Exact Analysis of the Age of Information in the Multi-Source M/GI/1 Queueing System
Yoshiaki Inoue, Tetsuya Takine
TL;DR
This work tackles the open problem of the exact AoI distribution in a multi-source FCFS M/GI/1 queue with competing sensor-monitor pairs. It delivers an explicit Laplace-Stieltjes transform for the stationary AoI, $f_A^*(s)$, by harnessing the double Laplace transform (Takagi transform) of the transient workload and expresses the result in a product form that separates delay and peak-AoI contributions. A closed-form mean AoI, $E[A]$, is provided, featuring a gamma parameter $\gamma=\psi(\lambda)$ that solves $\gamma-\lambda-\lambda^++\lambda^+ f_{H^+}^*(\gamma)=0$, with the special case $\lambda^+\to0$ recovering the single-source M/GI/1 result. Overall, the paper offers a tractable analytic framework for analyzing AoI in multi-source shared-resource networks, with implications for sensing and IoT systems where freshness of information is critical.
Abstract
We consider a situation that multiple monitoring applications (each with a different sensor-monitor pair) compete for a common service resource such as a communication link. Each sensor reports the latest state of its own time-varying information source to its corresponding monitor, incurring queueing and processing delays at the shared resource. The primary performance metric of interest is the age of information (AoI) of each sensor-monitor pair, which is defined as the elapsed time from the generation of the information currently displayed on the monitor. Although the multi-source first-come first-served (FCFS) M/GI/1 queue is one of the most fundamental model to describe such competing sensors, its exact analysis has been an open problem for years. In this paper, we show that the Laplace-Stieltjes transform (LST) of the stationary distribution of the AoI in this model, as well as the mean AoI, is given by a simple explicit formula, utilizing the double Laplace transform of the transient workload in the M/GI/1 queue.