As machine learning models increasingly replace traditional business logic in the production system, their lifecycle management is becoming a significant concern. Once deployed into production, the machine learning models are constantly evaluated on new streaming data. Given the continuous data flow, shifting data, also known as concept drift, is ubiquitous in such settings. Concept drift usually impacts the performance of machine learning models, thus, identifying the moment when concept drift occurs is required. Concept drift is identified through concept drift detectors. In this work, we assess the reliability of concept drift detectors to identify drift in time by exploring how late are they reporting drifts and how many false alarms are they signaling. We compare the performance of the most popular drift detectors belonging to two different concept drift detector groups, error rate-based detectors and data distribution-based detectors. We assess their performance on both synthetic and real-world data. In the case of synthetic data, we investigate the performance of detectors to identify two types of concept drift, abrupt and gradual. Our findings aim to help practitioners understand which drift detector should be employed in different situations and, to achieve this, we share a list of the most important observations made throughout this study, which can serve as guidelines for practical usage. Furthermore, based on our empirical results, we analyze the suitability of each concept drift detection group to be used as alarming system.