with finite buffers. The main differentiation compared to [@KRICHAR] is that we have additional ways of transmission through successive relaying; in other words, we have additional links from a certain buffer state to others. This will be more clearly reflected in the illustrative example.

Construction of the state transition matrix of the MC {#sec:outageA}
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We first formulate the state transition matrix of the Markov Chain (MC), denoted as $\mathbf{A}$, $\mathbf{A}\in \mathbb{R}^{{\left( {L + 1} \right)^K} \times {\left( {L + 1} \right)^K}}$. More specifically, ${{\bf{A}}_{i,j}} = {P}\left( {{s_i} \to {s_j}} \right) = {P}\left( {{X_{t + 1}} = {s_j}|{X_t} = {s_i}} \right)$ are the transition probabilities to move from a state $s_i$ to a state $s_j$. The transition probability depends on the number of relays that are available for cooperation.

As we consider finite buffers, relays that have full buffers cannot compete in the selection of the best relay that will receive the source’s signal. Also, relays with empty buffers are not able to transmit and as a result they are excluded from the best transmitting relay selection. Moreover, when there is no possibility of transmitting successively through two selected relays our system reduces to the $\maxlink$ relay selection scheme. The 