Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Error analysis of Abate--Whitt methods for Inverse Laplace Transforms and a new algorithm for queuing theory applications

Nikita Deniskin, Federico Poloni

TL;DR

This work analyzes the accuracy of Abate–Whitt inverse Laplace transform methods by linking ILT error to the quality of rational approximants of the exponential on a function-specific region. It introduces TAME, a family of Abate–Whitt methods generated via the AAA rational-approximation algorithm, with parameters tailored to ME/SE/LS function classes and to queuing-theory applications such as phase-type distributions and Markov-modulated fluid queues. The authors derive error bounds, propose moment-based estimates, and address practical numerical issues, including floating-point stability and region/point selection strategies. Numerical experiments demonstrate that TAME achieves comparable or better accuracy with significantly fewer function evaluations and improved stability relative to classical AW methods. The framework provides actionable guidance for applying ILT in queuing contexts while controlling computation cost and precision loss.

Abstract

We study the accuracy of a class of methods to compute the Inverse Laplace Transform, the so-called \emph{Abate--Whitt methods} [Abate, Whitt 2006], which are based on a linear combination of evaluations of $\widehat{f}$ in a few points. We provide error bounds which relate the accuracy of a method to the rational approximation of the exponential function. We specialize our analysis to applications in queuing theory, a field in which Abate--Whitt methods are often used; in particular, we study phase-type distributions and Markov-modulated fluid models (or \emph{fluid queues}). We use a recently developed algorithm for rational approximation, the AAA algorithm [Nakatsukasa, Sète, Trefethen 2018], to produce a new family of methods, which we call TAME. The parameters of these methods are constructed depending on a function-specific domain $Ω$; we provide a quasi-optimal choice for certain families of functions. We discuss numerical issues related to floating-point computation, and we validate our results through numerical experiments which show that the new methods require significantly fewer function evaluations to achieve an accuracy that is comparable (or better) to that of the classical methods.

Error analysis of Abate--Whitt methods for Inverse Laplace Transforms and a new algorithm for queuing theory applications

TL;DR

This work analyzes the accuracy of Abate–Whitt inverse Laplace transform methods by linking ILT error to the quality of rational approximants of the exponential on a function-specific region. It introduces TAME, a family of Abate–Whitt methods generated via the AAA rational-approximation algorithm, with parameters tailored to ME/SE/LS function classes and to queuing-theory applications such as phase-type distributions and Markov-modulated fluid queues. The authors derive error bounds, propose moment-based estimates, and address practical numerical issues, including floating-point stability and region/point selection strategies. Numerical experiments demonstrate that TAME achieves comparable or better accuracy with significantly fewer function evaluations and improved stability relative to classical AW methods. The framework provides actionable guidance for applying ILT in queuing contexts while controlling computation cost and precision loss.

Abstract

We study the accuracy of a class of methods to compute the Inverse Laplace Transform, the so-called \emph{Abate--Whitt methods} [Abate, Whitt 2006], which are based on a linear combination of evaluations of in a few points. We provide error bounds which relate the accuracy of a method to the rational approximation of the exponential function. We specialize our analysis to applications in queuing theory, a field in which Abate--Whitt methods are often used; in particular, we study phase-type distributions and Markov-modulated fluid models (or \emph{fluid queues}). We use a recently developed algorithm for rational approximation, the AAA algorithm [Nakatsukasa, Sète, Trefethen 2018], to produce a new family of methods, which we call TAME. The parameters of these methods are constructed depending on a function-specific domain ; we provide a quasi-optimal choice for certain families of functions. We discuss numerical issues related to floating-point computation, and we validate our results through numerical experiments which show that the new methods require significantly fewer function evaluations to achieve an accuracy that is comparable (or better) to that of the classical methods.
Paper Structure (16 sections, 19 theorems, 109 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 16 sections, 19 theorems, 109 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Inverse Laplace Transforms and the Abate--Whitt framework
  4. Accuracy of AW methods through rational approximants
  5. Bounds based on moments
  6. Laplace transforms in queuing theory
  7. Continuous-time Markov chains
  8. Fluid queues
  9. AAA approximation
  10. AAA algorithm
  11. Modifications to AAA
  12. Computing TAME parameters
  13. Floating-point precision issues
  14. Choice of $\Omega$
  15. Choice of $Z$
  16. ...and 1 more sections

Key Result

Theorem 3.2

Let $f(t)$ be a SE function Let $(w_n, \beta_n)_{n=1}^N$ be an Abate--Whitt method that is $\varepsilon$-accurate on a region $\Omega$ which contains $\alpha_1t, \dots, \alpha_Mt$. Then, the error of the method at point $t$ is bounded by

Figures (3)

  • Figure 1: Approximation error $\varepsilon$ and magnitude of weights $\max\lvert w_n\rvert$ when running AAA with $\Omega=B(-5,5)$ in either binary64 or VPA with 100 significant digits.
  • Figure 2: Errors $\varepsilon$ obtained by TAME on $\Omega=B(-r,r)$, with the corresponding values of $n$ and $n'$.
  • Figure 3: Error estimator $\eta$ obtained by TAME on $\Omega=B(-r,r)$, with the corresponding values of $n$ and $n'$.

Theorems & Definitions (44)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Definition 3.4
  • Lemma 3.5
  • ...and 34 more