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Analysis of Age of Information for A Discrete-Time hybrid Dual-Queue System

Zhengchuan Chen, Yi Qu, Nikolaos Pappas, Chaowei Tang, Min Wang, Tony Q. S. Quek

TL;DR

This work analyzes the Age of Information (AoI) and Peak AoI (PAoI) for a discrete-time dual-queue status updating system (Geo-D) with zero-wait policy, where one sensor has geometric service times and the other has deterministic service times. Using a graphical/state-based analysis, closed-form expressions for both average AoI and PAoI are derived as functions of the Bernoulli parameter $p$ and the period length $T$, capturing the interaction of random and deterministic services in parallel queues. The results reveal that the dual-queue geometry can significantly reduce AoI/PAoI compared to single-queue systems (up to about 39.74% for AoI and 27.71% for PAoI in certain regimes), and that randomness can aid freshness in many scenarios. The paper also establishes a rigorous connection to continuous-time models, proving that continuous-time AoI results are the limiting case of the discrete-time formulas as the slot length tends to zero, thereby providing a general framework that encompasses both discrete- and continuous-time analyses and guiding design of timely status updates in IoT systems.

Abstract

Using multiple sensors to update the status process of interest is promising in improving the information freshness. The unordered arrival of status updates at the monitor end poses a significant challenge in analyzing the timeliness performance of parallel updating systems. This work investigates the age of information (AoI) of a discrete-time dual-sensor status updating system. Specifically, the status update is generated following the zero-waiting policy. The two sensors are modeled as a geometrically distributed service time queue and a deterministic service time queue in parallel. We derive the analytical expressions for the average AoI and peak AoI using the graphical analysis method. Moreover, the connection of average AoI between discrete-time and continuous-time systems is also explored. It is shown that the AoI result of the continuous-time system is a limit case of that of the corresponding discrete-time system. Hence, the AoI result of the discrete-time system is more general than the continuous one. Numerical results validate the effectiveness of our analysis and further show that randomness of service time contributes more AoI reduction than determinacy of service time in dual-queue systems in most cases, which is different from what is known about the single-queue system.

Analysis of Age of Information for A Discrete-Time hybrid Dual-Queue System

TL;DR

This work analyzes the Age of Information (AoI) and Peak AoI (PAoI) for a discrete-time dual-queue status updating system (Geo-D) with zero-wait policy, where one sensor has geometric service times and the other has deterministic service times. Using a graphical/state-based analysis, closed-form expressions for both average AoI and PAoI are derived as functions of the Bernoulli parameter and the period length , capturing the interaction of random and deterministic services in parallel queues. The results reveal that the dual-queue geometry can significantly reduce AoI/PAoI compared to single-queue systems (up to about 39.74% for AoI and 27.71% for PAoI in certain regimes), and that randomness can aid freshness in many scenarios. The paper also establishes a rigorous connection to continuous-time models, proving that continuous-time AoI results are the limiting case of the discrete-time formulas as the slot length tends to zero, thereby providing a general framework that encompasses both discrete- and continuous-time analyses and guiding design of timely status updates in IoT systems.

Abstract

Using multiple sensors to update the status process of interest is promising in improving the information freshness. The unordered arrival of status updates at the monitor end poses a significant challenge in analyzing the timeliness performance of parallel updating systems. This work investigates the age of information (AoI) of a discrete-time dual-sensor status updating system. Specifically, the status update is generated following the zero-waiting policy. The two sensors are modeled as a geometrically distributed service time queue and a deterministic service time queue in parallel. We derive the analytical expressions for the average AoI and peak AoI using the graphical analysis method. Moreover, the connection of average AoI between discrete-time and continuous-time systems is also explored. It is shown that the AoI result of the continuous-time system is a limit case of that of the corresponding discrete-time system. Hence, the AoI result of the discrete-time system is more general than the continuous one. Numerical results validate the effectiveness of our analysis and further show that randomness of service time contributes more AoI reduction than determinacy of service time in dual-queue systems in most cases, which is different from what is known about the single-queue system.

Paper Structure

This paper contains 25 sections, 5 theorems, 80 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. System Model
  3. Dual-Queue Model
  4. Performance Metric
  5. AoI of the Geo-D System
  6. System State Analysis
  7. State $(0,0)$
  8. State $(0,1)$
  9. State $(0,n)$, $2\leq n\leq T$
  10. State $(k,0)$, $1\leq k\leq T$
  11. State $(k,1)$, $1\leq k\leq T$
  12. State $(k,n)$, $1\leq k\leq T$, $2\leq n\leq T$
  13. Average AoI and PAoI of the Geo-D System
  14. Average PAoI of the Geo-D System
  15. Average AoI of the Geo-D System
  16. ...and 10 more sections

Key Result

Lemma 1

Consider two integers $n,~T\in \mathbb{N^+}$ satisfying $1\leq n \leq T$. It holds that where $1\leq i,j \leq n$, $i\neq j$, and

Figures (8)

  • Figure 1: The considered dual-queue real-time monitoring system.
  • Figure 2: Evolution examples of AoI for the Geo-D system in each state.
  • Figure 3: Theoretical and simulation results of the average AoI and PAoI of Geo-D system under different parameters.
  • Figure 4: Average AoI and reduction ratio of different systems versus service rate $\mu_{\text{A}}=\mu_{\text{B}}$.
  • Figure 5: Normalized average AoI of dual-queue systems versus service rate ratio $\mu_{\text{B}}/\mu_{\text{A}}$.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof