On the Distributions of the State Sizes of Closed Continuous Time Homogeneous Markov Systems

The evolution of the state sizes of a closed continuous-time homogeneous Markov system is determined by the convolution of multinomial distributions expressing the number of transitions between the states of the system. In order to investigate the distributions of the state sizes, we provide the computation of their moments, at any time point, via a recursive formula concerning the derivative of the moments. The basic result is given by means of a new vector product which is similar to the Kronecker product. Finally, a formula for the computation of the state sizes distributions is given.