## 1 Introduction  
Ultra-Reliable Low-Latency Communication (URLLC) is expected to be an important feature of 5G and beyond [1, 2, 3]. Typically, the data payload of URLLC is small but the latency or delay and reliability requirements are stringent. For instance, in an example given by 3 GPP [1], the data packet has a size of 32 bytes, and the reliability and latency requirements are that the packet is transmitted and received at the user/application plane within 1 ms with success rate \(1-10^{-5}\). In fact, there is a broad range of scenarios where even more stringent latency and reliability guarantees are needed [3]. In the literature, a lot of results, covering different aspects of URLLC, have emerged. Excellent reviews, e.g., on physical layer [4], coding [5], access control [6] and network level [7] and journal special issues are available, e.g., [8]. However, the existing results, when treating the reliability requirement, take the error contribution due to finite length coding [4, 5] and the delay violation probability due to queueing [6, 7, 8] separately or indeed often ignoring the other part. In addition, few of these results take into account the impact of data duplication, which is an important technique proposed for URLLC to increase packet success rate while keeping the latency low [9]. Furthermore, a crucial aspect for URLLC to be implemented in real systems has surprisingly little been touched, which is the selection of a modulation and coding scheme (MCS). To the best of our knowledge, refs. [10, 11] are the only attempts investigating how MCS selection should be considered in URLLC. More specifically, the authors of [10, 11] proposed to measure the error probability of packet transmission and accordingly adapt the select of MCS. A fundamental limitation of such an approach is that, with a targeted error probability of \(10^{-5}\) or lower, for the measurement to be meaningful, it must measure at least 1 million packets. Unfortunately, this implies the measurement time span will be in the order of minutes or hours (e.g., 1 packet per 1 ms) or even longer, when the channel condition may have changed significantly. This motivates us to take on a different approach where the error probability can be directly estimated from channel quality measures, e.g., Signal-to-Noise Ratio (SNR) that can be easily obtained. The objective of this paper is to, through analysis, reveal a connection between the URLLC \((k,d,\epsilon)\)-tuple, where \(k\) denotes the data packet length, \(d\) the latency requirement and \(\epsilon\) the reliability requirement and channel modulation, coding length and quality. In addition, this connection is extended to consider data duplication in the transmission. This connection is finally expressed via inequations, which can be used in many aspects, such as to estimate the maximum delay, to allocate minimum bandwidth and to find the most suitable modulation and coding scheme. In this work, the results of optimal MCS selection and spectrum allocation for URLLC are presented and discussed. These results will not only be useful for MCS selection and spectrum allocation, but also add to the literature, providing additional insights into quantifying and understanding the fundamental limits of URLLC. Specifically, we focus our study on a system with block fading channel using Quadrature Amplitude Modulation (QAM), where the coding length effect, due to small data packet size and low latency requirement, cannot be ignored. Our contributions are several-fold. First, by extending the channel dispersion results [12; 13] to the QAM channel, the error probability due to finite block length coding is estimated. Second, queueing analysis is conducted based on the network calculus theory particularly the stochastic branch [14], which has been identified as an important analytical tool for URLLC [7], to reveal the relation between delay requirement and its violation probability. Third, the effect of using data duplication for URLLC is investigated.