Characterization Problems Associated With the Exponential Distribution.

In this article statistical bounds for certain output characteristics of the $M/GI/1/n$ and $GI/M/1/n$ loss queueing systems are derived on the basis of large samples of an input characteristic of these systems, such as service time in the $M/GI/1/n$ queueing system or interarrival time in the $GI/M/1/n$ queueing system. The analysis of this article is based on application of Kolmogorov's statistics for empirical probability distribution functions.

Statistical analysis of single-server loss queueing systems

This paper studies a closed queueing network containing a hub (a state dependent queueing system with service depending on the number of units residing here) and k satellite stations, which are GI/M/1 queueing systems. The number of units in the system, N, is assumed to be large. After service completion in the hub, a unit visits a satellite station j, 1 � jk, with probability pj, and, after the service completion there, returns to the hub. The parameters of service times in the satellite stations and in the hub are proportional to 1 N. One of the satellite stations is assumed to be a bottleneck station, while others are non-bottleneck. The paper establishes the continuity of the queue-length processes in non-bottleneck satellite stations of the network when the service times in the hub are close in certain sense (exactly defined in the paper) to the exponential distribution.

Continuity of large closed queueing networks with bottlenecks

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), 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 the Kolmogorov (uniform) metric is not greater than κ. 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.

Vyacheslav M. Abramov

Papers

Classes of probability distributions and their applications