Bounds for the loss probability in large loss queueing systems

Details Bounds for the loss probability in large loss queueing systems Bounds for the loss probability in large loss queueing systems

2008-04-15

arxiv.org/abs/0804.2310v7

Mathematics

Probability

27 pages, double spaced, completely rewritten

Scientific paper

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.

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