We consider a FIFO multiplexer fed by flows that are individually constrained by arrival curves, and look for the best possible arrival curve for every output flow. This problem arises in scenarios where aggregate multiplexing is performed, such as differentiated services or front ends to optical switches. We obtain an exact result for a fluid model and for piecewise linear concave arrival curves, which are common in practice and correspond to combinations of leaky buckets. Our results show that previous methods do not obtain tight bounds, unlike our finding here.