# User-Perceived Capacity: Theory, Computation, and Achievable Policies
## 1 Introduction
However, there is also a considerable amount of research that concentrates on soft real-time communications, specifically addressing the issue of delay violation probabilities over wireless channels [18]. Refs. [19; 20] contributed to the field by formulating a CMDP to minimize the queue-length-bound violation probability, utilizing linear programming to obtain the optimal probabilistic policy, balancing violation probability with transmission power. The authors of [21] proposed a framework for cross-layer optimization to ensure the required quality-of-service with finite transmit power, considering the transmission error probability, queueing delay violation probability, and packet dropping probability. [PERSON] et al. proposed a URLLC-centric task offloading and resource allocation framework, incorporating the statistical characteristics' extreme queue length events [22]. In [23], the authors focused on reducing packet delay violation probabilities within periodic traffic models and, by utilizing Markov decision processes and nonlinear Knapsack Problems, presented an asymptotically optimal policy. In the context of real-time communication, wireless networks are responsible for transmitting substantial amounts of data in a specified time frame, with an increasing emphasis on ensuring high service quality. A significant portion of data has strict latency requirements, which arise from various application scenarios involving real-time communication, such as autonomous driving, health monitoring, and industrial automation. To meet these latency requirements, data transmissions must be completed within a given deadline constraint. Consequently, [PERSON] and [PERSON] proposed a BT problem in [24], addressing the challenge of transmitting B units of data within a deadline T. They employed continuous-time stochastic control theory to derive the optimal policy for the BT problem. In addition, [PERSON] et al. provided an overview of research related to packet delay and energy consumption in [25], including the BT problem and delay-constrained issues. For discrete-time point-to-point communication systems, refs. [26; 27] investigated the optimal data transmission for BT problems under constraints such as random data arrival and Quality of Service (QoS), without considering time-varying channels. To specifically address the impact of time-varying channel conditions on the BT problem, several studies [11; 28; 29] extended the BT problem from static channels to fading channels. Furthermore, studies [24; 30; 31; 32] have considered the BT problem in continuous-time systems, exploring the optimal data transmission policies in the presence of time-varying channels. In addition to single-user scenarios, the academic community has also conducted research on multi-user scenarios. For instance, the authors of [33] investigated the BT problem in a time-shared multi-user system by considering random arrivals and time-varying channels for multiple users. Furthermore, study [34] extended the BT problem to multi-access scenarios and examined power minimization scheduling with deadline constraints in a multi-access channel environment. Moreover, under the framework of the BT problem, studies such as [35; 36; 37; 38; 39] further explored deadline-constrained communications in scenarios such as multi-hop networks, computation offloading, lossy networks, and multi-robot task allocation, providing optimal scheduling policies. Cross-layer design is widely recognized as a crucial methodology for ensuring real-time performance for delay-constrained services. Notably, [PERSON] and [PERSON] pioneered the integration of network and physical layer designs to optimize low-latency wireless transmissions, resulting in the formulation of a cross-layer scheduling framework [40]. After that, there has been an increasing focus on interactions across different layers [41; 42; 43], leading to the proposal of various approaches for cross-layer scheduling in diverse application scenarios.