Links in practical systems, such as satellite-terrestrial integrated networks, exhibit distinct delay distributions, intermittent availability, and heterogeneous energy costs. These characteristics pose significant challenges to maintaining timely and energy-efficient status updates. While link availability restricts feasible transmission routes, routing decisions determine the actual delay and energy expenditure. This paper tackles these challenges by jointly optimizing sampling and routing decisions to minimize monotonic, nonlinear Age of Information (AoI). The proposed formulation incorporates key system features, including multiple routes with correlated random delays, stochastic link availability, and route-dependent energy consumption. We model the problem as an infinite-horizon constrained semi-Markov decision process (CSMDP) with a hybrid state-action space and develop an efficient nested algorithm, termed Bisec-ReaVI, to solve this problem. We reveal a well-defined jointly optimal policy structure: (i) the optimal routing policy is a monotonic handover policy that adapts to the availability of routes and their mean delays; and (ii) the optimal sampling policy is a piecewise linear waiting policy, with at most "N choose 2 + N" breakpoints given N routes. Numerical experiments in a satellite-terrestrial integrated routing scenario demonstrate that the proposed scheme efficiently balances energy usage and information freshness, and reveal a counter-intuitive insight: even routes with higher average delay, higher delay variance, or lower availability can still play a critical role in minimizing monotonic functions of AoI.

翻译：实际系统（如星地一体化网络）中的链路具有不同的延迟分布、间歇性可用性以及异构的能量成本。这些特性对维持及时且能量高效的状态更新构成了显著挑战。链路可用性限制了可行的传输路径，而路由决策则决定了实际的延迟和能量消耗。本文通过联合优化采样与路由决策，以最小化单调非线性的信息年龄（Age of Information, AoI），应对这些挑战。所提出的建模框架融合了关键系统特征，包括具有相关随机延迟的多条路径、随机链路可用性以及路径依赖的能量消耗。我们将该问题建模为一个具有混合状态-动作空间的无限时域约束半马尔可夫决策过程（CSMDP），并开发了一种高效的嵌套算法（称为Bisec-ReaVI）来求解。我们揭示了一种结构清晰的联合最优策略：（i）最优路由策略是一种单调切换策略，能够适应路径的可用性及其平均延迟；（ii）最优采样策略是一种分段线性等待策略，在给定N条路径时，其断点数量最多为“N选2 + N”。在星地一体化路由场景中的数值实验表明，所提方案能有效平衡能量使用与信息新鲜度，并揭示了一个反直觉的洞见：即使平均延迟更高、延迟方差更大或可用性更低的路径，在最小化AoI的单调函数方面仍可发挥关键作用。