As an inexpensive and scalable access network to the Internet, we have extensively studied the Wireless Internet-access Mesh Network (WIMNET) that is composed of multiple access points (APs) connected through wireless links. Because most traffic in WIMNET passes through the gateway (GW) to the Internet, the links around the GW are much crowded so that they should be activated with higher priorities than others. In this paper, we propose the Fixed Back off-time Switching (CSMA-FBS) method for WIMNET using the CSMA/CA protocol to improve the performance by giving necessary activation chances to such links. Before starting communications, the target link activation rate and the active/passive back off-times for collision avoidance are calculated from the expected traffic load for each link. During communications, these back off-times are switched by comparing the actual link activation rate with the target one. By assigning different fixed values to these back off-times, the conflicts among the interfered links can be avoided. Through extensive simulations using two types of network topologies, we verify the effectiveness of our proposal.