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We investigate the packet-level and the connection-level performances in a power-constrained (e.g., solar-powered) wireless mesh router with packet scheduling and admission control. The system model under consideration is compatible with the IEEE 802.16a standard with orthogonal frequency division multiple access (OFDMA) air interface. In the medium access control (MAC) layer, transmission frames are grouped into super-frames and the allocation of subchannel and number of frames to each connection (i.e., scheduling) is performed to satisfy the traffic requirement of each connection and also the power supply constraint at a mesh router/802.16a base station (BS). A queueing analytical model based on discrete-time Markov chain (DTMC) is used to analyze the packet-level performances. Based on this scheduling, router capacity in terms of the maximum number of ongoing connections is obtained. Subsequently, a threshold-based admission control method is proposed for both relay and local connections in a router so that connection-level performances are satisfied. To this end, we optimize the admission control over multiple time periods in which the amount of supplied power and the traffic load at a mesh router are time-dependent.