Extreme values of mixed Erlang random variables

In this paper, we examine the limiting distributional behaviour of extreme values of mixed Erland random variables. We show that, in the finite mixture of Erlang distributions, the component distribution with an asymptotically dominant tail has a critical effect on the asymptotic extreme behaviour of the mixture distribution and it converges to the Gumbel extreme-value distribution. Normalizing constants are also established. We apply this result to characterize the asymptotic distribution of maxima of sojourn times in M/M/s queueing system. We also show that Erlang mixtures with continuous mixing may converge to the Gumbel or Type II extreme-value distribution depending on their mixing distributions, considering two special cases of uniform mixing and exponential mixing.