The rise of smart applications in wireless devices increasingly relies on mobile edge computing (MEC), where longterm system energy efficiency holds crucial significance for both green computing and application vendors. This paper focuses on long-term energy efficiency in a wireless power transferenabled MEC system. This system faces the challenges of timevarying channel states and stochastic task arrivals. We first formulate this problem to simultaneously optimize offloading, power transfer duration, and energy consumption, while ensuring device queue stability. We then introduce a novel algorithm based on Lyapunov-guided deep reinforcement learning, referred to as LyCNN-DRL. This approach efficiently handles the mixed integer non-linear programming problem by transforming it into a deterministic per-slot problem for online optimization, without needing prior knowledge of future conditions. Specifically, we tackle the problem by dividing it into resource allocation and binary offloading components, applying a convolutional neural network model for near-optimal offloading decisions, and obtaining the optimal solution for resource allocation. Simulation results show that LyCNN-DRL outperforms baseline algorithms, stabilizing MEC network task queues. Furthermore, we quantitatively derive the trade-off between energy efficiency and queue length, represented as$[O(1/V), O(V)]$with the variable$V$.

The rise of high-resolution cameras and deep learning models has propelled video analytics but also poses new challenges, especially in crowded outdoor scenes. Processing high-resolution video frames demands substantial computational resources, often surpassing edge device capabilities, resulting in high latency and energy costs. Detecting small objects in crowded regions further hinders accuracy in critical applications such as traffic surveillance and monitoring. To address these issues, we propose PAROD, a real-time system for high-resolution object detection in crowded outdoor environments. PAROD integrates three key technologies: a background modeling-based adaptive frame partitioning algorithm that dynamically identifies regions of interest (RoIs), parallel offloading of partitioned RoIs to multiple edge servers for inference, and an object counting model to detect crowded regions and enable differentiated inference strategies. By strategically allocating simpler models to ordinary regions and more complex models to crowded regions, PAROD enhances accuracy while minimizing latency. Furthermore, by resizing the input partitions to a fixed size before inference, PAROD reduces latency fluctuations and optimizes overall performance. Evaluations on a public dataset show that PAROD achieves a 3.4× speed-up over traditional edge offloading methods, with only a 1% accuracy reduction, offering a scalable solution for real-time, high-resolution video analytics in crowded outdoor scenes.

Virtual reality (VR) applications have revolutionized digital interaction by providing immersive experiences. 360$^{\circ }$VR video streaming has experienced significant growth and popularity as a pivotal VR application. However, the combination of limited network bandwidth and the demand for high-quality videos frequently hinders the achievement of a satisfactory quality of experience (QoE). Although prior methods have enhanced QoE, the effects of decoding latency have been poorly studied. It is technically challenging to design a quality adaptation algorithm that can balance the pursuit of high-quality videos and the limitation of limited bandwidth resources. To address this challenge, we propose an edge-end architecture for 360$^{\circ }$VR video streaming and aim to enhance overall QoE by solving a performance optimization problem. Specifically, our experiments on commercial mobile devices in real-world situations reveal that decoding latency significantly influences QoE. First, decoding latency plays a major role in contributing to end-to-end latency, which exceeds the transmission latency. Second, decoding latency can differ considerably between devices with varying computational capabilities. Building on this insight, we propose a novellatency-awarequalityadaptation (LAQA) algorithm. LAQA lies in developing a solution that can allocate video quality in real-time and enhance overall QoE. LAQA involves not only the quality of the received content, the transmission latency and the quality variance, but also the decoding latency and the fairness of the user quality. Subsequently, we formulate a combinatorial optimization problem to maximize overall QoE. Through extensive validation with experimental data from real-world situations, LAQA offers a promising approach to enhance QoE and ensure fairness performance in different devices. In particular, LAQA achieves 16.77% and 10.66% enhancement over the state-of-the-art combinatorial optimization and reinforcement learning algorithm, respectively, in terms of QoE at 4K resolution. Furthermore, LAQA ensures excellent scalability by simulating the number of users ranging from 15 to 60, making it a robust solution for diverse and growing user scales.

