The busy period in the fluid queue

Consider a fluid queue fed by $N$ on/off sources. It is assumed that the silence periods of the sources are exponentially distributed, whereas the activity periods are generally distributed. The inflow rate of each source, when active, is at least as large as the outflow rate of the buffer. We make two contributions to the performance analysis of this model. Firstly, we determine the Laplace-Stieltjes transforms of the distributions of the busy periods that start with an active period of source $i$, $i=1,dots,N$, as the unique solution in $[0,1]^N$ of a set of $N$ equations. Thus we also find the Laplace-Stieltjes transform of the distribution of an arbitrary busy period. Secondly, we relate the tail behaviour of the busy period distributions to the tail behaviour of the activity period distributions. We show that the tails of all busy period distributions are regularly varying of index $-nu$ iff the heaviest of the tails of the activity period distributions are regularly varying of index $-nu$. We provide explicit equivalents of the former in terms of the latter, which show that the contribution of the sources with lighter associated tails is equivalent to a simple reduction of the outflow rate. These results have implications for the performance analysis of networks of fluid queues.