# A Cross-Layer Optimized Opportunistic Routing Scheme for Loss-and-Delay Sensitive WSNs
## 4 System Model
### Problem Statement  
The study of Cross-layer optimized Opportunistic Routing (COOR) is a multiple target optimization problem. The goal in this paper is to maximize the network lifetime to minimize the probability of packet loss and delay. Just like in [29], the Cross-layer optimization of a WSN can be characterized by several performance indicators as explained below:  
(1) End-to-sink delay (\(\mathcal{T}^{x}_{sink}\)). End-to-sink delay refers to the period from the moment a node with distance \(x\) m to sink starts sending a packet until the sink finally receives it successfully. \(\mathcal{T}^{x}_{sink}=\sum_{i=0}^{k}\mathcal{T}^{x}_{i}\), where \(k\) indicates the hops during the entire transmission and \(\mathcal{T}^{x}_{i}\) refers to the time of the \(i^{th}\) hop. Obviously, the lower the end-to-sink delay the better, which can be expressed as:  
\[min(\mathcal{T}^{x}_{sink})=min(\sum_{i=1}^{k}\mathcal{T}^{x}_{i}) \tag{13}\]  
(2) Packet transmission reliability (\(\mathcal{P}^{x}_{sink}\)). Packet transmission reliability refers to the probability that the packet send by a node with distance \(x\) m can be received by the sink. Clearly, the higher the \(\mathcal{P}^{x}_{sink}\) the more reliable of the network:  
\[\left\{\begin{array}{l}max(\mathcal{P}^{x}_{sink})=\sum_{i=1}^{k}par_{(d_{i })}\\ s.t.\ x\leq\sum_{i=1}^{k}d_{i}\end{array}\right. \tag{14}\]Here, \(k\) has the same meaning as mentioned above. Meanwhile \(d_{i}\) and \(par_{(d_{i})}\) indicate the distance and the success rate of the \(i\)th transmission.  
(3) Network lifetime (\(\mathcal{L}\)). Like Reference [29], lifetime is defined as the death time of the first node in the network. Here, we assume the initial energy of node with distance \(x\) being \(E_{INT}^{x}\) and the energy consumption being \(\varepsilon_{x}\) per unit time. In conclusion, the lifetime is then given by the following equation:  
\[max(\mathcal{L})=max(min\bigg{(}\frac{E_{INT}^{x}}{\varepsilon_{x}}\bigg{)}) \tag{15}\]  
Because of the death time of the first node is defined as network lifetime, minimizing the energy consumption of the node spends the most is equivalent to maximize the lifetime. Therefore Equation (15) can also be expressed as:  
\[max(\mathcal{L})=min(max(E_{i}^{TOT})) \tag{16}\]  
Obviously, the goal of Cross-layer optimization can be stated as follow Equation (17):  
\[\left\{\begin{array}{c}min(\mathcal{T}_{sink}^{x})=min(\sum\limits_{i=1}^{k }\mathcal{T}_{i}^{x})\\ max(\mathcal{P}_{sink}^{x})=\prod\limits_{i=1}^{k}par_{(d_{i})}\\ max(\mathcal{L})=min(max(E_{i}^{TOT}))\\ s.t.\ \mathcal{T}_{sink}^{x}\leq\mathcal{T}^{\theta},\ \mathcal{P}_{sink}^{x}\geq \mathcal{P}^{\theta},L\geq\mathcal{L}^{\theta},x\leq\sum\limits_{i=1}^{k}d_ {i}\end{array}\right. \tag{17}\]  
Here, \(\mathcal{T}^{\theta},\mathcal{P}^{\theta},\mathcal{L}^{\theta}\) represent the minimum requirements of the transmission performance thresholds in applications. The goal of Equation (16) is to minimize the transmit performance \(\mathcal{T}_{sink}^{i}\), and keep it not less than the minimum requirements of the application performance. Meanwhile, make the network lifetime and reliability maximized.