Correctness of a real-time system depends on its computation as well as its timeliness. In recent years, research has been focusing on verifying the correctness of a real-time system during runtime by monitoring its runtime execution and checking it against its formal specifications. Such verification method is called Runtime Verification. While a few existing runtime verification tools verify both computational correctness and timeliness correctness, those that provide timeliness correctness fail to detect timeliness violations as soon as violations occur. In this paper, we investigate the verification of timeliness correctness by providing quantitative property specifications, address the problem why those tools fail to detect as soon as violations occur, provide an efficient solution, and present how to implement it in Real-Time Java.