A flexible scheduler in an ATM switch. The scheduler enables each connection to be served fairly according to associated quality of service parameters, while enabling several other features. A connection can be shaped while minimizing additional memory and processing requirements. Specifically, the conformance time of cells of a connections need not be stored when significant backlog exists in the transmission of the cells. The shaping rate can be dynamically varied. Sequence of cells forming a frame are buffered in the ATM switch until the end of frame cell is received. All the cells of a frame are then sent in quick succession.