Convergence time to equilibrium for large finite Markov chains

For a sequence of finite Markov chains $\mathcal L(N)$ we introduce a notion of convergence time to equilibrium $T(N)$. For sequences that are constructed by truncation of some countable Markov chain $\mathcal L$ we find the convergence time to equilibrium in terms of Lyapunov function of the chain $\mathcal L$. We apply this result to queueing systems with limited number of customers: a priority system with several types of customers and Jackson network.