We consider in this paper packets which arrive according to a Poisson process into a finite queue. Often, a group of consecutive packets forms a frame, and loss of one packet results in the loss of the whole frame. This is the case in ATM where a transport layer protocol (the AAL layer) is responsible for this grouping. Thus, packets of a frame, that arrive after a packet is lost from the same frame, have no use anymore, and it is advantageous to discard them. We derive exact expressions for the performan- ce of such a policy and obtain simple approximations for the heavy traffic regime. Our analytical results may be quite useful in dimensioning the buffer size that is necessary for a given required bound on loss rate.