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

TL;DR

The paper broadens AoI analysis by allowing non-zero initial ages for arriving updates, a realistic feature in multi-hop and retrial systems. It derives a general closed-form decomposition where the AAoI equals the standard AAoI plus a correction term capturing the interaction between initial age and inter-departure times, with a simple additive form under independence. The framework unifies and extends results for forwarding, tandem queues, and novel models such as heterogeneous M/M/1/1 → HE/M/1/∞ tandems and M/M/1 retrial queues, and provides practical bounds when dependence is unknown. It offers practical guidance for bounding AAoI in complex networks and demonstrates the approach via analytical results and simulations, enabling broader applicability and future optimization of status-update policies.

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.

Figures (9)

• Figure 1: Motivating example for the proposed aged updates framework with some application examples.
• Figure 2: Example of age profile in a model with aged updates. The packet arriving at epoch $t_3$ carries a high age. When the packet is delivered at $t'_3$, it produces an upper jump on the age at the monitor. Disjoint areas $H_n$ are used to calculate the Average Age of Information.
• Figure 3: Error-prone, zero-wait model as a system with aged updates. a) Error-prone, zero-wait standard model. Blue triangles are successful deliveries and generation of a fresh packet, and red triangles ($t_3$ and $t_5$) mark failed deliveries and subsequent retransmission of the same packet. b) Equivalent error-free, zero-wait model with aged updates. Failed transmissions in the standard model can be thought of as inducing an independent process of initial ages $A_n$.
• Figure 4: Correction term and bounds provided by \ref{['cor::interval']} in the error-prone, zero-wait model.
• Figure 5: Sample path for age in a tandem of two queues M/M/1/$\infty$$\rightarrow$ M/M/1/$\infty$. Packets arriving at the second queue have a positive age equal to the system time in the first queue. Note that $Y_n^{(1)}$ are inter-departure times in the first queue and inter-arrival times in the second queue.

Theorems & Definitions (23)

• Theorem 1
• Proof
• Remark
• Remark
• Corollary 1
• Proposition 1
• Theorem 2: kam:2022, Theorem 3
• Lemma 1
• Remark
• Remark