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Augmenting Automatic Differentiation for a Single-Server Queue via the Leibniz Integral Rule

Michael C. Fu

Abstract

New recursive estimators for computing higher-order derivatives of mean queueing time from a single sample path of a first-come, first-served single-server queue are presented, derived using the well-known Lindley equation and applying the Leibniz integral rule of differential calculus. Illustrative examples are provided.

Augmenting Automatic Differentiation for a Single-Server Queue via the Leibniz Integral Rule

Abstract

New recursive estimators for computing higher-order derivatives of mean queueing time from a single sample path of a first-come, first-served single-server queue are presented, derived using the well-known Lindley equation and applying the Leibniz integral rule of differential calculus. Illustrative examples are provided.

Paper Structure

This paper contains 8 sections, 8 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. Estimation Problem
  4. Single-Server Queue Inputs
  5. FCFS Dynamics via the Lindley Equation
  6. Derivatives
  7. Higher-Order Derivative Estimators Derived via the Leibniz Integral Rule
  8. Conclusions and Future Research

Key Result

Lemma 1

Under Assumptions ass1 and ass2, except at the point $W_i + S_i = A_i$, where they are undefined. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (23)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Remark 2
  • Example 1
  • Example 2
  • Example 3
  • Remark 3
  • ...and 13 more