# Coverage Strategy for Small-Cell UAV-Based Networks in IoT Environment
## 3 Problem Formulation
### Computations
In the context of two unmanned aerial vehicles (UAVs) with an encounter rate, beaconing period, and the goal of serving mobiles, latency can be defined as the time delay between a mobile's request for service and the moment when UAVs successfully serve that mobile as given by Equation (9). We here ignore the transmission time, and we mainly focus on the latency resulting from the limited coverage of UAVs. This latency is influenced by several factors, including the encounter rate, the beaconing period, and the time it takes for a UAV to reach and serve a mobile. The expected number of trials before a successful encounter with a UAV indexed as \(i\) is \(\frac{1}{P_{i}(\tau_{i},\tau_{j})}\). Note that the duration of each trial is \(m\). Thus, we can deduce the average latency (\(L_{i}\)) as a function of encounter probability as per Equation (9):  
\[u_{i}(\tau_{i},\tau_{j})=(P(T_{i}\leq T_{j})+P(T_{i}\geq T_{j})P_{j}^{slp})P_{i }^{bcn}(T-\tau_{i}) \tag{3}\]1  
**Theorem 1**. : _([PERSON], [PERSON], Fan): Consider a strategic form game_  
\[u_{i}(\tau_{i},\tau_{j})=(P(T_{i}\leq T_{j})+P(T_{i}\geq T_{j})P_{j}^{slp})P_{i }^{bcn}(T-\tau_{i}) \tag{3}\]2  
_such that for each \(i\in N\), the following apply:_  
* \(A_{i}\) _is compact and convex._
* \(u_{i}(\tau_{i},\,\tau_{-i})\) _is continuous and quasi-concave in_ \(\tau_{-i}\)_._  
_Then, a pure strategy Nash equilibrium exists._  
Proof of Theorem 1.: Let \(\eta\), given by Equation (10), be a strategic form game, where \(N\) is the set of players, \(A\) is the strategy space, and \(U\) is the utility function. Assume that strategy space \(A\) for each player is non-empty, compact, and convex. Consider utility function \(U\) for each player. Assume that \(U\) is quasi-concave in the player's own strategy and convex in the strategies of other players. Since \(A\) is compact, \(U\) is quasi-concave, and \(U\) is continuous, we can apply the Weierstrass Extreme Value Theorem. By the Extreme Value Theorem, for each player \(i\), there exists a strategy profile \(a_{i}\)\({}^{*}\) in \(A\) such that \(U(a_{i}\)\({}^{*},a_{-i})\) is maximized for all \(a_{i}\) in \(A_{i}\), the strategy space of all other players except player \(i\). Let \(a^{*}=(a_{1}\)\({}^{*},a_{2}\)\({}^{*},\ldots,a_{n}\)\({}^{*})\) be the strategy profile consisting of the best response strategies for all players. Now, we need to show that \(a^{*}\) is a Nash equilibrium, where no player has an incentive to unilaterally deviate. Consider any player \(i\) in \(N\). Since \(a_{i}\)\({}^{*}\) maximizes \(U(a_{i}\)\({}^{*},a_{-i})\) for all \(a_{-i}\) in \(A_{-i}\), player \(i\) has no incentive to unilaterally deviate from \(a_{i}\)\({}^{*}\). If they were to choose a different strategy, their utility would be weakly lower due to the convexity assumption. Thus, for each player \(i\), \(a_{i}\)\({}^{*}\) is a best response to the strategies of all other players. Therefore, \(a^{*}\) is a Nash equilibrium. Therefore, we have proven the existence of a Nash equilibrium in the given strategic form game \(\eta\). The structural characteristics of the game, such as quasi-concavity, play a pivotal role in shedding light on the existence and uniqueness of its Nash equilibrium. Given that the second-order derivative is negative, \(u_{i}(\tau_{i},\tau_{j})\) exhibits concavity, and as a result, it is also quasi-concave. Therefore, based on Theorem 1, the game possesses at least one pure Nash equilibrium. In the scenario where the drones share identical encounter rates, denoted by \(\lambda_{i}=\lambda_{j}=\lambda\), the symmetric nature of the game meets the dominance solvability conditions. Consequently, it also satisfies [PERSON]'s conditions, which guarantee the uniqueness of the Nash equilibrium. To ascertain this, we numerically solve the first-order condition, ensuring that both solutions are negative, indicating sub-modularity. To demonstrate the existence of an equilibrium, a sufficient condition is the quasi-concavity of the utility function.