# User-Perceived Capacity: Theory, Computation, and Achievable Policies
## 1 Introduction
Specifically, Lyapunov optimization and Markov decision processes are commonly used in cross-layer designs to enhance system throughput and reduce latency. Inspired by the existing literature on cross-layer scheduling approaches [40; 41; 42; 43], researchers utilized Lyapunov optimization to describe data arrivals at the network layer and channel state information at the physical layer, thereby exploring power-constrained cross-layer scheduling for low-latency wireless communications. In [44], the authors introduced online dynamic control algorithms using Lyapunov optimization theory for multi-hop networks. In [45], [PERSON] provided a systematic summary of Lyapunov optimization theory, focusing primarily on communication and queuing systems. Building upon this, refs. [46; 47; 48] demonstrated the throughput optimality of the Lyapunov drift method under the maximum weight rule, the logarithmic rule, and the exponential rule in multi-user scenarios over time-varying wireless channels. Subsequently, the Lyapunov drift method has been widely applied in packet switching [49; 50; 51], queuing networks [52], wireless communication [53; 54], and satellite communication systems [55]. When the system exhibits Markovian properties, a Markov decision process (MDP) can be employed to achieve a more rigorous analysis and optimization for metrics such as delay and power in wireless communication systems [56; 57]. [PERSON] and [PERSON] conducted an analysis on the average delay and power for a single-user scenario over block-fading channels in discrete-time systems [58]. Building upon this, ref. [59] studied the trade-off among the average power, delay, and the packet drop ratio in a single-user link over fading channels. It provided deterministic policies for unconstrained problems and probabilistic policies for constrained problems using value iteration and linear programming, respectively. Refs. [60; 61] further investigated the trade-off between delay and power for random arrivals in AWGN and fading channels based on the constrained MDP (CMDP) framework. Specifically, the authors showed structural properties of the optimal scheduling policy, such as a threshold-based structure and piecewise linearity, and consequently developed an efficient algorithm to search for the optimal delay-power tradeoff. As an extension of [60; 61], ref. [62] considered a Markovian arrival process, and showed similar threshold and piecewise linear characteristics for the optimal scheduling policy. Under a CMDP framework with cross-layer designs, refs. [63; 63] presented the optimal delay-power trade-off for two-user systems over AWGN and block-fading channels, respectively. In the aforementioned references, the studies from [40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64] focus on the research of cross-layer design and do not conduct specific studies on real-time communication. Although the references from [4; 4;transmitting B units of data within a deadline T. The studies from [24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39], while focusing on the BT problem, do not take into account the user-perceived capacity, that is, they do not consider how to meet the requirement for the continuous reading of the user. Based on this, this paper focuses on real-time communication, utilizing the framework of cross-layer design to improve the user-perceived capacity under the constraint of average power. This paper primarily investigates single-user communication systems over wireless fading channels. The objective is to maximize the user-perceived capacity under the constraint of average power. Specifically, we consider periodic data arrivals, with B packets of data arriving at the beginning of each period. The user reads the received data at a certain rate at the receiver. To satisfy the user-perceived capacity, we need to transmit B units of data within a deadline T. In particular, the main contributions of the paper are as follows.