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Wireless sensor networks have been used for many delay-sensitive applications, e.g., emergency response and plant automation. In such networks, delay measurement is important for a number of reasons, e.g., real-time control of the networked system, and abnormal delay detection. In this paper, we propose a measurement architecture using distributed air sniffers, which provides convenient delay measurement, and requires no clock synchronization or instrumentation at the sensor nodes. One challenge in deploying this architecture is how to place the sniffers for efficient delay measurement. We prove the sniffer placement problem is NP-hard and develop two algorithms to solve it. Using a combination of small-scale testbed experiments and large-scale simulation, we demonstrate that our architecture leads to accurate delay monitoring and is effective in detecting abnormal delays, and furthermore, the number of sniffers required by our sniffer placement algorithms is close to the minimum required value.