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Live streaming media are delay-sensitive, and their end-to-end allowable delay is an important factor. The proposed QMLS protocol has focused on an allowable delay, and QMLS routers placed on the multicast tree detect and recover lost packets. These functions reduce retransmission delays, and therefore, the number of packet losses caused by delays exceeding the allowable delay is reduced. However, when the traffic conditions of a certain link on the multicast tree get worse and a bottleneck develops, even the QMLS protocol cannot recover the QoS. Therefore, we propose an efficient path reconfiguration method for a QMLS multicast tree to avoid this inappropriate link. We consider not only the delay on the path but also the number of QMLS routers for path selection. Our method provides a multicast tree, which has tolerance to variations in traffic after path reconfiguration.