  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 server 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.
