scholarly journals Strong Truncation Approximation in Tandem Queues with Blocking

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Karima Adel-Aissanou ◽  
Karim Abbas ◽  
Djamil Aïssani
Keyword(s):  
State Space ◽  
Error Bounds ◽  
Numerical Solutions ◽  
Markov Models ◽  
Practical Interest ◽  
Strong Stability ◽  
Tandem Queue ◽  
Closed Form Solutions ◽  
Infinite State

Markov models are frequently used for performance modeling. However most models do not have closed form solutions, and numerical solutions are often not feasible due to the large or even infinite state space of models of practical interest. For that, the state-space truncation is often demanded for computation of this kind of models. In this paper, we use the strong stability approach to establish analytic error bounds for the truncation of a tandem queue with blocking. Numerical examples are carried out to illustrate the quality of the obtained error bounds.

A weak perturbation theory for approximations of invariant measures in M/G/1 model

2018 ◽  
Vol 52 (4-5) ◽  
pp. 1411-1428
Author(s):  
Badredine Issaadi ◽  
Karim Abbas ◽  
Djamil Aïssani
Keyword(s):  
Perturbation Theory ◽  
Closed Form ◽  
Stationary Distribution ◽  
Error Bounds ◽  
Invariant Measures ◽  
Infinite Matrix ◽  
Numerical Examples ◽  
Closed Form Solutions ◽  
Weak Perturbation

The calculation of the stationary distribution for a stochastic infinite matrix is generally difficult and does not have closed form solutions, it is desirable to have simple approximations converging rapidly to this distribution. In this paper, we use the weak perturbation theory to establish analytic error bounds for the M/G/1 model. Numerical examples are carried out to illustrate the quality of the obtained error bounds.

scholarly journals Novel domain expansion methods to improve the computational efficiency of the Chemical Master Equation solution for large biological networks

2020 ◽  
Vol 21 (1) ◽  
Author(s):  
Rahul Kosarwal ◽  
Don Kulasiri ◽  
Sandhya Samarasinghe
Keyword(s):  
Master Equation ◽  
State Space ◽  
Performance Management ◽  
Numerical Solutions ◽  
De Novo ◽  
Exact Analytical Solution ◽  
Chemical Master Equation ◽  
The State ◽  
Biochemical System ◽  
Biochemical Systems

Abstract Background Numerical solutions of the chemical master equation (CME) are important for understanding the stochasticity of biochemical systems. However, solving CMEs is a formidable task. This task is complicated due to the nonlinear nature of the reactions and the size of the networks which result in different realizations. Most importantly, the exponential growth of the size of the state-space, with respect to the number of different species in the system makes this a challenging assignment. When the biochemical system has a large number of variables, the CME solution becomes intractable. We introduce the intelligent state projection (ISP) method to use in the stochastic analysis of these systems. For any biochemical reaction network, it is important to capture more than one moment: this allows one to describe the system’s dynamic behaviour. ISP is based on a state-space search and the data structure standards of artificial intelligence (AI). It can be used to explore and update the states of a biochemical system. To support the expansion in ISP, we also develop a Bayesian likelihood node projection (BLNP) function to predict the likelihood of the states. Results To demonstrate the acceptability and effectiveness of our method, we apply the ISP method to several biological models discussed in prior literature. The results of our computational experiments reveal that the ISP method is effective both in terms of the speed and accuracy of the expansion, and the accuracy of the solution. This method also provides a better understanding of the state-space of the system in terms of blueprint patterns. Conclusions The ISP is the de-novo method which addresses both accuracy and performance problems for CME solutions. It systematically expands the projection space based on predefined inputs. This ensures accuracy in the approximation and an exact analytical solution for the time of interest. The ISP was more effective both in predicting the behavior of the state-space of the system and in performance management, which is a vital step towards modeling large biochemical systems.

Unit Mobility Ratio Displacement Calculations for Pattern Floods in Homogeneous Medium

1966 ◽  
Vol 6 (03) ◽  
pp. 217-227 ◽  
Author(s):  
Hubert J. Morel-Seytoux
Keyword(s):  
Oil Recovery ◽  
Numerical Solutions ◽  
Mobility Ratio ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Numerical Techniques ◽  
Sweep Efficiency ◽  
Closed Form Solutions ◽  
Regular Patterns ◽  
Quantities Of Interest

