## 3 Linear Programming Formulation for Maximizing the Event Communication Time in the Ideal Scenario  
As mentioned above, the UAV is a scarce resource for recovering communication of the broken link in disaster. How CN dispatches UAVs to serve all sub-events of each event with the EAMUD algorithm depends on which UAV in the UAV pool is available when the event happens. Since each event may occur at different time in the real disaster scenario, it is complex to model the real UAV dispatching process, not to mention the subsequent UAV handover processes and UAV dispatch exceptions. In the following, for maximizing the aggregate event communication time of all given events for a disaster duration, the UAV dispatching process in the _ideal scenario_ is formulated and solved by the linear programming technique under four constraints to find its optimal solution. Please note that the ideal scenario here assumes all events occur simultaneously when the simulation begins, which is rare in the real scenario. Further, it only considers how CN dispatches UAVs to serve each event in the first round and ignores the subsequent UAV handover processes and UAV dispatch exceptions. Assume \(Ev=\{Ev_{1},Ev_{2}\ldots,Ev_{j}\}\) is the set of all given events, where \(Ev_{j}\) denotes the \(j\)th event in this set. \(N_{j}\) is the number of sub-events in \(Ev_{j}\) and the \(i\)th sub-event of \(Ev_{j}\) is denoted as \(Ev_{j,i}\). Assume \(UAV_{A}\) is the set of all available UAVs.  
The objective function to maximize the sum of the event communication time \(\chi_{j}^{t}\) of each given \(Ev_{j}\) at each time \(t\) in the ideal scenario is formulated as follows:  
\[max\sum_{t}\chi_{j}^{t} \tag{10}\]  
Subject to:  
\[E_{u}^{Bat}\geq E^{DTH},\forall U_{u}\in UAV_{A}\]  
\[com_{j,i}=\left(\begin{array}{l}\frac{E_{u}^{Bat}-2\times E_{j,i}^{ Fly}}{E_{u}^{Com}+E_{u}^{Ho}}\\ \end{array}\right)>0,\ \ \forall U_{u}\in UAV_{A}\]  
\[\chi_{j,i}^{t}=\left\{\begin{array}{l}1,\ \text{if }UAV\ U_{u}\text{ has been dispatched}\\ \text{to serve }Ev_{j,i}\text{ at time }t\\ 0,\ Otherwise\end{array}\right.\]  
\[\chi_{j}^{t}=\left\{\begin{array}{l}1,\ \text{if }\sum\limits_{i}\chi_{j,i}^{t}= N_{j},\ \forall Ev_{j,i}\in Ev_{j}\\ 0,\ \text{Otherwise}\end{array}\right.\]  
Equation (10) has four constraints. Constraint (1) is to guarantee that the energy \(E_{u}^{Bat}\) of any available UAV \(U_{u}\) has been charged to an energy level equal to or higher than threshold \(E^{DTH}\). Constraint (2) formulates the second constraint, where means the maximum communication time \(com_{j,i}\) of any available UAV must be larger than zero. As mentioned in Section 2, the energy \(E_{u}^{Rem}\) for UAV \(U_{u}\) to consume on communication and hovering at the position of \(Ev_{j,i}\) is the difference of \(E_{u}^{Bat}\) and \(2\times E_{j,i}^{Fly}\). Hence, the maximum communication time \(com_{j,i}\) of \(U_{u}\) for \(Ev_{j,i}\) is equal to the quotient of \(E_{u}^{Rem}\) divided by the unit energy, \(E_{u}^{Com}+E_{u}^{Ho}\). Here, \(\chi_{j,i}^{t}\) in Constraint (3) is a decision variable. If an available UAV, i.e., \(U_{u}\in UAV_{A}\), has been dispatched to serve sub-event \(Ev_{j,i}\) at time \(t\), \(\chi_{j,i}^{t}\) is set as 1. The last constraint is formulated in Constraint (4). If the sum of \(\chi_{j,i}^{t}\) for each sub-event \(Ev_{j,i}\) at time \(t\) is equal to the number of sub-events of \(Ev_{j}\), i.e., \(N_{j}\), which means that each sub-event \(Ev_{j,i}\) at time \(t\) has been served by a dispatched UAV, the communication link of \(Ev_{j}\) at time \(t\) has been recovered. Hence, \(\chi_{j}^{t}\) is set to 1. Otherwise, \(\chi_{j}^{t}\) is 0.