## Appendix
### Problem Solving Steps
\\ Step 2: & Sort these break points and denote as \(g_{l}^{{}^{\prime}}\), \(g_{l}^{{}^{\prime}}\),...,\(g_{l}^{{}^{\prime}}\). There are at most \(|W|\) break points. \\ Step 3: & At each interval \(g_{l}^{{}^{\prime}}\leq g_{l}\leq g_{l}^{{}^{\prime}i+1}\), \(y_{w}^{*}(g_{l})\) is 1 if \(\mu_{l}^{{}^{\prime}}D_{l}(g_{l})-\mu_{wl}^{3}\leq 0\) and is 0 otherwise. \\ Step 4: & Denote \(\sum\limits_{w\in W}\mu_{w}^{*}y_{w}^{*}(g_{l}^{{}^{\prime}})\) as \(a_{l}\) and \(\sum\limits_{w\in W}\mu_{w}^{*}y_{w}^{*}(g_{l}^{{}^{\prime}})\) as \(b_{l}\), within the interval \(g_{l}^{{}^{\prime}}\Finally, this problem can be solved by the algorithm developed by [PERSON] and [PERSON] in [28]. The solution steps and graphics of the algorithm are briefly described in Table 1 and Figure 1.  
Step 3: The Dual Problem and the Subgradient Method  
According to the Weak Lagrangian Duality Theorem [29], for the \(\mu_{\ast}^{i}\), \(\mu_{i}^{i}\), \(\mu_{\ast}^{j}\geq 0\), the objective value of the Lagrangian Relaxation problem \(Z_{\iota\kappa}(\mu_{\ast}^{i},\,\mu_{i}^{j},\,\mu_{\ast}^{j})\) is a lower bound of the primal problem \(Z_{\textit{rp}}\). Based in problem (LR), the following dual problem \(Z_{\textit{D}}\) is constructed to calculate the tightest lower bound. **Dual Problem (D):**  
\(Z_{\textit{D}}\!=\!\max\,Z_{\textit{D}}(\mu_{\ast}^{i},\,\mu_{i}^{j},\,\mu_{ \ast}^{j})\) (D)  
**Subject to:**  
\(\mu_{\ast}^{i},\,\mu_{i}^{j},\,\mu_{\ast}^{j}\geq 0\). We use the Subgradient Method is used to solve that [26, 27] to solve the dual problem (D). First, let the vector \(S\) be a subgradient of \(Z_{\textit{D}}(\mu_{\ast}^{i},\,\mu_{i}^{j},\,\mu_{\ast}^{j})\). In iteration \(k\) of the Subgradient Optimization Procedure, the multiplier vector \(\pi^{k}\!=\! (\mu_{\ast}^{i},\,\mu_{i}^{j},\,\mu_{\ast}^{j})\) is updated by \(\pi^{k+1}\!=\!\pi^{k}\!+\!t^{k}S^{k}\,\). The step size \(t^{k}\) is determined by \(t^{k}=\lambda\frac{(Z_{\textit{rp}}^{k}-Z_{  
Step 5: Routing Metric  
The cost of passing through the network can be used as a metric. By virtue of the routing algorithm, each router chooses the path with the smallest (shortest) sum of the metrics. Different routing protocols define the metric distinctly. The metric can be distance, number of hops, delay, and so on. Here, the metric is defined as a combination of the link mean delay and the derivative of queue length for each link. Step 6: Estimation of Routing Metric Parameters  
Perturbation Analysis  
In [25], [PERSON] _et al_. proposed \"Perturbation Analysis\" (PA), a useful technique for estimating the so-called \"marginal delay\" defined in [24]. More specifically, this method is used to estimate the derivative of queue length. As previously mentioned, the derivative of queue length is composed of link mean delay, aggregate flow, and the partial derivative of link mean delay. The PA approach can also estimate the value of link mean delay, a step that precedes the calculation of the derivative of queue length. One can use the PA technique to determine the link mean delay and the derivative of queue length, both major components of the distributed routing protocol. Time Stamping for Packets  
Other approaches can also estimate critical information of each link in a conceptually straightforward way. During packet reception, packet delay time is defined as the difference between sending time and receiving time. For each adjacent link mean delay, it the sum of each packet delay divided by the number of received packets corresponding to each neighbor node. Approximation Function  
For the derivative of queue length, queue length and the corresponding aggregate flow over the link can be recorded at each time interval. Figure 10 is illustrated a monotonically increasing and convex function such as the power regression function (_i.e._, \(y=ax^{b}\), where \(a>0\) and \(b>1\)) can be used to approximate the recording results, and the newest record has the most impact to this approximation.