We consider a general discrete-time Markov chain (X(n),Y(n)), n=0,1,... where where the X-component forms a Markov chain itself. Assuming that (X(n)) is ergodic, we may formulate the following ``naive'' conjecture.
Consider an auxiliary Markov chain (Ŷ(n)) 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. The conjecture is: if the Ŷ-chain is positive recurrent, then the (X,Y)-chain should be positive recurrent too.
We first show that,under appropriate technical assumptions, such a general result indeed holds, and then apply it to two versions of a multi-access wireless model governed by two protocols.
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