Kempa Wojciech M, Marjasz Rafał
Department of Mathematics Applications and Methods for Artificial Intelligence, Faculty of Applied Mathematics, Silesian University of Technology, 23 Kaszubska Str., 44-100 Gliwice, Poland.
Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, 5 Bałtycka Str., 44-100 Gliwice, Poland.
Entropy (Basel). 2021 Oct 27;23(11):1410. doi: 10.3390/e23111410.
The transient behavior of the finite-buffer queueing model with batch arrivals and generally distributed repeated vacations is analyzed. Such a system has potential applications in modeling the functioning of production systems, computer and telecommunication networks with energy saving mechanism based on cyclic monitoring the queue state (Internet of Things, wireless sensors networks, etc.). Identifying renewal moments in the evolution of the system and applying continuous total probability law, a system of Volterra-type integral equations for the time-dependent queue-size distribution, conditioned by the initial buffer state, is derived. A compact-form solution for the corresponding system written for Laplace transforms is obtained using an algebraic approach based on Korolyuk's potential method. An illustrative numerical example presenting the impact of the service rate, arrival rate, initial buffer state and single vacation duration on the queue-size distribution is attached as well.
分析了具有批量到达和一般分布的重复休假的有限缓冲区排队模型的瞬态行为。这样的系统在对生产系统、基于循环监测队列状态的节能机制的计算机和电信网络（物联网、无线传感器网络等）的运行进行建模方面具有潜在应用。通过识别系统演化中的更新时刻并应用连续全概率定律，推导了一个以初始缓冲区状态为条件的、关于时间相关队列大小分布的沃尔泰拉型积分方程组。使用基于科罗柳克势方法的代数方法，获得了针对拉普拉斯变换编写的相应系统的紧凑形式解。还附上了一个说明性数值示例，展示了服务率、到达率、初始缓冲区状态和单次休假持续时间对队列大小分布的影响。
