On an exact solution of the rate matrix of G∕M∕1-type Markov process with small number of phases

In this research paper we consider the matrix polynomial equation arising naturally in the equilibrium analysis of a structured G∕M∕1-type Markov process. We obtain an explicit expression for the unknown rate matrix R being 2 × 2 matrix. The method is based on symbolic solution of the determinantal polynomial equation. Using Cayley-Hamilton theorem, the matrix polynomial equation for the matrix R is reduced to the system of linear equations. Motivated by applications in Edge Computing by means of Internet of Things devices having tight constraints in energy consumption, we demonstrate the applicability of the method by a novel approach to energy efficiency of a single-server computing system. 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.