# Capacity and Delay Analysis for Large Social-Aware Mobile Ad Hoc Wireless Networks
## 5 Scheduling Scheme and Achieved Throughput Capacity  
In this section, we propose a multi-hop relay and slots allocation scheme, as shown in Figure 2. Then, the achieved throughput capacity based on the proposed scheme is derived. Multi-hop relay and slots allocation scheduling scheme is as follow:(1) Spatially, we tessellate the network area \(\mathcal{O}\) into equal square cells. A straight line segment (\(\mathbb{SD}\) line) connecting the source node and the destination node denote as the reference line. Each source node forwards data approximate this \(\mathbb{SD}\) line by multi-hop. Nodes in each super-cell which intersect the \(\mathbb{SD}\) lines act as relay nodes. (2) Temporally, we adopt time division multiple access (TDMA) for transmission scheduling. Time is divided into slots, all of the cells scheduled to be active in time slot sequence. Each cell becomes active in every \(c^{2}(c\) is determined in the following) time slots. In an active cell, transmission is always between two nodes within the same cell or in the neighboring cells. Further, each time-slot is divided into three sub time-slots. (3) In sub-slot A, the source node, which is located in the interior active cell, gets a transmission opportunity then randomly selects one node from in the same cell or in neighboring cells. The source node maybe transmit data to a relay node, or a destination node. If the interior active cell contains two or more source nodes, randomly select one source node. Otherwise, the interior cell remains idle. (4) In sub-slot B, the destination node, which is located in the interior active cell, gets a transmission opportunity, then randomly selects one node from in the same cell or in neighboring cells. The transmission maybe happened between the relay node and the destination node, or maybe between the source node and destination node. If the cell contains two or more nodes, randomly select one destination node at random. Otherwise, the interior cell remains idle. (5) In sub-slot C, the relay node, which is located in the border active cell, gets a transmission opportunity, then randomly selects one node from in the same cell or in neighboring cells. The transmission maybe happened between the relay node and the relay node, or maybe the relay node and the destination node. If the cell contains two or more nodes, randomly select one destination node at random. Otherwise, the border cell remains idle. As illustrated by the shaded cells in Figure 1, active cell can transmit without interference at the same time. Let us assume that the transmission radius of each node is equal to \(r(n)\). Any two active cells should have a vertical and horizontal distance of multiple of \(r(n)\). Successful transmission occurs, when the following equations are satisfied. \[\mathcal{L}(X_{k},X_{i}) \geq (1+\Delta)r(n);\] \[\mathcal{L}(X_{j},X_{i}) \leq r(n). \tag{17}\]  
Figure 2: Multi-hop relay and slots allocation scheme. According to multi-hop scheduling and slots allocation scheme, transmission occur when two nodes are located in the same cell or adjacent cell. So, transmission radius to be \(\sqrt{8}\) times the side length of the cell \(\sqrt{b(n)}\), i.e., \(r(n)=\sqrt{8b(n)}\). Each cell alternately becomes active in \(c^{2}\) time slots. The protocol interference model should be hold to guarantee the simultaneous transmissions, i.e., \(c\sqrt{b(n)}=(1+\Delta)r(n)\). We obtain that \(c=\sqrt{8}(1+\Delta)\). If there is at least one node in an active cell, randomly select one node to conduct transmission. So, multi-hop scheduling that each cell becomes active in each \(c^{2}\) time slots is possible. Now, we bound the number of \(\mathbb{SD}\) lines passing through each super-cell. For each super-cell, the number of \(\mathbb{SD}\) lines that multi-hop scheduling and slots allocation scheme should pass through is \(O(nE\{\mathcal{L}(S,D)\}\sqrt{m})\).