An optimal approximation of the characteristics of the GI/M/1 queue with two-stage service policy

In this paper, we consider an GI/M/1 system with two-stage service policy, having a service rate (μ1,μ2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mu _1, \mu _2)$$\end{document}, for which we determine its global transition operator. After that, with using the strong stability method we establish the approximation conditions for the stationary characteristics of this system by those of the standard GI/M/1 system. Under assumption that the approximation conditions are satisfied, we give the estimate of the deviation (stability inequalities) between the stationary distribution of the GI/M/1 system with two-stage service policy and those of the standard GI/M/1 system for three considered cases: the standard system has a service rate μ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _1$$\end{document} (minimal threshold policy), the standard system has a service rate μ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _2$$\end{document} (maximal threshold policy) and the standard system has a service rate μ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu ^*$$\end{document} minimizing the deviation (optimal policy). To calculate these deviations, the situation is modeled by a mathematical optimization problem which belongs to the minimization of a constrained nonlinear multi-variable function. Finally, numerical studies are performed to support the theoretical obtained results.