Bridging ETAQA and Ramaswami’s formula for the solution of M/G/1-type processes

Abstract

Abstract.For some time, the method of Ramaswami has been the established way to analyze M/G/1-type processes. The ETAQA method, proposed previously in [15], has offered a more efficient alternative for the exact computation of a general class of metrics for M/G/1-type processes. However, the stability of ETAQA and its relation to Ramaswami's method were not well understood. In this paper, we derive a new formulation that improves the numerical stability and computational performance of ETAQA. As with ETAQA, the resulting methodology, newETAQA, solves a homogeneous system of equations to obtain the aggregate probability of a finite set of classes of states from the state space. In contrast to ETAQA, newETAQA constructs its matrix X in a way similar to the method of Ramaswami, decoupling the computation of the probabilities of the first two initial classes of states from the computation of the aggregate probability. Because direct methods are used to solve this system, the decoupling implies an often significant speedup over ETAQA. In addition, we show that the matrix X is an M-matrix, and under certain conditions, X is also diagonally dominant and thus can be factored stably. More importantly, we show that the newETAQA method is just an efficient way to implement Ramaswami's method. We also discuss alternative normalization conditions for Ramaswami's method. Our numerical experiments demonstrate the stability of our method for both stiff and well behaved processes, and for both low and high system utilizations.

ETAQA Solutions for Infinite Markov Processes with Repetitive Structure

“…Once subtractions are involved, the possibility of numerical instability increases because of the loss of significance (as discussed in Neuts 1989, p. 165). Our construction of X in (16) does introduce subtraction, but in Stathopoulos et al (2005), we provide theoretical and experimental evidence that ETAQA is numerically stable.…”

“…Here, we do not focus on numerical stability, but we instead illustrate that the method generalizes to the solution of M/G/1-type, GI/M/1-type, and QBD processes of any type. Numerical stability of ETAQA and its connection to matrix-analytic methods is explored formally in Stathopoulos et al (2005), where ETAQA's numerical stability is proved and shown to be often superior to the alternative matrix-analytic solutions.…”

jMarkov

“…Further, we plan to implement the ETAQA algorithm [4,24], and the algorithm proposed by Riska and Smirni in [21].…”

Clearing analysis on phases: Exact limiting probabilities for skip-free, unidirectional, quasi-birth-death processes

“…Although originally applicable to a narrow class of chains (see [7,6] for details), ETAQA can be generalized so as to be applicable to M/G/1-type, GI/M/1-type, and QBD Markov chains, including those in class M (see the work of Riska and Smirni [23,24]). Stathopoulos et al [27] show that ETAQA is also well suited for numerical computations; ETAQA can be adapted to avoid the numerical problems alluded to in Section 1.1. Unlike the CAP method, ETAQA (like RRR) cannot be used to determine a formula for a chain's limiting probability distribution (across all states) in finitely many operations.…”