  We consider a discrete-time Markov chain $(X^t,Y^t)$, $t=0,1,2,...$, where
the $X$-component forms a Markov chain itself. Assume that $(X^t)$ is
Harris-ergodic and consider an auxiliary Markov chain ${\hat{Y}^t}$ whose
transition probabilities are the averages of transition probabilities of the
$Y$-component of the $(X,Y)$-chain, where the averaging is weighted by the
stationary distribution of the $X$-component.
  We first provide natural conditions in terms of test functions ensuring that
the $\hat{Y}$-chain is positive recurrent and then prove that these conditions
are also sufficient for positive recurrence of the original chain $(X^t,Y^t)$.
The we prove a "multi-dimensional" extension of the result obtained. In the
second part of the paper, we apply our results to two versions of a
multi-access wireless model governed by two randomised protocols.
