## 1 Introduction
The authors in [22] re-evaluated frame design principles involving finite blocklength coding in cellular wireless systems. In [23], the scheduling of downlink non-orthogonal multiple access (NOMA) was examined in the context of short-packet communications. The work in [24] proposed a fast HARQ protocol, substantially improving throughput, delay, and error probability with finite blocklength coding. Numerous studies have implemented finite blocklength coding in low-latency communication systems. As demonstrated in [25], there exists a significant disparity between the current channel coding for URLLC and the standard approximation of finite blocklength coding [26]. Extensive research efforts have examined the design of low-latency transmission using finite blocklength coding in single-user systems. For instance, [PERSON]. [27] proposed an adaptive blocklength transmission framework and two novel schemes to optimize the end-to-end delay in 6G networks. Refs. [28, 29] have investigated finite blocklength coding in low-latency transmission systems. The exploration of low-latency transmission design using finite blocklength coding has extended to multiple access systems as well. A multi-access transmission design utilizing finite blocklength coding has been examined in low-latency vehicular communications, rate-splitting multiple access, and Internet of Things systems [30, 31, 32, 33, 34]. Furthermore, the authors in [35, 36] have investigated the variable coding policy used in low-latency communications. The aforementioned work [4, 5, 6, 7, 8, 13, 14, 15, 16] adopts an common assumption that the transmission power and rate analysis is based on the classical Shannon capacity formula. This as sumption is valid only when the channel coding adopts an extremely long blocklength [17]. Nonetheless, the future 6G system is expected to support a wide range of services for emerging applications, such as autonomous driving and industrial automation, where the blocklength is short and sometimes subject to hard delay and reliability constraints [3]. This implies that the long codewords required by the Shannon capacity formula may become impractical. Consequently, the previous designs in [4, 5, 6, 7, 8, 13, 14, 15, 16] are unsuitable for these 6G applications. In the previously mentioned research [3, 22, 23, 24], although finite blocklength coding is utilized, its application to the scheduling of communications with stringent delay constraints is not considered. While [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] have investigated the use of finite blocklength coding in low-latency communications, they do not adequately address the enforcement of hard delay constraints. These oversights render the aforementioned studies ill-suited for real-time communication systems. In our previous work [37], we focused on optimizing the average delay in short-packet transmission systems. However, in URLLC, we not only consider the average delay but also ensure that all data packets do not exceed a hard delay constraint [38]. Therefore, in this paper, we delve into hard-delay-constrained communication in short-packet transmission systems, utilizing variable-length coding. In conventional URLLC systems, the focus has primarily been placed on minimizing the average delay or outage probability. However, these studies have not addressed the issues pertaining to hard delay constraints and finite blocklength coding. To this end, we propose a cross-layer design policy tailored to short-packet transmission systems with hard delay constraints. In particular, we encode randomly arriving packets via variable-length coding and transmit the encoded symbols by truncated channel inversion over parallel channels, with the channel-side information at the transmitter (CSIT). Furthermore, we model the data queue and symbol queue as a two-dimensional Markov chain.