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 $x=\widehat{G}(\mu-\mu x)$, where $\widehat{G}(s)$ 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 $x=\widehat{G}(\mu-\mu x)$ 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, as well as to study the continuity of the loss probabilities in M/M/1/$n$ queueing systems when $n$ is large.