Certain Classes of Probability Distributions and Their Applications to 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 = b G(μ− μx) (or similar functional equations appearing in queueing problems), where b 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 g1 and g2, and μ is a positive parameter satisfying the condition μg1 > 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 κ. 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.