Calculation of one-type non-markov
generation-recombination process
O.L. Sirotkin
Saratov State University, Saratov, Russia
Generation-recombination stochastic processes ( or one-step ones) occurring within uninterrupted time are often used as models of various phenomena in physics, chemistry radio- engineering. Thus, e.g. they are used while analysing fluctuations of LPD generators, reliability theory problems solution and superconductivity research [1].
As a rule, the processes of these type have lots of various values representing arrays of integers. The lifetime of each such state is considered to be a random value, having exponential independently of time parameters. In view of "memory absence" of exponential distribution, determined in such a way a one-step process.
In view of exponential distribution "memory absence", determined in such a way one-step process is a Markov one and its mathemetical description is given by Kolmogorov's differential equtions that are well known. While infinitesmal birth and death intensity are dependent on a state number, the mean value, dispersion, autocorrelation function and energetic spectrum may be computed for applied problems solutions.
Apparently, that njt all of the systems can be attributed to markov ones and, therefore, it is of certain interest to develop the methods of computation the moments of the first and second order, including those of non-stationary and non-markov ones. In this report it is intended to solve a specific problem of asymetric random diffuse in each of the states is an Erlang distribution of the second order. The problem solution is made by dummy phases methods [2] for transition intensity time constants.
The proposed development dynamics of the process is essentially in that, that the lifetime in any of the states are composed by the two independently exponentially distributed random values - those of the first phase lifetime and the second phase lifetime. The transition intensity level from one state to another is considered to be the same. Following this approach the problem is reduced to the two-dimensional markov process analysis and a corresponding graph of the states is plotted and Kolmogorov equations are written. From these equations the independent from phase number the first and the second stationary regime moments are determined, which could be simply expressed by generation and recombination intensity relation. The non-stationary mean value equations are non-closed, because there is probability of the process to come to the zero state. The present work deals with the analysis of the generating function. The equations obtained for non-stationary mean values enable to write an autocorrelation function and compute the process energetic spectrum with the aid of Viner transformation.
The report deals with the process, the states of which have a lifetime separated into three phases. The average (mean) value and dispersion expressions in non-stationary regime have been obtained. They are expressed through generation and recombination intensity relation, as has been done in the first case. The probable location of the zero state has also been computed and formulated.