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.

翻译：本研究探讨了发射机（源节点）与两个进行闲话通信的接收器之间的最优传输与调度策略，旨在利用信息年龄（AoI）指标追踪源节点随时间变化的状态。闲话通信使得节点能够在去中心化模式下进行本地信息交换，不完全依赖发射机的直接通信（我们假设直接通信会产生传输成本）。另一方面，闲话通信可能传播过时信息，因此需要发射机进行干预。在通信链路具有特定成功概率的条件下，我们构建了一个平均成本马尔可夫决策过程（MDP），以在时隙化场景中联合最小化此类系统的总信息年龄与传输成本。我们采用相对值迭代（RVI）算法评估发射机的最优策略，随后证明了若干结构性特征：在闲话通信可靠性相对较高的场景中，该策略呈现具有最小年龄激活特性的年龄差阈值结构。具体而言，仅当接收器的最小信息年龄足够大且其年龄差低于特定阈值时，直接传输才是最优选择；否则发射机将保持空闲状态，以有效利用闲话通信机制并降低直接传输成本。数值评估表明，相较于多种基线策略，我们提出的最优策略具有显著优势。本研究结果为刻画闲话网络中信息新鲜度与传输成本权衡的最优特性迈出了第一步。