On extreme values of orbit lengths in $M/G/1$ queues with constant retrial rate

Abstract.

In the design of waiting facilities for the units in a retrial queue, it is of interest to know probability distributions of extreme values of the orbit length. The purpose of this paper is to investigate the asymptotic behavior of the maximum orbit length in the $M/G/1$ queue with constant retrial rate, as the time interval increases. From the classical extreme value theory, we observe that, under standard linear normalizations, the maximum orbit length up to the nth time the positive recurrent queue becomes empty does not have a limit distribution. However, by allowing the parameters to vary with n, we prove the convergence of maximum orbit lengths to three possible limit distributions when the traffic intensity $\rho_n$ approaches 1 from below and n approaches infinity.