We consider futuristic, intelligent reflecting surfaces (IRS)-aided communication between a base station (BS) and a user equipment (UE) for two distinct scenarios: a single-input, single-output (SISO) system whereby the BS has a single antenna, and a multi-input, single-output (MISO) system whereby the BS has multiple antennas. For the considered IRS-assisted downlink, we compute the effective capacity (EC), which is a quantitative measure of the statistical quality-of-service (QoS) offered by a communication system experiencing random fading. For our analysis, we consider the two widely-known assumptions on channel state information (CSI) -- i.e., perfect CSI and no CSI, at the BS. Thereafter, we first derive the distribution of the signal-to-noise ratio (SNR) for both SISO and MISO scenarios, and subsequently derive closed-form expressions for the EC under perfect CSI and no CSI cases, for both SISO and MISO scenarios. Furthermore, for the SISO and MISO systems with no CSI, it turns out that the EC could be maximized further by searching for an optimal transmission rate $r^*$, which is computed by exploiting the iterative gradient-descent method. We provide extensive simulation results which investigate the impact of the various system parameters, e.g., QoS exponent, power budget, number of transmit antennas at the BS, number of reflective elements at the IRS etc., on the EC of the system.