{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:50:45Z","timestamp":1760230245180,"version":"build-2065373602"},"reference-count":57,"publisher":"MDPI AG","issue":"14","license":[{"start":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T00:00:00Z","timestamp":1657756800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["62001264","L192025","2018YFB1801102"],"award-info":[{"award-number":["62001264","L192025","2018YFB1801102"]}]},{"name":"Natural Science Foundation of Beijing","award":["62001264","L192025","2018YFB1801102"],"award-info":[{"award-number":["62001264","L192025","2018YFB1801102"]}]},{"name":"National Key R&amp;D Program","award":["62001264","L192025","2018YFB1801102"],"award-info":[{"award-number":["62001264","L192025","2018YFB1801102"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Sixth-generation (6G) wireless communication scenarios are complex and diverse. Small-scale fading is a key part of wireless channels and its impact on performance in scenarios with time sensitivity and 6G ultrareliable and low latency communications (URLLC) quality-of-service requirements cannot be ignored. Therefore, it is necessary to accurately characterize small-scale fading when designing wireless communication systems. In this paper, we derive approximate closed form expressions for the probability density function, cumulative distribution function and moment-generating function of the postprocessing signal-to-noise ratio following the zero-forcing detector in a cell-free massive multiple-input multiple-output (CF mMIMO) system. CF mMIMO system is a nonorthogonal multiple access (NOMA) system that enables users to share all channel uses and can ensure the fairness of the communication quality experienced by different users. Our key contributions include the extension of the \u03ba\u2013\u03bc shadowed fading model to a CF mMIMO system and the proposal of theoretical tools (the derived closed-form expression) to improve its mathematical tractability. By exploiting the statistical characterizations of the arrival and service processes, another important contribution is the exploitation of the upper bound of the queuing delay violation probability (UB-QDVP) over the Mellin transforms of the arrival and service processes in the proposed CF mMIMO system under the \u03ba\u2013\u03bc shadowed fading model. Corroborated by extensive simulations, our analyses validate that the CF mMIMO system outperforms the orthogonal multiple access and power-domain NOMA systems and reveal the relationships among different small-scale fading types, energy efficiency, delay and the UB-QDVP, as well as the accuracy and effectiveness of the proposed theoretical tools based on the \u03ba\u2013\u03bc shadowed fading model.<\/jats:p>","DOI":"10.3390\/s22145279","type":"journal-article","created":{"date-parts":[[2022,7,15]],"date-time":"2022-07-15T01:57:11Z","timestamp":1657850231000},"page":"5279","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Guaranteeing QoS for NOMA-Enabled URLLC Based on \u03ba\u2013\u03bc Shadowed Fading Model"],"prefix":"10.3390","volume":"22","author":[{"given":"Jie","family":"Zeng","sequence":"first","affiliation":[{"name":"School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China"},{"name":"Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuxin","family":"Song","sequence":"additional","affiliation":[{"name":"School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China"},{"name":"Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Teng","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China"},{"name":"Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4127-2419","authenticated-orcid":false,"given":"Tiejun","family":"Lv","sequence":"additional","affiliation":[{"name":"School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shidong","family":"Zhou","sequence":"additional","affiliation":[{"name":"Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China"},{"name":"Department of Electronic Engineering, Tsinghua University, Beijing 100084, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1109\/JIOT.2021.3103320","article-title":"6G Internet of Things: A comprehensive survey","volume":"9","author":"Nguyen","year":"2022","journal-title":"IEEE Internet Things J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3939","DOI":"10.1109\/TIT.2013.2253194","article-title":"Optimal power-delay tradeoffs in fading channels-Small-delay asymptotics","volume":"59","author":"Berry","year":"2013","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1109\/JSAC.2016.2550338","article-title":"A survey of energy-efficient techniques for 5G networks and challenges ahead","volume":"34","author":"Buzzi","year":"2016","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"816","DOI":"10.1109\/TGCN.2021.3059206","article-title":"Towards optimal energy efficiency in cell-free massive MIMO systems","volume":"5","author":"Papazafeiropoulos","year":"2021","journal-title":"IEEE Trans. Green Commun. Netw."