Equivalence of two absorption problems with Markovian transitions and continuous or discrete time parameters
R. A. Sack · Mathematical Proceedings of the Cambridge Philosophical Society · 1959
1. Introduction. Ledermann(1) has treated the problem of calculating the asymptotic probabilities that a system will be found in any one of a finite number N of possible states if transitions between these states occur as Markov processes with a continuous time parameter t. If we denote by pi(t) the probability that at time t the system is in the ith state and by aij ( ≥ 0) the constant probability per unit time for transitions from the jth to the ith state, the rate of change of pi is given by where the sum is to be taken over all j ≠ i. This set of equations can be written in matrix form as where P(t) is the vector with components pi(t) and the constant matrix A has elements