In this paper we consider a generalization of the so-called M=G=1 model where M types of long-tailed sessions enter a buffer . The instantaneous rates of the sessions are functions of the occupancy of an M=G=1 system with long-tailed G distributions. In particular we assume that a session of type i transmits ri cells per unit time and lasts for a random time with long tailed distribution given ...

The asymptotic solution for the Painleve-2 equation with small parameter is considered. The solution has algebraic behavior before point t∗ and fast oscillating behavior after the point t∗. In the transition layer the behavior of the asymptotic solution is more complicated. The leading term of the asymptotics satisfies the Painleve-1 equation and some elliptic equation with constant coefficient...

We develop a stochastic mean-value method for the derivation of delay asymptotics in Processor-Sharing (PS) type systems with heavy-tailed service requirements. In order to demonstrate the strength of the approach, we apply the method to obtain the sojourn time asymptotics for a multi-class G/G/1 queue operating under the Discriminatory Processor-Sharing (DPS) discipline. Besides interesting fr...