利用倍增算法实现组件级准确的流队列计算
Componentwise accurate fluid queue computations using doubling algorithms
AI总结：
本文提出了一种基于倍增算法的组件级准确流队列计算方法，解决了传统算法在计算稳态密度时的精度问题，提供了显式表达式和误差分析，提高了计算的准确性。
摘要：
Markov-modulated fluid queues 是一种广泛应用的随机过程，用于建模现实应用。评估这些应用的重要性能指标是其稳态行为，这由稳态密度决定。计算它需要求解一个（非对称）M-矩阵代数Riccati方程，确实计算稳态密度是此类方程最重要的应用。Xue, Xu 和 Li [Numer. Math., 2012] 提供了该方程的组件级一阶扰动分析，证明即使在最小的条目中也能以高相对精度计算解，并建议了几种计算它的算法。所有提出算法中的重要步骤是使用所谓的三元组表示，这些是M-矩阵的特殊表示，允许高精度的高斯消去变种，即GTH-like算法。然而，所有算法中所需的M-矩阵的三元组表示并未显式找到。这可能导致精度损失，从而阻止算法在组件级收敛。本文重点研究了结构倍增算法，这是Xue等人提出的最有效的方法之一，并在此基础上提供（i）所需的三元组表示的显式和无抵消表达式，使算法能够真正无抵消地进行；（ii）一种计算最终部分以获得稳态密度的算法；以及（iii）对所得到算法的组件级误差分析，这是此类算法中首次明确的分析。我们还展示了数值结果，以说明我们的方法在准确性方面优于标准（规范准确）算法。
英文摘要：
Markov-modulated fluid queues are popular stochastic processes frequently used for modelling real-life applications. An important performance measure to evaluate in these applications is their steady-state behaviour, which is determined by the stationary density. Computing it requires solving a (nonsymmetric) M-matrix algebraic Riccati equation, and indeed computing the stationary density is the most important application of this class of equations. Xue, Xu and Li [Numer. Math., 2012] provided a componentwise first-order perturbation analysis of this equation, proving that the solution can be computed to high relative accuracy even in the smallest entries, and suggested several algorithms for computing it. An important step in all proposed algorithms is using so-called triplet representations, which are special representations for M-matrices that allow for a high-accuracy variant of Gaussian elimination, the GTH-like algorithm. However, triplet representations for all the M-matrices needed in the algorithm were not found explicitly. This can lead to an accuracy loss that prevents the algorithms to converge in the componentwise sense. In this paper, we focus on the structured doubling algorithm, the most efficient among the proposed methods in Xue et al., and build upon their results, providing (i) explicit and cancellation-free expressions for the needed triplet representations, allowing the algorithm to be performed in a really cancellation-free fashion; (ii) an algorithm to evaluate the final part of the computation to obtain the stationary density; and (iii) a componentwise error analysis for the resulting algorithm, the first explicit one for this class of algorithms. We also present numerical results to illustrate the accuracy advantage of our method over standard (normwise-accurate) algorithms.
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