When a G/D/1/B + 1 queue's B position buffer over-flows, arriving or buffered jobs must be dropped. Assuming a nonbatch arrival process with bounded increments, we use sample path arguments to identify the overflow dropping policy that minimizes the average number of consecutively dropped jobs (the average service gap). This policy, which creates gaps whenever possible, as close to the front of the buffer as possible, remains optimal drop by drop for batch arrivals, blocking servers, and time-varying buffer sizes.