We consider an m server system in which each server can service at most one update packet at a time. The system designer controls (1) scheduling - the order in which the packets get serviced, (2) routing - the server that an arriving update packet joins for service, and (3) the service time distribution with fixed service rate. Given a fixed update generation process, we prove a strong age-delay and age-delay variance tradeoff, wherein, as the average AoI approaches its minimum, the packet delay and its variance approach infinity. In order to prove this result, we consider two special cases of the m server system, namely, a single server system with last come first served with preemptive service and an infinite server system. In both these cases, we derive sufficient conditions to show that three heavy tailed service time distributions, namely Pareto, log-normal, and Weibull, asymptotically minimize the average AoI as their tail gets heavier, and establish the age-delay tradeoff results. We provide an intuitive explanation as to why such a seemingly counter intuitive age-delay tradeoff is natural, and that it should exist in many systems.

中文翻译：

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排队系统中的年龄延迟权衡


我们考虑一个 m 服务器系统，其中每个服务器一次最多可以服务一个更新数据包。系统设计者控制 (1) 调度 - 数据包获得服务的顺序，(2) 路由 - 到达的更新数据包加入服务的服务器，以及 (3) 具有固定服务速率的服务时间分配。给定固定的更新生成过程，我们证明了强大的年龄延迟和年龄延迟方差权衡，其中，当平均 AoI 接近其最小值时，数据包延迟及其方差接近无穷大。为了证明这个结果，我们考虑m服务器系统的两种特殊情况，即后到先服务的抢占式服务的单服务器系统和无限服务器系统。在这两种情况下，我们得出了充分的条件来证明三种重尾服务时间分布（即帕累托分布、对数正态分布和威布尔分布）随着尾部变重而渐近地最小化平均 AoI，并建立了年龄延迟权衡结果。我们提供了一个直观的解释，解释为什么这种看似违反直觉的年龄延迟权衡是自然的，并且它应该存在于许多系统中。