A functional inequality is used to investigate the behaviour of a decreasing function in the neighbourhood of zero and the results are applied to estimate the mean number of steps of a Markov chain to reach the ε{lunate} -neighbourhood of the boundary as ε{lunate}→0. In particular, the bound O(ln(|lnε{lunate}|)) is obtained on the time complexity of the Monte-Carlo solution of the internal and external Dirichlet problems for the Laplace operator.