M|D|$\infty$ Queue Busy Period and Busy Cycle Distributions Computational Calculus

TL;DR

This paper develops a FORTRAN implementation of an algorithm to compute the distribution functions of busy periods and cycles in M|D|∞ queues, leveraging their Laplace transform properties, which are not available for other M|G|∞ queues.

Contribution

It provides the first practical implementation of an algorithm for calculating busy period and cycle distributions in M|D|∞ queues using Laplace transforms.

Findings

Successful implementation of the algorithm in FORTRAN

Enables calculation of tail probabilities for M|D|∞ queues

Provides a computational tool for queuing system analysis

Abstract

Given the busy period and busy cycle major importance in queuing systems, it is crucial the knowledge of the respective distribution functions that is what allows the calculation of the important probabilities. For the M|G|$\infty$ queue system, there are no round form formulae for those distribution functions. But, for the M|D|$\infty$ queue, due the fact that its busy period and busy cycle have both Laplace transform expression round forms, what does not happen for any other M|G|$\infty$ queue system, with an algorithm created by Platzman, Ammons and Bartholdi III, that allows the tail probabilities computation since the correspondent Laplace transform in round form is known, those distribution functions calculations are possible. Here, we will implement the algorithm through a FORTRAN program.