Neutropenia (low neutrophil counts) is a significant concern for patients undergoing chemotherapy and often necessitates dose reduction or treatment interruption. Neutrophils, the most common white blood cell in the body, are particularly susceptible to cytotoxicity from chemotherapeutic drugs since their short lifespans in the blood translate to a constant production in the bone marrow. Granulocyte colony-stimulating factor (G-CSF), an endogenous cytokine regulating the process of granulopoiesis, is given in its exogenous form during chemotherapeutic treatment to boost neutrophil counts in the blood. To help mitigate neutropenia during chemotherapy, we developed a physiological delay differential equation model of the neutrophil production system which incorporates the variable maturation speed of the neutrophils. Together with PK/PD models of endo- and exogenous G-CSF and a chemotherapeutic drug, we show how the timing of the administration of G-CSF influences the degree of neutropenia experienced during 14 day periodic chemotherapy (based on CHOP14 timing protocols). Due to the physiological detail of the model, our results are motivated by examining the origins of neutropenia during periodic chemotherapy with G-CSF support.