Componentwise accurate fluid queue computations using doubling algorithms

Abstract

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 et al. (Accurate solutions of M-matrix algebraic Riccati equations, 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 from converging 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.

Highly accurate doubling algorithms for M-matrix algebraic Riccati equations

“…The sketch applies to the newer doubling algorithms: SDA-ss [6] and ADDA [24], too, and usually work well but the sketch is not completely satisfactory as cancellations persist in calculating triple representations of the involved M-matrices in the doubling iteration kernel. Recently, Nguyen and Poloni [20] discovered a very robust implementation of the doubling algorithms for the special case where W satisfies Our cancellation-free construction is, however, made possible by an introduction of novel recursively computable auxiliary nonnegative vectors that were not needed in the case of (1. which can be symbolically obtained by multiplying both sides of (1.1) by X −1 and then setting Y = X −1 . This action is only symbolic because X may not be square, not to mention being possibly singular.…”

“…Consequently, the accuracy in the computed (I − Y k X k ) −1 and (I − X k Y k ) −1 may not achieve the best possible or come to anywhere close to that. As pointed out in [20], such accuracy loss may not be reparable and thus can prevent doubling algorithms from converging in the componentwise sense.…”

“…For the special case where W in (1.2) is an irreducible singular M-matrix with W1 1 1 m+n = 0, (3.2) it is discovered in [20] that explicit triplet representations of I − X k Y k and I − Y k X k can be constructed without cancellation. Consequently, the doubling algorithms can compute to high componentwise relative accuracy, independent of the conditioning in inverting all I − X k Y k and I − Y k X k .…”

“…In the case when it is nonsingular, W has no (right) null vector at all. So the assumption (3.2) limits the applicability of the accurate implementation of SDAs in [20].…”

“…Our method of construction of the triplet representations for I − X k Y k and I −Y k X k relies on introducing recursively computable auxiliary nonnegative vectors which are far from obvious from the construction of Nguyen and Poloni [20].…”