Waiting time distributions in an M/G/1 retrial queue with two classes of customers

We consider an M/G/1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{M/G/1}$$\end{document} retrial queueing system with two classes of customers, in which the service time distributions are different for both classes of customers. When the server is unavailable, an arriving class-1 customer is queued in the queue with infinite capacity, whereas class-2 customer enters the retrial group. In this paper, we are concerned with the analysis of the waiting time distribution. We obtain the joint transform of the waiting time of a class-2 customer and the number of class-2 customers as well as the Laplace–Stieltjes transform of the waiting time of a class-1 customer. We also obtain all the moments of the waiting time distributions of class-1 and class-2 customers.