  While the notion of age of information (AoI) has recently emerged as an
important concept for analyzing ultra-reliable low-latency communications
(URLLC), the majority of the existing works have focused on the average AoI
measure. However, an average AoI based design falls short in properly
characterizing the performance of URLLC systems as it cannot account for
extreme events that occur with very low probabilities. In contrast, in this
paper, the main objective is to go beyond the traditional notion of average AoI
by characterizing and optimizing a URLLC system while capturing the AoI tail
distribution. In particular, the problem of vehicles' power minimization while
ensuring stringent latency and reliability constraints in terms of
probabilistic AoI is studied. To this end, a novel and efficient mapping
between both AoI and queue length distributions is proposed. Subsequently,
extreme value theory (EVT) and Lyapunov optimization techniques are adopted to
formulate and solve the problem. Simulation results shows a nearly two-fold
improvement in terms of shortening the tail of the AoI distribution compared to
a baseline whose design is based on the maximum queue length among vehicles,
when the number of vehicular user equipment (VUE) pairs is 80. The results also
show that this performance gain increases significantly as the number of VUE
pairs increases.
