Iterative Aggregation-disaggregation Research Articles

Error propogation formula of multi-level aggregation-disaggregation methods for non-symmetric problems

Iterative aggregation-disaggregation methods for numerical computing of stationary probability distribution vectors of stochastic matrices are studied. The methods can use arbitrary numbers of levels and of smoothing steps. A formula for the error propagation is derived. Using this formula, some asymptotic convergence properties of these methods for non-symmetric problems are demonstrated.

Physiology and pathology of iterative aggregation–disaggregation methods

SUMMARYWe found characteristics that can help us predict the convergence or divergence of iterative aggregation–disaggregation methods. We provided two results for spectral radii of asymptotic error propagation matrices: (i) the spectral radius is bounded by unity for symmetric Markov chains and (ii) the spectral radius can be arbitrarily large for a certain class of sparse Markov chains. Surprisingly, permuting states of cyclic Markov chains by algorithms usually used for reducing bandwidth of matrices leads to the latter case. We proposed a sorting method that prevents divergence for this class of Markov chains.Copyright © 2011 John Wiley & Sons, Ltd.

Convergence of multi-level iterative aggregation–disaggregation methods

This paper introduces an error propagation formula of a certain class of multi-level iterative aggregation–disaggregation (IAD) methods for numerical solutions of stationary probability vectors of discrete finite Markov chains. The formula can be used to investigate convergence by computing the spectral radius of the error propagation matrix for specific Markov chains. Numerical experiments indicate that the same type of the formula could be used for a wider class of the multi-level IAD methods. Using the formula we show that for given data there is no relation between convergence of two-level and of multi-level IAD methods.

Necessary and sufficient local convergence condition of one class of iterative aggregation–disaggregation methods

AbstractThis paper concludes one part of the local convergence analysis of a certain class of iterative aggregation–disaggregation methods for computing a stationary probability distribution vector of an irreducible stochastic matrix B. We show that the local convergence of the algorithm is determined only by the sparsity pattern of the matrix and by the choice of the aggregation groups. We introduce the asymptotic convergence rates of the normalized components of approximations corresponding to particular aggregation groups and we also specify an upper bound on the rates. Copyright © 2008 John Wiley & Sons, Ltd.

On the Convergence of a Class of Multilevel Methods for Large Sparse Markov Chains

This paper investigates the theory behind the steady state analysis of large sparse Markov chains with a recently proposed class of multilevel methods using concepts from algebraic multigrid and iterative aggregation-disaggregation. The motivation is to better understand the convergence characteristics of the class of multilevel methods and to have a clearer formulation that will aid their implementation. In doing this, restriction (or aggregation) and prolongation (or disaggregation) operators of multigrid are used, and the Kronecker-based approach for hierarchical Markovian models is employed, since it suggests a natural and compact definition of grids (or levels). However, the formalism used to describe the class of multilevel methods for large sparse Markov chains has no influence on the theoretical results derived.

Local convergence analysis of iterative aggregation–disaggregation methods with polynomial correction

The paper introduces some new results on local convergence analysis of one class of iterative aggregation–disaggregation methods for computing a stationary probability distribution vector of an irreducible stochastic matrix. We focus on methods, where the basic iteration on the fine level corresponds to a multiplication by a polynomial of order one with nonnegative coefficients in the original matrix. We show that this process is locally convergent for matrices with positive diagonals or when the coefficients of the polynomial are positive. On the other hand there are examples for which the process may diverge in a local sense for higher degree polynomials even if it converges for a polynomial of a lower degree for the same matrix.

A note on local and global convergence analysis of iterative aggregation–disaggregation methods

The purpose of the paper is to present some convergence properties of the iterative aggregation–disaggregation method for computing a stationary probability distribution vector of a column stochastic matrix. A sufficient condition for the local convergence property and the corresponding rate of convergence are established. Some global convergence considerations are presented. Several illustrative examples are included.

Comparison of Partitioning Techniques for Two-Level Iterative Solvers on Large, Sparse Markov Chains

Experimental results for large, sparse Markov chains, especially the ill-conditioned nearly completely decomposable (NCD) ones, are few. We believe there is need for further research in this area, specifically to aid in the understanding of the effects of the degree of coupling of NCD Markov chains and their nonzero structure on the convergence characteristics and space requirements of iterative solvers. The work of several researchers has raised the following questions that led to research in a related direction: How must one go about partitioning the global coefficient matrix into blocks when the system is NCD and a two-level iterative solver (such as block SOR) is to be employed? Are block partitionings dictated by the NCD form of the stochastic one-step transition probability matrix necessarily superior to others? Is it worth investigating alternative partitionings? Better yet, for a fixed labeling and partitioning of the states, how does the performance of block SOR (or even that of point SOR) compare to the performance of the iterative aggregation-disaggregation (IAD) algorithm? Finally, is there any merit in using two-level iterative solvers when preconditioned Krylov subspace methods are available? We seek answers to these questions on a test suite of 13 Markov chains arising in 7 applications.

