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In a typical sensor network, nodes around the sink consume more energy than those further away. It is not unusual that limited energy resources available at the nodes around the sink become the bottleneck which confines the performance of the whole network. In this letter, we firstly present our considered bottleneck zone in a general sensor network scenario. Then, the effect of the bottleneck zone on network performance is investigated by deducing performance bounds imposed by the energy resources available inside the bottleneck zone. In this letter, both the performance bound in terms of network lifetime and the performance bound in terms of information collection are explored. Finally, the ways by which network deployment variables may affect the performance bounds are analyzed.