This paper considers a two-hop status update system, in which an information source aims for the timely delivery of status updates to the destination with the aid of a relay. The relay is assumed to be an energy-constraint device and our goal is to devise scheduling policies that adaptively switch between the information decoding and information forwarding to minimize the long-term average Age-of-Information (AoI) at the destination, under a resource constraint on the average number of forwarding operations at the relay. We first identify an optimal scheduling policy by modelling the considered scheduling problem as a constrained Markov decision process (CMDP) problem. We resolve the CMDP problem by transforming it into an unconstrained Markov decision process (MDP) using a Lagrangian method. The structural properties of the optimal scheduling policy is analyzed, which is shown to have a multiple threshold structure. For implementation simplicity, based on the structural properties of the CMDP-based policy, we then propose a low-complexity double threshold relaying (DTR) policy with only two thresholds, one for relay's age and the other one for the age gain between destination and relay. We manage to derive approximate closed-form expressions of the average AoI at the destination, and the average number of forwarding operations at the relay for the DTR policy, by modelling the tangled evolution of age at the relay and destination as a Markov chain (MC). Numerical results are provided to verify all the theoretical analysis, and show that the low-complexity DTR policy can achieve near optimal performance compared with the optimal scheduling policy derived from the CMDP problem. The simulation results also unveil that only one threshold for the relay's age is needed in the DTR policy when there is no resource constraint or the resource constraint is loose.

中文翻译：

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在资源约束下优化两跳状态更新系统中的信息新鲜度

本文考虑了一个两跳状态更新系统，其中信息源旨在借助中继将状态更新及时传送到目的地。假设中继是一个能量约束设备，我们的目标是设计调度策略，在信息解码和信息转发之间自适应切换，以最小化目的地的长期平均信息年龄 (AoI)，在中继转发操作平均次数的资源限制。我们首先通过将考虑的调度问题建模为受约束的马尔可夫决策过程 (CMDP) 问题来确定最佳调度策略。我们通过使用拉格朗日方法将 CMDP 问题转换为无约束马尔可夫决策过程 (MDP) 来解决该问题。分析了最优调度策略的结构特性，表明其具有多阈值结构。为了实现简单，基于基于 CMDP 的策略的结构特性，我们然后提出了一种低复杂性双阈值中继 (DTR) 策略，只有两个阈值，一个是中继的年龄，另一个是目的地和目的地之间的年龄增益。中继。通过将中继和目的地的年龄复杂演化建模为马尔可夫链（MC），我们设法推导出目的地平均 AoI 的近似封闭形式表达式，以及 DTR 策略的中继转发操作的平均数量）。提供了数值结果来验证所有的理论分析，并表明与从 CMDP 问题导出的最优调度策略相比，低复杂度 DTR 策略可以实现接近最优的性能。仿真结果还表明，当没有资源约束或资源约束松散时，DTR 策略中只需要一个中继年龄阈值。