Non-orthogonal multiple access (NOMA) emerges as a superior technology for enhancing spectral efficiency, reducing latency, and improving connectivity compared to orthogonal multiple access. In NOMA networks, successive interference cancellation (SIC) plays a crucial role in decoding user signals sequentially. The challenge lies in the joint optimization of SIC ordering and power allocation, a task complicated by the factorial nature of ordering combinations. This study introduces an innovative solution, the Attention-based SIC Ordering and Power Allocation (ASOPA) framework, targeting an uplink NOMA network with dynamic SIC ordering. ASOPA aims to maximize weighted proportional fairness by employing deep reinforcement learning, strategically decomposing the problem into two manageable subproblems: SIC ordering optimization and optimal power allocation. We use an attention-based neural network to process real-time channel gains and user weights, determining the SIC decoding order for each user. A baseline network, serving as a mimic model, aids in the reinforcement learning process. Once the SIC ordering is established, the power allocation subproblem transforms into a convex optimization problem, enabling efficient calculation of optimal transmit power for all users. Extensive simulations validate ASOPA’s efficacy, demonstrating a performance closely paralleling the exhaustive method, with over 97% confidence in normalized network utility. Compared to the current state-of-the-art implementation, i.e., Tabu search, ASOPA achieves over 97.5% network utility of Tabu search. Furthermore, ASOPA has two orders of magnitude less execution latency than Tabu search when$N=10$and even three orders magnitude less execution latency less than Tabu search when$N=20$. Notably, ASOPA maintains a low execution latency of approximately 50 milliseconds in a ten-user NOMA network, aligning with static SIC ordering algorithms. Furthermore, ASOPA demonstrates superior performance over baseline algorithms besides Tabu search in various NOMA network configurations, including scenarios with imperfect channel state information, multiple base stations, and multiple-antenna setups. These results underscore the robustness and effectiveness of ASOPA, demonstrating its ability to ability to achieve good performance across various NOMA network environments.

This paper addresses long-term energy efficiency in a wireless power transfer-enabled mobile-edge computing (MEC) system, facing challenges from time-varying channels and stochastic task arrivals. We formulate the problem to optimize offloading, power transfer duration, and energy consumption while ensuring queue stability. We propose a novel Lyapunov-guided deep reinforcement learning (LyCNN-DRL) algorithm to efficiently solve the long-term mixed integer non-linear programming problem without prior knowledge of future conditions. The approach decomposes the problem into resource allocation and binary offloading components, using a convolutional neural network for near-optimal offloading decisions and the Lagrange dual function for optimal resource allocation. Extensive simulations show that LyCNN-DRL outperforms benchmark algorithms in energy efficiency and latency, achieving over 97% of the optimal utility while reducing execution latency to approximately 50 milliseconds in ten-WD networks. Additionally, we derive the trade-off between energy efficiency and queue length as [O(1/V),O(V)], where V is the Lyapunov control parameter.

For real-time monitoring applications, the age of information (AoI) is used as a key metric to quantify the freshness of updated information. In this paper, we consider the wireless powered networks where multiple source nodes observe processes and send update packets to the base station. Time is divided into slots which are equal duration. At each slot, either wireless energy transfer or packet update via non-orthogonal multiple access (NOMA) communication is scheduled. We aim to minimize the long-term average weighted sum of AoI of processes at the base station. Particularly, we formulate the AoI minimization problem as a multi-stage stochastic non-linear integer programming subject to the battery energy constraints. By adopting the Lyapunov optimization, we decouple the multi-stage stochastic problem into perframe deterministic subproblems and solve it with a low computational complexity algorithm. Simulation results show that our proposed scheme can achieve much smaller average weighted AoI than the benchmark algorithm.

The Internet of Things (IoT) is a highly integrated application of the advanced information technology, which is expected to bring convenience for daily life and improve the efficiency of industrial production. Owing to the limitation of battery capacity and the broadcast nature of IoT nodes, IoT networks face the bottleneck of energy shortage and security vulnerability. In recent years, the emerging technologies of wireless powered communication network (WPCN) and physical-layer security (PLS) are regarded as potential solutions to allow IoT nodes to harvest energy from radio frequency (RF) and ensure the secure data delivery. How to efficiently allocate energy for IoT devices to improve throughput while guaranteeing secure data transmission is a challenging problem. In this article, we consider a WPCN with the existence of an eavesdropper, who is trying to eavesdrop the data transmitted from a certain node to the hybrid sink ($H$-sink). In the proposed system, the nodes first harvest energy from the$H$-sink, then transmit the information to the$H$-sink and generate interference to the eavesdropper. We first formulate the sum-throughput maximization problem as the nonlinear optimization problem and find its closed-form solution by the Lagrangian method. We further design an efficient algorithm to obtain the optimal numerical results to make up for the defect that the closed-form solution may not meet the explicit constraints. Furthermore, we propose a simple and reasonable method, the ratio method (RM), based on the observation of the optimal solution and make a comparison between the proposed method and the most common method, the same interference power method (SIPM).

