## 4 EMOR Protocol Design for WSN in NAN
### Optimal Multi-Disjoint Path Selection for Real-Time and Non-Real-Time Traffic
Primary multi-path connectivity for real-time traffic based on first best hop decision criteria in link connectivity function
2. Alternate multi-path connectivity for non-real-time based on second best hop decision criteria in link connectivity function \[\left\{ \begin{array}{l}L_{VIVj}^{RT}=\max\limits_{Vj\in N}\{\alpha TF_{ViVj}+ \beta R_{E,Vj}+\gamma B_{S,Vj}+\sigma(S/N)_{VIVj}\}\dots\dots\dots\mathrm{H1:real- time\\ L_{VIVj}^{NT}=2 nd\max\limits_{Vj\in N}\{\alpha TF_{ViVj}+\beta R_{E,Vj}+ \gamma B_{S,Vj}+\sigma(S/N)_{VIVj}\}\dots\mathrm{H2:non-real-time}\end{array} \right.\] (15)  
\[\left\{ \begin{array}{l}PC_{V_{i}Vi+t}^{RT}=L_{V_{i}V_{i+1}}^{RT}\times\prod \limits_{k=1}^{t-2}L_{V_{i+k}V_{i+(k+1)}}^{RT}\times L_{V_{i+t-1}V_{i+t}}^{RT} \dots\dots\mathrm{H1:real-time}\\ PC_{V_{i}Vi+t}^{NT}=L_{V_{i}V_{i+1}}^{NT}\times\prod\limits_{k=1}^{t-2}L_{V_{ i+k}V_{i+(k+1)}}^{NT}\times L_{V_{i+t-1}V_{i+t}}^{NT}\dots\mathrm{H2:non-real-time}\end{array} \right. \tag{16}\]  
where \(L_{V_{i}Vj}^{RT}\) is the link connectivity for real-time traffic, \(L_{V_{i}Vj}^{NT}\) is the link connectivity for non-real-time traffic, \(PC_{V_{i}Vi+t}^{RT}\) is the path connectivity for real-time traffic from node \(V_{i}\) to any other node \(V_{i+t}\) and \(PC_{V_{i}Vi+t}^{NT}\) is the path connectivity for non-real-time traffic from node \(V_{i}\) to any other node \(V_{i+t}\) in Equations (15) and (16). Foregoing in view, the constructed paths are node-disjoint paths which have no rendezvous point except source and destination. Node-disjoint paths are also preferred because they utilize most available network resources while avoiding the bottle necks by keeping energy balance. If an intermediate node fails in a node-disjoint path, only the path containing that failed node will be affected, thus maintaining the diversity of the routes intact with minimum impact. To avoid misuse of energy resources, we limit each sensor node to involve either in first-best hop decision or second-best hop decision, so that no sensor node is involved in constructing paths for both real-time and non-real-time traffic. Algorithm 3 provides details about the multi-disjoint path selection for real-time and non-real-time traffic. After the construction of multi-disjoint paths, we need to divide our total paths for real-time and non-real-time traffic (i.e., out of N available paths, let us assume that there are \(\mu\) paths that correspond to a probability of successfully delivering data to a destination) [46]. For real-time traffic, we need \(\tau\) paths and for non-real-time traffic, we need \(\epsilon\) paths, where the total traffic \(\mu=\tau+\varepsilon\). Assuming that the traffic size of real-time and non-real-time data is known, we can easily calculate \(\tau\) and \(\varepsilon\). If \(R_{T}\) represents the real-time traffic size and \(N_{T}\) represents non-real-time traffic size, we can have:  
\[\tau=(\frac{R_{T}}{R_{T}+N_{T}})\mu \tag{17}\]  
\[\varepsilon=(\frac{N_{T}}{R_{T}+N_{T}})\mu \tag{18}\]  
Moreover, the path connectivity from mobile sink \(V_{SINK}\) to \(V_{j}\) for real-time and non-real-time traffic can be expressed in Equations (19) and (20). The path information from mobile sink \(V_{SINK}\) to \(V_{j}\) for real-time and non-real-time traffic can be seen in Equations (21) and (22). \[PC_{V_{SINK}V_{j}}^{RT}=PC_{V_{SINK}V_{i}}^{RT}\times\prod\limits_{k=0}^{t-2}L _{V_{i+k}V_{i+(k+1)}}^{RT}\times L_{V_{i+t-1}V_{i+t}}^{RT} \tag{19}\]  
\[PC_{V_{SINK}V_{j}}^{NT}=PC_{V_{SINK}V_{i}}^{NT}\times\prod\limits_{k=0}^{t-2}L _{V_{i+k}V_{i+(k+1)}}^{NT}\times L_{V_{i+t-1}V_{i+t}}^{NT} \tag{20}\]  
\[PATH(V_{SINK},V_{j},*)=\left\{V_{SINK},V_{j},t+1\right\}=\left\{V_{SINK},V_{i} ^{RT}.\dots V_{i+t-1}^{RT},V_{j}\right\} \tag{21}\]  
\[PATH(V_{SINK},V_{j},*)=\left\{V_{SINK},V_{j},t+1\right\}=\left\{V_{SINK},V_{i} ^{NT}.\dots V_{i+t-1}^{NT},V_{j}\right\} \tag{22}\]where \(t\) is the hops on the path between node \(V_{i}\) to \(V_{j}\) and \(t+1\) hops on the path between \(V_{SINK}\) to \(V_{j}\). ```
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