[Paper Review] Correlation Coefficient Analysis of the Age of Information in Multi-Source Systems

Yukang Jiang, Kiichi Tokuyama+2 more|arXiv (Cornell University)|Sep 23, 2020

Age of Information Optimization7 references4 citations

TL;DR

This paper proposes a closed-form expression for the correlation coefficient of age of information (AoI) in a multi-source M/M/1/1 queueing system with preemption, where multiple sources share a single server. By deriving the Laplace-Stieltjes transform of the stationary AoI distribution and analyzing two-source systems, the authors reveal nontrivial structural dependencies between AoI processes, enabling deeper system-level understanding and management of information freshness in real-time monitoring systems.

ABSTRACT

This paper studies the age of information (AoI) on an information updating system such that multiple sources share one server to process packets of updated information. In such systems, packets from different sources compete for the server, and thus they may suffer from being interrupted, being backlogged, and becoming stale. Therefore, in order to grasp structures of such systems, it is crucially important to study a metric indicating a correlation of different sources. In this paper, we aim to analyze the correlation of AoIs on a single-server queueing system with multiple sources. As our contribution, we provide the closed-form expression of the correlation coefficient of the AoIs. To this end, we first derive the Laplace-Stieltjes transform of the stationary distribution of each AoI for the multiple sources. Some nontrivial properties on the systems are revealed from our analysis results.

Motivation & Objective

• To analyze the correlation between age of information (AoI) values from different sources in a shared-server real-time monitoring system.
• To address the lack of analytical studies on AoI correlation, despite extensive work on average AoI in multi-source systems.
• To develop a framework for quantifying inter-source dependencies in AoI dynamics using stochastic analysis.
• To provide a closed-form expression for the correlation coefficient of AoI in a two-source M/M/1/1 system with preemption.
• To reveal nontrivial system-level properties through analytical results on AoI correlation and distribution.

Proposed method

• Model the system as an M/M/1/1 queue with multiple sources and preemption, where only one packet is processed at a time.
• Derive the Laplace-Stieltjes transform (LST) of the stationary distribution of AoI for each source using renewal theory and memoryless properties of exponential random variables.
• Analyze the inter-arrival and service time structure by conditioning on the number of competing packets before a valid packet transmission.
• Use the memoryless property of exponential distributions to compute conditional expectations of inter-packet intervals and service times.
• Apply the LST results to compute the mean and variance of AoI for each source.
• Derive the joint distribution of AoI for two sources and compute the correlation coefficient using the covariance and marginal variances.

Experimental results

Research questions

• RQ1 What is the analytical expression for the correlation coefficient between the ages of information from two different sources in a shared-server system with preemption?
• RQ2 How do the arrival rates and service rates of different sources affect the correlation of their respective AoI processes?
• RQ3 What structural properties of the multi-source system are revealed by analyzing AoI correlation beyond average AoI metrics?
• RQ4 How does preemption in an M/M/1/1 queue influence the temporal dependence between AoI values of different sources?
• RQ5 Can the stationary distribution of AoI be fully characterized using the Laplace-Stieltjes transform in a multi-source setting?

Key findings

• The correlation coefficient between AoI processes of two sources in an M/M/1/1 system with preemption is derived in closed form, providing a precise measure of inter-source dependency.
• The Laplace-Stieltjes transform of the stationary AoI distribution is derived as $\frac{\lambda_k \mu}{(s + \lambda)(s + \mu) - (\lambda - \lambda_k)\mu}$ for source $k$, enabling exact computation of mean and variance.
• For a two-source system, the joint AoI process is shown to follow a distribution that is the convolution of two exponential random variables with rates $\lambda$ and $\mu$, indicating a specific memory structure.
• The correlation coefficient depends on the relative arrival rates $\lambda_k$ and the service rate $\mu$, revealing non-monotonic behavior under varying load conditions.
• Nontrivial system-level properties emerge from the analysis, such as asymmetric correlation behavior when source rates differ, even under symmetric service conditions.
• The derived correlation coefficient enables better system design and optimization for real-time monitoring systems by quantifying freshness interdependencies across sources.