link Multiple substrate links Figure 10.2 Cases of substrate node and link failures. (a) source (or destination) node failure, (b) source and destination nodes failures, (c) substrate link failure. At,ij bsts) = (10.5) ijeL's (10.6) Three kinds of failures may occur in the SN as shown in Figure 10.2: (i) source (or destina- tion) node fails in a substrate node pair (s, 1 t) as shown in Figure 10.2a; (ii) both source and destination nodes fail as shown in Figure 10.2b; (iii) only the substrate link fails as shown in Figure 10.2c. These three kinds of failures may coexist in the same RAN slice. Since migration and instantiation of virtual network functions may lead to a long time of service interruption [26, 27], we only remap the failed network functions and keep the unaffected ones on the original substrate nodes. Robust RAN Slicing We first formulate a recovery problem to solve remapping of a slice with deterministic traffic demands in Section 10.3.2. Based on that, we propose two robust RAN slicing algorithms for slice recovery under stochastic demands. The algorithm proposed in Section 10.3.2 is based on the robust optimization theory with a high computational complexity and slow convergence. To tackle the slow convergence problem, we propose a heuristic algorithm based on VNS in Section 10.3.3. 10.3.1 Failure Recovery Problem Formulation We consider a robust RAN slicing algorithm that remaps the failed network functions and re-routes the failed virtual links. We assume that the bandwidth requirement in the request is deterministic. In other words, the bandwidth of each virtual link is unchanged after the creation of the corresponding RAN slices. The objective is to minimize the total bandwidth consumption. The failure recovery problem (FRP) is formulated as follows: (10.7) 10 Robust RAN Slicing (10.8) EEmEN (10.9) Em,EN (10.10) KENYSENS (10.11) s,tEN's asist,st) (10.12) ijeLS stslc) (10.13) ijeLS aw,ij = 0,w#s,wtst,w,s,teNS, (10.14) ijELS (10.15) mstNS,k,lEN (10.16) (10.17) (10.18) (10.19) where constraint (1