Abstract
A remarkably versatile dynamical system is given by
where n ≥ 2, {p 1, p 2,..., p n } are system variables, and {a ij } is a matrix of nonnegative real numbers. Diagonal entries in {a{ do not occur in the model in the sense that they are both added and subtracted in the given sum; for secretarial purposes we set a ii = 0. The system variables themselves arise from probabilities, so we also assume throughout that at time t = 0, all p i are nonnegative and the component sum p 1 + p 2 +... + p n is equal to 1.
The goal of this paper is to describe the trajectories of the model in terms of a Lyapunov-like function. Doing so involves the use of theorems of Brualdi and Gersgorin, the graph-theoretic notion of balanced cycles, and the qualitative solvability of an equation involving a Hadamard product.
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© 1993 Springer-Verlag New York, Inc.
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Jeffries, C. (1993). Some Matrix Patterns Arising in Queuing Theory. In: Brualdi, R.A., Friedland, S., Klee, V. (eds) Combinatorial and Graph-Theoretical Problems in Linear Algebra. The IMA Volumes in Mathematics and its Applications, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8354-3_7
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DOI: https://doi.org/10.1007/978-1-4613-8354-3_7
Publisher Name: Springer, New York, NY
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