This article investigates an unmanned aerial vehicle (UAV)-assisted wireless-powered mobile-edge computing (MEC) system, where the UAV powers the mobile terminals by wireless power transfer (WPT) and provides computation service for them. We aim to maximize the computation bits of terminals while ensuring fairness among them. Considering the random trajectories of mobile terminals, we propose a soft actor–critic (SAC)-based UAV trajectory planning and resource allocation (SAC-TR) algorithm, which combines off-policy and maximum entropy reinforcement learning to improve the convergence of the algorithm. We design the reward as a heterogeneous function of computation bits, fairness, and destination. Simulation results show that SAC-TR can quickly adapt to varying network environments and outperform representative benchmarks in various situations.

The advanced Internet of Things (IoT) enables more and more interactions between people and machines in the emerging applications, which rely on real-time communication and computing. However, the limited battery capacity and low computing capacity of IoT nodes can hardly support high-performance computing applications. The integration of wireless power transmission (WPT) and mobile edge computing (MEC) is a feasible and promising solution to address the energy shortage and computing capacity limitation of IoT nodes by harvesting radio frequency signal’s energy and offloading the nodes’ computation tasks to edge computing servers (ECSs). In this work, we focus on the wireless powered MEC network with an ECS and multiple edge devices (EDs), and study the joint optimization of WPT duration, transmission time allocation of each ED and partial offloading decision to maximize the sum computation rate. First, we formulate this as a non-convex problem which is hard to solve. Second, to conquer this problem, we decompose the original offloading problem into the sub-problem of optimizing the offloading time allocation among EDs and the proportion of harvested energy allocated for offloading at each ED under a given WPT duration and the top-problem of optimizing the WPT duration. Finally, we design an online DRL-based framework where one DNN together with its exploration strategy and training strategy is adopted to learn the near-optimal WPT duration and an efficient optimal algorithm is designed to solve the sub-problem. Numerical results show that the DRL-based offloading algorithm achieves the near-maximal sum computation rate while greatly reducing the processing time by at least three orders of magnitude compared with using the solver CVX for the sub-problem and the DNN for the top-problem.

In this paper, we consider a multi-user mobile-edge computing (MEC) network with time-varying wireless channels and stochastic user task data arrivals in sequential time frames. In particular, we aim to design an online computation offloading algorithm to maximize the network data processing capability subject to the long-term data queue stability and average power constraints. The online algorithm is practical in the sense that the decisions for each time frame are made without the assumption of knowing future channel conditions and data arrivals. We formulate the problem as a multi-stage stochastic mixed integer non-linear programming (MINLP) problem that jointly determines the binary offloading (each user computes the task either locally or at the edge server) and system resource allocation decisions in sequential time frames. To address the coupling in the decisions of different time frames, we propose a novel framework, named LyDROO, that combines the advantages of Lyapunov optimization and deep reinforcement learning (DRL). Specifically, LyDROO first applies Lyapunov optimization to decouple the multi-stage stochastic MINLP into deterministic per-frame MINLP subproblems of much smaller size. Then, it integrates model-based optimization and model-free DRL to solve the per-frame MINLP problems with very low computational complexity. Simulation results show that the proposed LyDROO achieves optimal computation performance while satisfying all the long-term constraints. Besides, it induces very low execution latency that is particularly suitable for real-time implementation in fast fading environments.