Analyzing closed Kanban-controlled assembly systems by iterative aggregation–disaggregation

Abstract In recent years, many companies have begun using WIP-limiting strategies such as CONWIP, Kanban or drum-buffer-rope to control parts flows in complex production systems. This paper analyzes assembly systems with a tree structure, random processing times and a constant WIP control system. We present heuristic version of the exact aggregation–disaggregation theory for finite Markov chains to evaluate the performance of these closed Kanban-controlled assembly systems. Because the approximation is theory based, it provides a framework for further model development, with some possible extensions described in the paper. The approximation has the novel feature of doing simultaneous multiple partitions of the state space, in such a way that the associated aggregate transition rates are mutually consistent. The methodology is a novel approach towards extending aggregation ideas to fork-and-join queueing networks, and it provides several useful operational and analytical insights. Numerical comparisons with simulation show that the proposed approximation computes accurate estimates of the plant throughput. It provides a fast way to assess the performance and economic impact of changes in the total WIP level (or the number of Kanbans), or in the part routes, on the throughput rate of the assembly system. Scope and purpose We develop a new methodology for rapid performance analysis of assembly systems with a tree structure, random processing times, and a constant work in process Kanban control system. Extensive evaluations show that this methodology computes accurate estimates of the plant throughput. It provides a fast way for managers to assess the performance and economic impact of changes in the total WIP level (or the number of Kanbans) and in the part routes, on the overall throughput rate of the assembly system.

On the Effects of Using the Grassmann–Taksar–Heyman Method in Iterative Aggregation–Disaggregation

Iterative aggregation–disaggregation (IAD) is an effective method for solving finite nearly completely decomposable (NCD) Markov chains. Small perturbations in the transition probabilities of these chains may lead to considerable changes in the stationary probabilities; NCD Markov chains are known to be ill-conditioned. During an IAD step, this undesirable condition is inherited by the coupling matrix and one confronts the problem of finding the stationary probabilities of a stochastic matrix whose diagonal elements are close to l. In this paper, the effects of using the Grassmann–Taksar–Heyman (GTH) method to solve the coupling matrix formed in the aggregation step are investigated. Then the idea is extended in such a way that the same direct method can be incorporated into the disaggregation step. Finally, the effects of using the GTH method in the IAD algorithm on various examples are demonstrated, and the conditions under which it should be employed are explained.

On a two-level multigrid solution method for finite Markov chains

We study a two-level algebraic multigrid scheme for computing the stationary distribution of a homogeneous Markov chain with a finite state space. We reveal its relationship to an iterative aggregation-disaggregation method, provide an error analysis, and prove its local convergence. Moreover, we discuss the stochastic meaning of a block SOR procedure that can be used as iterative smoother of the multigrid scheme.

A new aggregation-disaggregation algorithm

Abstract This paper proposes a new algorithm for solving the Kolmogoroff equations for large Markov chains. The algorithm consists of two stages. At the first stage, it finds the optimal partition of states into blocks which makes the Markov chain as completely lumpable or decomposable as possible. We define an objective function for evaluating the lumpability or decomposability of a Markov chain with respect to a partition. A lower bound of the objective function is presented and the Branch-and-Bound method is proposed for obtaining the optimal partition. At the second stage, the Kolmogoroff equations are solved by the iterative aggregation-disaggregation method with the optimal partition. Numerical results are presented in order to demonstrate the effectiveness of the proposed algorithm.

A multi-level solution algorithm for steady-state Markov chains

A new iterative algorithm, the multi-level algorithm, for the numerical solution of steady state Markov chains is presented. The method utilizes a set of recursively coarsened representations of the original system to achieve accelerated convergence. It is motivated by multigrid methods, which are widely used for fast solution of partial differential equations. Initial results of numerical experiments are reported, showing significant reductions in computation time, often an order of magnitude or more, relative to the Gauss-Seidel and optimal SOR algorithms for a variety of test problems. It is shown how the well-known iterative aggregation-disaggregation algorithm of Takahashi can be interpreted as a special case of the new method.

An iterative aggregation-disaggregation algorithm for solving linear equations

A general procedure is given for solving large sets of linear equations by first rewriting them in a form suitable for aggregation of both the variables and equations, followed by disaggregation. A computational algorithm which iteratively aggregates and disaggregates is shown to converge geometrically to the exact solution. Provided the original problem has a structure suitable for such aggregation, the algorithm exhibits fast computation times, small main-memory requirements, and robustness to the starting point. A rigorous foundation for aggregation and disaggregation is provided by the equations employed by this algorithm.

Iterative Aggregation-Disaggregation Procedures for Discounted Semi-Markov Reward Processes

The equation v = q + Mv, where M is a matrix with nonnegative elements and spectral radius less than one, arises in Markovian decision processes and input-output models. In this paper, we solve the equation using an iterative aggregation-disaggregation procedure that alternates between solving an aggregated problem and disaggregating the variables, one block at a time, in terms of the aggregate variables of the other blocks. The disaggregated variables are then used to guide the choice of weights in the subsequent aggregation. Computational experiments on randomly generated and inventory problems indicate that this algorithm is significantly faster than successive approximations when the spectral radius of M is near one, and is slower in unstructured problems with spectral radii in the neighborhood of 0.8. The algorithm appears promising for large structured problems, where it can often reduce computational time and main memory storage requirements and offer greater robustness to initial values.