# Capacity and Delay Analysis for Large Social-Aware Mobile Ad Hoc Wireless Networks
## 6 Network Delay of Our Scheduling Scheme  
In this section, we provide an analysis on network delay of multi-hop relay and slots allocation scheduling scheme based on queueing theory. The delay consists of two components: hop delay (processing delay) and network delay (queuing delay). Noted that the processing time (such as coding/decoding) spent is negligible. Like many important work, hop delay is ignored in this paper. It worth noting that the value of \(E\{\mathcal{L}(S,D)\}\) is larger than value of super-cell \(m\) for social parameter \(0\leq\delta\leq 3/2\). The S-D pairs are not located in the same super-cell. For social parameter \(\delta>3/2\), the S-D pairs are located in the same super-cell. Thus, we solve the problem in two cases. Let \(\lambda_{i}\) (\(\lambda_{i}=O(\mu)\)) denote the input stream for source node \(i\). The total rate of each node \(\mu\) can be divided into two parts: \(r1\) represents the rate that the node is scheduled to transmit to the relay node; \(r2\) represents the rate that the node is scheduled to transmit to the destination node. For social parameter \(\delta\geq 3/2\), there are two possible routings from source node to its destination node: one is single hop from source node to destination node directly; the other is two hop path - source node to relay node transmission and relay node to destination node transmission. Since data transmission processes is independent of mobility processes, the network delay can compute from the perspective of source node \(i\). Decoupled transmission path is illustrated in Figure 3, which can be modeled as discrete queue. Source node generates data at rate of \(\lambda_{i}\), and service opportunity arises with some fixed rate \(\mu\). Thus, source node have an expected number of data given by  
\[\overline{L}_{s}=\frac{\rho(1-\lambda_{i})}{1-\rho},\]  
where \(\rho=\frac{\lambda_{i}}{\mu}\). Figure 3: Queue in network for \(\delta>3/2\). It is a reversible queue, the output stream is also a discrete stream at rate \(\lambda_{i}\). Data from this output process is transmitted to relay node. Because the total number of relay nodes in each super-cell is \(nm-2\), data delivered to a relay node with probability \(\frac{\
u 1}{u}\) equally. Hence, each relay node independently receives data with probability \(\lambda_{r}=\frac{\lambda_{r}\Gamma 1}{\mu(nm-2)}\). Because the total number of relay nodes in each super-cell is \(nm-2\), the relay node is scheduled for data transmission to the destination node with probability \(\frac{\rho}{u}\). Hence, each relay node independently sends data with probability \(\mu_{r}=\frac{\
u 2}{nm-2}\). However, receiving and sending data is mutually exclusive events in cache queue of each relay node. The queue of the relay node can be seen as a continuous time queue with input rate \(\lambda_{r}\) and service rate \(\mu_{r}\), where \(\frac{\lambda_{r}}{\mu_{r}}=\rho\). This holds for each relay node, the expected number of data is  
\[\overline{L}_{r}=\frac{\rho}{1-\rho}.\]  
Finally, when social parameter \(\delta>3/2\), using Little's Theorem, we can get the average waiting time for S-D pairs--the network delay \(D(n)\) is  
\[D(n) = \frac{\overline{L_{s}}+(nm-2)\overline{L_{r}}}{\lambda_{i}} \tag{25}\] \[= \frac{nm-1-\lambda_{i}}{\mu-\lambda_{i}}.\]  
For social parameter \(\delta>3/2\), the network delay \(D(n)\) is \(\Theta(nm)\). For social parameter \(\delta\leq 3/2\), data is forward via a multi-hop path. Three kinds of transmissions are allowed: transmissions between source node and relay node, transmissions between relay node and relay node, transmissions between relay node and destination node. As illustrated in Figure 4, the process can be modeled as a open queuing network just as \(\delta>3/2\). The super-cell where the source node is located is defined as the source super-cell; the super-cell where the relay node is located is defined as the relay super-cell; and the destination super-cell where the destination node is located is defined as the destination super-cell.