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：我们研究多跳多播缓存网络中的信息版本年龄，其中源端的更新标记有递增的版本号，且链路上的更新间隔时间不一定服从指数分布。我们聚焦于非算术分布集合（包含连续概率分布作为子集），这类分布的更新间隔时间具有有限一阶矩和二阶矩。首先，我们对任意网络中每个节点的瞬时信息版本年龄进行刻画。随后，我们阐明多跳网络中瞬时信息版本年龄的递归方程，并利用更新过程的半鞅表示，推导出终端用户期望信息版本年龄的闭式表达式。研究表明，多跳网络中的期望年龄呈现可加结构。此外，我们发现每个用户的期望年龄与用户到源端之间所有链路的更新间隔时间方差成正比。因此，终端用户节点应以恒定间隔请求数据包更新。