Hybrid Automatic Repeat Request (HARQ) is a high performance communication protocol, leading to effective use of the wireless channel and the resources with only limited feedback about the channel state information (CSI) to the transmitter. In this paper, the throughput of HARQ with incremental redundancy (IR) and fixed transmission rate is studied in the presence of quality of service (QoS) constraints imposed as limitations on buffer overflow probabilities. In particular, tools from the theory of renewal processes and stochastic network calculus are employed to characterize the maximum arrival rates that can be supported by the wireless channel when HARQ-IR is adopted. Effective capacity is employed as the throughput metric and a closed-form expression for the effective capacity of HARQ-IR is determined for small values of the QoS exponent. The impact of the fixed transmission rate, QoS constraints, and hard deadline limitations on the throughput is investigated and comparisons with regular ARQ operation are provided.