We are interested in a single machine scheduling problem where jobs can neither start nor end on some specified instants, and the aim is to minimize the makespan. This problem may model the situation where an additional resource, subject to unavailability constraints, is required to start and to finish a job. We consider in this paper the High-Multiplicity version of the problem, when the input is given using a compact encoding. We present a polynomial time algorithm for large diversity instances (when the number of different processing times is greater than the number of forbidden instants). We also show that this problem is Fixed-Parameter Tractable when the number of forbidden instants is fixed, regardless of jobs characteristics.