Asymptotic bounds for the fluid queue fed by sub-exponential on/off sources

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 output 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.