A zero-order Markov process with multiple final sequences of states represents a stochastic system with independent transitions that stops its evolution as soon as one of the given final sequences of states is reached. The transition time of the system is unitary and the transition probability depends only on the destination state. It is proved that the distribution of the evolution time is a homogeneous linear recurrent sequence and a polynomial algorithm to determine the initial state and the generating vector of this recurrence is developed. Using the generating function, the main probabilistic characteristics are determined.