Study about a Differential Equation in an Infinite Servers Queue System with Poisson Arrivals Busy Cycle Distribution Study

TL;DR

This paper analyzes an infinite servers queue with Poisson arrivals, deriving simple distribution functions for busy periods and cycles using Riccati equations, aiding practical probabilistic modeling.

Contribution

It introduces a novel approach using Riccati equations to obtain explicit distributions for busy periods and cycles in infinite server queues.

Findings

Busy period and cycle lengths often follow exponential distributions.

The method simplifies the analysis of queue busy cycle distributions.

Results are applicable to real-world systems with Poisson arrivals.

Abstract

In the infinite servers queue with Poisson arrivals real life practical applications, the busy period and the busy cycle probabilistic study is of main importance. But it is a very difficult task. In this text, we show that by solving a Riccati equation induced by this queue transient probabilities monotony study as time functions, we obtain a collection of service length distribution functions, for which both the busy period and the busy cycle have lengths with quite simple distributions, generally given in terms of exponential distributions and the degenerate at the origin distribution.