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Peak Age of Information under Tandem of Queues

Ashirwad Sinha, Shubhransh Singhvi, Praful D. Mankar, Harpreet S. Dhillon

TL;DR

This work addresses the age of information in a multi-hop setting of N tandem unit-capacity queues with memoryless arrivals and services. It introduces a recursive analytical framework to compute the mean peak age of information (PAoI) under preemptive and non-preemptive disciplines, and also yields the mean AoI for the preemptive policy. Key contributions include a detailed state-based analysis with reach probabilities and conditional moments, plus closed-form PAoI expressions for small N (notably N=2) and a scalable method for general N. The results illuminate how service rates and discipline choice influence information freshness, offering a computationally efficient alternative to SHS-based approaches with practical implications for network design.

Abstract

This paper considers a communication system where a source sends time-sensitive information to its destination via queues in tandem. We assume that the arrival process as well as the service process (of each server) are memoryless, and each of the servers has no buffer. For this setup, we develop a recursive framework to characterize the mean peak age of information (PAoI) under preemptive and non-preemptive policies with $N$ servers having different service rates. For the preemptive case, the proposed framework also allows to obtain mean age of information (AoI).

Peak Age of Information under Tandem of Queues

TL;DR

This work addresses the age of information in a multi-hop setting of N tandem unit-capacity queues with memoryless arrivals and services. It introduces a recursive analytical framework to compute the mean peak age of information (PAoI) under preemptive and non-preemptive disciplines, and also yields the mean AoI for the preemptive policy. Key contributions include a detailed state-based analysis with reach probabilities and conditional moments, plus closed-form PAoI expressions for small N (notably N=2) and a scalable method for general N. The results illuminate how service rates and discipline choice influence information freshness, offering a computationally efficient alternative to SHS-based approaches with practical implications for network design.

Abstract

This paper considers a communication system where a source sends time-sensitive information to its destination via queues in tandem. We assume that the arrival process as well as the service process (of each server) are memoryless, and each of the servers has no buffer. For this setup, we develop a recursive framework to characterize the mean peak age of information (PAoI) under preemptive and non-preemptive policies with servers having different service rates. For the preemptive case, the proposed framework also allows to obtain mean age of information (AoI).
Paper Structure (7 sections, 16 theorems, 40 equations, 3 figures)

This paper contains 7 sections, 16 theorems, 40 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Age under Preemptive Queues in Tandem
  4. Mean Age of Information
  5. Age under Non-preemptive Queues in Tandem
  6. Numerical Results and Discussion
  7. Conclusion

Key Result

Lemma 1

The transition probability from state $(a,b)$ to $(a+1,b)$ and $(a,b+1)$ is equal to $\frac{\mu_a}{\mu_a+\mu_b}$ and $\frac{\mu_b}{\mu_a+\mu_b}$, respectively.

Figures (3)

  • Figure 1: System model with $N$ queues in tandem.
  • Figure 2: Illustration of the sample path of age $\delta(t)$ for $N=3$ under preemptive policy. The green up arrow markers indicate the arrival instants of updates at server 1 whereas light-blue, magenta, and orange down arrow markers indicate the service instants of the updates from the first, second, and third servers, respectively. The red cross markers represent the older updates replaced with newer ones. The three different shades represent the AoIs observed at the output of three servers.
  • Figure 3: Mean service time $\bar{\rm T}_1$ vs $\lambda$ for $N=3,4,5$ servers with service times $\mu_i=1.5$ for $i=1,\dots,N-1$ and $\mu_N=1.5,~5,~ \text{and}~10$ for $N=3, 4,~\text{and}~5$, respectively. The solid and dashed lines indicate the mean service times under preemption and non-preemption, respectively.

Theorems & Definitions (27)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 17 more