This letter presents two results for multiuser wireless systems employing dirty-paper coding strategies along with greedy scheduling over the broadcast multiple-input multiple-output channel. Specifically, an efficient and suboptimal downlink scheduler is proposed to approximate the maximum sum rate using equal power allocation, and it is shown to approach the maximum sum rate of optimal power allocation for a large number of users under optimal scheduling. The second result demonstrates that the average maximum sum rate can be tightly upper bounded when spatial multiplexing is maximized.