Time-dependent distribution function, stochastic

More sophisticated calculations (14,20), using either stochastic Monte Carlo or deterministic methods, are able to consider not only different Irradiating particles but also reactant diffusion and variations In the concentration of dissolved solutes, giving the evolution of both transient and stable products as a function of time. The distribution of species within the tracks necessitates the use of nonhomogeneous kinetics (21,22) or of time dependent kinetics (23). The results agree quite well with experimental data. 

Chvosta et alP consider another case where the work probability distribution function can be determined. They study a two energy-level system, modelled as a stochastic, Markovian process, where the transition rates and energies depend on time. Like the previous examples it provides an exact model that can be used to assist in identifying the accuracy of approximate, numerical studies. Ge and Qian extended the stochastic derivation for a Markovian chain to a inhomogeneous Markov chain. 

In timed PN both untimed as well as timed transitions are used. To fire, a timed transition has to be enabled for a specific time duration. This duration may be deterministic or stochastic, depending on the transition-specific distribution function and on further parameters (see German (2000), Trivedi (2001)). In this paper we only use exponential transitions in order to be in the position to compare our results with the analytical ones and with the MC-simulation. 

In the case of time-dependent processes, the question concerning the life of the components of a system arises frequently. The distribution function F(t) is then assigned to the stochastic variable lifetime T". 

A statistical description of the theoretical plate model leads to the stochastic model [46], which is based on Gaussian distribution functions of the nonintercon-verted stereoisomers 4>(t) and uses a time-dependent probability density function... 

In postulating the stationarity of the stochastic process, very strong assumptions regarding the structure of the process are made. Once these assumptions are dropped, the process can become nonstationary in many different ways. In the framework of the spectral analysis of nonstationary processes, Priestley (see, e.g., Priestley 1999) introduced the evolutionary power spectral density (EPDS) function. The EPSD function has essentially the same type of physical interpretation of the PSD function of stationary processes. The main difference is that whereas the PSD function describes the power-frequency distribution for the whole stationary process, the EPSD function is time dependent and describes the local power-frequency distribution at each instant time. The theory of EPSD function is the only one which preserves this physical interpretation for the nonstationary processes. Moreover, since the spectrum may be estimated by fairly simple numerical techniques, which do not require any specific assumption of the structure of the process, this model, based on the EPSD function, is nowadays the most adopted model for the analysis of structures subjected to nonstationary processes as the seismic motion due to earthquakes. 

As an example, performed to a CT-specimen the distribution function of J-integral can be determined in only one computational step. The amount of hardware resources depends on the approximation order and the number of stochastic variables. In many cases it will be only two— or three—times higher compared with a single deterministic computation. 

A discrete state stochastic Markov process simulates the movement of the evacuees. Transition from node-to-node is simulated as a random process where the prohahility of transition depends on the dynamically changed states of the destination and origin nodes and on the link between them. Solution of the Markov process provides the expected distribution of the evacuees in the nodes of the area as a function of time. 

Mathematically, a stochastic process X t) could be regarded as a family of random variables on a parametric set, say f e [0, T]. Eor a continuous-parameter process, to characterize the probabilistic information of the stochastic process, the hnite-dimensional distributions of PDEs, i.e., p xi, h), p(xi, tp, X2, 2), pixi, tp X2, tp x , f ), , should be specihed. By doing so a stochastic process is regarded as a random function of time, but the dependence of X on f is specihed not by an explicit expression of f but in an indhect way of specifying the complete cross-probabilistic information of X at all possible different time instants. This description is complete in mathematics. However, two dehciencies exist... 

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