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We present a linear programming based method to design time varying routing strategies for positive conservative compartmental systems to satisfy piecewise constant capacity constraints. Such systems can be used to describe the flow of material through a network of interconnected reservoirs and have become a popular method of modeling air traffic flows. While most dynamic models for such systems use flow rates which depend linearly on the amount of material present in each reservoir, we focus on flow rates which are nonlinear in the amount of material. Nonlinear, saturating outflow rates can more accurately describe air traffic flow in the limit of dense traffic than linear outflow models.