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On the Uncertainty of a Simple Estimator for Remote Source Monitoring over ALOHA Channels

Andrea Munari

TL;DR

This work studies the uncertainty at the receiver when tracking two-state Markov sources over slotted ALOHA without feedback, using $H(X_n\,|\,\Delta_n,\hat{X}_n)$ as the performance metric. An analytical framework based on a terminating Markov chain yields closed-form expressions for $p(x_n\,|\,\delta_n,\hat{x}_n)$, the per-slot entropy $\mathsf h(\delta_n,\hat{x}_n)$, and the joint distribution $p(\delta_n,\hat{x}_n)$, enabling evaluation of three access policies: throughput-maximizing, change-driven (reactive), and uncertainty-minimizing balanced. Key findings include that purely maximizing throughput does not necessarily reduce uncertainty, reactive policies are optimal for symmetric sources, and mixed strategies that allow transmissions during persistence improve performance for asymmetric sources. These results provide practical design insights for IoT networks with limited cross-layer feedback and guide cross-layer protocol design to control receiver uncertainty.

Abstract

Efficient remote monitoring of distributed sources is essential for many Internet of Things (IoT) applications. This work studies the uncertainty at the receiver when tracking two-state Markov sources over a slotted random access channel without feedback, using the conditional entropy as a performance indicator, and considering the last received value as current state estimate. We provide an analytical characterization of the metric, and evaluate three access strategies: (i) maximizing throughput, (ii) transmitting only on state changes, and (iii) minimizing uncertainty through optimized access probabilities. Our results reveal that throughput optimization does not always reduce uncertainty. Moreover, while reactive policies are optimal for symmetric sources, asymmetric processes benefit from mixed strategies allowing transmissions during state persistence.

On the Uncertainty of a Simple Estimator for Remote Source Monitoring over ALOHA Channels

TL;DR

This work studies the uncertainty at the receiver when tracking two-state Markov sources over slotted ALOHA without feedback, using as the performance metric. An analytical framework based on a terminating Markov chain yields closed-form expressions for , the per-slot entropy , and the joint distribution , enabling evaluation of three access policies: throughput-maximizing, change-driven (reactive), and uncertainty-minimizing balanced. Key findings include that purely maximizing throughput does not necessarily reduce uncertainty, reactive policies are optimal for symmetric sources, and mixed strategies that allow transmissions during persistence improve performance for asymmetric sources. These results provide practical design insights for IoT networks with limited cross-layer feedback and guide cross-layer protocol design to control receiver uncertainty.

Abstract

Efficient remote monitoring of distributed sources is essential for many Internet of Things (IoT) applications. This work studies the uncertainty at the receiver when tracking two-state Markov sources over a slotted random access channel without feedback, using the conditional entropy as a performance indicator, and considering the last received value as current state estimate. We provide an analytical characterization of the metric, and evaluate three access strategies: (i) maximizing throughput, (ii) transmitting only on state changes, and (iii) minimizing uncertainty through optimized access probabilities. Our results reveal that throughput optimization does not always reduce uncertainty. Moreover, while reactive policies are optimal for symmetric sources, asymmetric processes benefit from mixed strategies allowing transmissions during state persistence.

Paper Structure

This paper contains 5 sections, 5 theorems, 15 equations, 7 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Analysis
  4. Results and Discussion
  5. Conclusions

Key Result

Proposition 1

The conditional probability of the reference source being in state $x_n$ given that its current AoI is $\delta_n$ and the last received updated contained state $\hat{x}_n$ is where, for any $\hat{x}_n$ and $x_n$ in $\mathcal{X}$, $\mathbf e_{\hat{x}_n}$ and $\mathbf e_{x_n}$ are defined as $\mathbf e_0 = [1,0]^{\mathsf T}$; $\mathbf e_1 = [0,1]^{\mathsf T}$.

Figures (7)

  • Figure 1: (a) Markov chain $X_n$ describing a monitored source; (b) terminating Markov chain $Y_n$ used to characterize $H(X_n\, | \,\Delta_n,\hat{X}_n)$. The process enters the absorbing state $\mathsf d$ when an update from the reference node is received. Conversely, it moves between $0$ and $1$, describing the corresponding current source value, so long as no update message from the reference terminal arrives.
  • Figure 2: Example of time evolution of the entropy $\mathsf h(\delta_n,\hat{x}_n)$. In this case, $\alpha=0.1$, $\beta=0.01$, $\mathsf m=50$, $\lambda_{x_{n-1}x_n} = 1/\mathsf m$, $\forall \, (x_{n-1},x_n)$.
  • Figure 3: Comparison of analysis (under the approximation \ref{['eq:ps']}) and simulation (considering the exact interference behavior of the network). Results obtained for $\mathsf m=50$ nodes, $\alpha=\beta=0.02$, $\bm\lambda = [0,1,1,0]$.
  • Figure 4: CDF of $\mathsf h(\delta_n,\hat{x})$, considering symmetric sources with $\alpha=0.02$.
  • Figure 5: Example of time evolution of the entropy $\mathsf h(\delta_n,\hat{x}_n)$ for symmetric sources, with $\alpha=0.02$, $\mathsf m=50$. For the random strategy (dash-dotted lines), the transmission probability has been set to $\lambda=1/\mathsf m$. The steeper rise of the receiver uncertainty of this approach compared to the reactive one (solid line) is evident at the very beginning, as well as after the second reset of $\mathsf h(\delta_n,\hat{x}_n)$ for the random scheme.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Proposition 1
  • proof
  • Lemma 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof