Explicit Convergence Rates of the Embedded M/G/1 Queue

This paper investigates the explicit convergence rates to the stationary distribution π of the embedded M/G/1 queue; specifically, for suitable rate functions r(n) which may be polynomial with r(n) = nl, l > 0 or geometric with r(n) = αn, α > 1 and "moments" f = 1, we find the conditions under which\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\sum\nolimits_{n = 0}^\infty {r{\left( n \right)}} }{\left\| {P^{n} {\left( {i, \cdot } \right)} - \pi {\left( \cdot \right)}} \right\|}_{f} \leqslant M{\left( i \right)} $$\end{document} for all i ∈ E. For the polynomial case, the explicit bounds on M(i) are given in terms of both "drift functions" and behavior of the first hitting time on the state 0; and for the geometric case, the largest geometric convergence rate α* is obtained.