Discrete models transition function

Conventional CA models are defined on particular lattice-networks, the sites of which are populated with discrete-valued dynamic elements evolving under certain local transition functions. Such a network with N sites is simply a general (undirected) graph G of size N and is completely defined by the NxN) connectivity matrix... 

Deval Let us consider a model that consists of a developmental space, D a collection of cell types (or colors), C a set of actions, A and a transition function, discrete time. Each point in the lattice is a cell, possibly the empty cell each nonempty cell may be viewed as an independent agent. Cells change in time by executing one of several actions. Which action is executed is determined by the cell s genome and the transition function. 

The surveyed DRM methods are Dynamic Event Trees Dynamic Flowgraph Methodology Discrete state-transition approaches (Markov chains, Petri Nets extensions) Dynamic Bayesian Networks Direct system simulation DRM for aircraft certification TOPAZ (Traffic Organization and Perturbation Analyzer) SoTeRia (Socio-Technical Risk Analysis) FRAM (Functional Resonance Analysis Method) STPA Hazard analysis (Systems-Theoretic Process Analysis) Collision Risk Modelling Encounter-based model methodology. For references and brief descriptions of all these methods, refer to (DRM D04 2012). 

CA are mathematical models of dynamic systems in which space and time are discrete and quantities take on a finite set of discrete values. CA are often represented as a regular array with a variable at each site, metaphorically referred to as a cell. The state of the CA is defined by the values of the variables at each cell. The automata evolve according to an algorithm, called a transition function, that determines the value of each cell based on the value of its... 

A Markov process model describes several discrete health states in which a person can exist at time t, as well as the health states into which the person may move at time t +1. A person can reside in just one health state at any given time. The progression from time t to time t +1 is known as a cycle. All clinically important events are modeled as transitions in which a person moves from one health state to another. The probabilities associated with each change between health states are known as transition probabilities. Each transition probability is a function of the health state and the treatment. 

Except the kinetic equations, now various numerical techniques are used to study the dynamics of surfaces and gas-solid interface processes. The cellular automata and MC techniques are briefly discussed. Both techniques can be directly connected with the lattice-gas model, as they operate with discrete distribution of the molecules. Using the distribution functions in a kinetic theory a priori assumes the existence of the total distribution function for molecules of the whole system, while all numerical methods have to generate this function during computations. A success of such generation defines an accuracy of simulations. Also, the well-known molecular dynamics technique is used for interface study. Nevertheless this topic is omitted from our consideration as it requires an analysis of a physical background for construction of the transition probabilities. This analysis is connected with an oscillation dynamics of all species in the system that is absent in the discussed kinetic equations (Section 3). 

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