Opportunistic computation offloading is an effective method to improve the computation performance of mobile-edge computing (MEC) networks under dynamic edge environment. In this paper, we consider a multi-user MEC network with time-varying wireless channels and stochastic user task data arrivals in sequential time frames. In particular, we aim to design an online computation offloading algorithm to maximize the network data processing capability subject to the long-term data queue stability and average power constraints. The online algorithm is practical in the sense that the decisions for each time frame are made without the assumption of knowing the future realizations of random channel conditions and data arrivals. We formulate the problem as a multi-stage stochastic mixed integer non-linear programming (MINLP) problem that jointly determines the binary offloading (each user computes the task either locally or at the edge server) and system resource allocation decisions in sequential time frames. To address the coupling in the decisions of different time frames, we propose a novel framework, named LyDROO, that combines the advantages of Lyapunov optimization and deep reinforcement learning (DRL). Specifically, LyDROO first applies Lyapunov optimization to decouple the multi-stage stochastic MINLP into deterministic per-frame MINLP subproblems. By doing so, it guarantees to satisfy all the long-term constraints by solving the per-frame subproblems that are much smaller in size. Then, LyDROO integrates model-based optimization and model-free DRL to solve the per-frame MINLP problems with very low computational complexity. Simulation results show that under various network setups, the proposed LyDROO achieves optimal computation performance while stabilizing all queues in the system. Besides, it induces very low computation time that is particularly suitable for real-time implementation in fast fading environments.

In this paper, we consider a mobile-edge computing (MEC) system, where an access point (AP) assists a mobile device (MD) to execute an application consisting of multiple tasks following a general task call graph. The objective is to jointly determine the offloading decision of each task and the resource allocation (e.g., CPU computing power) under time-varying wireless fading channels and stochastic edge computing capability, so that the energy-time cost (ETC) of the MD is minimized. Solving the problem is particularly hard due to the combinatorial offloading decisions and the strong coupling among task executions under the general dependency model. To address the issue, we propose a deep reinforcement learning (DRL) framework based on the actor-critic learning structure. In particular, the actor network utilizes a deep neural network (DNN) to learn the optimal mapping from the input states (i.e., wireless channel gains and edge CPU frequency) to the binary offloading decision of each task. Meanwhile, for the critic network, we show that given the offloading decision, the remaining resource allocation problem becomes convex, where we can quickly evaluate the ETC performance of the offloading decisions output by the actor network. Accordingly, we select the best offloading action and store the state-action pair in an experience replay memory as the training dataset to continuously improve the action generation DNN. Numerical results show that for various types of task graphs, the proposed algorithm achieves up to 99.5% of the optimal performance while significantly reducing the computational complexity compared to the existing optimization methods.

Non-orthogonal multiple access (NOMA) has been considered as a promising scheme for enabling ultra-high throughput transmission and massive-connectivity in next generation wireless systems. In this paper, we investigate the secrecy-based NOMA transmission for encountering the eavesdropping attack. Exploiting the NOMA-users simultaneous transmission as an artificial jamming, we investigate the joint optimization of NOMA-users' power allocations and the secrecy-provisioning, with the objective of the effective secure throughput of NOMA-users while ensuring the fairness among them. Despite the non-convexity of the formulated joint optimization problem, we explore its hidden feature and design a search algorithm to compute the optimal solution. Numerical results are provided to validate the performance of our proposed algorithm.1

This study presents a mobile edge computing (MEC)‐enabled UAV communication system, where a number of UAVs are served by terrestrial base stations (TBSs) equipped with computation resource in the non‐orthogonal multiple access manner. Each UAV has to offload its computing tasks to the proper TBS due to the limited energy supply. For this, the authors aim at minimising the sum of transmission energy of UAVs and computation energy of TBSs through jointly optimising the UAV transmit power, computation resource allocation, and UAV grouping. Considering the non‐convexity of this optimisation problem, they obtain the optimal solution in the coupled steps: the convex resource allocation optimisation and the combinatorial UAV grouping optimisation. By exploiting the convex nature of the resource allocation optimisation problem, they obtain the optimal transmit power and computation allocation based on the KKT conditions and the idea of gradient descent method when considering a single TBS. Then, they adopt the simulated annealing to obtain the optimal UAV grouping and TBS selection based on the proposed resource allocation optimisation algorithm. Finally, simulation results show that the proposed joint optimisation of transmit power, computation resource allocation, and UAV grouping can effectively reduce the energy consumption of MEC‐aware UAV communication system.

