Timely and informative data dissemination in communication networks is essential for enhancing system performance and energy efficiency, as it reduces the transmission of outdated or redundant data. Timeliness metrics, such as Age of Information (AoI), effectively quantify data freshness; however, these metrics fail to account for the intrinsic informativeness of the content itself. To address this limitation, content-based metrics have been proposed that combine both timeliness and informativeness. Nevertheless, existing studies have predominantly focused on evaluating average metric values, leaving the complete distribution-particularly in multi-hop network scenarios-largely unexplored. In this paper, we provide a comprehensive analysis of the stationary distribution of the Version Age of Information (VAoI), a content-based metric, under various scheduling policies, including randomized stationary, uniform, and threshold-based policies, with transmission constraints in single-hop and multi-hop networks. We derive closed-form expressions for the stationary distribution and average VAoI under these scheduling approaches. Furthermore, for threshold-based scheduling, we analytically determine the optimal threshold value that minimizes VAoI and derive the corresponding optimal VAoI in closed form. Numerical evaluations verify our analytical findings, providing valuable insights into leveraging VAoI in the design of efficient communication networks.

翻译：通信网络中及时且信息丰富的数据分发对于提升系统性能和能源效率至关重要，因为它减少了过时或冗余数据的传输。信息年龄等时效性指标能有效量化数据新鲜度；然而，这些指标未能考虑内容本身的内在信息价值。为弥补这一不足，已有研究提出了结合时效性与信息价值的基于内容的指标。尽管如此，现有研究主要集中于评估指标的平均值，而完整分布——尤其是在多跳网络场景下——在很大程度上仍未得到探索。本文对基于内容的指标——版本信息年龄的稳态分布进行了全面分析，考虑了单跳与多跳网络中具有传输约束的多种调度策略，包括随机稳态调度、均匀调度以及基于阈值的调度。我们推导了这些调度策略下稳态分布与平均版本信息年龄的闭式表达式。此外，对于基于阈值的调度，我们通过解析方法确定了最小化版本信息年龄的最优阈值，并以闭式形式推导了相应的最优版本信息年龄。数值评估验证了我们的分析结果，为在高效通信网络设计中利用版本信息年龄提供了有价值的见解。