  In this paper we propose a model for open Markov chains that can be
interpreted as a system of non-interacting particles evolving according to the
rules of a Markov chain. The number of particles in the system is not constant,
because we allow the particles to arrive or leave the state space according to
prescribed protocols. We describe this system by looking at the population of
particles on every state by establishing the rules of time-evolution of the
distribution of particles. We show that it is possible to describe the
distribution of particles over the state space through the corresponding moment
generating function. Such a description is given through the dynamics ruling
the behavior of such a moment generating function and we prove that the system
is able to attain the stationarity under some conditions. We also show that it
is possible to describe the dynamics of the two first cumulants of the
distribution of particles, which in some way is a simpler technique to obtain
useful information of the open Markov chain for practical purposes. Finally we
also study the behavior of the time-dependent correlation functions of the
number of particles present in the system. We give some simple examples of open
chains that either, can be fully described through the moment generating
function or partially described through the exact solution of the cumulant
dynamics.