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2208","DOI":"10.1109\/TWC.2019.2963185","article-title":"DeepNOMA: A unified framework for NOMA using deep multi-task learning","volume":"19","author":"Ye","year":"2020","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Pan, J., Ye, N., Yu, H., Hong, T., Al-Rubaye, S., Mumtaz, S., Al-Dulaimi, A., and Chih-Lin, I. (2022). AI-Driven Blind Signature Classification for IoT Connectivity: A Deep Learning Approach. IEEE Trans. Wirel. Commun., early access.","DOI":"10.1109\/TWC.2022.3145399"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1109\/MWC.001.2000472","article-title":"Deep-Learning-Enhanced NOMA Transceiver Design for Massive MTC: Challenges, State of the Art, and Future Directions","volume":"28","author":"Ye","year":"2021","journal-title":"IEEE Wirel. Commun."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.dcan.2021.07.008","article-title":"Help from space: Grant-free massive access for satellite-based IoT in the 6G era","volume":"8","author":"Ye","year":"2021","journal-title":"Digit. Commun. Netw."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1109\/MCOM.2017.1500657CM","article-title":"Application of non-orthogonal multiple access in LTE and 5G networks","volume":"55","author":"Ding","year":"2017","journal-title":"IEEE Commun. Mag."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1786","DOI":"10.1109\/TWC.2019.2957745","article-title":"Joint power and blocklength optimization for URLLC in a factory automation scenario","volume":"19","author":"Ren","year":"2019","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2347","DOI":"10.1109\/JPROC.2017.2768666","article-title":"Non-orthogonal multiple access for 5G and beyond","volume":"105","author":"Liu","year":"2017","journal-title":"Proc. IEEE"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1834","DOI":"10.1109\/TWC.2017.2655515","article-title":"Cell-free massive MIMO versus small cells","volume":"16","author":"Ngo","year":"2017","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"99878","DOI":"10.1109\/ACCESS.2019.2930208","article-title":"Cell-free massive MIMO: A new next-generation paradigm","volume":"7","author":"Zhang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_14","first-page":"77","article-title":"Making cell-free massive MIMO competitive with MMSE processing and centralized implementation","volume":"19","author":"Sanguinetti","year":"2019","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Interdonato, G., Frenger, P., and Larsson, E.G. (2019, January 20\u201324). Scalability aspects of cell-free massive MIMO. Proceedings of the IEEE International Conference on Communications (ICC), Shanghai, China.","DOI":"10.1109\/ICC.2019.8761828"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1109\/COMST.2021.3135119","article-title":"User-centric cell-free massive MIMO networks: A survey of opportunities, challenges and solutions","volume":"24","author":"Ammar","year":"2022","journal-title":"IEEE Commun. Surv. Tutor."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2822","DOI":"10.1109\/JSYST.2020.2993048","article-title":"Spectral and energy efficiency in cell-free massive MIMO systems over correlated Rician fading","volume":"15","author":"Jin","year":"2020","journal-title":"IEEE Syst. J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"8384","DOI":"10.1109\/JIOT.2020.2990378","article-title":"Wirelessly powered cell-free IoT: Analysis and optimization","volume":"7","author":"Wang","year":"2020","journal-title":"IEEE Internet Things J."},{"key":"ref_19","unstructured":"Elwekeil, M., Zappone, A., and Buzzi, S. (2021). Power control in cell-free massive MIMO networks for UAVs URLLC under the finite blocklength regime. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"9625","DOI":"10.1109\/JIOT.2021.3130026","article-title":"Performance analysis and optimization of NOMA-based cell-free massive MIMO for IoT","volume":"9","author":"Zhang","year":"2021","journal-title":"IEEE Internet Things J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2188","DOI":"10.1109\/TCOMM.2020.2969351","article-title":"Efficient angle-domain processing for FDD-based cell-free massive MIMO systems","volume":"68","author":"Abdallah","year":"2020","journal-title":"IEEE Trans. Commun."