## Appendix A
As the optimal solution to **(MILP)**, \((\theta^{*},C^{*},\alpha^{*})\) satisfies the constraint (6). \[\theta^{*}_{i,t+1}=\theta^{*}_{i,t}+\alpha^{*}_{i,t}-\theta^{*}_{i,t},\quad \forall\ i,t\]0 (A19) \[\theta^{*}_{i,t+1}=\theta^{*}_{i,t}+\alpha^{*}_{i,t}-\theta^{*}_{i,t},\quad \forall\ i,t\]1 (A20) \[\theta^{*}_{i,t+1}=\theta^{*}_{i,t}+\alpha^{*}_{i,t}-\theta^{*}_{i,t},\quad \forall\ i,t\]2 (A21) \[\theta^{*}_{i,t+1}=\theta^{*}_{i,t}+\alpha^{*}_{i,t}-\theta^{*}_{i,t},\quad \forall\ i,t\]3 (A22) \[\theta^{*}_{i,t+1}=\theta^{*}_{i,t}+\alpha^{*}_{i,t}-\theta^{*}_{i,t},\quad \forall\ i,t\]4 (A23) \[\theta^{*}_{i,t+1}=\theta^{*}_{i,t}+\alpha^{*}_{i,t}-\theta^{*}_{i,t},\quad \forall\ i,t\]5 (A24)  
Equations (A20)-(A23) imply that:  
\[\alpha_{i,t}\leq C_{i,t}\cdot\theta_{0,t}\cdot\lambda\cdot\min_{ \begin{subarray}{c}j_{i}(1,2,2\\ j_{i}(2,t)-1\end{subarray} ,\\ j_{i}(2,t)-1\end{subarray}}\big{\{}F\big{(}X_{i,j2}\big{)}-F\big{(}X_{j1,t} \big{)},F\big{(}X_{i,j2}\big{)}-F(X_{0,i}),1-F\big{(}X_{j1,t}\big{)},1-F(X_{0, i})\big{\}}\] (A25)  
For any \(C_{t}\), we have:  
\[p_{i,t} = \Pr\Bigg{(}C_{i,t}=1,r_{i}(\tilde{D})\leq\min_{j:C_{j}=1}r_{j}( \tilde{D})\Bigg{)}\] (A26) \[= C_{i,t}\cdot\min_{ \begin{subarray}{c}i(1,2,2\\ j_{i}(2,t)-1\end{subarray} ,\\ j_{i}(2,t)-1\end{subarray}}\big{\{}F\big{(}X_{i,j2}\big{)}-F\big{(}X_{j1,t} \big{)},F\big{(}X_{i,j2}\big{)}-F(X_{0,i}),1-F\big{(}X_{j1,t}\big{)},1-F(X_{0, j})\big{\}}\]  
Combining (A25) and (A26), we have \(\alpha_{i,t}\leq\theta_{0,t}\cdot\lambda\cdot p_{i,t}\). Then, with (A24) and the property that \(\sum_{i=1}^{m}p_{i,t}=1\), we have:  
\[y\alpha_{i,t}=\theta_{0,t}\cdot\lambda\cdot p_{i,t},\quad\forall\ i,t.\] (A27)With an analogous proof for the constraints (8) and (9) in Step 1, we can show that if \(\sum_{i=1}^{m}\theta_{i,t}\cdot\mathbb{E}[U_{i}^{f}]>\tau\), then \(G_{t}=1\), and if \(\sum_{i=1}^{m}\theta_{i,t}\cdot\mathbb{E}[U_{i}^{f}]\leq\tau\), then \(G_{t}=0\). No matter whether network congestion occurs or not in period \(t\), the equation \(\beta_{i,t}=\theta_{i,t}\cdot q_{i,t}\) always holds. Then,  
\[\theta_{i,t+1}^{*}=\theta_{i,t}^{*}+\alpha_{i,t}^{*}-\beta_{i,t}^{*},\quad \forall\ i,t,\]  
can be reformulated as:  
\[\theta_{t+1}^{*}=A_{t}(\theta_{t}^{*},C_{t}^{*})\cdot\theta_{t}^{*}.\]  
We conclude that given the optimal solution \((\theta^{*},C^{*},G^{*},\alpha^{*},\beta^{*})\) to **(MILP)**, there exists a feasible solution \((\theta^{*},C^{*})\) to **(DP2)**. Step 4, \(Z(\theta_{1})\leq\overline{V}_{1}(\theta_{1})\):  
Because \((\theta^{*},\alpha^{*})\) is the optimal solution to **(MILP)**,  
\[Z(\theta_{1})\] \[= \sum_{t=1}^{T}\sum_{i=1}^{m}(\theta_{i,t}^{*}+\alpha_{i,t}^{*}) \cdot[F_{i}(v_{i})+\bar{F}_{i}(v_{i})\cdot(1-w_{i})+\rho_{i}\cdot w_{i}] \cdot b_{i}\] \[= \sum_{t=1}^{T}\sum_{i=1}^{m}\theta_{i,t}^{*}\cdot[F_{i}(v_{i})+ \bar{F}_{i}(v_{i})\cdot(1-w_{i})+\rho_{i}\cdot w_{i}]\cdot b_{i}\] \[+\sum_{t=1}^{T}(1-\sum_{i=1}^{m}\theta_{i,t}^{*})\cdot\sum_{i=1}^ {m}\lambda p_{i,t}(C_{t}^{*})\cdot[F_{i}(v_{i})+\bar{F}_{i}(v_{i})\cdot(1-w_{ i})+\rho_{i}\cdot w_{i}]\cdot b_{i}\] \[= \sum_{i=1}^{m}\theta_{i,1}^{*}\cdot[F_{i}(v_{i})+\bar{F}_{i}(v_{ i})\cdot(1-w_{i})+\rho_{i}\cdot w_{i}]\cdot b_{i}\] \[+(1-\sum_{i=1}^{m}\theta_{i,1}^{*})\cdot\sum_{i=1}^{m}\lambda p _{i,1}(C_{1}^{*})\cdot[F_{i}(v_{i})+\bar{F}_{i}(v_{i})\cdot(1-w_{i})+\rho_{i} \cdot w_{i}]\cdot b_{i}+\overline{V}_{2}(\theta_{2}^{*})\]  
Because \((\theta^{*},C^{*})\) is a feasible solution to **(DP2)**, we have:  
\[\overline{V}_{1}(\theta_{1}) = \sum_{i=1}^{m}\theta_{i,1}^{*}\cdot[F_{i}(v_{i})+\bar{F}_{i}(v_{ i})\cdot(1-w_{i})+\rho_{i}\cdot w_{i}]\cdot b_{i}\] \[\theta_{i,t+1}^{*}=\theta_{i,t}^{*}+\alpha_{i,t}^{*}-\beta_{i,t}^{*},\quad \forall\ i,t,\]0 \[\theta_{i,t+1}^{*}=\theta_{i,t}^{*}+\alpha_{i,t}^{*}-\beta_{i,t}^{*},\quad \forall\ i,t,\]1  
In conclusion, we have \(Z(\theta_{1})=\overline{V}_{1}(\theta_{1})\). Theorem 2 holds.