Unpredictable two-stage retrial model featuring impatient customers pinned Bernoulli working vacations and starting failure

The present investigation establishes an unreliable single-server retrial queue with a starting failure, a mandatory first-phase service, an optional second-phase service, a Bernoulli working vacation, and an impatient customer. After completing the mandatory service, customers have the option to opt for the optional second-phase service. Aside from the probability beta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}, the system can be left with the probability beta<overline>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{\beta }$$\end{document}. When the orbit is empty, the server switches over to working vacation. The server will provide service at a discounted rate during the working vacation. At the end of every vacation, if any of the customers find themselves available in the system, then the server develops into inactive and takes access for serving the newly arriving customer with gamma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} as its probability (single working vacation) or will continue the vacation along probability gamma<overline>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{\gamma }$$\end{document} (multiple working vacation). An unreliable server faces starting failure with probability alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} on the occasion of the arrival of a new customer or customer from orbit. Next, we transferred the customer from the service station to the orbit, and he attempted to receive service again at a random interval. We used the supplementary variable technique to determine the steady-state probability-generating function of both the system and the orbit. This paper discussed several key performance metrics of the system. In the end, we produced numerical illustrations and models of 3D figures for various outcomes.