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"652","DOI":"10.1109\/JSAC.2020.3020109","article-title":"Statistical delay and error-rate bounded QoS provisioning for mURLLC over 6G CF M-MIMO mobile networks in the finite blocklength regime","volume":"39","author":"Zhang","year":"2020","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1109\/MNET.001.1900287","article-title":"A vision of 6G wireless systems: Applications, trends, technologies, and open research problems","volume":"34","author":"Saad","year":"2020","journal-title":"IEEE Netw."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1038\/s41928-019-0355-6","article-title":"What should 6G be?","volume":"3","author":"Dang","year":"2020","journal-title":"Nat. Electron."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3826","DOI":"10.1109\/TVT.2017.2787204","article-title":"An extension of the \u03ba-\u03bc shadowed fading model: Statistical characterization and applications","volume":"67","author":"Paris","year":"2018","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"570","DOI":"10.1109\/JIOT.2019.2951179","article-title":"A unified performance analysis of relaying communication system for IoT application with hybrid fading","volume":"7","author":"Chen","year":"2020","journal-title":"IEEE Internet Things J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"9630","DOI":"10.1109\/TVT.2016.2525721","article-title":"The \u03ba-\u03bc shadowed fading model: Unifying the \u03ba-\u03bc and \u03b7-\u03bc distributions","volume":"65","author":"Paris","year":"2016","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1109\/TVT.2013.2281213","article-title":"Statistical characterization of \u03ba-\u03bc shadowed fading","volume":"63","author":"Paris","year":"2013","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3597","DOI":"10.1109\/TAP.2016.2570235","article-title":"The \u03b1\u2013\u03b7\u2013\u03ba\u2013\u03bc Fading Model","volume":"64","author":"Yacoub","year":"2016","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"9112","DOI":"10.1109\/TVT.2020.2998895","article-title":"On the effective rate and energy detection based spectrum sensing over \u03b1\u2013\u03b7\u2013\u03ba\u2013\u03bc fading channels","volume":"69","year":"2020","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"7653","DOI":"10.1109\/TVT.2017.2678430","article-title":"The \u03ba-\u03bc shadowed fading model with integer fading parameters","volume":"66","author":"Paris","year":"2017","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1109\/TWC.2018.2880907","article-title":"Optimizing resource allocation in the short blocklength regime for ultra-reliable and low-latency communications","volume":"18","author":"Sun","year":"2018","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2307","DOI":"10.1109\/TIT.2010.2043769","article-title":"Channel coding rate in the finite blocklength regime","volume":"56","author":"Polyanskiy","year":"2010","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"3425","DOI":"10.1109\/JSAC.2021.3088625","article-title":"AoI-driven statistical delay and error-rate bounded QoS provisioning for mURLLC over UAV-multimedia 6G mobile networks using FBC","volume":"39","author":"Zhang","year":"2021","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1834","DOI":"10.1109\/JPROC.2018.2867029","article-title":"Ultrareliable and low-latency wireless communication: Tail, risk, and scale","volume":"106","author":"Bennis","year":"2018","journal-title":"Proc. IEEE"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1109\/MCOM.2017.1601092","article-title":"Radio resource management for ultra-reliable and low-latency communications","volume":"55","author":"She","year":"2017","journal-title":"IEEE Commun. Mag."},{"key":"ref_37","first-page":"204","article-title":"Network-layer performance analysis of multihop fading channels","volume":"24","author":"Liebeherr","year":"2014","journal-title":"IEEE\/ACM Trans. Netw."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"5861","DOI":"10.1109\/TWC.2021.3070836","article-title":"Cell-free massive MIMO in the short blocklength regime for URLLC","volume":"20","author":"Nasir","year":"2021","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1390","DOI":"10.1109\/OJCOMS.2021.3089281","article-title":"Cell-free and user-centric massive MIMO architectures for reliable communications in indoor factory environments","volume":"2","author":"Alonzo","year":"2021","journal-title":"IEEE Open J. Commun. Soc."