Positive recurrence for a single-server queueing system is established under generalised intensity conditions of service which does not assume existence of the density distribution function of service but a certain integral type lower bound as a sufficient condition. Positive recurrence implies existence of the invariant distribution and a guaranteed slow convergence to it in the total variation metric.

Polynomial recurrence bounds for a class of stochastic differential equations with a gradient type drift and an additive Wiener process are studied without Lyapunov functions.