A General Formula for the Stationary Distribution of the Age of Information and Its Application to Single-Server Queues

TL;DR

This paper derives a universal formula for the stationary distribution of the age of information (AoI) applicable to various systems and applies it to analyze AoI in different single-server queue disciplines, providing new insights into their performance.

Contribution

It introduces a general formula for the stationary distribution of AoI and demonstrates its application across multiple queue disciplines, expanding understanding of AoI behavior in these systems.

Findings

Derived a general formula linking AoI distribution to system delay and peak AoI.

Analyzed AoI in FCFS, preemptive LCFS, and non-preemptive LCFS queues.

Compared mean AoI across different queue disciplines and service time distributions.

Abstract

This paper considers the stationary distribution of the age of information (AoI) in information update systems. We first derive a general formula for the stationary distribution of the AoI, which holds for a wide class of information update systems. The formula indicates that the stationary distribution of the AoI is given in terms of the stationary distributions of the system delay and the peak AoI. To demonstrate its applicability and usefulness, we analyze the AoI in single-server queues with four different service disciplines: first-come first-served (FCFS), preemptive last-come first-served (LCFS), and two variants of non-preemptive LCFS service disciplines. For the FCFS and the preemptive LCFS service disciplines, the GI/GI/1, M/GI/1, and GI/M/1 queues are considered, and for the non-preemptive LCFS service disciplines, the M/GI/1 and GI/M/1 queues are considered. With these results, we further show comparison results for the mean AoI's in the M/GI/1 and GI/M/1 queues under those service disciplines.