# Efficient Resource Scheduling for Multipath Retransmission over Industrial WSAN Systems
## 3 Network Model
### System Model  
We describe the model of a WSAN, which consists of field devices, multiple sinks, a gateway, and a centralized network manager, as shown in Figure 1. We consider that the network can be modelled as a graph \(G=(N,L)\), in which the set \(N=\{n_{1},n_{2},\ldots,sk\}\) represents all the nodes, including field devices \(n_{i}\) and sinks \(sk\). \(L\) is the link set, where the element \(l_{ij}\) in \(L\) indicates the connection state between the nodes \(n_{i}\) and \(n_{j}\). \(l_{ij}=1\) means \(n_{i}\) and \(n_{j}\) can directly communicate with each other. \(x(n_{i},n_{j})\) represents a transmission with the direction from node \(n_{i}\) to \(n_{j}\). Thus, for a link \(l_{ij}\), \(x_{t,e}(l_{ij})\) represents the transmission allocated on the time slot \(t\) and channel offset \(c\). The length of a time slot is denoted by \(\kappa\). The actual channel can be calculated by \((ASN+c)\) mod \(|C|\), where \(ASN\) is the absolute counting of slot advance and \(C\) is an available channel set. Thus, the channel scheduling is only relevant to the assignment of channel offset. For example, \(x_{t,c}(l_{ij})=1\) represents that node \(n_{i}\) transmits a packet to node \(n_{j}\) using channel \(c\) at slot \(t\), otherwise, \(x_{t,c}(l_{ij})=0\). Based on the above network model, all transmissions need to be scheduled by the TSCH MAC protocol. Thus, some basic constraints expressed by Equations (1)-(4) should be considered, which seriously influence the performance of transmission scheduling. First, more than one transmissions cannot be occurred simultaneously in a certain slot and channel unless nodes use some contention-based mechanism. Otherwise, one transmission may generate wireless interference to interfere with the other transmissions. \[\sum_{n_{i}\in N}\sum_{n_{j}\in N}x_{t,c}(l_{ij})\leq 1,\forall t,\forall c\in C \tag{1}\]  
Second, the DL-ACK should be sent within one slot for reliability. Thus, a node can only transmit to one neighbor at any slot. \[\sum_{c\in C}\sum_{n_{j}\in N}x_{t,c}(l_{ij})\leq 1,\forall t \tag{2}\]  
Third, since a low-power node in the network only supports half-duplex wireless communication, obviously, it cannot be the transmitter and the receiver simultaneously. \[\sum_{c\in C}\sum_{n_{i}\in N}x_{t,c}(l_{ij})+x_{t,c}(l_{ij})\leq 1,\forall t \tag{3}\]  
Finally, multichannel transmission can make full use of slot and channel resources; however, the number of available channels is limited. Thus, the number of transmissions cannot be beyond the number of available channels. \[\sum_{c\in C}\sum_{n_{i}\in N}\sum_{n_{j}\in N}x_{t,c}(l_{ij})\leq|C|,\forall t \tag{4}\]  
We consider all the end-to-end deliveries in a network as a flow set, which is denoted by \(F=\{F_{0},F_{1},\ldots,F_{\gamma}\}\). The number of flows that need to be scheduled is denoted as \(\gamma\). A flow can be characterized by \(F_{k}:\langle X_{k},E_{k},p^{\prime}_{k},\phi_{k}\rangle\), where \(X_{k}=\{x^{k}(s_{1},d_{1}),\ldots,x^{k}(s_{\tau},d_{\tau}),\ldots\}\) is the collection of all transmissions in a flow, \(E_{k}\) is the collection of transmission release orders related to the routing order, and \(\phi_{k}\) is the routing path from the first source node to the last destination node. \(s_{\tau}\) and \(d_{\tau}\) are the source node and the destination node in a transmission, respectively. One of the elements \(e(\alpha,\beta)\) in \(E_{k}\) represents that the transmission \(x^{k}(s_{\beta},d_{\beta})\) is released after all the transmissions \(x^{k}(s_{\alpha},d_{\alpha})\). Each \(F_{k}\) periodically generates a packet with the period \(p^{\prime}_{k}\), and the flow should be finished before its deadline \(d_{k}\). All the slots for a flow construct a superframe which is repeated periodically. The supported periods are defined as \(2^{a}\cdot p_{m}\) based on WirelessHART, where \(a\) is an integer and \(p_{m}\) is the minimum period among all flows.