## 3 Proposed Quality of Service Routing Protocol (QoSRP) in UWSNs
### Packets Forwarding Scheme (UIWPF)
#### 3.4.3 Reliable Packets Forwarding Process
Finally, the entire gathered data is upload from the source to the destination over the best upstream and downstream multi-hop links over a set of best forwarding nodes, which numerically can be expressed as:  
\[\mathcal{D}_{i\to j}=\mathcal{D}(i,\mathcal{S}_{in\&})-\mathcal{D}\!\left((j )_{re\ell\_ay},\mathcal{S}_{in\&}\right)\leq 1\,\forall\,i,\mathcal{J}_{re\ell \_ay}\in\,\mathcal{S}_{in\&} \tag{34a}\] \[\mathcal{D}_{j\to\mathcal{S}_{in\&}}=\,\mathcal{D}\!\left(\mathcal{F}_{re\ell \_ay(|(j,\ell)\leq n)|}\right)\!>\mathcal{D}_{min}\,\&\&\leq\mathcal{D}_{m\, \alpha\,x}\,\forall\mathcal{F}_{re\ell\,y}\in\,\mathcal{S}_{in\&}\] (34b) \[\mathcal{D}_{j\to\mathcal{S}_{in\&}}=\,\mathcal{D}\!\left(\mathcal{F}_{re\ell \_ay(n)}\in\,\mathcal{U}_{\mathcal{D}\,\delta\,re\,m}\,\forall\mathcal{F}_{ re\ell\,y(n)}\in\,\mathcal{S}_{in\&}\] (34c) \[\mathcal{D}_{\mathcal{S}_{in\&}\,\to i}=\,\mathcal{D}\!\left(\mathcal{RF}_{ re\ell\,y(n)}\in\,\mathcal{D}\!\left(\mathcal{down}_{\delta\,re\,m}\,\left(1q \right)\,\forall\,\mathcal{BF}_{re\ell\,y(n)}\in i\right.\] (34d) \[\mathcal{D}_{sum}=\,\sum_{j\in\mathcal{F}_{re\ell\,y}>1}\mathcal{D}_{j\to \mathcal{S}_{in\&}}\!\left(\mathcal{F}_{re\ell\,y(j)}\right)\leq\mathcal{F}_ {re\ell\,y(m\,\alpha\,x)}\,\forall\mathcal{J}_{re\ell\,y}\in\,\mathcal{S}_{in\, \delta\,k}\] (34e) \[\rho_{b(\rho\,d(i))}(\mathcal{RP}_{i})=\beta_{1}\mathcal{Q}_{\
u i}-\beta_{2} \!\left(1-\sum_{j=1}^{n}\frac{\mathcal{J}\!\left(Q_{j}\right)}{Q_{\ell}} \right)+\rho_{b}(\rho\,d(i)_{in\,i})\,\forall\mathcal{J}_{re\ell\,y}\in\, \mathcal{S}_{in\,\delta\,k}\] (34f) \[\rho_{b}(\rho\,d(i)_{in\,k})=\,\left\{0,\,\text{if}:i\,\mathcal{Q}_{i}<1;\, \mathcal{N}(\mathcal{J}\!\,)^{2}\!\left(Q_{j}/Q_{\ell}\right)-1,\,\text{if}:i \,\mathcal{Q}_{i}\geq 1\right\}\] (34g) \[\mathcal{Q}_{\
u i}=\frac{Q_{i}^{n\,\varepsilon\,\omega}-Q_{i}^{\,\ell\,d}}{Q_{ \ell}/\mathcal{N}e(\mathcal{J}\!\,)}<\forall\mathcal{Q}_{i}\leq\mathcal{Q}_{ \ell}=1\] (34h) \[0\leq\rho_{b(\rho\,d(i))}<1\] (34i) \[\mathcal{D}_{j\to\mathcal{S}_{in\,\delta}}=\mathcal{D}\!\left(j,\mathcal{S}_{in\,\delta}\right)/\mathcal{A}_{eg}\!\left(\mathcal{F}_{re\ell \,y(\mathcal{J}\!\,)}\right)\text{where}\,\,\forall\,\mathcal{J}_{re\ell\,y}>1 \,\&\&\,\,\mathcal{J}_{re\ell\,y}\in\,\mathcal{S}_{in\,\delta}\] (34j) \[\mathcal{D}_{j\to\mathcal{S}_{in\&}}=\,\mathcal{D}\!\left(\mathcal{F}_{re\ell \_ay(|(j,\ell)\leq n)|}\right)\!>\mathcal{D}_{min}\,\&\&\leq\mathcal{D}_{m\, \alpha\,x}\,\forall\mathcal{F}_{re\ell\,y}\in\,\mathcal{S}_{in\&}\]0  
Constraints in (34a) state that in single-hop packet progress, the Euclidian distance (\(\mathcal{D}\)) of next-hop forwarder \(\mathcal{J}\) receiving data packets from an acoustic node \(i\) must be less than the sender to the sink. The constraints in (34b) states that the forwarding node distance must be greater than the minimum (\(\mathcal{D}_{m\,in}\)) and less than maximum Euclidian distance (\(\mathcal{D}_{m\,\alpha\,x}\)) in the network. Constraints in (34c) show that the maximum and minimum distance information of the forwarders during data transmission is bounded to the upward direction \(\mathcal{U}_{\mathcal{P}\,\delta\,stream}\) towards the sink. Similarly, the constraints in (34d) verify that the message forwarding from the sink towards the source node \(i\) in the reverse forwarding nodes \(\mathcal{RF}_{re\ell\,\omega\,y(n)}\) is bounded to downward direction (\(\mathcal{D}\!\left(\mathcal{down}_{\delta\,stream\,\delta\,re\,m}\right)\)). Constraints in (34e) guarantees that the number of forwarding nodes cannot be higher than the defined maximum forwarding nodes\(\mathcal{F}_{\sigma e\ell\,a\,g\,(m.\,a\,x)}\) along a routing path towards the sink. Equation (34f) indicates the probability of a packet drop \(\rho_{b}(\rho\,d(\iota))\) in the \(\iota\)th queue of a forwarding candidate along the selected routing path \(\mathcal{RP}_{\iota}\) in the network.