ИСТИНА

The problem of quality of service estimation is the most important one in telecommunication systems analysis. In our previous work (Khokhlov, Lukashenko, Morozov [2]) using the methodology proposed in the paper of Norros ([1]) we propose some lower asymptotic estimate of the overflow probability of large buffer when the input is a stream consisting of two independent components: the fractional Brownian motion and stable Levy motion with same Hurst parameters. Now we consider the case of different Hurst parameters. We consider the single-server fluid queue which is fed by the following input process: $A(t) = m t + \sigma_1 B_{H_1} (t) + \sigma_2 L_{\alpha} (t) ,\,\,t\ge 0,$ where where $m>0$ is the mean input rate; $B_{H_1} = (B_{H_1} (t) , t\in R )$ is a fractional Brownian motion (FBM) with Hurst parameter $H_1$, and $L_{\alpha} = (L_{\alpha} (t), t\in R)$ is symmetric $\alpha$-stable Levy motion. Both processes are self-similar with indexes $H_1$ and $H_2 = 1/\alpha$ respectively. In what follows we assume that $H_1 \not= H_2$, $1/2 < H_1 , H_2 <1$, $\sigma_1 = \sigma_2 = \sigma$, the processes $B_{H_1}$ and $L_{\alpha}$ are independent. We are interested in estimation of so-called {\it overflow probability}, i. e. the probability that stationary workload $Q$ exceeds some threshold level $b$, namely $\varepsilon (b) := P[ Q>b]$. Denote $H=\min (H_1 , H_2)$. Our main result is the following estimate: for large $b>0$ $$ \varepsilon (b) \geq C \cdot b^{-(1 - H)\cdot \alpha}. $$