With the introduction of the Integrated Access and Backhaul (IAB) technology by 3GPP, the last mile in beyond 5G cellular systems will be characterized by multiple wireless backhaul links between the user equipment and base station. In such multi-hop configurations, packet latency becomes a dominant factor characterizing the quality of service (QoS) delivered to the users. At the same time, directional communications at millimeter wave (mmWave) frequencies are prone to frequent outages caused by blockage and micromobility leading to periods of silence and subsequent rate compensation. In this paper, we are interested in assessing the impact of mmWave propagation specifics on the delay and medium access control (MAC) buffer overflow performance at a 5G NR IAB network node. To this aim, we first propose a continuous-time Markov chain model for the bitrate provided by such a system that explicitly accounts for mmWave-specific propagation. Then, by formulating a queuing model of packet transmission, we characterize the mean packet delay experienced by a traffic source. Our results illustrate that the mean delay and MAC buffer overflow probability at the NR BS interface is a step function that is not strictly decreasing due to the packet encapsulation specifics of the modulation and coding (MCS) adaptation mechanism. Further, these metrics are non-negligible even for the system having a sufficient amount of resources. For IAB architecture, where the access link may not be available the the mean delay increases exponentially even when the arrival rate is adjusted to the available resources.