Analysis of repairable discrete-time queueing systems with negative customers, disasters, balking customers and interruptible working vacations under Bernoulli schedule

Abstract

The study of discrete-time queueing systems is important for modeling and optimizing real-world systems that operate in fixed time intervals, such as telecommunications, computer networks, and manufacturing. This paper contributes to this field by analyzing two unreliable discrete-time $Geo/G/1$ queueing models that incorporate Bernoulli working vacation interruptions and balking customers under two different killing strategies, allowing for a more realistic representation of disruptions in service operations. After serving all currently present positive customers, the server promptly begins a working vacation. If a service is completed and there are still positive customers awaiting service during this vacation period, the server will either attend to the next customer at the normal speed with a probability of $p$, or continue to serve the existing customer at a reduced speed with a probability of $1-p$. Employing the supplementary variable method and the probability generating function technique, we obtain the steady-state queue length distributions and sojourn time distributions for both models. Besides, some crucial performance characteristics are presented. Finally, Sensitivity analysis is conducted through numerical examples to explore the operational characteristics and patterns of the systems under consideration. The findings of this study can be applied to optimizing operations in digital communication systems, minimizing customer waiting times and reducing the risk of server failures.