Stochastic models for interacting processes feature a dimensionality that grows exponentially with the number of processes. This state space explosion severely impairs the use of standard methods for the numerical analysis of such Markov chains. In this work, we develop algorithms for the approximation of steady states of structured Markov chains that consider tensor train decompositions, combined with wellestablished techniques for this problem - aggregation/disaggregation techniques. Numerical experiments demonstrate that the newly proposed algorithms are efficient on the determination of the steady state of a representative set of models.