Bounds for the loss probabilities of large loss queueing systems

The aim of this paper is to establish the bounds for the least root of the functional equation $x=\hat{G}(\mu-\mu x)$, where $\hat{G}(s)$ is the Laplace-Stieltjes transform of an unknown probability distribution function $G(x)$ of a positive random variable having the first two moments $\frak{g}_1$ and $\frak{g}_2$, and $\mu$ is a positive parameter satisfying the condition $\mu\frak{g}_1>1$. The additional information characterizing $G(x)$ is an empirical probability distribution function ${G}_{\mathrm{emp}}(x)$, and it is assumed that the distance in the uniform (Kolmogorov) metric between $G(x)$ and ${G}_{\mathrm{emp}}(x)$ is not greater than $\kappa$. The obtained bounds for the positive least root of the functional equation $x=\hat{G}(\mu-\mu x)$ are then used to find the asymptotic bounds for the loss probabilities in certain queueing systems with a large number of waiting places, when only an empirical probability distribution function of an interarrival or service time is known.