Abstract—This paper considers energy harvesting wireless sensor networks (EH-WSN) with multiple sinks supporting concurrent applications. Each application is associated with a set of sensor nodes that generate and send traffic to the associated application sink. Each node can relay any application traffic toward the application sink. In addition, each node uses an energy management unit to control the amount of traffic the node can relay based on its available energy. With the nodes’ energy levels fluctuations, it is essential to quantify the network's ability to fulfill the different applications’ quality of information and service requirements. Therefore, a novel multicommodity flow reliability problem (called MultiFlowRel) is formalized to estimate the likelihood that at least a certain amount of each application traffic is delivered to the associated application sink. The proposed problem is proven to be #P-hard, and an iterative bounding framework is proposed for deriving lower bounds on the exact reliability solutions. The proposed framework compute’s exact reliability solutions if allowed a sufficient number of iterations. Numerical results show the effectiveness of using the proposed solution to obtain good lower reliability bounds and exact solutions in reasonable running times. Furthermore, the results show examples of the use of the proposed framework in solving some interesting network design problems (e.g. optimal sink locations and appropriate transmission parameters).