Two main methods are used to solve continuous-time quasi birth-and-death processes: matrix geometric (MG) and probability generating functions (PGFs). MG requires a numerical solution (via successive substitutions) of a matrix quadratic equation A0 + RA1 + R2A2 = 0. PGFs involve a row vector $\vec{G}(z)$ of unknown generating functions satisfying $H(z)\vec{G}{(z)^\textrm{T}} = \vec{b}{(z)^\textrm{T}},$ where the row vector $\vec{b}(z)$ contains unknown “boundary” probabilities calculated as functions of roots of the matrix H(z). We show that: (a) H(z) and $\vec{b}(z)$ can be explicitly expressed in terms of the triple A0, A1, and A2; (b) when each matrix of the triple is lower (or upper) triangular, then (i) R can be explicitly expressed in terms of roots of $\det [H(z)]$ ; and (ii) the stability condition is readily extracted.

中文翻译：

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通过使用矩阵几何分析和概率生成函数方法之间的关系，为连续时间 QBD 过程提供显式解决方案

解决连续时间准生死过程的主要方法有两种：矩阵几何 (MG) 和概率生成函数 (PGF)。MG 需要矩阵二次方程的数值解（通过连续替换）一种0+类风湿关节炎1+R2一种2= 0. PGF 涉及行向量 $\vec{G}(z)$ 的未知生成函数满足 $H(z)\vec{G}{(z)^\textrm{T}} = \vec{b}{(z)^\textrm{T}},$ 行向量在哪里 $\vec{b}(z)$ 包含作为矩阵根函数计算的未知“边界”概率H(z）。我们表明：(a)H(z） 和 $\vec{b}(z)$ 可以用三元组明确表示一种0,一种1， 和一种2; (b) 当三元组的每个矩阵都是下（或上）三角形时，则 (i)R可以用根的形式明确表示 $\det [H(z)]$ ; (ii) 容易提取稳定条件。