In this paper, throughput of hybrid automatic repeat request (HARQ) schemes is studied in the presence of Markovian data arrivals and statistical queuing constraints. In particular, two queuing models are considered. Specifically, when outage occurs, the transmitter keeps the packet, lowers its priority, and attempts to retransmit it later in the first queue model while the packet is discarded and removed from the buffer in the second queue model. The throughput is investigated when outage constraints, statistical queuing constraints and deadline constraints are imposed. The deadline constraint provides a limitation on the number of retransmissions. Under these assumptions, throughput characterizations are obtained for HARQ chase combining (CC) scheme with three types of Markovian sources, namely the ON-OFF discrete-time and fluid Markov sources and Markov modulated Poisson source (MMPS). Our analytical results are verified via Monte Carlo simulations. In the numerical results, the impact of source randomness, deadline constraints, outage probability and queuing constraints on the throughput is analyzed.