A wireless-optical broadband access network (WOBAN) shows promise as potential 5G access infrastructure. Since network slicing allows efficient sharing of physical network resources, we consider the provisioning of availability-guaranteed slices in a WOBAN supporting mobile edge computing (MEC). A new definition for the availability of a slice is proposed accounting for a slice that functions only partially because of the failure of a fiber link, a microwave link, a base station (BS) node, and/or an optical line terminal (OLT). An integer linear programming (ILP) model and a simple but effective heuristic algorithm that balances the network traffic load and maximizes the slice availability are developed to maximally provision availability-guaranteed slices. Simulation results show the efficiency of the proposed approaches.