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Effects of the Auto-Correlation of Delays on the Age of Information: A Gaussian Process Framework

Atsushi Inoie, Yoshiaki Inoue

TL;DR

This work addresses freshness in information updates by allowing delays to be driven by a nonnegative continuous-time virtual delay process, capturing temporal dependence beyond i.i.d. assumptions. It derives the transient AoI distribution and uses stochastic orders to show that stronger delay dependence degrades AoI, with i.i.d. delays achieving the best performance for given marginals. The paper then specializes to a stationary Gaussian delay framework, providing a tractable characterization of the AoI distribution and leveraging the Markov property in OU-type processes to enable efficient computation. Numerical experiments reveal that AoI tails become heavier as delay autocorrelation grows, particularly for small packet-generation intervals, highlighting the practical impact of delay dependence on real-time monitoring systems. Overall, the framework offers a principled way to analyze and compare AoI under realistic, correlated delay dynamics and guides design choices for aging-sensitive networks and applications.

Abstract

The age of information (AoI) has been studied actively in recent years as a performance measure for systems that require real-time performance, such as remote monitoring systems via communication networks. The theoretical analysis of the AoI is usually formulated based on explicit system modeling, such as a single-server queueing model. However, in general, the behavior of large-scale systems such as communication networks is complex, and it is usually difficult to express the delay using simple queueing models. In this paper, we consider a framework in which the sequence of delays is composed from a non-negative continuous-time stochastic process, called a virtual delay process, as a new modeling approach for the theoretical analysis of the AoI. Under such a framework, we derive an expression for the transient probability distribution of the AoI and further apply the theory of stochastic orders to prove that the high dependence of the sequence of delays leads to the degradation of AoI performance. We further consider a special case in which the sequence of delays is generated from a stationary Gaussian process, and we discuss the sensitivity of the AoI to second-order statistics of the delay process through numerical experiments.

Effects of the Auto-Correlation of Delays on the Age of Information: A Gaussian Process Framework

TL;DR

This work addresses freshness in information updates by allowing delays to be driven by a nonnegative continuous-time virtual delay process, capturing temporal dependence beyond i.i.d. assumptions. It derives the transient AoI distribution and uses stochastic orders to show that stronger delay dependence degrades AoI, with i.i.d. delays achieving the best performance for given marginals. The paper then specializes to a stationary Gaussian delay framework, providing a tractable characterization of the AoI distribution and leveraging the Markov property in OU-type processes to enable efficient computation. Numerical experiments reveal that AoI tails become heavier as delay autocorrelation grows, particularly for small packet-generation intervals, highlighting the practical impact of delay dependence on real-time monitoring systems. Overall, the framework offers a principled way to analyze and compare AoI under realistic, correlated delay dynamics and guides design choices for aging-sensitive networks and applications.

Abstract

The age of information (AoI) has been studied actively in recent years as a performance measure for systems that require real-time performance, such as remote monitoring systems via communication networks. The theoretical analysis of the AoI is usually formulated based on explicit system modeling, such as a single-server queueing model. However, in general, the behavior of large-scale systems such as communication networks is complex, and it is usually difficult to express the delay using simple queueing models. In this paper, we consider a framework in which the sequence of delays is composed from a non-negative continuous-time stochastic process, called a virtual delay process, as a new modeling approach for the theoretical analysis of the AoI. Under such a framework, we derive an expression for the transient probability distribution of the AoI and further apply the theory of stochastic orders to prove that the high dependence of the sequence of delays leads to the degradation of AoI performance. We further consider a special case in which the sequence of delays is generated from a stationary Gaussian process, and we discuss the sensitivity of the AoI to second-order statistics of the delay process through numerical experiments.
Paper Structure (8 sections, 6 theorems, 39 equations, 26 figures)

This paper contains 8 sections, 6 theorems, 39 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. System model
  4. Transient analysis
  5. Special case: model with the sequence of delays composed from a stationary Gaussian process
  6. Numerical Experiments
  7. Conclusion
  8. Computation of the AoI distribution for virtual delay processes composed from the OU process

Key Result

Theorem 1

The complementary cumulative distribution function (CDF) of the AoI at time $t$ ($t \geq 0$) is given by

Figures (26)

  • Figure 1: An example of a monitoring system.
  • Figure 2: Top: An example of sample paths of $(X_t)_{t\ge 0}$ and $(D_n)_{n=0,1,\ldots}$. Bottom: An example of a sample path of $(A_t)_{t\ge 0}$.
  • Figure 3: Sample paths of virtual delay processes $(X_t)_{t\ge 0}$ composed from a stationary Gaussian process (Top: Example 1, Bottom: Example 2)
  • Figure 4: Simulation results of the AoI for Example 1 (left) and heat map of probability mass function (right): $s=0.75$, $c=10$, $\tau=2.0$.
  • Figure 6: $c=0.1$, $\tau=2.0$
  • ...and 21 more figures

Theorems & Definitions (16)

  • Remark 1
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Definition 1
  • Definition 2
  • Theorem 2
  • ...and 6 more