A General Formula for the Stationary Distribution of the Age of Information and Its Application to Single-Server Queues (1804.06139v2)

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.

• The paper introduces a unified analytical framework that derives the stationary distribution of AoI using system delay and peak AoI metrics.
• It applies the methodology across FCFS, preemptive LCFS, and non-preemptive LCFS queues, providing explicit formulas and comparative analyses.
• Numerical examples validate the approach and demonstrate practical strategies for optimizing information freshness in diverse network systems.

Analytical Framework for Stationary Distribution of the Age of Information

The paper under discussion explores the stationary distribution of the Age of Information (AoI) and applies its findings specifically to single-server queue systems exhibiting diverse service disciplines. The Age of Information has emerged as a significant metric in assessing the freshness of data in communication and information systems. The authors present a general formula that enables the determination of the stationary distribution of AoI across a broad class of systems, substantially broadening the scope and applicability of traditional queuing theory techniques.

Contribution and Methodology

Inoue, Masuyama, Takine, and Tanaka's paper provides a unifying analytical framework connecting AoI with standard queuing parameters, namely system delay and peak AoI. The core methodological achievement is the derivation of the AoI's stationary distribution in ergodic update systems, expressed in terms of the easily analyzable distributions of system delay and peak AoI. This synthesis significantly reduces the complexity involved in AoI analysis compared to prior approaches focusing primarily on mean AoI.

Application to Queueing Systems

The utility of this general framework is demonstrated through its application to single-server queues operating under multiple disciplines: First-Come First-Served (FCFS), preemptive Last-Come First-Served (LCFS), and two non-preemptive LCFS variants. The results for these queuing models provide insights into the intricacies of different service disciplines on AoI:

1. FCFS Queues: The stationary distribution is derived using established queueing results, providing bounds and explicit formulas for various cases like M/GI/1 and GI/M/1 queues.
2. Preemptive LCFS Queues: The authors tackle general GI/GI/1 queues under preemptive service, highlighting unique considerations when arriving packets preempt ongoing services. This analysis is particularly relevant, as it shows conditions under which service variability most influences AoI, offering comparisons that extend understanding of service-time distributions' impact.
3. Non-Preemptive LCFS Queues: Results for these queues include setups both with and without packet discarding, and offer comparisons situating these within wider system performance evaluations.

Numerical Results and Implications

The paper integrates numerical examples to explore the practical implications of the derived results. These include comparisons of mean AoI across service types with varied traffic intensity and service-time variability, playing a pivotal role in provisioning strategies and performance optimizations in systems where information freshness is critical.

Theoretical and Practical Implications

The introduced methodology transitions AoI analysis from isolated use-case studies to a broader applicable theorem grounded in classical queueing theory. This presents a paradigm that not only simplifies the computation of AoI but also informs optimal design considerations for systems where data freshness is pivotal. Practically, this research lays a foundational understanding crucial for optimizing system parameters in diverse fields, such as IoT networks, telecommunications, and status management systems.

Future Directions

Future advancements in this research area may include extending these models to more complex network structures or incorporating additional variables such as correlated arrivals and priority-based queuing. Potential expansions could also explore how these models perform under varying degrees of system congestion, offering richer insights into practical deployment scenarios.

This paper constitutes a significant step in AoI research, building a comprehensive, easily generalized analytical foundation applicable across a variety of real-world systems. Through explicit derivation of AoI distributions and comparative evaluation, it paves the way for subsequent inquiries and refinements in the optimization of informational systems.