In this paper, we are interested in the robustness analysis of an elementary logistic system having a fixed loss factor on the inventory level and uncertainties on the production delay. The problem is treated in control theory domain, where the model is considered as an input time delay system characterized by positivity and saturation constraints, and an external disturbance. Indeed, we apply a prediction state feedback control strategy using an affine control law, where the prediction of the future inventory level is based on a delay estimation of the delay uncertainty. Hence, the objective of the study is to quantify the impact of the uncertainty induced by this estimation on the performance of the controlled system. First, we use a frequency-domain technique to identify the robust stability condition in the set of parameters. In particular, we specify the range of the delay deviation such that the closed-loop system stability is guaranteed. Then, we move on to define the input-output flow variations that allows to check the system constraints, based on the invariance properties. Finally, a comparative simulation is provided to highlight the advantages of this study.