Some Matrix Patterns Arising in Queuing Theory

A remarkably versatile dynamical system is given by $$ d{p_i}/dt = \sum\limits_{{\begin{array}{*{20}{c}} {j = 1} \\ {j \ne i} \\ \end{array} }}^n {{a_{{ij}}}{p_j} - (\sum\limits_{{\begin{array}{*{20}{c}} {j = 1} \\ {j \ne i} \\ \end{array} }}^n {{a_{{ji}}}} )} {p_i} $$ where n ≥ 2, {p₁, p₂,..., 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 p₁ + p₂ +... + 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.