This paper considers a single server finite-buffer batch-size-dependent bulk service queue with queue-length dependent vacation. Customers arrive at the system according to the Poisson process and are served in batches of maximum size ‘b’ with a minimum threshold value ‘a’ following the ‘general bulk service’ rule. The service time distribution is assumed to be of general type which modulates depending on serving batch size. The server is allowed to take vacation, either single or multiple vacations, whenever permissible numbers of customers are not found in the queue at the beginning of the service. The vacation time distribution is assumed to be of general type and dynamically changes depending on the queue content at vacation initiation epoch. We use the supplementary variable and the embedded Markov chain techniques to obtain the steady-state joint distribution of the queue content service batch size, queue content and type of vacation taken by the server at various epochs. Several numerical results are presented at the end to bring out the qualitative aspect of the model, which reveals the fact that queue-length dependent vacation further reduces congestion in the batch-size-dependent bulk service queues.

本文考虑了一个具有与队列长度相关的休假的单服务器与有限缓冲区批量大小相关的批量服务队列。根据泊松过程客户系统到达和供应最大尺寸“的批次b ”与最小阈值'一遵循“一般批量服务”规则。服务时间分配被假定为一般类型，其根据服务批次大小进行调制。只要在服务开始时队列中找不到允许的客户数量，服务器就可以休假，无论是一次还是多次。休假时间分布被假定为一般类型，并且根据休假开始时期的队列内容而动态变化。我们使用补充变量和嵌入式Markov链技术来获取队列内容服务批处理大小，队列内容和服务器在各个时期采取的休假类型的稳态联合分布。最后给出了一些数值结果，以说明模型的定性方面，