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On An Exact Solution Of The Rate Matrix Of Quasi-Birth-Death Process With Small Number Of Phase

A new method of obtaining exact solution for the rate matrix R in the Matrix-Analytic method in case of the phase state of dimension two is proposed. The method is based on symbolic solution of the determinental polynomial equation, and obtaining a linear matrix equation for the unknown rate matrix R by Cayley–Hamilton theorem. The method is applied to analyze the EnergyPerformance tradeoff of an Internet-of-Things device. A new randomized regime switching scheme is proposed, which, as it is shown by means of numerical experiment, provides significant decrease of energy consumption of the system under study. INTRODUCTION Univariate polynomial equations naturally arise in many branches of mathematics. An exact symbolic solution of such an equation (in terms of arithmetic operations and radicals) is known to exist for polynomials of order less or equal to four. Moreover, the nonexistence of such a solution for a polynomial equation of order greater or equal to five was established in Abel–Ruffini theorem. The solvability concept of an arbitrary monic polynomial was provided by Galois group theory. A generalization of polynomial equations in which the coefficients and the argument are matrices (matrix polynomial equations) has been studied in a number of works, and the detailed theory of uni-variate matrix polynomial equations was developed [15]. In the research area of Queueing Theory, the matrix quadratic equation RA +RA +A = 0 (1) was first used to find a solution of a QBD process (by means of Complex Analysis-based method) in late 60’s. Wallace [34] and Evans [10] showed that in the case of QBD process, matrix geometric solution exists for the equilibrium distribution where the rate matrix R is a minimal nonnegative solution of matrix quadratic equation (1). M. Neuts generalized the result to arbitrary G/M/1-type Markov processes and showed [24] that R is the minimal non-negative solution of matrix power series equation