Bounds for the loss probability in large loss queueing systems

TL;DR

This paper derives bounds on the difference between roots of certain equations related to queueing systems, and uses these bounds to estimate loss probabilities in large loss queueing models.

Contribution

It introduces a method to bound the difference in roots of Laplace-Stieltjes transform equations, aiding in estimating loss probabilities in large queueing systems.

Findings

Derived an upper bound for the difference between roots of specific equations.

Provided lower and upper bounds for loss probabilities in large queueing systems.

Applied the bounds to different queueing models to estimate loss probabilities.

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.