We consider the case of a FIFO multiplexer fed by flows that are individually constrained by piecewise linear concave arrival curves. We show that, contrary to what happens at the input, at the output not all valid scenarios in accordance with the worst case arrival curves can occur. This implies that taking an iterative approach to characterize the arrival curves at the output when flows pass throughout several FIFO nodes is suboptimal (in the sense that, although valid, they do not necessarily have to be the best arrival curves that can be found).