In a wireless network, the efficiency of scheduling algorithms over time-varying channels depends heavily on the accuracy of the Channel State Information (CSI), which is usually quite ``costly'' in terms of consuming network resources. Scheduling in such systems is also subject to stringent constraints such as power and bandwidth, which limit the maximum number of simultaneous transmissions. In the meanwhile, communication channels in wireless systems typically fluctuate in a time-correlated manner. We hence design schedulers to exploit the temporal-correlation inherent in channels with memory and ARQ-styled feedback from the users for better channel state knowledge, under the assumption of Markovian channels and the stringent constraint on the maximum number of simultaneously active users. We model this problem under the framework of a Partially Observable Markov Decision Processes. In recent work, a low-complexity optimal solution was developed for this problem under a long-term time-average resource constraint. However, in real systems with instantaneous resource constraints, how to optimally exploit the temporal correlation and satisfy realistic stringent constraint on the instantaneous service remains elusive. In this work, we incorporate a stringent constraint on the simultaneously scheduled users and propose a low-complexity scheduling algorithm that dynamically implements user scheduling and dummy packet broadcasting. We show that the throughput region of the optimal policy under the long-term average resource constraint can be asymptotically achieved in the stringent constrained scenario by the proposed algorithm, in the many users limiting regime.