Modelling asymptotic stochastic models

By another model, obtained by the transformation of the original model towards one of its boundaries and which can also be solved by an analytical or numerical solution. These models are called limit stochastic models or asymptotic stochastic models . 

All other discrete stochastic models, obtained from polystochastic chains, attached to an investigated process, present the capacity to be transformed into an asymptotic model. When the original and its asymptotic model are calculated numerically, we can rapidly observe if they converge by direct simulation. In this case, the comparison between the behaviour of the original model and the generator function of the asymptotic stochastic model is not necessary. 

The performed calculations demonstrate that a type of the asymptotic solution of a complete set of the kinetic equations is independent of the initial particle concentrations, iVa(0) and 7Vb(0). Variation of parameters a and (3 does not also result in new asymptotical regimes but just modifies there boundaries (in t and k). In the calculations presented below the parameters 7Va(0) = 7Vb(0) =0.1 and a = ft = 0.1 were chosen. The basic parameters of the diffusion-controlled Lotka-Volterra model are space dimension d and the ratio of diffusion coefficients k. The basic results of the developed stochastic model were presented in [21, 25-27],... 

The numerical as well as the asymptotic model solutions are estimated solutions, which often produce characteristic outputs of the model in difterent forms when compared to the natural state of the exits. Both stochastic and transfer phenomenon models present the same type of resolution process. The analysis developed in the paragraphs below can be applied equally to both types of models. 

The Resolution of Stochastic Models by Means of Asymptotic Models... 

At the same time, it is known that, during exploitation of stochastic models, cases that show great difficulty concerning the selection and the choice of some parameters of the models frequently appear. As a consequence, the original models become unattractive for research by simulation. In these cases, the models can be transformed to equivalent models which are distorted but exploitable. The use of stochastic distorted models is also recommended for the models based on stochastic chains or polystocastic processes where an asymptotic behaviour is identified with respect to a process transition matrix of probabilities, process chains evolution, process states connection, etc. The distorted models are also of interest when the stochastic process is not time dependent, as, for example, in the stochastic movement of a marked particle occurring with a constant velocity vector, like in diffusion processes. 

The diffusion model can usually be used for the description of many stochastic distorted models. The equivalent transformation of a stochastic model to its associated diffusion model is fashioned by means of some limit theorems. The first class of limit theorems show the asymptotic transformation of stochastic models based on polystochastic chains the second class is oriented for the transformation of stochastic models based on a polystochastic process and the third class is carried out for models based on differential stochastic equations. 

Stochastic Models Based on Asymptotic Polystochastic Chains... 

For the derivation of one asymptotic variant of a given polystochastic model of a process, we can use the perturbation method. For this transformation, a new time variable is introduced into the stochastic model and then we analyze its behaviour. The new time variable is t = eT, which includes the time evolution t and an arbitrary parameter e, which allows the observation of the model behaviour when its values become very small (e—>0). Here, we study the changes in the operator 0(t, t) when e 0 whilst paying attention to having stable values for t/e or t/e. ... 

Some restrictions are imposed when we start the application of limit theorems to the transformation of a stochastic model into its asymptotic form. The most important restriction is given by the rule where the past and future of the stochastic processes are mixed. In this rule it is considered that the probability that a fact or event C occurs will depend on the difference between the current process (P(C) = P(X(t)e A/V(X(t))) and the preceding process (P (C/e)). Indeed, if, for the values of the group (x,e), we compute = max[P (C/e) — P(C)], then we have a measure of the influence of the process history on the future of the process evolution. Here, t defines the beginning of a new random process evolution and tIt- gives the combination between the past and the future of the investigated process. If a Markov connection process is homogenous with respect to time, we have = 1 or Tt O after an exponential evolution. If Tt O when t increases, the influence of the history on the process evolution decreases rapidly and then we can apply the first type limit theorems to transform the model into an asymptotic... 

To complete this short analysis, we can conclude that, for the asymptotic transformation of a stochastic model, we must identify (i) the infinitesimal generator (ii) what type of theorem will be used for the transformation procedure. 

Asymptotic Models Derived from Stochastic Models with Differential Equations... 

Studies of the transformation of a stochastic model characterized by an assembly of differential equations to its corresponding asymptotic form, show that the use of a perturbation method, where we replace the variable t by t = can be recommended without any restrictions [4.47, 4.48]. 

Very difficult problems occur with the asymptotic transformation of original stochastic models based on stochastic differential equations where the elementary states are not Markov connected. This fact will be discussed later in this chapter (for instance see the discussion of Eq. (4.180)). 

The second example discusses the numerical transposition of the asymptotic models based on poly stochastic chains (see Section 4.4.1.1) where to compute the limit transition probabilities, we must solve the system ePek =... 

The particularization of the limit theorem of the second type to model (4.267) (for instance see also Section 4.5.1.2, relations (4.132)-(4.134)) shows that the stochastic model of the process becomes asymptotic with the parabolic model. 

The discussed stochastic model presents the capacity to be converted into a steady state model in addition, an interesting asymptotic transformation can also be carried out. For the conversion ofthe model into a steady state one, we consider... 

The asymptotic transformation of the discussed stochastic model (see relation 4.292 and Section 4.5.1.2) is carried out with the identification of the operators ... 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

Considerable work has been focused on determining the asymptotic null distribution of -2 log-likelihood -ILL) when the alternative hypothesis is the presence of two subpopulations. In the case of two univariate densities mixed in an unknown proportion, the distribution of -ILL has been shown to be the same as the distribution of [max(0, Y)f, where Y is a standard normal random variable (28). Work with stochastic simulations resulted in the proposal that -2LL-c is distributed with d degrees of freedom, where d is equal to two times the difference in the number of parameters between the nonmixture and mixture model (not including parameters used for the probability models) and c=(n-l-p- gl2)ln (31). In the expression for c, n is the number of observations, p is the dimensionality of the observation, and g is the number of subpopulations. So for the case of univariate observations (p = 1), two subpopulations (g = 2), and one parameter distinguishing the mixture submodels (not including the mixing parameter), -2LL-(n - 3)/n with two... 

ABSTRACT This paper presents methods to evaluate the reliability and optimize the maintenance of engineering systems that are damaged by shocks or transients arriving randomly in time and overall degradation is modeled as a cumulative stochastic point process. The paper presents a conceptually clear and comprehensive derivation of formulas for computing the discoimted cost associated with a periodic inspection and maintenance policy. The proposed discounted cost model provides a more realistic basis for optimizing the maintenance policies than those based on the asymptotic (non-discoimted) cost rate criterion. 

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