This paper proposes an optimization model under uncertain traffic demands to design the backup network to minimize the total capacity of a backup network to protect the primary network from multiple link failures, where the probability of link failure is specified. The hose uncertainty is adopted to express uncertain traffic demands. The probabilistic survivability guarantee is provided by determining both primary and backup network routing, simultaneously. Robust optimization is introduced to provide probabilistic survivability guarantees for different link capacities in the primary network model under the hose uncertainty. Robust optimization in the proposed model handles two uncertain items: uncertain failed primary link with different capacities and uncertain traffic demands. We formulate an optimization problem for the proposed model. Since it is difficult to directly solve it, we introduce a heuristic approach for the proposed model. By using the heuristic approach, we investigate how the probability of link failure affects both primary and backup network routing. Numerical results show that the proposed model yields a backup network with lower total capacity requirements than the conventional model for the link failure probabilities examined in this paper. The results indicate that the proposed model reduces the total capacity of the backup network compared to the conventional model under the hose uncertainty. The proposed model shares more effectively the backup resources to protect primary links by determining routing in both primary and backup networks.

This paper proposes an optimization approach that designs the backup network with the minimum total capacity to protect the primary network from random multiple link failures with link failure probability. In the conventional approach, the routing in the primary network is not considered as a factor in minimizing the total capacity of the backup network. Considering primary routing as a variable when deciding the backup network can reduce the total capacity in the backup network compared to the conventional approach. The optimization problem examined here employs robust optimization to provide probabilistic survivability guarantees for different link capacities in the primary network. The proposed approach formulates the optimization problem as a mixed integer linear programming (MILP) problem with robust optimization. A heuristic implementation is introduced for the proposed approach as the MILP problem cannot be solved in practical time when the network size increases. Numerical results show that the proposed approach can achieve lower total capacity in the backup network than the conventional approach.