Nevertheless, we need to check separately whether $P_{S}^* \leq P_{S}^{\max}$.

-   assuming IC did not take place, $P_{R_t}^\dagger $ and $P_{S}^\dagger$ are as given in and , respectively. $P_{R_t}^\dagger \leq P_{R_t}^{\max}$ is satisfied by the feasibility check, but we need again to check separately whether $P_{S}^\dagger \leq P_{S}^{\max}$.

-   The minimum energy expenditure at a specific time slot for each pair is the minimum sum of the powers for the two cases, i.e., $\min \left\{ P_{S}^*+ P_{R_t}^*, P_{S}^\dagger + P_{R_t}^\dagger \right\}$. Note that if IC can take place, this might require $P_{R_t}^* > P_{R_t}^\dagger$, such that $P_{S}^*+ P_{R_t}^* > P_{S}^\dagger + P_{R_t}^\dagger$.

3\) If IC is *infeasible*, then we use case (ii) only from step 2).\
4) We compare the minimum energy expenditure for all possible relay pairs and we choose the minimum among them.

Note that in the worst case scenario (in which all the queues are neither empty nor full), there will be $K\times (K-1)$ combinations. Hence, the worst case complexity of the problem is $\mathcal{O}(K^2)$.

Model and outage probability analysis {#sec:outage}
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In this section, the outage probability behavior of the $\minpow$ relay selection scheme follows the theoretical framework of [@KRICHAR], which is also a relay network 