Extended network method for determining the main probabilistic characteristics of the evolution for stochastic discrete systems with positive rational transitions
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LAZARI, Alexandru. Extended network method for determining the main probabilistic characteristics of the evolution for stochastic discrete systems with positive rational transitions. In: International Conference of Young Researchers , Ed. 8, 11-12 noiembrie 2010, Chişinău. Chişinău: Tipogr. Simbol-NP SRL, 2010, Ediția 8, p. 93. ISBN 978-9975-9898-4-8..
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International Conference of Young Researchers
Ediția 8, 2010
Conferința "International Conference of Young Researchers "
8, Chişinău, Moldova, 11-12 noiembrie 2010

Extended network method for determining the main probabilistic characteristics of the evolution for stochastic discrete systems with positive rational transitions


Pag. 93-93

Lazari Alexandru
 
Moldova State University
 
Disponibil în IBN: 5 mai 2021


Rezumat

Consider a stochastic discrete system L with finite set of states { , , , } 1 2 E e e  e , with given distribution function ( ) * j p e on E and transition probability matrix u v E P p u v , ( ( , )) . In addition we assume that for arbitrary two states u,v E is given the positive rational transit-time (u,v) , i.e. for this system is defined the transit-time matrix u v E u v , ( ( , )) . The evolution of the system is considered finished if it realizes a given sequence m m X (x , x , , x ) E 1 2  . In this paper we describe how to calculate the main probabilistic characteristics, such as distribution, average value, variance and the moments of order n of the evolution time. We show that this problem can be reduced to the case of stochastic discrete systems with nonzero natural transit-time by changing the measurement unit. As the measurement unit can be used the least common multiple of denominators of matrix elements. The matrix * is the matrix of transit-time of the new system. Next, using extended network method, the obtained problem is reduced to the case of random discrete systems with unit transit-time. This case of the problem has been solved in [1] , where an efficient polynomial algorithm for determining probabilistic characteristics of the evolution time is proposed. Extended network is constructed in the following way. For each state v E we select a state u E that satisfies the equality ( , ) max ( , ) ( ) u v z v z E v , where E (v) {z E| p(z,v) 0} . Then, we divide the transition (u,v) into (u,v) units using the new states z u v u z u v z u v z u v z u v v u v u v ( , ) , ( , ), ( , ), , ( , ), ( , ) 0 1 2 ( , ) 1 ( , )  . The probability that the system initiate the evolution from one intermediate state z (u,v), j 1, (u,v) 1 j , is considered equal to zero. If (u,v) 1, then the transition probability matrix P is changed as follows: − Add the rows and the columns with zero components for the intermediate states z (u,v), j 1, (u,v) 1 j ; − Set ( ( , ), ( , )) 1 1 p z u v z u v j j , j 1, (u,v) 1; − Set ( , ( , )) ( , ) ( , ) ( , ) 1 p z z u v p z v u v z v for each z E (v) , that satisfies the condition (z,v) 1 and then put p(z,v) 0 . The solution of the initial problem can be determined using the solution of the reduced problem and the multiplier . So, if the evolution time in initial problem is equal to T and in reduced problem is equal to * T , then holds * T T . This implies that the moments of order n are expressed by relation n n n (T) (T ) * ; the average value is determined by * MT MT ; the variance is expressed by * 2 DT DT and the distribution law is obtained according to ( ) ( ), 0, * P T k P T k k .

Cuvinte-cheie
stochastic discrete system, extended network, probabilistic characteristics, rational transit-time.