A source provides status updates to monitors through a network with state defined by a continuous-time finite Markov chain. An age of information (AoI) metric is used to characterize timeliness by the vector of ages tracked by the monitors. Based on a stochastic hybrid systems (SHS) approach, first order linear differential equations are derived for the temporal evolution of both the moments and the moment generating function (MGF) of the age vector components. It is shown that the existence of a non-negative fixed point for the first moment is sufficient to guarantee convergence of all higher order moments as well as a region of convergence for the stationary MGF vector of the age. The stationary MGF vector is then found for the age on a line network of preemptive memoryless servers. From this MGF, it is found that the age at a node is identical in distribution to the sum of independent exponential service times. This observation is then generalized to linear status sampling networks in which each node receives samples of the update process at each preceding node according to a renewal point process. For each node in the line, the age is shown to be identical in distribution to a sum of independent renewal process age random variables.

中文翻译：

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网络信息时代：时刻、分布和采样

源通过具有由连续时间有限马尔可夫链定义的状态的网络向监视器提供状态更新。信息年龄 (AoI) 指标用于通过监视器跟踪的年龄向量来表征及时性。基于随机混合系统 (SHS) 方法，推导了年龄矢量分量的矩和矩生成函数 (MGF) 的时间演化的一阶线性微分方程。结果表明，一阶矩的非负不动点的存在足以保证所有高阶矩的收敛以及年龄的平稳 MGF 向量的收敛区域。然后在抢占式无记忆服务器的线路网络上找到固定 MGF 向量。从这个 MGF，发现节点的年龄在分布上与独立指数服务时间的总和相同。然后将此观察推广到线性状态采样网络，其中每个节点根据更新点过程接收每个先前节点的更新过程样本。对于行中的每个节点，年龄在分布上与独立更新过程年龄随机变量的总和相同。