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Optimizing the Trade-off Between Throughput and PAoI Outage Exponents

Tai-Chun Yeh, Yu-Pin Hsu

TL;DR

This work addresses the trade-off between throughput and PAoI reliability in a multi-sensor information-collection system sharing a radio resource. It develops a joint design of sampling delays and resource allocations to minimize a weighted delay cost while enforcing PAoI outage-exponent constraints, reformulating the outage condition via the log MGF $\Lambda_i(\theta)$ and deriving $b_i^* = \frac{\Lambda_i(\theta_i)}{\theta_i}$ for fixed resource. For exponential transmission times, it yields a convex resource-allocation problem with a capacity-like feasibility region $\sum_i \frac{\theta_i}{\mu_i} < 1$, solvable by KKT, and provides a large-scale, closed-form approximation: $r_i^* \approx \frac{\theta_i}{\mu_i} + \frac{C_i}{\theta_i} \frac{1 - \sum_j \frac{\theta_j}{\mu_j}}{\sum_j \frac{C_j}{\theta_j}}$ and $b_i^* \approx \frac{1}{\theta_i} \ln\left(1 + \frac{\theta_i^2}{C_i \mu_i} \frac{\sum_j \frac{C_j}{\theta_j}}{1 - \sum_j \frac{\theta_j}{\mu_j}}\right)$, enabling explicit throughput–PAoI trade-off expressions in large systems. Numerical results validate the approximation against the optimum and demonstrate scalability across sensor counts. The findings offer actionable design guidelines for systems requiring frequent updates with statistical timeliness guarantees.

Abstract

This paper investigates the trade-off between throughput and peak age of information (PAoI) outage probability in a multi-sensor information collection system. Each sensor monitors a physical process, periodically samples its status, and transmits the updates to a central access point over a shared radio resource. The trade-off arises from the interplay between each sensor's sampling frequency and the allocation of the shared resource. To optimize this trade-off, we formulate a joint optimization problem for each sensor's sampling delay and resource allocation, aiming to minimize a weighted sum of sampling delay costs (representing a weighted sum of throughput) while satisfying PAoI outage probability exponent constraints. We derive an optimal solution and particularly propose a closed-form approximation for large-scale systems. This approximation provides an explicit expression for an approximately optimal trade-off, laying a foundation for designing resource-constrained systems in applications that demand frequent updates and also stringent statistical timeliness guarantees.

Optimizing the Trade-off Between Throughput and PAoI Outage Exponents

TL;DR

This work addresses the trade-off between throughput and PAoI reliability in a multi-sensor information-collection system sharing a radio resource. It develops a joint design of sampling delays and resource allocations to minimize a weighted delay cost while enforcing PAoI outage-exponent constraints, reformulating the outage condition via the log MGF and deriving for fixed resource. For exponential transmission times, it yields a convex resource-allocation problem with a capacity-like feasibility region , solvable by KKT, and provides a large-scale, closed-form approximation: and , enabling explicit throughput–PAoI trade-off expressions in large systems. Numerical results validate the approximation against the optimum and demonstrate scalability across sensor counts. The findings offer actionable design guidelines for systems requiring frequent updates with statistical timeliness guarantees.

Abstract

This paper investigates the trade-off between throughput and peak age of information (PAoI) outage probability in a multi-sensor information collection system. Each sensor monitors a physical process, periodically samples its status, and transmits the updates to a central access point over a shared radio resource. The trade-off arises from the interplay between each sensor's sampling frequency and the allocation of the shared resource. To optimize this trade-off, we formulate a joint optimization problem for each sensor's sampling delay and resource allocation, aiming to minimize a weighted sum of sampling delay costs (representing a weighted sum of throughput) while satisfying PAoI outage probability exponent constraints. We derive an optimal solution and particularly propose a closed-form approximation for large-scale systems. This approximation provides an explicit expression for an approximately optimal trade-off, laying a foundation for designing resource-constrained systems in applications that demand frequent updates and also stringent statistical timeliness guarantees.
Paper Structure (12 sections, 2 theorems, 19 equations, 3 figures)

This paper contains 12 sections, 2 theorems, 19 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System overview
  3. Network model
  4. Problem formulation
  5. Optimal design for sampling delay
  6. Optimal design for resource allocation under a specific transmission distribution
  7. Feasibility region of outage exponent $\boldsymbol{\theta}$
  8. Optimal solution for resource allocation
  9. Approximate Solution
  10. Numerical results
  11. Conclusion
  12. Proof of Lemma \ref{['lemma:necessary-condition']}

Key Result

Lemma 1

The condition $\lim_{x \to \infty} \frac{\ln\mathbb{P}[A_{i,\infty} \geq x]}{x} \leq -\theta_i$ holds if and only if $\frac{\Lambda_i(\theta_i)}{\theta_i} \leq b_i$.

Figures (3)

  • Figure 1: An information collection system.
  • Figure 2: Sampling delay $b_1$ versus outage exponent $\theta_1$.
  • Figure 3: The total sampling delay cost versus the number $N$ of sensors.

Theorems & Definitions (4)

  • Lemma 1
  • proof : Proof (sketch)
  • Lemma 2
  • proof