We propose an efficient scheme for generating fake network traffic to disguise the real event notification in the presence of a global eavesdropper, which is especially relevant for the quality of service in delay-intolerant applications monitoring rare and spatially sparse events, and deployed as large wireless sensor networks with single data collector. The efficiency of the scheme that provides statistical source anonymity is achieved by partitioning network nodes randomly into several node groups. Members of the same group collectively emulate both temporal and spatial distribution of the event. Under such dummy-traffic framework of the source anonymity protection, we aim to better model the global eavesdropper, especially her way of using statistical tests to detect the real event, and to present the quality of the location protection as relative to the adversarys strength. In addition, our approach aims to reduce the per-event work spent to generate the fake traffic while, most importantly, providing a guaranteed latency in reporting the event. The latency is controlled by decoupling the routing from the fake traffic schedule. We believe that the proposed source anonymity protection strategy, and the quality evaluation framework, are well justified by the abundance of the applications that monitor a rare event with known temporal statistics, and uniform spatial distribution.