Wireless powered mobile-edge computing (MEC) has recently emerged as a promising paradigm to enhance the data processing capability of low-power networks, such as wireless sensor networks and internet of things (IoT). In this paper, we consider a wireless powered MEC network that adopts a binary offloading policy, so that each computation task of wireless devices (WDs) is either executed locally or fully offloaded to an MEC server. Our goal is to acquire an online algorithm that optimally adapts task offloading decisions and wireless resource allocations to the time-varying wireless channel conditions. This requires quickly solving hard combinatorial optimization problems within the channel coherence time, which is hardly achievable with conventional numerical optimization methods. To tackle this problem, we propose a Deep Reinforcement learning-based Online Offloading (DROO) framework that implements a deep neural network as a scalable solution that learns the binary offloading decisions from the experience. It eliminates the need of solving combinatorial optimization problems, and thus greatly reduces the computational complexity especially in large-size networks. To further reduce the complexity, we propose an adaptive procedure that automatically adjusts the parameters of the DROO algorithm on the fly. Numerical results show that the proposed algorithm can achieve near-optimal performance while significantly decreasing the computation time by more than an order of magnitude compared with existing optimization methods. For example, the CPU execution latency of DROO is less than 0.1 second in a 30-user network, making real-time and optimal offloading truly viable even in a fast fading environment.

In mobile edge computing (MEC) systems, edge service caching refers to pre-storing the necessary programs for executing computation tasks at MEC servers. Service caching effectively reduces the real-time delay/bandwidth cost on acquiring and initializing service applications when computation tasks are offloaded to the MEC servers. The limited caching space at resource-constrained edge servers calls for careful design of caching placement to determine which programs to cache over time. This is in general a complicated problem that highly correlates to the computation offloading decisions of computation tasks, i.e., whether or not to offload a task for edge execution. In this paper, we consider a single edge server that assists a mobile user (MU) in executing a sequence of computation tasks. In particular, the MU can upload and run its customized programs at the edge server, while the server can selectively cache the previously generated programs for future reuse. To minimize the computation delay and energy consumption of the MU, we formulate a mixed integer non-linear programming (MINLP) that jointly optimizes the service caching placement, computation offloading decisions, and system resource allocation (e.g., CPU processing frequency and transmit power of MU). To tackle the problem, we first derive the closed-form expressions of the optimal resource allocation solutions, and subsequently transform the MINLP into an equivalent pure 0-1 integer linear programming (ILP) that is much simpler to solve. To further reduce the complexity in solving the ILP, we exploit the underlying structures of caching causality and task dependency models, and accordingly devise a reduced-complexity alternating minimization technique to update the caching placement and offloading decision alternately. Extensive simulations show that the proposed joint optimization techniques achieve substantial resource savings of the MU compared to other representative benchmark methods considered.

Narrow-band Internet of Things (NB-IoT), one of the emerging paradigms of low power wide area networks (LPWAN) for Internet of Things (IoT), has been envisioned as a promising solution to enable massive connectivity, cost-efficient, and highly reliable Internet of Thing (IoT) systems in future smart cities. In this work, we build up an indoor environment-temperature monitoring system based on NB-IoT. We present a detailed design of our system and illustrate the key technologies. Based on our system and the collected data (i.e., the temperature data), we further design an abnormality-detection mechanism based on the support vector machine (SVM). We provide experimental results to show the performance of our designed system and the proposed abnormality-detection mechanism.

The emerging non‐orthogonal multiple access (NOMA) has been considered as a promising scheme to reach the goals of 5G cellular systems. By enabling a group of mobile users (MUs) to share a same frequency channel and adopting the successive interference cancellation to mitigate the co‐channel interference, NOMA can improve the spectrum efficiency compared with the orthogonal multiple access (OMA). This study proposes a joint optimisation scheme of the channel bandwidth and the transmit‐power allocations for the NOMA downlink transmission, which aims at minimising the overall resource consumption cost including both the spectrum consumption and the power consumption, while satisfying the MUs' traffic requirements. In spite of the non‐convexity nature of the joint optimisation problem, this study characterises the connection between the channel bandwidth and the associated transmit powers for the MUs. Based on this connection, this study transforms the joint optimisation problem into an equivalent bandwidth optimisation problem, and further proposes an efficient algorithm to compute the optimal bandwidth allocation (which enables us to derive the corresponding transmit powers for the MUs). Extensive numerical results are provided to validate the proposed algorithm and the advantage of the proposed joint channel bandwidth and power allocations for the NOMA transmission.

