On bounds of the stationary waiting time extremal index in $M/G/1$ system with mixture service times

It is proved that if the original stationary sequence has $m$-component mixture distribution with stochastically ordered components, there are limit distributions for the maxima of all components, and the normalizing sequences are ordered, then the extremal index of the original sequence is within the boundaries of the extremal indexes of the smallest and largest components. This result is used to estimate the extremal index of the stationary waiting time in a queuing system of type $M/G/1$ in which the queuing time is given by an $m$-component distribution mixture. An example of a system $M/H_m/1$ with hyperexponential service time is considered. Using the exact simulation approach, the results of estimating the extremal index of stationary waiting time in the system $M/H_2/1$ are obtained.