Analysis of Renewal Batch Arrival Queues with Multiple Vacations and Geometric Abandonment

We investigate the dynamics of customer geometric abandonment within a queueing framework characterized by renewal input batch arrivals and multiple vacations. Customers' impatience becomes evident when confronted with server vacations, triggering instances of abandonment. This phenomenon reduces the number of customers within the system during abandonment epochs following a geometric distribution. The probability of customers leaving the queue escalates with prolonged waiting times. We derive concise and closed-form expressions for system-length distributions at pre-arrival and arbitrary epochs by harnessing the power of supplementary variable and difference operator methods. Furthermore, we elucidate specific instances of our model, shedding light on its versatility. To substantiate our theoretical framework, we provide a series of illustrative numerical experiments presented through meticulously crafted tables and graphs, thereby showcasing the robustness and applicability of our methodology.