Poisson Arrivals, Exponential Service Time, and Infinite Servers Queue Busy Period and Busy Cycle Distribution Function Bounds

TL;DR

This paper derives simpler bounds for the busy period and cycle distribution functions in MGINF queues with two exponential service times, aiding analysis of complex queue systems.

Contribution

It introduces new bounds for the busy period and cycle distributions in MGINF queues with two exponential services, simplifying the complex exact expressions.

Findings

Derived bounds for busy period distribution function

Presented bounds for busy cycle distribution function

Simplified analysis of MGINF queue systems

Abstract

The busy period length distribution function knowledge is important for any queue system, and for the MGINF queue. But the mathematical expressions are in general very complicated, with a few exceptions, involving usually infinite sums and multiple convolutions. So, in this work are deduced some bounds for the MMINF system busy period length distribution function, meaning the second M exponential service time, which analytic expressions are simpler than the exact one. As a consequence, also some bounds for the MMINF system busy cycle length distribution function are presented.