Certain classes of probability distributions and their applications in queueing problems

The aim of this paper is a nontrivial application of certain classes of probability distribution functions with further establishing the bounds for the least root of the functional equation $x=\widehat{G}(\mu-\mu x)$ (or similar functional equations appearing in queueing problems), 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 that it belongs to the special class of distributions such that the difference between two elements of that class in Kolmogorov's metric is not greater than $\kappa$. The obtained result is then used to establish the lower and upper bounds for loss probabilities and continuity theorems in certain loss queueing systems with large buffers.