Chydzinski Andrzej
Department of Computer Networks and Systems, Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland.
Entropy (Basel). 2024 Sep 22;26(9):807. doi: 10.3390/e26090807.
A non-zero correlation between service times can be encountered in many real queueing systems. An attractive model for correlated service times is the Markovian service process, because it offers powerful fitting capabilities combined with analytical tractability. In this paper, a transient study of the queue length in a model with MSP services and a general distribution of interarrival times is performed. In particular, two theorems are proven: one on the queue length distribution at a particular time , where can be arbitrarily small or large, and another on the mean queue length at . In addition to the theorems, multiple numerical examples are provided. They illustrate the development over time of the mean queue length and the standard deviation, along with the complete distribution, depending on the service correlation strength, initial system conditions, and the interarrival time variance.
在许多实际排队系统中，服务时间之间可能存在非零相关性。相关服务时间的一个有吸引力的模型是马尔可夫服务过程，因为它结合了强大的拟合能力和解析易处理性。本文对具有马尔可夫服务过程（MSP）服务且到达间隔时间具有一般分布的模型中的队列长度进行了瞬态研究。特别地，证明了两个定理：一个是关于特定时刻的队列长度分布，其中该时刻可以任意小或大；另一个是关于该时刻的平均队列长度。除了这些定理之外，还提供了多个数值示例。它们展示了平均队列长度和标准差随时间的变化情况，以及完整的分布，这取决于服务相关强度、初始系统条件和到达间隔时间方差。
