We consider a fluid queue fed by a superposition of a finite number of On/Off sources, the distribution of the On period being subexponential for some of them and exponential for the others. We provide general lower and upper bounds for the tail of the stationary buffer content distribution in terms of the so-called minimal subsets of sources. We then show that this tail decays at exponential or subexponential speed according as a certain parameter is smaller or larger than the ouput rate. If we replace the subexponential tails by regularly varying tails, the upper bound and the lower bound are sharp in that they differ only by a multiplicative factor.

We consider degenerate limit laws for the sequence {Xn, n}n(N of successive maxima of identically distributed random variables. It turns out that the concentration of Xn, n for large n can be determined in terms of a tail ratio of the underlying distribution function F. Applications to the outlier-behaviour of probability distributions are given.

For the distribution function of the busy period in the M/G/l queueing system with traffic intensity less than one it is shown that the tail varies regularly at infinity iff the tail of the service time varies regularly at infinity.

We obtain results connecting the distributions of the random variables Z1 and W in the supercritical Galton-Watson process. For example, if a > 1, and converge or diverge together, and regular variation of the tail of one of Z1, W with non-integer exponent α > 1 is equivalent to regular variation of the tail of the other.