Geometric bounds for stationary distributions of infinite Markov chains via Lyapunov functions

In this paper, we develop a systematic method for deriving bounds on the stationary distribution of stable Markov chains with a countably infinite state space. We assume that a Lyapunov function is available that satisfies a negative drift property (and, in particular, is a witness of stability), and that the jumps of the Lyapunov function corresponding to state transitions have uniformly bounded magnitude. We show how to derive closed form, exponential type, upper bounds on the steady-state distribution. Similarly, we show how to use suitably defined lower Lyapunov functions to derive closed form lower bounds on the steady-state distribution. We apply our results to homogeneous random walks on the positive orthant, and to multiclass single station Markovian queueing systems. In the latter case, we establish closed form exponential type bounds on the tail probabilities of the queue lengths, under an arbitrary work conserving policy.