A Comparative Analysis of the Successive Lumping and the Lattice Path Counting Algorithms

TL;DR

This paper compares the successive lumping and lattice path counting algorithms for queueing models, showing that SL often outperforms LPCA in applicability and efficiency, especially for complex or finite state models.

Contribution

It provides a detailed comparison highlighting the advantages of the SL methodology over LPCA, including broader applicability and steady state computation capabilities.

Findings

SL algorithms outperform LPCA when both are applicable.

SL is applicable to models with non-homogeneous rates or finite states where LPCA fails.

SL includes a method for steady state distribution computation.

Abstract

This article provides a comparison of the successive lumping (SL) methodology with the popular lattice path counting algorithm in obtaining rate matrices for queueing models, satisfying the quasi birth and death structure. The two methodologies are compared both in terms of applicability requirements and numerical complexity by analyzing their performance for the same classical queueing models. The main findings are: i) When both methods are applicable SL based algorithms outperform the lattice path counting algorithm (LPCA). ii) There are important classes of problems (e.g., models with (level) non-homogenous rates or with finite state spaces) for which the SL methodology is applicable and for which the LPCA cannot be used. iii) Another main advantage of successive lumping algorithms over LPCAs is that the former includes a method to compute the steady state distribution using this rate matrix.