This paper studies energy harvesting sensor networks in green cities that transmit a variety of data packets with different reward values. With the aim to maximize its long-term average transmission reward, almost all the existing optimal energy management strategies are based on the policy iteration algorithm, which suffers from the curse of dimensionality. By contrast, we focus on developing low-complexity optimal policies that can lead to practical implementation. Our main contribution is to propose a threshold-based scheduling policy for energy harvesting sensor networks achieving long-term average rewards. As a result, a sensor node only requires limited memory to store a few optimal value thresholds to perform energy management. Specifically, we propose an algorithm to compute the optimal thresholds, whose complexity is linear with the size of data and energy storage. Numerical results are studied based on real solar radiation data measured at Queensland and show that the optimal expected reward of our proposed scheduling policy approaches its theoretical offline upper bound.

We study status updates over wireless fading channels in terms of the age of information, which measures the freshness of the last received update since its generation. In this paper, we model the wireless transmission systems with Poisson arrivals, a First-Come-First-Served (FCFS) packet buffer, and two-state Markov modulated service process (MMSP) as M/MMSP/1/K queueing model. We derive closed-form expressions of average age of information in two extreme cases when the packet buffer size is either infinite or zero. Our analysis and simulation results show that the average age of information for M/MMSP/1/∞queue greatly depends on channel variations and that M/MMSP/1/1 queue is more affected by packet arrival rate.

Recently, the newly emerging wireless powered communication network (WPCN) has drawn significant interests, where network nodes are powered by the energy harvested from the radio-frequency (RF) signal. This paper studies the WPCN where one hybrid sink (H-sink) coordinates the wireless energy/information transmissions to/from a set of one-hop nodes powered by the harvested RF energy only. The transmission completion time (TCT) minimization for the uplink (UL) transmissions of a given number of bits per node is considered. First, we prove that the harvest-then-transmit (HTT) transmission strategy is one of the transmission strategies able to achieve the minimal TCT, where all nodes first harvest the RF energy broadcast by the H-sink in the downlink and then send their independent information to the H-sink in the UL by time-division multiple access. Then for the HTT transmission, we prove that in order to achieve the minimal TCT, each node must transmit with constant power and consume all available energy, which helps to simplify the considered TCT minimization problem to be the optimization of time allocated for the H-sink's wireless energy transfer and the nodes' wireless information transmissions, and we formulate the optimal time allocation problem as a nonlinear optimization problem. Finally, we prove that it is a convex optimization problem. Due to the inexistence of explicit closed-form expressions of optimal time allocations to minimize TCT, one efficient algorithm is presented to obtain the optimal time allocations. Simulation results show that, compared with the available transmission strategies, the designed TCT-minimized transmission achieves a significantly smaller TCT.

This paper considers an energy harvesting sensor node with finite data and energy storage, which transmits data packets with different reward values to its corresponding receiver node. In this regard, we propose an optimal threshold-based transmission scheduling policy for maximizing the long-term average transmission reward. In particular, we first analyze the performance of the proposed threshold-based transmission scheduling policy by studying the steady states of the energy harvesting sensor node and derive its expected transmission reward. We then propose a polynomial-time algorithm to compute these optimal reward value thresholds that maximize the expected reward. Numerical results show that the system expected reward increases with the increase of data and energy storage capacity. Our analysis further shows that the expected reward increases exponentially with the increase of data or energy storage capacity.

The wireless fading channels with finite input buffer, Poisson arrivals and two-state Markov modulated service processes (MMSP) are modeled as M/MMSP/1/K queues in this paper. The existing performance analyses of Markov channels are almost all based on the matrix-geometric method, which provides little physical insights for system design. By contrast, we focus on deriving closed-form analytic expressions with physical interpretations in terms of system parameters of interest. Our main contribution is to derive the generalized Pollaczek-Khinchin (P-K) formula of M/MMSP/1/K queue from start-service probability to explore the impact of state transitions on the queueing behavior of Markov channels. This generalized P-K formula reveals that the performance of wireless channels with varying rates can be fully characterized by a newly defined system parameter, called state transition factor β, which clearly explains the reason that the channel with slow state transition rate owns a larger delay for the same channel capacity. In the extreme case when the state transition factor β approaches 0, we show that the channel under consideration can be approximately modeled as an M/G/1 queue. We use the Type I Hybrid ARQ system with a fixed data-rate as an example in this paper to illustrate our results.