We study the version age of information in a multi-hop multi-cast cache-enabled network, where updates at the source are marked with incrementing version numbers, and the inter-update times on the links are not necessarily exponentially distributed. We focus on the set of non-arithmetic distributions, which includes continuous probability distributions as a subset, with finite first and second moments for inter-update times. We first characterize the instantaneous version age of information at each node for an arbitrary network. We then explicate the recursive equations for instantaneous version age of information in multi-hop networks and employ semi-martingale representation of renewal processes to derive closed form expressions for the expected version age of information at an end user. We show that the expected age in a multi-hop network exhibits an additive structure. Further, we show that the expected age at each user is proportional to the variance of inter-update times at all links between a user and the source. Thus, end user nodes should request packet updates at constant intervals.

翻译：本文研究了多跳多播缓存启用网络中的信息版本年龄，其中源上的更新以递增版本号标记，而链路上的更新时间不一定服从指数分布。 我们集中研究非算术分布集合，其中包括连续概率分布作为子集，在链路之间的更新时间具有有限的第一和第二时刻。 我们首先刻画了任意网络中每个节点的瞬时版本信息年龄。 然后我们详细说明了多跳网络中瞬时版本信息年龄的递归方程，并利用更新过程的半鞅表示来导出期望的信息版本年龄的闭式表达式。 我们证明了多跳网络中的期望年龄具有加法结构。 此外，我们表明每个用户的期望年龄与用户和源之间所有链路上的更新时间的方差成正比。 因此，终端用户节点应等间隔请求分组更新。