*4.7. IoT Device Pairing Problem*

The optimal offloading policy in the previous subsection is obtained given the paired IoT devices *i* and *j*. In this subsection, we formulate an IoT device pairing problem whose objective is to minimize the summation of energy outage probabilities of all IoT devices.

Let *ζ<sup>E</sup> ij* denote an individual energy outage probability of IoT device *i* when it is paired with IoT device *j*. *ξ<sup>E</sup> ij* can be calculated as (We assume that the individual energy outage probability of IoT device *i* when it is paired with IoT device *j* is lower than that when it is not paired (i.e., *ξ<sup>E</sup> ij* < *<sup>E</sup><sup>E</sup> ii*). This assumption is reasonable because paired IoT devices operate by following the optimal policy obtained by CMDP:

$$\mathcal{L}\_{ij}^{E} = \sum\_{S} \sum\_{A} \pi^\* \left( \left[ T\_i^M \, , T\_i^O \, \_\prime E\_i = 0 , D\_{i\prime} S\_j \right] \, \_\prime A \right). \tag{58}$$

Then, the optimization problem for pairing IoT devices can be defined as

$$\min\_{\mathbf{x}\_{ij}} \sum\_{i} \mathbb{Z}\_{ij}^{E} \mathbf{x}\_{ij} \tag{59}$$

$$\text{s.t. } \sum\_{j} \mathbf{x}\_{ij} = \mathbf{1}\_{\prime} \; \forall i\_{\prime} \tag{60}$$

where *xij* is a decision variable that is 1 if IoT devices *i* and *j* are paired and 0 otherwise. The objective function in (59) is to minimize the summation of energy outage probabilities of all IoT devices. Meanwhile, the constraint in (60) to ensure that all IoT devices are paired with only one IoT device. This optimization problem is solved in the controller by using several algorithms (e.g., brute-force approach, LP relaxation, and branch-and-bound algorithm), and therefore there is no burden in IoT devices.
