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

M/G/1-type processes are often evaluated using matrix analytic methods. Specifically, Ramaswami's recursive formula has been established as the numerically stable solution tool. Additionally, the ETAQA method, proposed previously, offers a more efficient alternative for the exact computation of a general class of metrics for M/G/1-type processes. However, the theoretical 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. The resulting new method solves a homogeneous system of equations to obtain the aggregate probability of a finite set of classes of states from the state space of the underlying Markov chain. The new method, constructs this system of linear equations 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 speed-up over ETAQA. In addition, we show that the coefficients of the homogeneous system of equations define an M-matrix, which under certain conditions, is also diagonally dominant and thus can be factored stably. More importantly, we show that the new 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.

[1]  Beatrice Meini,et al.  Solving M/G/1 type Markov chains: recent advances and applications , 1998 .

[2]  B. Conolly.  Structured Stochastic Matrices of M/G/1 Type and Their Applications , 1991 .

[3]  V. Ramaswami.  A stable recursion for the steady state vector in markov chains of m/g/1 type , 1988 .

[4]  Beatrice Meini,et al.  An improved FFT-based version of Ramaswami's formula , 1997 .

[5]  Beatrice Meini,et al.  Analyzing M/G/1 paradigmsthrough QBDs: the role of the block structure in computing the matrixG , 2000 .

[6]  David M. Lucantoni,et al.  The BMAP/G/1 QUEUE: A Tutorial , 1993, Performance/SIGMETRICS Tutorials.

[7]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[8]  Nail Akar,et al.  Matrix-geometric solutions of M/G/1-type Markov chains: a unifying generalized state-space approach , 1998, IEEE J. Sel. Areas Commun..

[9]  Beatrice Meini,et al.  Using displacement structure for solving Non-Skip-Free M/G/1 type Markov chains , 1998 .

[10]  Mei Han An,et al.  accuracy and stability of numerical algorithms , 1991 .

[11]  Winfried K. Grassmann,et al.  Matrix Analytic Methods , 2000 .

[12]  Vaidyanathan Ramaswami,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 1999, ASA-SIAM Series on Statistics and Applied Mathematics.

[13]  Alma Riska,et al.  ETAQA Solutions for Infinite Markov Processes with Repetitive Structure , 2007, INFORMS J. Comput..

[14]  Gianfranco Ciardo,et al.  ETAQA: An Efficient Technique for the Analysis of QBD-Processes by Aggregation , 1999, Perform. Evaluation.

[15]  Alma Riska,et al.  M/G/1-Type Markov Processes: A Tutorial , 2002, Performance.

[16]  Marcel F. Neuts,et al.  Matrix-Geometric Solutions in Stochastic Models , 1981 .

[17]  William J. Stewart,et al.  Introduction to the numerical solution of Markov Chains , 1994 .

[18]  Attahiru Sule Alfa,et al.  Advances in matrix-analytic methods for stochastic models , 1998 .

[19]  Alma Riska,et al.  Exact aggregate solutions for M/G/1-type Markov processes , 2002, SIGMETRICS '02.