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We study the delay performance of a sensor network, whose nodes access the medium by using the unslotted MAC protocol specified by the IEEE 802.15.4 standard. Unlike previous works, which focus on the average throughput and delay analysis, we develop a detailed model that allows us to obtain the delivery delay distribution of messages sent by concurrently contending sensors toward a central controller. We carry out a transient analysis that is of particular interest when sensor networks are deployed to provide k-coverage for real-time applications, and we study both single- and multi-hop network topologies. We validate our analytical results against simulation results obtained through ns2.