Bidirectional Age of Incorrect Information: A Performance Metric for Status Updates in Virtual Dynamic Environments

Virtual dynamic environments (VDEs) such as the Metaverse and digital twins (DTs) require proper representation of the interacting entities to map their characteristics within the simulated or augmented space. Keeping these representations accurate and up-to-date is crucial for seamless interaction and system reliability. In this paper, we propose bidirectional age of incorrect information (BAoII) to address this aspect. BAoII quantifies the time-dependent penalty paid by an entity in a VDE due to incorrect or outdated knowledge about itself and the overall dynamically changing space. This extends the concept of age of incorrect information for a bidirectional information exchange, capturing that a VDE requires mutual awareness of the entity's own representation, measured in the virtual space, and what the other entities share about their representations. Using a continuous-time Markov chain model, we derive a closed-form expression for long-term BAoII and identify a transmission cost threshold for optimal update strategies. We describe a trade-off between communication cost and information freshness and validate our model through numerical simulations, demonstrating the impact of BAoII on evaluating system performance and highlighting its relevance for real-time collaboration in the Metaverse and DTs.