Monitoring network status such as end-to-end delay, jitter, and available bandwidth is important to support QoS-sensitive applications and timely detection of network anomalies like Denial of Service attacks. For this purpose, Internet Service Providers (ISPs) have started to instrument their networks with Network Measurement Infrastructures (NMIs) that periodically run active measurement tasks using measurement servers located at strategic points in their networks. However, one problem that most network engineers have overlooked is the measurement conflict problem. Since active measurement tasks actively inject test packets to collect measurements along network paths, running multiple active measurements at the same time over the same path could result in misleading reports of network performance. We call this phenomenon a measurement conflict. Our recent observation of such measurement conflict motivates us to form a measurement task scheduling problem of meeting periodicity requirements, where real-time scheduling algorithms can play a role. The scheduling problem, however, is not exactly same as any of the existing scheduling problems in the realtime literature, because the problem involves multiple measurement servers running multiple measurement tasks whose conflict dependency propagates along the chains of paths. For this problem, we propose to use an EDF (Earliest Deadline First) heuristic but allowing "Concurrent Executions" if possible, to construct an offline schedule for a given measurement task set. Also, we propose a novel mechanism to flexibly use the offline schedule for minimizing the response time of dynamic on-demand measurement jobs. Further, we implement and deploy our scheduling algorithms in a real working NMI for monitoring Internet 2 Abilene network.