Steady-State Performance Evaluation of an MX/G(a, b)/1 Queue with Low-Batch Service, Multiple Vacations and Uninterrupted Server Renovation

Abstract

This study examines the behaviour of uninterruptible server breakdowns in a bulk service queueing system with multiple vacations and low-batch service (LBS). The system consists of a single server that operates under two service modes: the General Bulk Service Rule (GBSR) and low-batch service (LBS). When each time bulk service is completed, the server may break
down with probability φ, in this case it is immediately sent for repair. Upon repair completion, or if no breakdown occurs (with probability 1 − φ), or at the moment of LBS completion, the server checks the queue length. If no customers are waiting, the server initiates multiple random-length vacations. Upon returning from a vacation, if fewer than ‘a’ customers are present, the server takes another vacation, repeating this process until at least ‘a’ customers are in the queue. Once this threshold is met, the server starts bulk service. After each service completion, the server start bulk service or low-batch service based on the queue length. This paper primarily analyses how LBS implementation influences key performance metrics and the breakdown probability ratio. Numerical results demonstrate the model’s effectiveness, and a cost analysis reveals that adopting LBS significantly reduces overall operational expenses.