This paper examines numerical issues in computing solutions to networks of stochastic automata. It is well-known 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 matrix-vector multiply, is given by ρ N = Π N i-1 n i × Σ N 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.

Abstract-This 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 state-transition 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.

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In this paper a new technique to handle complex Markov models is presented. This method is based on a description using stochastic automatas and is dedicated to distributed algorithms modelling. One example of a mutual exclusion algorithm in a distributed environment is extensively analysed. The mathematical analysis is based on tensor algebra for matrices.

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