## 3 Macroscopic Transmission Control: Virtual Hub-Based Transmission
### Transmission Link Decision and Route Reply  
The destination node \(n_{d}\) formulates the following optimization problem, QoS-constrained transmission:  
\[\text{minimize}\quad z=\sum_{(i,j)\in A}\left[c_{ij}x_{ij}\right] \tag{7}\] \[\text{subject to}\] \[\sum_{\{j:(i,j)\in A\}}x_{ij}-\sum_{\{j:(j,i)\in A\}}x_{ji}=1, \quad\text{$i$ is a starting hub node}\] (8) \[\sum_{\{j:(i,j)\in A\}}x_{ij}-\sum_{\{j:(j,i)\in A\}}x_{ji}=-1, \quad\text{$i$ is a terminating hub node} \tag{9}\]  
Figure 1: System model.  
\[\sum_{\{j:(i,j)\in A\}}x_{ij}-\sum_{\{j:(j,i)\in A\}}x_{ji}=0,\quad\quad i \text{ is an intermediate hub node} \tag{10}\] \[\Pr\left[\sum_{(i,j)\in A}(d_{ij}x_{ij})<DCT\right]\geq\xi,\] (11) \[\sum_{(i,j)\in A}x_{ij}\leq m,\] (12) \[\sum_{(i,j)\in A}x_{ij}\leq 1,\] (13) \[x_{ij}=0\text{ or }1,\text{ for }\forall(i,j)\in A. \tag{14}\]  
In Equation (7), the objective is to select the optimal hub route that will minimize the transmission path cost:  
\[c_{ij}=|v_{i}-v_{j}| \tag{15}\]  
where \(c_{ij}\) denotes the relative mobility between hub \(H_{i}\) and hub \(H_{j}\)--i.e., the link cost between \(H_{i}\) and \(H_{j}\). \(x_{ij}\) denotes the decision variable. The constraints in Equations (8)-(10) guarantee that a transmission path starts from the source and ends at the destination. The constraints in Equations (8) and (10) are the constraints for the hub node nearest the source node and the destination node, respectively. The constraint in Equation (9) is the constraint for the intermediate hub node in a hub path. The constraint in Equation (11) denotes the stochastic delay constraint. Here, \(d_{ij}\) denotes the delay between \(H_{i}\) and \(H_{j}\).  
Assume that the delay distribution between hub \(H_{i}\) and \(H_{j}\) follows a Gaussian distribution \((\mu_{ij},\sigma_{ij}^{2})\). Then, the constraint in Equation (11) with \(\xi=0.95\) can be interpreted as Equation (16), because the sum of Gaussian distributions also follows a Gaussian distribution:  
\[\Pr\left[\sum_{(i,j)\in A}(d_{ij}x_{ij})<\mu+2.33\sigma\right]\geq 95\% \tag{16}\]  
where  
\[\mu=\sum_{(i,j)\in A}\mu_{ij}x_{ij},\quad\sigma=\sum_{(i,j)\in A}\sigma_{ij}x_ {ij}.\]  
That is, we can check whether path A satisfies the delay constraint or not by comparing \(\mu+2.33\sigma\) and \(DCT\). The constraint in Equation (12) denotes that the length of the selected path cannot be longer than \(m\). The constraint in Equation (13) denotes that the transmission path includes no subtour path; that is, the transmission path should be a simple path encountering no vertex more than once. The problem is a stochastic integer programming model. It is hard to find a solution because it is an NP-hard problem [11]. In the following section, we suggest two heuristic search methods using a virtual flooding-based approach: they are performed virtually at the destination node using the constructed reduced-hub topology map.  
**Remark 1**.: _When we say the stochastic guarantees of the delay limit, it does not include delays caused by route failure. However, the proposed scheme provides very fast route-failure recovery using the proposed stable hubs and redundant local paths._  
**Remark 2**.: _In this work, we assume that the delay distribution follows a Gaussian distribution because: (1) some studies [12,13] have explained that the delay distribution follows an exponential distribution, such as Gamma distribution or Gaussian distribution, and, when comparing the distributions, they find Gamma distribution to be slightly more accurate most of the time, with Gaussian distribution infrequently being more accurate; (2) Gaussian distribution can be matched well because the delay is affected by such various causes as TCP/IP protocol, MAC protocol, physical-layer processing, propagation delay, noise and interference cancellation,and random movement, among others; and (3) Gaussian distribution makes the stochastic programming model easier to solve and tractable._