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Multi-sink wireless sensor networks (WSNs) are being increasingly deployed in an ever-widening range of application scenarios, especially as they are reliable and exhibit low power consumption. Providing a backhaul for WSN traffic has become an important issue because the sensor data must usually be sent to the Internet or the core network. Passive optical networks (PONs) represent one next-generation access network technology which is appropriate for such a backhaul, however existing research appears to have concentrated on either WSN performance or PON performance, without considering the overall performance of both networks converged together. This paper proposes a new architecture which converges multi-sink WSNs and PONs, then provides a novel queuing-theory analysis of the converged network performance. Results from this analytical model are then used to motivate a new a DBA algorithm which optimizes grant size allocation. Numerical results show that this algorithm outperforms existing proposals when minimizing the system queue length in the converged network, while providing shorter end-to-end packet delay and higher throughput. We believe that this first resource allocation algorithm which considers the performance of both networks as one converged unit.