A stochastic model for solitons

The soliton physics for the propagation of waves is represented by a stochastic model in which the particles of the wave can jump ahead according to some probability distribution. We demonstrate the presence of a steady state (stationary distribution) for the wavelength. It is shown that the stationary distribution is a convolution of geometric random variables. Approximations to the stationary distribution are investigated for a large number of particles. The model is rich and includes Gaussian cases as limit distribution for the wavelength (when suitably normalized). A sufficient Lindeberg‐like condition identifies a class of solitons with normal behavior. Our general model includes, among many other reasonable alternatives, an exponential aging soliton, of which the uniform soliton is one special subcase (with Gumbel's stationary distribution). With the proper interpretation, our model also includes the deterministic model proposed in Takahashi and Satsuma [A soliton cellular automaton, J Phys Soc Japan 59 (1990), 3514–3519]. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Numerical techniques for simulating a fractional mathematical model of epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider the numerical simulation of a fractional mathematical model of epidermal wound healing (FMM-EWH), which is based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in the advection and diffusion terms belong to the intervals (0,1) or (1,2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of Riemann-Liouville and Grünwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.     

A stochastic model for drilling optimization

Drilling optimization problems in oilfields are usually formulated and solved by using deterministic mathematical models, in which uncertain (indeterminate) factors or random issues are not taken into consideration. However, it has been widely experienced that random factors (such as those from soil layers, drill bits, and surface equipment) greatly affect the drilling performance. This paper introduces a new stochastic model for describing such random effects. This model, when used to optimization design, is more practical and provides a better characterization for real oilfield situations as compared with other deterministic models, and has been demonstrated to be more efficient in solving real design problems of drilling optimizations.     

A two-stage stochastic model for labour wastage

A stochastic model for labour wastage is presented which accounts for the employee's variability on the propensity to leave through dependence on both his length of service and his tenure-in-current-state. The basic assumption underlying the model is that the employee's personal characteristics, the job characteristics and the external labour market conditions are stochastic over time, therefore affecting his decision to stay or leave the company. It is shown that the model provides a good fit to a variety of observed leaving patterns for several companies reported in the literature, explains the relationships among a number of important occupational variables, and is useful for planning purposes in predicting future developments.     

A stochastic model for internal HIV dynamics

In this paper we analyse a stochastic model representing HIV internal virus dynamics. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling. We show that the model established in this paper possesses non-negative solutions as this is essential in any population dynamics model. We also carry out analysis on the asymptotic behaviour of the model. We approximate one of the variables by a mean reverting process and find out the mean and variance of this process. Numerical simulations conclude the paper.     

A stochastic block model for interaction lengths

Advances in Data Analysis and Classification - We propose a new stochastic block model that focuses on the analysis of interaction lengths in dynamic networks. The model does not rely on a...     

A stochastic model for two interacting species

An exactly solvable stochastic model for two interacting species with a prey-predator relationship in the presence of self-interaction is investigated. The parameters describing the interactions are assumed to be represented by a dichotomic Markov process. Explicit expressions are derived for the time development of the average of the logarithm of the population size in respect of each species and their asymptotic behaviours are discussed.     

A stochastic inventory model

Summary In some cases arising in certain industries or military installations not only the demand for a particular commodity is a stochastic variable but its supply as well. In these cases it is convenient to consider the inventory level resulting from the interaction of supply and demand as a third stochastic variable. The variation of the inventory level in time can then be considered as a stochastic process. If this process is ergodic, the total inventory cost over a certain timeT may be represented as a function of the mean inventory level. This mean level can then be manipulated in such a way as to minimize the total inventory cost.

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Zusammenfassung Es kommt vor, daß in gewissen Industriezweigen sowohl der Verbrauch, als auch die Anlieferung eines bestimmten Gutes stochastische Variable sind. In solchen Fällen ist es zweckmäßig, wenn man die aus der Zusammenwirkung von Verbrauch und Anlieferung resultierende Vorratsmenge als eine dritte stochastische Variable einführt. Man kann dann die Oszillationen der Vorratsmenge in der Zeit als einen stochastischen Prozeß auffassen. Falls dieser Prozeß ergodisch ist, können die gesamten Vorratshaltungskosten für eine bestimmte ZeitT dargestellt werden als eine Funktion der mittleren Vorratsmenge. Diese mittlere Vorratsmenge kann dann so bestimmt werden, daß sie die gesamten Vorratshaltungskosten minimalisiert.

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SHAPE Air Defence Technical Centre. Formerly with Tidewater Oil Company, Los Angeles, California, where the present problem was originally investigated.

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Vorgel. v.:J. Nitsche.     

A stochastic back-up model

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 52–63, 1991.     