Abstract The influence of pattern geometry on assisted oil recovery for a particular displacement mechanism is the object of investigation in this paper. The displacement is assumed to be of unit mobility ratio and piston-like. Fluids are assumed incompressible and gravity and capillary effects are neglected. With these assumptions it is possible to calculate by analytical methods the quantities of interest to the reservoir engineer for a great variety of patterns. Specifically, this paper presentsvery briefly, the methods and mathematical derivations required to obtain the results of engineering concern, andtypical results in the form of graphs or formulae that can be used readily without prior study of the methods. Results of this work provide checks for solutions obtained from programmed numerical techniques. They also reveal the effect of pattern geometry and, even though the assumptions of piston-like displacement and of unit mobility ratio are restrictive, they can nevertheless be used for rather crude but quick, cheap estimates. These estimates can be refined to account for non-unit mobility ratio and two-phase flow by correlating analytical results in the case M=1 and the numerical results for non-Piston, non-unit mobility ratio displacements. In an earlier paper1 it was also shown that from the knowledge of closed form solutions for unit mobility ratio, quantities called "scale factors" could be readily calculated, increasing considerably the flexibility of the numerical techniques. Many new closed form solutions are given in this paper. INTRODUCTION BACKGROUND Pattern geometry is a major factor in making water-flood recovery predictions. For this reason many numerical schemes have been devised to predict oil recovery in either regular patterns or arbitrary configurations. The numerical solutions, based on the method of finite difference approximation, are subject to errors often difficult to evaluate. An estimate of the error is possible by comparison with exact solutions. To provide a variety of checks on numerical solutions, a thorough study of the unit mobility ratio displacement process was undertaken. To calculate quantities of interest to the reservoir engineer (oil recovery, sweep efficiency, etc.), it is necessary to first know the pressure distribution in the pattern. Then analytical procedures are used to calculate, in order of increasing difficulty: injectivity, breakthrough areal sweep efficiency, normalized oil recovery and water-oil ratio as a function of normalized PV injected. BACKGROUND Pattern geometry is a major factor in making water-flood recovery predictions. For this reason many numerical schemes have been devised to predict oil recovery in either regular patterns or arbitrary configurations. The numerical solutions, based on the method of finite difference approximation, are subject to errors often difficult to evaluate. An estimate of the error is possible by comparison with exact solutions. To provide a variety of checks on numerical solutions, a thorough study of the unit mobility ratio displacement process was undertaken. To calculate quantities of interest to the reservoir engineer (oil recovery, sweep efficiency, etc.), it is necessary to first know the pressure distribution in the pattern. Then analytical procedures are used to calculate, in order of increasing difficulty: injectivity, breakthrough areal sweep efficiency, normalized oil recovery and water-oil ratio as a function of normalized PV injected.

scholarly journals Combining experimental and simulation data of molecular processes via augmented Markov models

2017 ◽  
Vol 114 (31) ◽  
pp. 8265-8270 ◽  
Author(s):  
Simon Olsson ◽  
Hao Wu ◽  
Fabian Paul ◽  
Cecilia Clementi ◽  
Frank Noé
Keyword(s):  
Force Field ◽  
Structural Changes ◽  
Markov Models ◽  
Force Fields ◽  
Divide And Conquer ◽  
Simulation Data ◽  
Wide Range ◽  
Simulation And Experiment ◽  
Kinetics Of

Accurate mechanistic description of structural changes in biomolecules is an increasingly important topic in structural and chemical biology. Markov models have emerged as a powerful way to approximate the molecular kinetics of large biomolecules while keeping full structural resolution in a divide-and-conquer fashion. However, the accuracy of these models is limited by that of the force fields used to generate the underlying molecular dynamics (MD) simulation data. Whereas the quality of classical MD force fields has improved significantly in recent years, remaining errors in the Boltzmann weights are still on the order of a few kT, which may lead to significant discrepancies when comparing to experimentally measured rates or state populations. Here we take the view that simulations using a sufficiently good force-field sample conformations that are valid but have inaccurate weights, yet these weights may be made accurate by incorporating experimental data a posteriori. To do so, we propose augmented Markov models (AMMs), an approach that combines concepts from probability theory and information theory to consistently treat systematic force-field error and statistical errors in simulation and experiment. Our results demonstrate that AMMs can reconcile conflicting results for protein mechanisms obtained by different force fields and correct for a wide range of stationary and dynamical observables even when only equilibrium measurements are incorporated into the estimation process. This approach constitutes a unique avenue to combine experiment and computation into integrative models of biomolecular structure and dynamics.

Remaining useful life estimation based on the joint use of an observer and a hidden Markov model

2021 ◽  
pp. 1748006X2110443
Author(s):  
Toufik Aggab ◽  
Pascal Vrignat ◽  
Manuel Avila ◽  
Frédéric Kratz
Keyword(s):  
Markov Models ◽  
Hidden Markov ◽  
Practical Interest ◽  
Markov Property ◽  
Rolling Bearing ◽  
Industrial Applications ◽  
Remaining Useful Life ◽  
Discrete Observations ◽  
Useful Life ◽  
The Difference

We propose an approach for failure prognosis based on the estimation of the Remaining Useful Life (RUL) of a system in a situation in which monitoring signals providing information about its degradation evolution are not measured and no operating model of the system is available. These conditions are of practical interest for industrial applications such as mechanical (e.g. rolling bearing) or electrical (e.g. wind turbine) devices or equipment-critical components (e.g. batteries) in which the addition of sensors to the system is not feasible (e.g. space limitations for sensors, cost, etc.). The approach is based on an estimation of the system degradation using residual generation (where the difference between the system and the observer outputs is processed) and Hidden Markov Models with discrete observations. The prediction of the system RUL is given by the Markov property concerning the mean time before absorption. The approach comprises two phases: a training phase to model the degradation behavior and an “on-line” use phase to estimate the remaining life of the system. Two case studies were conducted for RUL prediction to verify the effectiveness of the proposed approach.

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