Browsing by Subject "Near complete decomposability"

Now showing 1 - 4 of 4

Open Access
Analysis of large Markov chains using stochastic automata networks
(2001-07) Oleg, Gusak
This work contributes to the existing research in the area of analysis of finite Markov chains (MCs) modeled as stochastic automata networks (SANs). First, this thesis extends the near complete decomposability concept of Markov chains to SANs so that the inherent difficulty associated with solving the underlying MC car! be forecasted and solution techniques based on this concept car! be investigated. A straightforward approach to finding a nearly completely decomposable (NCD) partitioning of the MC underlying a SAN requires the computation of the nonzero elements of its global generator. This is not feasible for very large systems ever! in sparse matrix representation due to memory and execution time constraints. In this thesis, an efficient decompositional solution algorithm to this problem that is based on analyzing the NCD structure of each component of a giver! SAN is introduced. Numerical results show that the giver! algorithm performs much better than the straightforward approach. Second, this work specifies easy to check lumpability conditions for the generator of a SAN. When there exists a lumpable partitioning induced by the tensor representation of the generator, it is shown that an efficient iterative aggregation-disaggregation algorithm (IAD) may be employed to compute the steady state distribution of the MC underlying the SAN model. The results of experiments with continuous-time arid discrete-time SAN models show that the proposed algorithm performs better than the highly competitive block Gauss- Seidel (BGS) in terms of both the number of iterations arid the time to converge to the solution. having relatively large blocks in lurnpable partitionings is investigated. To overcome difficulties associated with solving large diagonal blocks at each iteration of the IAD algorithm, the recursive implementation of BGS for SANs is employed. The performance of IAD is compared with that of BGS. The results of experiments show that it is possible to tune IAD so that it outperforms BGS.
Open Access
Componentwise bounds for nearly completely decomposable Markov chains using stochastic comparison and reordering
(Elsevier, 2005) Pekergin, N.; Dayar T.; Alparslan, D. N.
This paper presents an improved version of a componentwise bounding algorithm for the state probability vector of nearly completely decomposable Markov chains, and on an application it provides the first numerical results with the type of algorithm discussed. The given two-level algorithm uses aggregation and stochastic comparison with the strong stochastic (st) order. In order to improve accuracy, it employs reordering of states and a better componentwise probability bounding algorithm given st upper- and lower-bounding probability vectors. Results in sparse storage show that there are cases in which the given algorithm proves to be useful. © 2004 Elsevier B.V. All rights reserved.
Open Access
Experiments with two-stage iterative solvers and preconditioned Krylov subspace methods on nearly completely decomposable Markov chains
(1997) Gueaieb, Wail
Preconditioned Krylov subspace methods are state-of-the-art iterative solvers developed mostly in the last fifteen years that may be used, among other things, to solve for the stationary distribution of Markov chains. Assuming Markov chains of interest are irreducible, the ¡problem amounts to computing a positive solution vector to a homogeneous system of linear algebraic equations with a singular coefficient matrix under a normalization constraint. That is, the (n X 1) unknown stationary vector x in Ax = 0, ||a:||^ = 1 (0.1 ) is sought. Here A = I — , an n x n singular M-matrix, and P is the one-step stochastic transition probability matrix. Albeit the recent advances, practicing performance analysts still widely prefer iterative methods based on splittings when they want to compare the performance of newly devised algorithms against existing ones, or when they need candidate solvers to evaluate the performance of a system model at hand. In fact, experimental results with Krylov subspace methods on Markov chains, especially the ill-conditioned nearly completely decomposable (NCD) ones, are few. We believe there is room for research in this area siDecifically to help us understand the effect of the degree of coupling of NCD Markov chains and their nonzero structure on the convergence characteristics and space requirements of preconditioned Krylov subspace methods. The work of several researchers have raised important and interesting questions that led to research in another, yet related direction. These questions are the following: “How must one go about partitioning the global coefficient matrix A in equation (0.1) into blocks if the system is NCD and a two-stage iterative solver (such as block successive overrelaxation— SOR) is to be employed? Are block partitionings dictated by the NCD normal form of F necessarily superior to others? Is it worth investing alternative partitionings? Better yet, for a fixed labelling 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 (lAD) algorithm? Finally, is there any merit in using two-stage iterative solvers when preconditioned Krylov subspace methods are available?” Experimental results show that in most of the test cases two-stage iterative solvers are superior to Krylov subspace methods with the chosen preconditioners, on NCD Markov chains. For two-stage iterative solvers, there are cases in which a straightforward partitioning of the coefficient matrix gives a faster solution than can be obtained using the NCD normal form.
Open Access
Stochastic comparison on nearly completely decomposable Markov chains
(2000) Alparslan, Denizhan N.
This thesis presents an improved version of a componentwise bounding algorithm for the steady state probability vector of nearly completely decomposable Markov chains. The given two-level algorithm uses aggregation and stochastic comparison with the strong stochastic (st) order. In order to improve accuracy, it employs reordering of states and a better componentwise probability bounding algorithm given st upper- and lower- bounding probability vectors. A thorough analysis of the algorithm is implemented in sparse storage and its implementation details are given. Numerical results on an application of wireless Asynchronous Transfer Mode network show that there are cases in which the given algorithm proves to be useful in computing bounds on the performance measures of the system. An improvement in the algorithm that must be considered to obtain better bounds on performance measures is also presented at the end.