Queueing theory is implemented for modeling and analyzing actual conditions in industries and real-world problems. In many cases, the input is converted to the desired output after several successive steps. Lack of space, feedback and vacation are the main characters of these processes. This article deals with the modeling and analyzing the steady-state behavior of an \(M/G/1\) retrial queueing system with first essential and \(k-1\) optional phases of service. Also, the probabilistic feedback to orbit at each phase and Bernoulli vacation at the end of \(k\)-th phase may occur in this system. If the customers find the server busy or on vacation, they join to the orbit. In this article, after finding the probability generating functions of the system and orbit sizes, some important performance measures are found. Also, the system reliability is defined. Eventually, to demonstrate the capability of the proposed model, the sensitivity analysis of cost indices and performance measures via some model parameters (arrival/retrial/vacation rate) in different reliability levels are investigated in two applicable examples. Additionally, for optimizing the performance of the system, some technical suggestions are presented.

排队理论用于建模和分析行业中的实际情况和现实世界的问题。在许多情况下，输入会在几个连续步骤后转换为所需的输出。缺乏空间、反馈和假期是这些过程的主要特征。本文涉及对\(M/G/1\)重试排队系统的稳态行为进行建模和分析，该系统具有第一个基本和\(k-1\) 个可选服务阶段。此外，每个阶段的轨道概率反馈和\(k\)结束时的伯努利假期-th 阶段可能发生在这个系统中。如果客户发现服务器忙碌或在休假，他们就会加入轨道。在本文中，在找到系统的概率生成函数和轨道大小之后，找到了一些重要的性能度量。此外，还定义了系统可靠性。最后，为了证明所提出模型的能力，在两个适用的例子中研究了在不同可靠性水平下通过一些模型参数（到达/重试/休假率）对成本指标和性能度量的敏感性分析。此外，为了优化系统的性能，提出了一些技术建议。