The size order of the state vector of a continuous-time homogeneous Markov system with fixed size

Abstract

The variation of the state vectors p(t) = (p_i(t)) of a continuous-time homogeneous Markov system with fixed size is examined. A specific time t₀ after which the size order of the elements p_i(t) becomes stable provides a criterion of the system's convergence rate. A method is developed to find t₀ and a quickly evaluated lower bound for t₀. This method is based on the geometric characteristics and the volumes of the attainable structures. Moreover, a condition concerning the selection of starting vectors p(0) is given so that the vector functions p(t) retain the same size order for every time greater than a given time t.