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Transmit or Idle: Efficient AoI Optimal Transmission Policy for Gossiping Receivers

Irtiza Hasan, Ahmed Arafa

TL;DR

We address the problem of minimizing the long-term sum of AoI and transmission cost in a two-receiver gossiping network. The authors model the system as an average-cost MDP over AoI states $a_i$ and solve it with Relative Value Iteration, revealing an age-difference threshold structure with minimum age activation when gossiping is reliable. Key contributions include proving monotonicity and symmetry of the relative value function and establishing a rigorous age-difference threshold policy that activates only once the minimum AoI crosses a bound, yielding significant cost savings over several baselines. This work provides a principled framework for balancing information freshness and transmission cost in gossip-based networks and offers actionable insights for threshold-based policy design in similar settings.

Abstract

We study the optimal transmission and scheduling policy for a transmitter (source) communicating with two gossiping receivers aiming at tracking the source's status over time using the age of information (AoI) metric. Gossiping enables local information exchange in a decentralized manner without relying solely on the transmitter's direct communication, which we assume incurs a transmission cost. On the other hand, gossiping may be communicating stale information, necessitating the transmitter's intervention. With communication links having specific success probabilities, we formulate an average-cost Markov Decision Process (MDP) to jointly minimize the sum AoI and transmission cost for such a system in a time-slotted setting. We employ the Relative Value Iteration (RVI) algorithm to evaluate the optimal policy for the transmitter and then prove several structural properties showing that it has an age-difference threshold structure with minimum age activation in the case where gossiping is relatively more reliable. Specifically, direct transmission is optimal only if the minimum AoI of the receivers is large enough and their age difference is below a certain threshold. Otherwise, the transmitter idles to effectively take advantage of gossiping and reduce direct transmission costs. Numerical evaluations demonstrate the significance of our optimal policy compared to multiple baselines. Our result is a first step towards characterizing optimal freshness and transmission cost trade-offs in gossiping networks.

Transmit or Idle: Efficient AoI Optimal Transmission Policy for Gossiping Receivers

TL;DR

We address the problem of minimizing the long-term sum of AoI and transmission cost in a two-receiver gossiping network. The authors model the system as an average-cost MDP over AoI states and solve it with Relative Value Iteration, revealing an age-difference threshold structure with minimum age activation when gossiping is reliable. Key contributions include proving monotonicity and symmetry of the relative value function and establishing a rigorous age-difference threshold policy that activates only once the minimum AoI crosses a bound, yielding significant cost savings over several baselines. This work provides a principled framework for balancing information freshness and transmission cost in gossip-based networks and offers actionable insights for threshold-based policy design in similar settings.

Abstract

We study the optimal transmission and scheduling policy for a transmitter (source) communicating with two gossiping receivers aiming at tracking the source's status over time using the age of information (AoI) metric. Gossiping enables local information exchange in a decentralized manner without relying solely on the transmitter's direct communication, which we assume incurs a transmission cost. On the other hand, gossiping may be communicating stale information, necessitating the transmitter's intervention. With communication links having specific success probabilities, we formulate an average-cost Markov Decision Process (MDP) to jointly minimize the sum AoI and transmission cost for such a system in a time-slotted setting. We employ the Relative Value Iteration (RVI) algorithm to evaluate the optimal policy for the transmitter and then prove several structural properties showing that it has an age-difference threshold structure with minimum age activation in the case where gossiping is relatively more reliable. Specifically, direct transmission is optimal only if the minimum AoI of the receivers is large enough and their age difference is below a certain threshold. Otherwise, the transmitter idles to effectively take advantage of gossiping and reduce direct transmission costs. Numerical evaluations demonstrate the significance of our optimal policy compared to multiple baselines. Our result is a first step towards characterizing optimal freshness and transmission cost trade-offs in gossiping networks.
Paper Structure (7 sections, 6 theorems, 24 equations, 6 figures)

This paper contains 7 sections, 6 theorems, 24 equations, 6 figures.

Table of Contents

  1. Introduction
  2. System Model and MDP Formulation
  3. Optimal Policy Visualization and Performance
  4. Structural Analysis of the Optimal Policy
  5. Properties of the Relative Value Function
  6. Age-Difference Threshold Structure with Minimum Age Activation
  7. Conclusion

Key Result

Lemma 1

The relative value function $U(s)$ is component-wise non-decreasing: for any states $(a_1, a_2)$ and $(a_1', a_2')$ in $\mathcal{S}$ such that $(a_1, a_2) \preceq (a_1', a_2')$, it holds that $U(a_1, a_2) \le U(a_1', a_2')$.

Figures (6)

  • Figure 1: Transmitter updating two receivers via direct links (solid lines) and receivers gossiping (dashed lines). Symbols above the lines indicate channels' successful communication probabilities.
  • Figure 2: (a) At each time slot, the transmitter has three action options: transmit to RX$_1$ (blue) or RX$_2$ (purple), or idle (gray). (b) If the TX idles in any time slot (e.g., in time slot $t$), both RX$_1$ and RX$_2$ can gossip. Whichever receiver has a successful update from the other lets the transmitter know by an ACK. (c) If both receivers have successful gossip updates, TX receives ACK's from both of them. TX maintains perfect RX age knowledge in all cases.
  • Figure 3: Optimal policy visualization using RVI. Here, $C_{\mathrm{tx}}=1$.
  • Figure 4: Performance against varying direct transmission reliability.
  • Figure 5: Performance against varying gossiping reliability.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Lemma 1: Monotonicity
  • Lemma 2: Symmetry
  • Proposition 1
  • Remark 1
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof