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It is well known that cross-layer scheduling can boost the spectral efficiency of multi-user OFDMA systems through multi-user selection diversity but existing designs usually have two important assumptions - users are delay-insensitive and channel state information at the transmitter (CSIT) is perfect. In practice, users have heterogeneous delay requirements and CSIT usually becomes outdated in time varying channel, which in turns leads to systematic packet errors and hence results in significant degradation on the throughput and delay performance in the OFDMA systems. In this paper, a novel cross-layer design problem is formulated as a convex optimization problem, on which a delay-sensitive jointly optimal power, rate and subcarrier allocation scheme is proposed so as to maintain heterogeneous users' delay requirement as well as achieving a target packet outage probability through combining queueing theory and information theory. Furthermore, we obtain closed-form asymptotic performance of the proposed delay-sensitive scheduler. Unlike the well-known SNR gain of Theta(log - sigma2 DeltaH) in conventional cross-layer scheduler with perfect CSIT, we demonstrate a cross-layer SNR gain of Theta((1 - sigma2 DeltaH)log K)can still be achieved under heterogeneous delay constraints and outdated CSIT with error variance sigma2 DeltaH. Finally, simulation results show that our proposed delay-sensitive CSIT error considerate schemes provide robust system performance enhancement over conventional CSIT error inconsiderate opportunistic scheduler and naive queue length based MAX-Weight scheduler while satisfying heterogeneous delay requirements even at moderate to high CSIT errors.