First Order Differential Equations Induced by the Infinite Servers Queue with Poisson Arrivals Transient Behavior Probability Distribution Parameters Study as Time Functions

TL;DR

This paper investigates the transient probability distributions of an infinite servers queue with Poisson arrivals, deriving differential equations that describe how mean and variance evolve over time based on service time hazard rates.

Contribution

It introduces differential equations linking service time hazard rates to the time-dependent behavior of mean and variance in the queue's transient state.

Findings

Derived differential equations for mean and variance evolution.

Identified specific service time distributions with particular time-dependent behaviors.

Analyzed the impact of hazard rate functions on queue dynamics.

Abstract

The infinite servers queue with Poisson arrivals state transient probabilities, considering the time origin at the beginning of a busy period, mean and variance monotony as time functions is studied. These studies, for which results it is determinant the hazard rate function service time length, induce the consideration of two differential equations, one related with the mean monotony study and another with the variance monotony study, which solutions lead to some particular service time distributions, for which those parameters present specific behaviors as time functions.