0.11) ensures that the allocated bandwidth on a substrate link does not exceed its link capacity; constraint (10.15) ensures that all the failed VNFs are remapped onto the new ones; constraint (10.16) is to keep the unaffected VNFs; and constraint (10.17) ensures that the number of VNFs requested is within the capacity of the substrate node. (10.18) guarantees that the allocated VNFs are integers. (10.19) ensures that the allocated bandwidth is nonnegative. Problem in (10.7) is a mixed integer program (MIP). To simplify the description, we reformulate (10.7) into an equivalent compact form, i.e. the matrix form. Let H repre- sent the coefficient matrix of the constraints in FRP. H is a IXJ dimensional matrix where1C2and = denote the component of H. Let the variables of the compact form a problem denoted by a vector X = (b y m) ² = where vectors b, s, and m are composed by and ms, respectively. c is a J dimensional constant vector, where the first |b| components of c are all ones and the rest of the components are all zeros. d is a I dimensional constant vector. Thus, the FRP can be rewritten as (10.20) Hx < d (10.21) (10.22) where X = {xjlx,>0, for ..., 10.3 Robust RAN Slicing We use the Matlab solver to solve FRP. Since FRP is somewhat idealized, we take it as a benchmark for comparing with the robust RAN slicing algorithms. In Section 10.3.2, we extend FRP with more realistic assumptions. 10.3.2 Robust RAN Slicing Problem Formulation The assumption that the bandwidth of virtual and substrate links is deterministic in (10.20) can be too ideal. In particular, the substrate link resources are likely to be under-utilized, since the request usually requires its link bandwidth according to the peak hour demand; if traffic shaping is not enabled, any increase in traffic demands can lead the mapped request to be infeasible; the availability of the substrate link is time-varying. The reasons can be, for instance, the time-varying switch port/interface failures in fiber connected network [15], or the time-varying interfe