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The inherent measurement support in routers (SNMP counters or NetFlow) is not sufficient to diagnose performance problems in IP networks, especially for flow-specific problems where the aggregate behavior within a router appears normal. Tomographic approaches to detect the location of such problems are not feasible in such cases as active probes can only catch aggregate characteristics. To address this problem, in this paper, we propose a Consistent NetFlow (CNF) architecture for measuring per-flow delay measurements within routers. CNF utilizes the existing NetFlow architecture that already reports the first and last timestamps per flow, and it proposes hash-based sampling to ensure that two adjacent routers record the same flows. We devise a novel Multiflow estimator that approximates the intermediate delay samples from other background flows to significantly improve the per-flow latency estimates compared to the naive estimator that only uses actual flow samples. In our experiments using real backbone traces and realistic delay models, we show that the Multiflow estimator is accurate with a median relative error of less than 20% for flows of size greater than 100 packets. We also show that Multiflow estimator performs two to three times better than a prior approach based on trajectory sampling at an equivalent packet sampling rate.