# Delay-Sensitive Multi-Sensor Routing Scheduling Method for Underground IoT in Mines
## 3 System Modeling
### Multi-Sensor Real-Time Routing Scheduling Problem  
Assume that there are a total of \(I\) base stations; for any \(0\leq i\leq I-1\), \(nbr(i)\) denotes the set of all nodes connected to node \(i\). Obviously, for \(0\leq i^{\prime}\leq I-1\), if \(i\in nbr(i^{\prime})\), then \(i^{\prime}\in nbr(i)\). Let \(e\) be the set of export nodes. Suppose that there are a total of \(J\) data packets in the periodic time of \(T\) time slots (assuming that all nodes can transmit a data packet within a time slot). For \(0\leq j\leq J-1\), use \(B_{j}\) to represent the base station to which the data packet \(j\) belongs (issued by which base station), the arrival time of the data packet \(j\) is \(A_{j}\) (the arrival time of the data packet \(j\) to the base station), the deadline is \(D_{j}\), and the absolute deadline is \(A_{j}+D_{j}\). \(Q_{i}\) is used to represent the maximum buffer queue length of base station \(i\). Suppose that \(s[j][t][i]\) is used to represent the state of the packet \(j\) at time \(t\) on the base station \(i\). If \(s[j][t][i]=1\), this means that the packet \(j\) arrives (stays) on the base station \(i\) at time \(t\); otherwise, \(s[j][t][i]=0\). \(r[j]\) is used to represent the time when the packet \(j\) reaches the target node, that is, if \(s[j][r][t][i]=1\), then \(i\in e\). \(c[t][i]\) is used to represent the number of packets that the base station \(i\) waits to process in the cache queue at time \(t\). In this paper, for the data packet \(j\), the purpose of the transmission is to send it to the target node as soon as possible. Therefore, the objective function \(f(j)\) is defined as follows:  
\[f(j)=\frac{r[j]-A_{j}}{D_{j}} \tag{1}\]  
Obviously, \(r[j]>A_{j}\). At the same time, in order to complete the scheduling before the deadline, we need to satisfy \(r[j]<A_{j}+D_{j}\); therefore, \(0<\left(r[j]-A_{j}\right)/D_{j}\leq 1\). That is, for any \(j\), this satisfies \(f(j)\in(0,\,1]\). Therefore, we define the multi-sensor real-time routing scheduling problem for a given \(I\), \(J\), \(T\), \(A\), \(e\), and \(nbr\). The multi-sensor real-time routing scheduling problem may contain many feasible solutions. To find an optimal solution among multiple feasible solutions, this paper takes the minimization of the objective function value in a feasible solution as the optimization objective. That is to say, for the scheduling results of \(J\) sensors, the largest \(f(j)\) is selected as the representative of this set of solutions, and then the smallest of multiple feasible solutions is selected as the final problem solution. The mathematical model of the multi-sensor real-time routing scheduling problem \(\mathcal{S}\) is as follows:\[\min\left(\max_{j=0}^{I-1}f(j)\right)\] \[s.t.\] \[\left\{ \begin{array}{l}(1)\forall j\in\{0,1,\ldots,J-1\},\\ r[j]\leq T-1\wedge r[j]-A_{j}\leq D_{j}\\ \\ (2)\forall j\in\{0,1,\ldots,J-1\},\\ s[j][r[j]][i_{e}]=1\wedge s[j][A_{j}][B_{j}]=1\\ \\ (3)\forall j\in\{0,1,\ldots,J-1\},i,i^{\prime}\in\{0,1,\ldots,I-1\},t\in\{0,1, \ldots,T-2\},\\ \text{if}\,s[j][t][i]=s[j][t+1][i^{\prime}]=1,\,\text{then}\,i^{\prime}\in nbr (i)\\ \\ (4)\forall t\in\{0,1,\ldots,T-1\},i\in\{0,1,\ldots,I-1\},\\ c[t][i]\leq Q_{i}\\ \\ (5)\forall j\in\{0,1,\ldots,J-1\},t\in\{0,1,\ldots,T-1\},\\ \sum\limits_{i=0}^{I-1}s[j][t][i]\leq 1\end{array} \right.\]  
The following explains the constraints of the problem \(\mathcal{S}\). Constraints (1) and (2) show that, if there is \(r[j]\), it must be scheduled between the deadlines, and all scheduling beyond the deadline is not satisfied. If there is a packet that does not reach the endpoint, this condition is not satisfied. Therefore, as long as Constraints (1) and (2) are satisfied, all the data packets on the surface reach the destination node within the specified time. Constraint Condition (3) indicates that data transmission can only be performed when two base stations are adjacent. Constraint Condition (4) indicates that the cache of all base stations cannot overflow.