Classes of probability distributions and their applications

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 $x=\hat{G}(\mu-\mu x)$, 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 that it belongs to the special class of distributions such that the difference between two elements of that class in the Kolmogorov (uniform) metric is not greater than $\kappa$. The obtained result is then used to establish the lower and upper bounds for loss probabilities in certain loss queueing systems with large buffers as well as continuity theorems in large $M/M/1/n$ queueing systems.