如何解决具有无限状态空间的二维马尔可夫链

Question

我有二维马尔可夫链，我想计算稳态概率，然后计算基本性能测量值，如预期客户数量，预计等待时间等。您可以查看下面的转换率图表链接：

http://tinypic.com/view.php?pic=2n063dd&s=8

当我搜索解决方法时，会出现矩阵几何和光谱扩展方法。我试过矩阵几何方法，但是因为我的马尔可夫链不是重复的，所以它不起作用。

我读了一些论文（例如，一类马尔可夫模型的谱扩展解：应用和矩阵几何方法的比较），但我无法弄清楚如何创建矩阵以及什么是稳态概率。

1. 光谱扩展方法是否需要“重复过程”作为矩阵几何方法呢？如果不是，如何申请我的问题？
2. 还有其他计算方法吗？

感谢您的帮助！

阿里

Answer 1

First, there is no stable solution method for two-way infinite lattice strip. At least one variable should be capacitated. Second, the following are the most known solution methods for two-dimensional Markov chains with semi-infinite or finite state space:

• Spectral Expansion Method
• Matrix Geometric Method
• Block Gauss-Seidel Method
• Seelen's Method

All methods require high computational work. Experimental studies show that for semi-infinite lattice strip, as the capacitated variable exceeds 50, solution may not be trustable. Also there is a state explosion problem beyond that threshold. To overcome the state explosion problem, iterative methods are used such Gauss-Seidel and Seelen's methods.

Regarding my problem, I determined capacity for both variables. After a search in the literature, block Gauss-Seidel Iterative method seems to be the most appropriate method to apply my problem.

Thank you.