Asymptotic loss probability in a finite buffer fluid queue with hetergeneous heavy-tailed on--off processes

Abstract

Consider a fluid queue with a finite buffer $B$ and capacity $c$ fed by a superposition of $N$ independent On--Off processes. An On--Off process consists of a sequence of alternating independent periods of activity and silence. Successive periods of activity, as well as silence, are identically distributed. The process is active with probability $p$ and during its activity period produces fluid at constant rate $r$. For this queueing system, under the assumption that the excess activity periods are intermediately regularly varying, we derive explicit and asymptotically exact formulas for approximating the stationary overflow probability and loss rate. In the case of homogeneous processes with excess activity periods equal in distribution to $\tau^e$, the queue loss rate is asymptotically, as $B\rightarrow \infty$, equal to \[ \Lambda^B = (r_0 -c) \pmatrix{N\cr m} \Bigl(p \,\Pr\Bigl[\tau^e > \frac{B}{r_0-c}\Bigr]\Bigr)^{m} (1+o(1)), \] where $m$ is the smallest integer greater than $(c-N\rho)/(r-\rho)$, $r_0= m r + (N -m)\rho$, $\rho=r p$ and $N \rho < c$; the results require a mild technical assumption that $(c-N\rho)/(r-\rho)$ is not an integer. The analyzed queueing system represents a standard model of resource sharing in telecommunication networks. The derived asymptotic results are shown to provide accurate approximations to simulation experiments. Furthermore, the results offer insight into qualitative tradeoffs between the overflow probability, offered traffic load, capacity and buffer space.