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In wireless sensor networks (WSNs), congestion will cause packet loss which in turn wastes energy and reduces the lifetime of WSNs, and therefore congestion in WSNs must be controlled or avoided in either a fairness or a weighted-fairness way. It is very important to achieve weighted fairness for many WSN applications, and this problem becomes more complicated when the data flow is forwarded to multiple routing paths. In this paper we propose a joint priority-based algorithm (JPA) that eliminates congestion and achieves weighted fairness in multi-path and multi-hop wireless sensor networks. Weighted fairness is achieved when the source node with high source priority (SP) sends more packets than the one with low SP in response to congestion. JPA defines a new variable, joint priority (JP) for each node and link, as the expected value of SP. The JP of a node or link indicates the arithmetic means of SP of source nodes whose data flow passes through that particular node or link, and the sending rate of each node is adjusted based on the value of JP when congestion occurs. The JPA algorithm is simulated and evaluated in different scenarios.