# DISCRETE-STOCHASTIC MODELS (P-SCHEMES), Basic relations. - Modeling of systems

## DISCRETE STOCHASTIC MODELS (P-SCHEMES)

Consider the peculiarities of constructing mathematical schemes for a discrete stochastic approach to the formalization of the process of functioning of the system under study. Since the essence of discretization of time in this approach remains analogous to the finite automata considered in § 2.3, the influence of the stochasticity factor is traced also on a variety of such automata, namely, on probabilistic (stochastic) automata.

## Basic relations.

In general, the probabilistic automat (English, probabilistic automat) can be defined as a discrete on-line information converter with memory, the operation of which in each clock cycle depends only on the memory state in it and can be described statistically.

The application of schemes of probability automata (P-schemes) is important for the development of methods for designing discrete systems exhibiting statistically regular random behavior, to elucidate the algorithmic capabilities of such systems and to justify the boundaries of their expediency, and for the solution of synthesis problems by the chosen criterion of discrete stochastic systems satisfying the given constraints.

We introduce the mathematical concept of a P-automaton , using the concepts introduced for the F-automaton. Consider the set G, whose elements are all possible pairs ( x z,) t where x { and z s are elements of the input subset X and subsets of the states of Z, respectively. If there exist two such functions q> and φ, then with their help the maps G- + Z and G - * F = y y Y, & lt; p, i/r) defines an automaton of the deterministic type.

We introduce a more general mathematical scheme. Let Φ be the set of all possible pairs of the form ( z ki yj), where y t is an element of the output subset Y. We require that any element of the set G induce on the set Φ some distribution law of the following form:

In this case

where bkj are the probabilities of the transition of the automaton to the state r to and the appearance of the signal y sub> if it was in the state r 3 and the signal Xb entered its input at that time. The number of such distributions represented in the form of tables, is equal to the number of elements of the set & lt; 7. Denote the set of these tables by B. Then the four elements P = y X y Y, B ) is called a probability automaton (P-automaton).

Let the elements of the set U induce some distribution laws on the subsets Y and Z, which can be represented respectively in the form:

yes of the P-automaton to the state r to and the appearance of the output signal y to provided that the P-automaton was in the state r x and its input received the input signal x { .

If for all к and у the relation then such

A P-automaton is called a Mili probability machine. This requirement means that the independence condition for the new state of the P-automaton and of its output signal is met.

Suppose now that the definition of the output signal P-automaton depends only on the state in which the automaton is in the given work cycle. In other words, let each element of the output subset Y induce a probability distribution of outputs, having the following form:

Here

where 5, is the probability of appearance of an output signal

y ( , provided that the r-automaton is in the state r *.

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