# Coverage Strategy for Small-Cell UAV-Based Networks in IoT Environment
## 3 Problem Formulation
### Computations  
For UAV \(i\) to first encounter an IoT station, it should be first beaconing during \(\theta_{i}\), and the other competing UAVs should have unsuccessful encounters before \(\theta_{i}\), meaning that competing UAVs need to be inactive. Thus, the encounter probability for UAV \(i\) is  
\[P_{i}(\tau_{i},\tau_{j})=(P(T_{i}\leq T_{j})+P(T_{i}\geq T_{j})P_{j}^{slp})P_{i }^{bcn} \tag{2}\]  
By using the above expression of each probability, the utility function is given by (3):  
\[u_{i}(\tau_{i},\tau_{j})=(P(T_{i}\leq T_{j})+P(T_{i}\geq T_{j})P_{j}^{slp})P_{i }^{bcn}(T-\tau_{i}) \tag{3}\]  
From the previous calculation, the probability expressions become  
\[P_{i}^{bcn}=-e^{\lambda_{i}T}\frac{e^{m\lambda_{i}}-e^{-\lambda_ {i}(m+\tau_{i})}-1+e^{\lambda_{i}\tau_{i}}}{e^{\lambda_{i}T}-1} \tag{4}\] \[P_{j}^{slp}=e^{\lambda_{j}T}\frac{-e^{\lambda_{j}(m+\tau_{j})}+ e^{\lambda_{j}(m+T)}+e^{\lambda_{j}\tau_{j}}-e^{\lambda_{j}T}}{e^{\lambda_{j}T}-1}\] (5) \[P(T_{i}\leq T_{j})=\frac{\lambda_{j}e^{-m(\lambda_{i}+\lambda_{j })}+(-\lambda_{i}-\lambda_{j})e^{-m\lambda_{j}}+\lambda_{i}}{\lambda_{i}+ \lambda_{j}}\] (6) \[P(T_{i}\geq T_{j})=\frac{\lambda_{i}e^{-m(\lambda_{i}+\lambda_{j })}+(-\lambda_{i}-\lambda_{j})e^{-m\lambda_{i}}+\lambda_{j}}{\lambda_{i}+ \lambda_{j}} \tag{7}\]  
So, the utility function becomes \[u_{i}(\tau_{i},\tau_{j})= \Bigg{(}\frac{\lambda_{j}e^{-m(\lambda_{i}+\lambda_{j})}+(-\lambda_ {i}-\lambda_{j})e^{-m\lambda_{j}}+\lambda_{i}}{\lambda_{i}+\lambda_{j}}\] \[+\frac{\lambda_{i}e^{-m(\lambda_{i}+\lambda_{j})}+(-\lambda_{i}- \lambda_{j})e^{-m\lambda_{i}}+\lambda_{j}}{\lambda_{i}+\lambda_{j}}e^{\lambda_ {j}T}\] \[\frac{-e^{\lambda_{i}(m+\tau_{j})}+e^{\lambda_{j}(m+T)}+e^{ \lambda_{j}\tau_{j}}-e^{\lambda_{j}T}}{e^{\lambda_{j}T}-1}\Bigg{)}\] \[\Bigg{(}e^{\lambda_{i}T}\frac{e^{m\lambda_{i}}-e^{-\lambda_{i}(m +\tau_{i})}-1+e^{\lambda_{i}\tau_{i}}}{e^{\lambda_{i}T}-1}\Bigg{)}(T-\tau_{i})\]  
We need to check \(\frac{\partial^{2}u_{j}}{\partial\tau_{0}\partial\tau_{j}}\). If it is positive, the game is super-modular. The Nash equilibrium represents the operational state (duty-cycling configuration) in which none of the drones can individually alter its strategy to improve its outcomes. In the context of the beaconing-scheduling game, it exhibits sub-modularity and possesses at least one pure Nash equilibrium. Sub-modular games possess highly appealing properties, as they do not rely on concavity or convexity assumptions to guarantee the existence of a Nash equilibrium. In simpler terms, the sub-modularity of the game implies that if one UAV shortens its beaconing period, it becomes advantageous for the other UAV to do the same. Put differently, a UAV's best response is a function that does not increase with the beaconing duration of another UAV. From a single-UAV perspective, there is a trade-off between the encounter rate and energy consumption. On one hand, as the beaconing duration increases, the encounter rate (Ps) grows. On the other hand, energy consumption is proportional to the beaconing period duration. We define the energy efficiency metric as the ratio of the successful probability encounter and the consumed energy. Hence, an efficient beaconing strategy is reached by increasing the encounter rate while reducing the associated energy consumption, equivalently reducing the beaconing duration. Namely, we measure the individual energy efficiency with the following metric, \(EE_{i}\):  
\[u_{i}(\tau_{i},\tau_{j})=(P(T_{i}\leq T_{j})+P(T_{i}\geq T_{j})P_{j}^{slp})P_{i }^{bcn}(T-\tau_{i}) \tag{3}\]0  
We denote by \(C_{b}\) (respectively, \(C_{s}\)) the energy cost per slot for sending beacons (respectively, remaining and switching the transceiver state), i.e., the payoff of UAV i under the beaconing strategy profile \((\tau_{i},\tau_{j})\).