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In our earlier works, we proposed rate adaptive modulation assisted two-user opportunistic scheduling schemes. The proposed schemes are innovative in the sense that they include second user in a transmission opportunistically using rate adaptive hierarchical modulations. As such the frequency of information access of the users increases without any degradation of the system link spectral efficiency (SE) compared to the classical opportunistic scheduling. In this paper, we analyze delay performance of our earlier proposed schemes. Specifically, using a queuing analytic model we derive queuing delay as well as buffer distributions of the packets that wait at the base station (BS) buffer for downlink (DL) transmission. Using this model, we compare performance of the two-user opportunistic scheduling schemes with that of the classical single user opportunistic schemes. These single user opportunistic scheduling schemes are the absolute carrier-to-noise ratio (CNR) based scheduling and normalized CNR based scheduling which is equivalent to the proportional fair scheduling (PFS). For an interdependent and identically distributed (i.i.d.) fading environment, our proposed scheme can improve delay performance significantly. Selected numerical results in an independent but non-identically distributed (i.n.d.) fading environment, our proposed scheme achieves overall good delay performance.