This paper studies a discrete-time batch arrival GI/Geo/1 queue where the server may take multiple vacations depending on the state of the queue/system. However, during the vacation period, the server does not remain idle and serves the customers with a rate lower than the usual service rate. The vacation time and the service time during working vacations are geometrically distributed. Keeping note of the specific nature of the arrivals and departures in a discrete-time queue, we study the model under late arrival system with delayed access and early arrival system independently. We formulate the system using supplementary variable technique and apply the theory of difference equation to obtain closed-form expressions of steady-state system content distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the associated characteristic equations. We discuss the stability conditions of the system and develop few performance measures as well. Through some numerical examples, we illustrate the feasibility of our theoretical work and highlight the asymptotic behavior of the probability distributions at pre-arrival epochs. We further discuss the impact of various parameters on the performance of the system. The model considered in this paper covers a wide class of vacation and non-vacation queueing models which have been studied in the literature.

中文翻译：

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迟到和早到系统下具有多个工作假期的离散GIX / Geo / 1队列

本文研究了离散时间批量到达G I / G e o/ 1队列，其中服务器可能根据队列/系统的状态进行多个休假。但是，在休假期间，服务器不会保持空闲状态，而是以低于正常服务费率的价格为客户提供服务。工作假期的休假时间和服务时间是几何分布的。注意到离散时间队列中到达和离开的特定性质，我们独立研究了具有延迟访问的延迟到达系统和早期到达系统下的模型。我们使用补充变量技术来公式化系统，并应用差分方程的理论，根据相关特征方程的根，同时获得到达前和任意时期的稳态系统含量分布的闭式表达式。我们讨论了系统的稳定性条件，并很少开发性能指标。通过一些数值例子，我们说明了我们理论研究的可行性，并强调了到达前时期概率分布的渐近行为。我们将进一步讨论各种参数对系统性能的影响。本文考虑的模型涵盖了文献中已研究的一类休假和非休假排队模型。