Relaxations for the numerical solutions of some stochastic problems

Mitra, Debasis; Tsoucas, Pantelis

doi:10.1080/15326348808807087

Cited by 20 publications

(16 citation statements)

References 20 publications

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“…Unfortunately, the result requires the knowledge of all the cycles in Γ(P). Several, more practical sufficient conditions on Γ(P) or Γ(Q) for γ(H GS ) to be < 1 have been derived [2,17] …”

Section: Convergence Resultsmentioning

confidence: 99%

Numerical iterative methods for Markovian dependability and performability models: new results and a comparison

Suñé

Domingo

Carrasco

2000

Performance Evaluation

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In this paper we deal with iterative numerical methods to solve linear systems arising in continuous-time Markov chain (CTMC) models. We develop an algorithm to dynamically tune the relaxation parameter of the successive over-relaxation method. We give a sufficient condition for the Gauss-Seidel method to converge when computing the steady-state probability vector of a finite irreducible CTMC, an a suffient condition for the Generalized Minimal Residual projection method not to converge to the trivial solution 0 when computing that vector. Finally, we compare several splitting-based iterative methods an a variant of the Generalized Minimal Residual projection method.

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Section: Convergence Resultsmentioning

confidence: 99%

Numerical iterative methods for Markovian dependability and performability models: new results and a comparison

Suñé

Domingo

Carrasco

2000

Performance Evaluation

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“…Unfortunately, the number of states grows very rapidly with N, q, and (if relevant) k, which prevents solving cases significantly larger than those presented here. (By using an antilexicographic ordering suggested by Mitra and Tsoucas 1987 for raster convergence, we are able to solve cases with several tens of thousands or states on a CONVEX-240 machine with vectorized code. ) We assume here that N, q, and c are fixed, whereas the w j are continuous decision variables.…”

Section: Formulationmentioning

confidence: 99%

Some data for applying the bowl phenomenon to large production line systems

HILLlER

1993

International Journal of Production Research

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The bowl phenomenon provides a way of increasing the throughput of some production line systems with variable processing times by purposely unbalancing the line in a certain manner. However, previously available numerical results for applying the bowl phenomenon have been quite limited. We extend these numerical results here by obtaining the optimal allocation of work for somewhat larger cases than previously considered. We also develop some guidelines and data for extrapolating these results to estimate the optimal allocation of work for even larger production lines that are beyond the reach of exact solution methods. The results cover a broad cross-section of values of both buffer capacities and the coefficientof variation of processing times. I. IntroductionHillier and Boling (1966, 1979) investigated the optimal allocation of work in unpaced production line systems with variable processing times. Their 1966 paper discovered the 'bowl phenomenon', whereby a plot of the optimal allocation vs. station number has the shape of the cross-section of the interior of a bowl. Their 1979 paper obtained numerical results on the optimal allocation for a variety of small cases, and then gave guidelines for extrapolating these results to larger cases. (Also see Rao 1976 for the case where processing times at different stations differ substantially in their variability, Magazine and Silver 1978 for heuristics related to the bowl phenomenon, and Muth and Alkaff 1987 for further analysis of the bowl phenomenon.)Advances in computing power now make it possible to obtain exact results on the optimal allocation for somewhat larger cases than in the 1979 paper. The purposes of the present paper are to present these extended numerical results, and then to use them to refine the 1979 extrapolation guidelines. These results and guidelines are given in Sections 3 and 4, respectively, after first summarizing the model and notation in

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“…We then use the Gauss-Seidel method to solve the resulting system of linear equations to obtain the stationary distribution, from which the mean output rate R(q) can' be easily calculated. (An anti-lexicographic ordering, as suggested by Mitra and Tsoucas [10] for faster converging results, is used to scan the states of the Markov process in each iteration of the Gauss-Seidel method. )…”

Section: Model Formulationmentioning

confidence: 99%

The Effect of the Coefficient of Variation of Operation Times on the Allocation of Storage Space in Production Line Systems

Hillier

1991

IIE Transactions

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Relaxations for the numerical solutions of some stochastic problems

Cited by 20 publications

References 20 publications

Numerical iterative methods for Markovian dependability and performability models: new results and a comparison

Numerical iterative methods for Markovian dependability and performability models: new results and a comparison

Some data for applying the bowl phenomenon to large production line systems

The Effect of the Coefficient of Variation of Operation Times on the Allocation of Storage Space in Production Line Systems

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