Bounds for the loss probability in large loss queueing systems

Abstract

Let $\mathcal{G}(\frak{g}_1,\frak{g}_2)$ be the class of all probability distribution functions of positive random variables having the given first two moments $\frak{g}_1$ and $\frak{g}_2$. Let $G_1(x)$ and $G_2(x)$ be two probability distribution functions of this class satisfying the condition $|G_1(x)-G_2(x)|<\epsilon$ for some small positive value $\epsilon$ and let $\widehat{G}_1(s)$ and, respectively, $\widehat{G}_2(s)$ denote their Laplace-Stieltjes transforms. For real $\mu$ satisfying $\mu\frak{g}_1>1$ let us denote by $\gamma_{G_1}$ and $\gamma_{G_2}$ the least positive roots of the equations $z=\widehat{G}_1(\mu-\mu z)$ and $z=\widehat{G}_2(\mu-\mu z)$ respectively. In the paper, the upper bound for $|\gamma_{G_1}-\gamma_{G_2}|$ is derived. This upper bound is then used to find lower and upper bounds for the loss probabilities in different large loss queueing systems.