Abstract We apply physical interpretations to construct algorithms for the key matrix G of discrete-time quasi-birth-and-death (dtQBD) processes which records the probability of the process reaching level for the first time given the process starts in level n. The construction of G and its z-transform was motivated by the work on stochastic fluid models (SFMs). In this methodology, we first write a summation expression for by considering a physical interpretation similar to that of an algorithm. Next, we construct the corresponding iterative scheme, and prove its convergence to We construct in detail two algorithms for one of which we show is Newton’s Method. We then generate a comprehensive set of algorithms, an additional one of which is quadratically convergent and has not been seen in the literature before. Using symmetry arguments, we generate analogous algorithms for and again find that two are quadratically convergent. One of these can be seen to be equivalent to applying Newton’s Method to evaluate and the other is again novel.

中文翻译：

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通过物理解释构建离散时间准生死过程算法

摘要 我们应用物理解释来构造离散时间准生死 (dtQBD) 过程的关键矩阵 G 的算法，该算法记录了给定过程在第 n 级开始的过程中首次达到水平的概率。G 及其 z 变换的构建受到随机流体模型 (SFM) 工作的启发。在这种方法中，我们首先通过考虑类似于算法的物理解释来编写求和表达式。接下来，我们构造相应的迭代方案，并证明其收敛性。我们详细构造了两种算法，其中一种是牛顿法。然后我们生成了一套全面的算法，其中一个额外的算法是二次收敛的，以前在文献中没有见过。使用对称参数，我们生成了类似的算法，并再次发现两个是二次收敛的。可以看出其中之一相当于应用牛顿方法进行评估，而另一个又是新颖的。