  We introduce a unified framework to estimate the convergence of Markov chains
to equilibrium using Wasserstein distance. The framework provides convergence
bounds with various rates, ranging from polynomial to exponential, all derived
from a single contractive drift condition. This approach removes the need for
finding a specific set with drift outside and contraction inside. The
convergence bounds are explicit, as they can be estimated based on one-step
expectations and do not rely on equilibrium-related quantities. To enhance the
applicability of the framework, we introduce the large M technique and the
boundary removal technique. We illustrate these methods in queueing models and
algorithms in stochastic optimization.
