Age of Information (AoI) has proven to be a useful metric in networked systems where timely information updates are of importance. In the literature, minimizing “average age” has received considerable attention. However, various applications pose stricter age requirements on the updates which demand knowledge of the AoI distribution. Furthermore, the analysis of AoI distribution in a multi-hop setting, which is important for the study of Wireless Networked Control Systems (WNCS), has not been addressed before. Toward this end, we study the distribution of AoI in a WNCS with two hops and devise a problem of minimizing the tail of the AoI distribution with respect to the frequency of generating information updates, i.e., the sampling rate of monitoring a process, under first-come-first-serve (FCFS) queuing discipline. We argue that computing an exact expression for the AoI distribution may not always be feasible; therefore, we opt for computing upper bounds on the tail of the AoI distribution. Using these upper bounds, we formulate Upper Bound Minimization Problems (UBMP), namely, Chernoff-UBMP and $\alpha $ -relaxed Upper Bound Minimization Problem ( $\alpha $ -UBMP), where $\alpha > 1$ is an approximation factor, and solve them to obtain “good” heuristic rate solutions for minimizing the tail. We demonstrate the efficacy of our approach by solving the proposed UBMPs for three service distributions: geometric, exponential, and Erlang. Simulation results show that the rate solutions obtained are near optimal for minimizing the tail of the AoI distribution for the considered distributions.

中文翻译：

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具有周期性到达的单跳和两跳 FCFS 系统中 AoI 的统计保证优化

信息时代 (AoI) 已被证明是网络系统中一个有用的指标，在​​这些系统中，及时的信息更新很重要。在文献中，最小化“平均年龄”受到了相当多的关注。然而，各种应用程序对需要 AoI 分布知识的更新提出了更严格的年龄要求。此外，在多跳设置中分析 AoI 分布，这对于无线网络控制系统 (WNCS) 的研究很重要，以前尚未解决。为此，我们研究了具有两跳的 WNCS 中的 AoI 分布，并设计了一个问题，即根据生成信息更新的频率（即监控过程的采样率）最小化 AoI 分布的尾部，首先-先到先得 (FCFS) 排队规则。我们认为计算 AoI 分布的精确表达式可能并不总是可行的。因此，我们选择计算 AoI 分布尾部的上限。使用这些上限，我们制定了上限最小化问题 (UBMP)，即 Chernoff-UBMP 和 $\alpha $ - 宽松的上限最小化问题（ $\alpha $ -UBMP)，其中 $\alpha > 1$ 是一个近似因子，并求解它们以获得最小化尾部的“良好”启发式速率解决方案。我们通过为三种服务分布求解提议的 UBMP 来证明我们的方法的有效性：几何、指数和 Erlang。仿真结果表明，对于最小化所考虑分布的 AoI 分布的尾部，获得的速率解接近最佳。