On the Age of Information in Single-Server Queues with Aged Updates

TL;DR

This paper extends the analysis of Age of Information (AoI) to scenarios where packets have a non-zero initial age, deriving new formulas and bounds for average AoI in various queueing systems, including complex tandem queues.

Contribution

It introduces a generalized AoI expression accounting for initial packet age and dependency structures, applicable to multiple queueing scenarios, including previously unsolved tandem queues.

Findings

Derived an expression for average AoI with non-zero initial age.

Established bounds for the correction term involving packet age and inter-departure times.

Validated the approach on complex queueing models, including tandem queues.

Abstract

The Age of Information (AoI) is a performance metric that quantifies the freshness of data in systems where timely updates are critical. Most state-of-the-art methods typically assume that packets enter the monitored system with zero age, neglecting situations, such as those prevalent in multi-hop networks or distributed sensing, where packets experience prior delays. In this paper, the AoI is investigated when packets have a non-zero initial age. We derive an expression for the average AoI in this setting, showing that it equals the standard AoI plus a correction term involving the correlation between packet age and inter-departure times. When these variables are independent, the expression simplifies to an additive correction equal to the mean initial age. In cases where the dependency structure is unknown, we also establish lower and upper bounds for the correction term. We demonstrate the applicability of our approach across various queueing scenarios, such as forwarding, tandem, and retrial queues. Additionally, we explore the accuracy of the derived bounds on a tandem composed of several queues, a model that has not yet been analytically solved from an age perspective.