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Taming the Heavy Tail: Age-Optimal Preemption

Aimin Li, Yiğit İnce, Elif Uysal

TL;DR

This work addresses aged-of-information optimization in a continuous-time setting with joint sampling and preemption under general, potentially heavy-tailed service times. It formulates the problem as an impulse-controlled PDMP and derives coupled integral average-cost equations, reducing the busy-phase decision to a one-dimensional optimal-stopping problem on a busy-start boundary. For exponential service times, it proves a threshold structure for both sampling and preemption, while for general heavy-tailed distributions it develops a heavy-tail-accelerated policy-iteration algorithm with a hybrid action grid and far-field closure, demonstrating large gains in simulations. The results reveal that delaying strategies and preemption can dramatically reduce information-age costs, with a surprising insight that delay variance can become advantageous for freshness under preemption, highlighting practical implications for real-time systems with variable delays.

Abstract

This paper studies a continuous-time joint sampling-and-preemption problem, incorporating sampling and preemption penalties under general service-time distributions. We formulate the system as an impulse-controlled piecewise-deterministic Markov process (PDMP) and derive coupled integral average-cost optimality equations via the dynamic programming principle, thereby avoiding the smoothness assumptions typically required for an average-cost Hamilton-Jacobi-Bellman quasi-variational inequality (HJB-QVI) characterization. A key invariance in the busy phase collapses the dynamics onto a one-dimensional busy-start boundary, reducing preemption control to an optimal stopping problem. Building on this structure, we develop an efficient policy iteration algorithm with heavy-tail acceleration, employing a hybrid (uniform/log-spaced) action grid and a far-field linear closure. Simulations under Pareto and log-normal service times demonstrate substantial improvements over AoI-optimal non-preemptive sampling and zero-wait baselines, achieving up to a 30x reduction in average cost in heavy-tailed regimes. Finally, simulations uncover a counterintuitive insight: under preemption, delay variance, despite typically being a liability, can become a strategic advantage for information freshness.

Taming the Heavy Tail: Age-Optimal Preemption

TL;DR

This work addresses aged-of-information optimization in a continuous-time setting with joint sampling and preemption under general, potentially heavy-tailed service times. It formulates the problem as an impulse-controlled PDMP and derives coupled integral average-cost equations, reducing the busy-phase decision to a one-dimensional optimal-stopping problem on a busy-start boundary. For exponential service times, it proves a threshold structure for both sampling and preemption, while for general heavy-tailed distributions it develops a heavy-tail-accelerated policy-iteration algorithm with a hybrid action grid and far-field closure, demonstrating large gains in simulations. The results reveal that delaying strategies and preemption can dramatically reduce information-age costs, with a surprising insight that delay variance can become advantageous for freshness under preemption, highlighting practical implications for real-time systems with variable delays.

Abstract

This paper studies a continuous-time joint sampling-and-preemption problem, incorporating sampling and preemption penalties under general service-time distributions. We formulate the system as an impulse-controlled piecewise-deterministic Markov process (PDMP) and derive coupled integral average-cost optimality equations via the dynamic programming principle, thereby avoiding the smoothness assumptions typically required for an average-cost Hamilton-Jacobi-Bellman quasi-variational inequality (HJB-QVI) characterization. A key invariance in the busy phase collapses the dynamics onto a one-dimensional busy-start boundary, reducing preemption control to an optimal stopping problem. Building on this structure, we develop an efficient policy iteration algorithm with heavy-tail acceleration, employing a hybrid (uniform/log-spaced) action grid and a far-field linear closure. Simulations under Pareto and log-normal service times demonstrate substantial improvements over AoI-optimal non-preemptive sampling and zero-wait baselines, achieving up to a 30x reduction in average cost in heavy-tailed regimes. Finally, simulations uncover a counterintuitive insight: under preemption, delay variance, despite typically being a liability, can become a strategic advantage for information freshness.
Paper Structure (26 sections, 8 theorems, 64 equations, 1 figure, 1 table)

This paper contains 26 sections, 8 theorems, 64 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related Works
  4. Contributions and Novelties of This Work
  5. System Model and Problem Formulation
  6. Residual Life and Hazard Rate
  7. Impulse Control Formulation
  8. Problem Formulation
  9. Average Cost Optimality Equations
  10. Coupled HJB-QVI Heuristic
  11. Optimal Stopping-Time
  12. Case Study
  13. Optimal Policy Structure without Preemption
  14. Exponential Service Times: Explicit Solutions
  15. Numerical Solutions and Simulation Results
  16. ...and 11 more sections

Key Result

Proposition 1

Let $h_I(\Delta)\triangleq V(\Delta,0,I)$ and $h_B(\Delta,b)\triangleq V(\Delta,b,B)$ denote the relative value functions in the idle and busy modes, respectively. Heuristically, they satisfy: I. Idle mode (sampling vs. waiting): Define the (heuristic) waiting and sampling regions by the complementarity conditions Thus, sampling is optimal when $\Delta\in\mathcal{R}_I$, while waiting is optimal w

Figures (1)

  • Figure 1: System Model. The source takes action $s$ (sample a fresh update) or $p$ (drop in-service update + sample a fresh update).

Theorems & Definitions (19)

  • Definition 1: Residual Life and Hazard Rate shortle2018fundamentals
  • Proposition 1: Heuristic Coupled HJB-QVI
  • proof
  • Theorem 1: Integral ACOEs and Optimal-stopping Reduction
  • proof
  • Remark 1
  • Theorem 2: Structure of Optimal Sampling and Preemption
  • proof
  • Remark 2
  • Lemma 2: Idle-phase reduction to deterministic thresholds
  • ...and 9 more