Hyperbolastic modeling of wound healing

A new mathematical model for wound healing is introduced and applied to three sets of experimental data. The model is easy to implement but can accommodate a wide range of factors affecting the wound healing process. The data sets represent the areas of trace elements, diabetic wounds, growth factors, and nutrition within the field of wound healing. The model produces an explicit function accurately representing the time course of healing wounds from a given data set. Such a function is used to study variations in the healing velocity among different types of wounds and at different stages in the healing process. A new multivariable model of wound healing capable of analyzing the effects of several variables on accelerating the wound healing process is also introduced. Such a model can help to formulate appropriate strategies to treat wounds. It also would enable us to evaluate the efficacy of different treatment modalities during the inflammatory, proliferative, and tissue remodeling phases.     

A stochastic model for cell survival after irradiation

This paper deals with the theory of cell survival after irradiation. It is based on the assumption that the cell is composed of two regions with different sensitivities and takes into account phenomena with general repair-time distribution different for each region. Expressions are obtained for the probability that the cell will be damaged and for the mean life time of the cell.     

A fuzzy stochastic single-period model for cash management

The major purpose of this paper is to apply a stochastic single-period inventory management approach to analyze optimal cash management policies with fuzzy cash demand based on fuzzy integral method so that total cost is minimized. We will find that, after defuzzification, the cash-raising amounts and the total costs between the fuzzy case and the crisp case are slightly different when the variation of cash demand is small. As a result, we point out that the fuzzy stochastic single-period model is one extension of the crisp models. In any case, one may conclude that a conscientious analysis in fuzzy mathematics like that presented in this paper provides a financial decision maker with a deeper insight into the more real cash management problem.     

A stochastic model for fronts in porous media

We analyze a stochastic model for the motion of fronts in two-phase fluids and derive upscaled equations for the capillary pressure. This extends results of [11], where the same law for the capillary pressure was derived under an assumption on typical explosion patterns. With the work at hand we remove that assumption and show that in the stochastic case the upscaled equations hold almost surely.     

A modified stochastic model for closed birth interval

"Based on some simple assumptions regarding [the] human reproduction process, a continuous time stochastic model for describing the variation in any closed birth interval of a woman of marital duration (t) has been developed. The model incorporates the possibility of the variation of the amenorrhea period among the women under observation. For illustration, the model is applied to an observed closed birth interval between first and second births of the women with marital duration of seven years."     

A stochastic programming model for water resource planning

A framework is offered for the evaluation of electricity generation and water supply for agricultural irrigation. This assessment is conducted through the construction of an appropriate stochastic optimization model. A recursive least squares algorithm is incorporated in the model which enables more accurate estimation of model parameters.     

A valuation model for firms with stochastic earnings

A model is proposed to value a firm with stochastic earnings. It is assumed that the earnings of the firm follow a time‐varying mean reverting stochastic process. It is shown that the value of the firm satisfies a boundary value problem of a second‐order partial differential equation, which can be solved numerically. Some special cases are discussed. An analytic solution is found for one special case. Moreover, it is shown that the analytic solution is consistent with a previous result obtained by other researchers. Numerical solutions are obtained for the other special cases. Finally, the model is also applied to value the debt issued by the firm.     

A stochastic programming model for aggregate production planning

This paper looks at a problem of production planning at the level of the work centre, rather than of the individual machine. The approach is based on methodology which combines aspects of decision analysis with mathematical programming procedures.A model is developed which can take into account various real life factors in the planning phase, including delaying production, and the probability of production being scrapped, or of extra work being required. An extension to the multiple objective situation is outlined, and some computational results are reported.     

A stochastic delay financial model

We compute the logarithmic utility of an insider when the financial market is modelled by a stochastic delay equation. Although the market does not allow free lunches and is complete, the insider can draw more from his wealth than the regular trader. We also offer an alternative to the anticipating delayed Black-Scholes formula, by proving stability of European call option prices when the delay coefficients approach the nondelayed ones.

     

A stochastic multiscale model for electricity generation capacity expansion

Long-term planning for electric power systems, or capacity expansion, has traditionally been modeled using simplified models or heuristics to approximate the short-term dynamics. However, current trends such as increasing penetration of intermittent renewable generation and increased demand response requires a coupling of both the long and short term dynamics. We present an efficient method for coupling multiple temporal scales using the framework of singular perturbation theory for the control of Markov processes in continuous time. We show that the uncertainties that exist in many energy planning problems, in particular load demand uncertainty and uncertainties in generation availability, can be captured with a multiscale model. We then use a dimensionality reduction technique, which is valid if the scale separation present in the model is large enough, to derive a computationally tractable model. We show that both wind data and electricity demand data do exhibit sufficient scale separation. A numerical example using real data and a finite difference approximation of the Hamilton–Jacobi–Bellman equation is used to illustrate the proposed method. We compare the results of our approximate model with those of the exact model. We also show that the proposed approximation outperforms a commonly used heuristic used in capacity expansion models.