# Asymmetric Propagation Delay-Aware TDMA MAC Protocol for Mobile Underwater Acoustic Sensor Networks
## The APD-TDMA
### Transmission Phase
#### Scheduling to Avoid Collisions at the SN  
At the end of the \(j\)th cycle, the SN determines each UN's transmission time at the \((j+1)\)th cycle such that data packets of UNs arrive at the SN in the order specified in ascending order of \(\delta_{i}(T_{Brx,i}^{j+1})\). The transmission time of UN is assigned by waiting time after the reception of the \((j+1)\)th beacon packet. For convenience, we assume that the UN index \(i\) is sorted in ascending order of \(\delta_{i}(T_{Brx,i}^{j+1})\), which means that \(\delta_{m}(T_{Brx,m}^{j+1})>\delta_{k}(T_{Brx,k}^{j+1})\) if \(m>k\). The principle to avoid packet collision at the SN is to make the start time of the \((j+1)\)th data packet (i.e., \(UN_{i+1}\)'s data packet) reception larger than the end time of the \(i\)th data packet reception. The first node, \(UN_{1}\), transmits the data packet immediately  
Figure 4: The transmission phase of the APD-TDMA. if the \((j+1)\)th beacon packet is received. Therefore, the start time and the end time for the data packet reception of \(UN_{1}\), denoted by \(T_{Drx,1}^{j+1}\cdot start\) and \(T_{Drx,1}^{j+1}\cdot end\), respectively, are given by  
\[ \begin{split} T_{Drx,1}^{j+1}\cdot start=T_{CYC}^{j}+\tau_{B}+2 \delta_{1}(T_{Brx,1}^{j+1}),\\ T_{Drx,1}^{j+1}\ \cdot\ end=T_{CYC}^{j}+\tau_{B}+2\delta_{1}(T_{ Brx,1}^{j+1})+L_{1}^{j+1}.\end{split}  \tag{10}\]  
If \(UN_{2}\) transmits the data packet as soon as the reception of the beacon packet, the start time for the reception of this data packet, denoted by \(T_{Drx,2}^{j+1}\cdot start_{pre}\), is given by  
\[T_{Drx,2}^{j+1}\cdot start_{pre}=T_{CYC}^{j}+\tau_{B}+2\delta_{2}(T_{Brx,2}^{ j+1}). \tag{11}\]  
If \(T_{Drx,2}^{j+1}\cdot start_{pre}>T_{Drx,1}^{j+1}\cdot end\), the data packet of \(UN_{1}\) does not collide with the data packet of \(UN_{2}\). However, if \(T_{Drx,2}^{j+1}\cdot start_{pre}<T_{Drx,1}^{j+1}\cdot end\), the data packets of \(UN_{1}\) and \(UN_{2}\) overlap and the data packet collision occurs. In order to avoid this collision, \(UN_{2}\) should postpone the data packet transmission such that the arrival time of the data packet of \(UN_{2}\), denoted by \(T_{Drx,2}^{j+1}\cdot start\), is greater than \(T_{Drx,1}^{j+1}\cdot start\). If this intentional wait time is denoted by \(\Delta w_{2}^{j+1}\), we assign \(\Delta w_{2}^{j+1}\) such that the following condition is satisfied:  
\[T_{Drx,2}^{j+1}\cdot start\triangleq T_{Drx,2}^{j+1}\cdot start_{pre}+ \Delta w_{2}^{j+1}>T_{Drx,1}^{j+1}\cdot end. \tag{12}\]  
Note that, in the case of \(T_{Drx,2}^{j+1}\cdot start_{pre}>T_{Drx,1}^{j+1}\cdot end\),  
\[T_{Drx,2}^{j+1}\cdot start=T_{Drx,2}^{j+1}\cdot start_{pre}. \tag{13}\]  
The above two equations can be merged as follows:  
\[T_{Drx,2}^{j+1}\cdot start=\max\Bigl{[}T_{Drx,1}^{j+1}\cdot end,\ T_{Drx,2}^{ j+1}\cdot start_{pre}\Bigr{]}. \tag{14}\]  
The propagation delay of \(UN_{2}\) at the time of data packet transmission, denoted by \(\delta_{2}^{j+1}\cdot up\), is different with \(\delta_{2}(T_{Brx,2}^{j+1})\) in the case of \(T_{Drx,2}^{j+1}\cdot start_{pre}<T_{Drx,1}^{j+1}\cdot end\) due to the mobility of \(UN_{2}\). This asymmetry of the propagation delay between \(\delta_{2}(T_{Brx,2}^{j+1})\) and \(\delta_{2}^{j+1}\cdot up\) should be carefully treated to avoid collision. The velocity of \(UN_{2}\) at the \((j+1)\)th cycle cannot be known when the schedule for the \((j+1)\)th cycle is computed, and thus, we estimate \(\delta_{2}^{j+1}\cdot up\) assuming the maximum mobility at the time interval from the \((j+1)\)th beacon packet reception to the data packet transmission that corresponds to \(T_{Drx,2}^{j+1}\cdot start-T_{Drx,2}^{j+1}\cdot start_{pre}\). In other words, \(\delta_{2}^{j+1}\cdot up\) is determined from the location estimation of \(UN_{2}\) that moves from \(X_{2}^{j+1}=X_{i}(T_{Brx,2}^{j+1})\) at the maximum speed of UN, denoted by \(Vmax\), during the time interval of \(T_{Drx,2}^{j+1}\cdot start-T_{Drx,2}^{j+1}\cdot start_{pre}\).