Analysis of Age of Incorrect Information under Generic Transmission Delay

TL;DR

The paper investigates the Age of Incorrect Information (AoII) in a slotted communication system with random transmission delays. It analyzes a threshold-based transmission policy using a Markov chain framework, reducing the problem to solving a finite set of linear equations to obtain the stationary distribution and AoII. Closed-form results are provided for particular thresholds, and numerical results show that simply transmitting updates at every opportunity is not optimal. The methodology enables exact AoII computation under bounded-delay assumptions and offers insights for designing better update strategies in networks with random delays, with future work on preemptive policies and more complex source models.

Abstract

This paper investigates the Age of Incorrect Information (AoII) in a communication system whose channel suffers a random delay. We consider a slotted-time system where a transmitter observes a dynamic source and decides when to send updates to a remote receiver through the communication channel. The threshold policy, under which the transmitter initiates transmission only when the AoII exceeds the threshold, governs the transmitter's decision. In this paper, we analyze and calculate the performance of the threshold policy in terms of the achieved AoII. Using the Markov chain to characterize the system evolution, the expected AoII can be obtained precisely by solving a system of linear equations whose size is finite and depends on the threshold. We also give closed-form expressions of the expected AoII under two particular thresholds. Finally, calculation results show that there are better strategies than the transmitter constantly transmitting new updates.

Figures (4)

• Figure 1: Two-state symmetric Markov chain with state transition probability $p$.
• Figure 2: An illustration of the system model.
• Figure 3: A sample path of $\Delta_k$, where $T_i$ and $D_i$ are the transmission start time and the delivery time of the $i$-th update, respectively. At $T_1$, the transmitted update is $X_3$. The estimate at time slot $6$ (i.e., $\hat{X}_6$) changes due to the reception of the update transmitted at $T_2$.
• Figure 4: Illustrations of the expected AoII in the function of $p$ and $\tau$. In the figure, lines represent the simulation results, and markers represent the calculation results. We set the upper limit on the transmission time $t_{max}=5$, the success probability in Geometric distribution $p_s = 0.7$, and the constant in Zipf distribution $a=3$. The simulation results are the average of $15$ runs, each run containing $25000$ epochs.

Theorems & Definitions (25)

• Remark 1
• Remark 2
• Definition 1: Threshold policy
• Remark 3
• Remark 4
• Lemma 1
• proof
• Corollary 1
• proof
• Lemma 2