Unstable Asymptotics for Nonstationary Queues

We relate laws of large numbers and central limit theorems for nonstationary counting processes to corresponding limits for their inverse processes. We apply these results to develop approximations for queues that are unstable in a nonstationary manner. We obtain unstable nonstationary analogs of the queueing relation L = λ W and associated central-limit- theorem versions. For modeling and to obtain the first limits, we can construct nonstationary point processes as random time-transformations of familiar point processes, such as renewal processes and stationary point processes. We deduce the asymptotic behavior of the nonstationary point process from the asymptotic behavior of the familiar point process and the time transformation.