This paper presents a fundamental design trade-off applicable to a class of linear positive systems. The result connects the maximum and minimum output response peaks due to a disturbance for all feasible inputs. A core consequence of the result is that, when the disturbance pulse response peaks faster than the input pulse response, then attempts to minimise the maximum peak response to the disturbance are necessarily accompanied by unavoidable undershoot at a later time. The result has potential application in many areas. For example, it provides a defensible benchmark for comparison of all possible insulin treatment strategies for Type 1 diabetes patients. This includes any control strategy implemented in the context of an Artificial Pancreas (AP). The lack of such a benchmark has been a deficiency in the existing AP literature. (C) 2015 Elsevier Ltd. All rights reserved.