论文标题
计算$ gi^x/m/c/n $型号的固定概率的有效方法
An Efficient Method to Compute the Stationary Probabilities of the $GI^X/M/c/N$ Model
论文作者
论文摘要
考虑使用$ C $服务器，一般到达批处理时间，有限缓冲区和指数服务时间的批处理$ gi^x/m/c/n $型号。到达批处理时间，批处理大小和服务时间为$ i.i.d。$，彼此独立。在本文中，我们给出了一种简单的有效算法，以得出稳态系统尺寸概率的精确解决方案。起点是计算在相应的$ g/m/c $模型的系统到达时期观察到的嵌入的马尔可夫链的一步过渡概率。一步过渡概率是通过将数值集成问题转换为有限总和来准确计算的。另一个关键贡献是通过使用简单而直观的方法来生成批处理模型的过渡概率，以扩展标准的$ GI/m/c $模型的结果，以便在有和没有有限缓冲区的情况下，并在有限的情况下使用有限的缓冲区，并具有部分和完整的批次拒绝。此外，我们开发了一种有效的稳定算法，该算法可以准确地解决比以前已知的更多服务器的问题。我们给出数值示例以证明我们的方法的性能。}
Consider the batch-arrival $GI^X/M/c/N$ model with $c$ servers, general inter-arrival batch times, finite buffer, and exponential service times. Inter-arrival batch times, batch sizes, and service times are $i.i.d.$ and independent of each other. In this article we give a simple efficient algorithm to derive an exact solution for the steady state system size probabilities. The starting point is computing the one-step transition probabilities of the imbedded Markov chain observed at the system arrival epochs of the the corresponding $G/M/c$ model. The one-step transition probabilities are computed exactly by converting a numerical integration problem into a finite sum. Another key contribution is generating the transition probabilities of the batch-arrival model by using a simple and intuitive method to extend the results of the standard $GI/M/c$ model to batch arrivals with and without a finite buffer, and in the case of finite buffer with partial and full batch rejection. Moreover, we develop an efficient stable algorithm that can accurately solve problems with a larger number of servers than previously known. We give numerical examples to demonstrate the performance of our method.}
