The comparison of waiting time extremal indexes in $M/G/1$ queueing systems

The theorem which states that if the initial stationary sequences are stochastically ordered, there are limiting distributions for maxima and the normalizing sequences are ordered, then their extreme indexes are also ordered is proved. This result is applied to compare the extreme indexes of stationary waiting times in two $M/G/1$ systems with the same input flows and stochastically ordered service times. Three examples of queueing systems with exponential distribution, Pareto distribution, and Weibull distribution of service times are considered. For these distributions, the relations between the parameters guaranteeing the stochastic ordering of the distributions and the normalizing sequences are obtained.