Browsing by Subject "Markov Processes"

Now showing 1 - 8 of 8

Open Access
Comparison of multilevel methods for kronecker-based Markovian representations
(Springer, 2004) Buchholz, P.; Dayar T.
The paper presents a class of numerical methods to compute the stationary distribution of Markov chains (MCs) with large and structured state spaces. A popular way of dealing with large state spaces in Markovian modeling and analysis is to employ Kronecker-based representations for the generator matrix and to exploit this matrix structure in numerical analysis methods. This paper presents various multilevel (ML) methods for a broad class of MCs with a hierarchcial Kronecker structure of the generator matrix. The particular ML methods are inspired by multigrid and aggregation-disaggregation techniques, and differ among each other by the type of multigrid cycle, the type of smoother, and the order of component aggregation they use. Numerical experiments demonstrate that so far ML methods with successive over-relaxation as smoother provide the most effective solvers for considerably large Markov chains modeled as HMMs with multiple macrostates.
Open Access
Comparison of partitioning techniques for two-level iterative solvers on large, sparse Markov chains
(SIAM, 2000) Dayar T.; Stewart, W. J.
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.
Open Access
Iterative methods based on splittings for stochastic automata networks
(1998) Uysal, E.; Dayar T.
This paper presents iterative methods based on splittings (Jacobi, Gauss-Seidel, Successive Over Relaxation) and their block versions for Stochastic Automata Networks (SANs). These methods prove to be better than the power method that has been used to solve SANs until recently. With the help of three examples we show that the time it takes to solve a system modeled as a SAN is still substantial and it does not seem to be possible to solve systems with tens of millions of states on standard desktop workstations with the current state of technology. However, the SAN methodology enables one to solve much larger models than those could be solved by explicitly storing the global generator in the core of a target architecture especially if the generator is reasonably dense. © 1998 Elsevier Science B.V. All rights reserved.
Open Access
Lumpable continuous-time stochastic automata networks
(Elsevier, 2003-07-16) Gusak, O.; Dayar, T.; Fourneau, J. M.
The generator matrix of a continuous-time stochastic automata network (SAN) is a sum of tensor products of smaller matrices, which may have entries that are functions of the global state space. This paper specifies easy to check conditions for a class of ordinarily lumpable partitionings of the generator of a continuous-time SAN in which aggregation is performed automaton by automaton. When there exists a lumpable partitioning induced by the tensor representation of the generator, it is shown that an efficient aggregation-iterative disaggregation algorithm may be employed to compute the steady-state distribution. The results of experiments with two SAN models show that the proposed algorithm performs better than the highly competitive block Gauss-Seidel in terms of both the number of iterations and the time to converge to the solution. © 2002 Elsevier Science B.V. All rights reserved.
Open Access
M/M/1 polling models with two finite queues
(1995) Daşçı, Abdullah
Polling models are special kinds of queueing models where multiple-customer type single-stage is considered. In this thesis, first an overview and a classification of polling models will be given. Then two-costomer one server M /M /l polling models will be analyzed and the performance of models will be developed for exhaustive, gated, and G-limited service policies. We give analytical methods for a special type of polling model where we solve the system to get mean queue lengths and thruput rates by three methods. The first one is based on solving the steady state distribution of the Markov Process. The second is a decompositon aiming to decrease the size of the problem. The third one is an approximation method that uses the earlier results and it is very accurate. The thesis will be concluded with possible future extensions.
Open Access
Modelling and analysis of pull production systems
(1995) Kırkavak, Nureddin
A variety of production systems appearing in the literature are reviewed in order to develop a classification scheme for production systems. A number of pull production systems appearing in the classification are found to be equivalent to a tandem queue so that accurate tandem queue decomposition methods can be used to find the performance of such systems. The primary concern of this dissertation is to model and analyze non-tandem queue equivalent periodic pull production systems. In this research, an exact performance evaluation model is developed for a singleitem periodic pull production system. The processing and demand interarrival times are assumed to be Markovian. For large systems, which are difficult to evaluate exactly because of large state spaces involved, an approximate decomposition method is proposed. A typical approximate decomposition procedure takes individual stages or pairs of stages in isolation to analyze the system and then it aggregates the results to obtain an approximate performance for the whole system. An experiment is designed in order to investigate the general behavior of the decomposition. The results are worth attention. A second aspect of this study is to investigate an allocation methodology to achieve the maximum throughput rate with providing two sets of allocation parameters regarding the number of kanbans and the workload at each stage of the system. Together with some structural properties, the experimental results provide some insight into the behavior of pull production systems and also provide a basis for the proposed allocation methodology. Finally, we conclude our findings together with some directions for future research.
Open Access
Prosody-based automatic segmentation of speech into sentences and topics
(Elsevier, 2000) Shriberg, E.; Stolcke, A.; Hakkani-Tür, D.; Tür, G.
A crucial step in processing speech audio data for information extraction, topic detection, or browsing/playback is to segment the input into sentence and topic units. Speech segmentation is challenging, since the cues typically present for segmenting text (headers, paragraphs, punctuation) are absent in spoken language. We investigate the use of prosody (information gleaned from the timing and melody of speech) for these tasks. Using decision tree and hidden Markov modeling techniques, we combine prosodic cues with word-based approaches, and evaluate performance on two speech corpora, Broadcast News and Switchboard. Results show that the prosodic model alone performs on par with, or better than, word-based statistical language models-for both true and automatically recognized words in news speech. The prosodic model achieves comparable performance with significantly less training data, and requires no hand-labeling of prosodic events. Across tasks and corpora, we obtain a significant improvement over word-only models using a probabilistic combination of prosodic and lexical information. Inspection reveals that the prosodic models capture language-independent boundary indicators described in the literature. Finally, cue usage is task and corpus dependent. For example, pause and pitch features are highly informative for segmenting news speech, whereas pause, duration and word-based cues dominate for natural conversation.
Open Access
Transforming stochastic matrices for stochastic comparison with the st-order
(E D P Sciences, 2003) Dayar T.; Fourneau, J. M.; Pekergin, N.
We present a transformation for stochastic matrices and analyze the effects of using it in stochastic comparison with the strong stochastic (st) order. We show that unless the given stochastic matrix is row diagonally dominant, the transformed matrix provides better st bounds on the steady state probability distribution. © EDP Sciences 2003.