Algebraic multigrid method for queueing networks

Abstract

A modified algebraic multigrid (AMG) method for queueing networks is presented. The method keeps the singularity of queueing networks in the coarse grid by modifying the restriction operators. Numerical results demonstrate that this method is more efficient and robust than conventional AMG method.

Multigrid algorithm from cyclic reduction for Markovian queueing networks

“…Therefore, the GMG is not applicable and we should turn to the AMG method. Although the AMG methods have already been proposed to solve the steady and transient problems in [8,6] respectively, both of them have not made use of the special structure of the coefficient matrices. In [8], the authors just modify the general AMG method to cope with the singular linear systems to get the steady state solution.…”

“…Although the AMG methods have already been proposed to solve the steady and transient problems in [8,6] respectively, both of them have not made use of the special structure of the coefficient matrices. In [8], the authors just modify the general AMG method to cope with the singular linear systems to get the steady state solution. The construction of the prolongation and restriction operators is complicated and expensive: it requires separating grid points into two sets according to the matrix graphs of the coefficient matrices on each level.…”

“…We note that the coefficient matrix A in (7) is singular with a one dimensional null-space and a positive null-vector. Therefore there exists a unique solution of (7), see [4,8] for details. Although (7) is a singular linear system, we can transform it into a nonsingular one by the trick introduced in [4].…”

“…The other is the transient state for analysis in a finite horizon. The steady state probability distribution of Markovian queueing network can be obtained by solving a nonsymmetric singular linear system [3,4,6,8,11,16], while the transient solutions can be obtained by solving a system of ordinary differential equations (ODEs). In this paper, we are interested in finding both the steady state probability distribution vectors and the transient solutions of Markovian queueing networks.…”

Application of splitting scheme and multigrid method for TV-Stokes denoising

“…The method is nearly optimal in the sense that the computational work required to achieve a fixed accuracy is proportional to the number of discrete unknowns [9]. The algebraic multigrid (AMG) method is designed to utilize the principle of the geometric multigrid (GMG) method to obtain a fast and automatic solution procedure for linear algebraic system of equations (see [10,11]). This method is particularly suitable to denoising problem, where the coefficients may vary dramatically.…”