Stochastic Service Systems, Random Interval Graphs and Search Algorithms

TL;DR

This paper analyzes stochastic service systems and their relation to random interval graphs and search algorithms, deriving asymptotic behaviors of moments involving Lambert series and providing detailed expansions.

Contribution

It introduces new asymptotic analysis of moments in stochastic service systems, linking them to Lambert series and advancing understanding of related algorithms.

Findings

Asymptotic expansions for moments involving Lambert series.

Connection between stochastic service systems and random interval graphs.

Complete asymptotic analysis for specific cases.

Abstract

We consider several stochastic service systems, and study the asymptotic behavior of the moments of various quantities that have application to models for random interval graphs and algorithms for searching for an idle server or empty waiting station. In two cases the moments turn out to involve Lambert series for the generating functions for the sums of powers of divisors of positive integers. For these cases we are able to obtain complete asymptotic expansions for the moments of the quantities in question.