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

TLDR: A new formulation is derived that improves the numerical stability and computational performance of ETAQA and shows that the new method is just an efficient way to implement Ramaswami's method.

About: This article is published in Performance Evaluation.The article was published on 2005-10-01 and is currently open access. It has received 10 citations till now. The article focuses on the topics: Matrix analytic method & Numerical stability.

Figures

Fig. 5.3. Numerical accuracy of the methods of Ramaswami, ETAQA and newETAQA for a well conditioned case. We plot the difference from the accurate solution ‖x − xaccurate‖2 for each method, under various machine precisions. The methods are implemented in Maple, which allows for software simulated arbitrary machine precision, by setting the digits in the mantissa of floating point numbers. 100 digits are used for the accurate solution.

Fig. 5.2. Numerical accuracy of the methods of Ramaswami, ETAQA and newETAQA for two ill conditioned cases. We plot the difference from the accurate solution ‖x − xaccurate‖2 for each method, under various machine precisions. The methods are implemented in Maple, which allows for software simulated arbitrary machine precision, by setting the digits in the mantissa of floating point numbers. 100 digits are used for the accurate solution.

Fig. 1.1. Aggregation of an infinite S into a finite number of classes of states.

Fig. 4.1. The table compares of computational costs for two typical values of m and for each of the three methods: Ramaswami, old ETAQA, and newETAQA. The graph below shows for two cases m = 1, n = 4 and m = 1, n = 40 that the dominant cost in Ramaswami’s method is the iterative process, more so as the utilization increases.

Fig. 5.1. The norm-2 condition number of X33 as a function of system utilization (1 − π(0)) for four different cases with increasing coefficient of variation (CV) in the matrix elements. The stiffness caused by high CVs has a larger impact on the condition number.

Table 5.1 The choices of the vector z in creating smooth or highly stiff coefficients in B.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Bridging etaqa and ramaswami’s formula for the solution of m/g/1-type processes" ?

In this paper, the authors derive a new formulation that improves the numerical stability and computational performance of ETAQA. In addition, the authors 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, the authors show that the newETAQA method is just an efficient way to implement Ramaswami ’ s method. The authors also discuss alternative normalization conditions for Ramaswami ’ s method.