markov models. The paper presents an approach where these process types are used to describe the timing behavior in quantitative models like queueing networks, stochastic Petri nets or stochastic automata networks. The resulting stochastic process, which is called a rational process, is defined and it is shown that the matrix governing the behavior of the process has a structured representation which allows one to represent the matrix in a very compact form. The main emphasis of the paper is on numerical techniques to compute the stationary distribution. It is shown which techniques from Markov chain analysis may also be applied for rational processes. Aggregation steps can be used to build an aggregated process which is a Markov chain and allows one to compute certain marginal probabilities for the detailed rational process. Furthermore, a block iteration method is presented which exploits the matrix structure and is an extension of previously published techniques for Markov chains. Finally, we present some examples showing that the extension of Markov models results in models which may have a significantly smaller state space than equivalently or similar behaving Markov chains.