In order to ease the impact of the packet fragmentation problem and to avoid network congestion in TCP over UBR, packet discard schemes in ATM layer (such as PPD and EPD) have been proposed. These schemes drop packets before they reach their intended destinations if the network is congested and the packets are to be partially discarded. On the other hand, TCP also regulates data flow with its own flow control method. Due to restriction of data flow at the TCP layer, buffer space is not fully used in an ATM switch. In order to make use of more buffer resources, this paper generalizes the PPD and EDP schemes. From this generalization, an optimistic packet discard scheme named the "Probability-Based Delayed Packet Discard" (PDPD) scheme is proposed. Depending on a particular probability, this scheme sets a discard flag to delay actual discard operation. This paper presents the results of several simulated models to find out the potential of improvement of goodput by PDPD. The results of these simulations indicate that PDPD obtains higher goodput than ordinary schemes when the packet size is large and the input load is not light. This author concludes that a PDPD scheme should achieve effective goodput and link utilization while using more buffer resources effectively.