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1272","DOI":"10.1109\/JSTSP.2021.3097898","article-title":"Statistical delay and error-rate bounded QoS provisioning for SWIPT over CF M-MIMO 6G mobile wireless networks using FBC","volume":"15","author":"Zhang","year":"2021","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1046","DOI":"10.1109\/TVT.2019.2953109","article-title":"Performance analysis of downlink NOMA systems over \u03ba-\u03bc shadowed fading channels","volume":"69","author":"ElHalawany","year":"2019","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"6995","DOI":"10.1109\/TWC.2017.2734080","article-title":"A comprehensive analysis of 5G heterogeneous cellular systems operating over \u03ba-\u03bc shadowed fading channels","volume":"16","author":"Chun","year":"2017","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"6302","DOI":"10.1109\/TVT.2021.3086026","article-title":"Information-theoretic security of MIMO networks under \u03ba-\u03bc shadowed fading channels","volume":"70","author":"Osorio","year":"2021","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1109\/JSAC.2019.2898759","article-title":"Delay performance of the multiuser MISO downlink under imperfect CSI and finite-length coding","volume":"37","author":"Schiessl","year":"2019","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"3879","DOI":"10.1109\/TWC.2020.2979114","article-title":"NOMA in the uplink: Delay analysis with imperfect CSI and finite-length coding","volume":"19","author":"Schiessl","year":"2020","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"780","DOI":"10.1109\/JSAC.2019.2898785","article-title":"Downlink MIMO-NOMA for ultra-reliable low-latency communications","volume":"37","author":"Xiao","year":"2019","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3526","DOI":"10.1109\/TCOMM.2018.2817199","article-title":"Low-latency millimeter-wave communications: Traffic dispersion or network densification?","volume":"66","author":"Yang","year":"2018","journal-title":"IEEE Trans. Commun."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1109\/COMST.2021.3123267","article-title":"Cell-free massive MIMO: A survey","volume":"24","author":"Elhoushy","year":"2022","journal-title":"IEEE Commun. Surv. Tutor."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"116195","DOI":"10.1109\/ACCESS.2020.3004499","article-title":"Fronthaul-constrained cell-free massive MIMO with low resolution ADCs","volume":"8","author":"Femenias","year":"2020","journal-title":"IEEE Access"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1109\/MCOM.001.2000697","article-title":"Hierarchical scheduling for suppression of fronthaul delay in C-RAN with dynamic functional split","volume":"59","author":"Becvar","year":"2021","journal-title":"IEEE Commun. Mag."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"6798","DOI":"10.1109\/TWC.2020.3006083","article-title":"Joint power allocation and load balancing optimization for energy-efficient cell-free massive MIMO networks","volume":"19","author":"Larsson","year":"2020","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1604","DOI":"10.1109\/LCOMM.2015.2449317","article-title":"On minimizing the MSE in the presence of channel state information errors","volume":"19","author":"Fodor","year":"2015","journal-title":"IEEE Commun. Lett."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1109\/TWC.2019.2948841","article-title":"Spectral efficiency analysis of cell-free massive MIMO systems with zero-forcing detector","volume":"19","author":"Liu","year":"2019","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Arnau, J., and Kountouris, M. (2018, January 7\u201311). Delay performance of MISO wireless communications. Proceedings of the 2018 16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), Shanghai, China.","DOI":"10.23919\/WIOPT.2018.8362864"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1109\/TWC.2017.2762684","article-title":"Cross-layer optimization for ultra-reliable and low-latency radio access networks","volume":"17","author":"She","year":"2017","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1436","DOI":"10.1109\/TCOMM.2013.020413.110848","article-title":"Energy and spectral efficiency of very large multiuser MIMO systems","volume":"61","author":"Ngo","year":"2013","journal-title":"IEEE Trans. Commun."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1612","DOI":"10.1109\/TWC.2020.3034877","article-title":"Accurate variable-order approximations to the sum of \u03b1\u2013\u03bc variates with application to MIMO systems","volume":"20","author":"Payami","year":"2020","journal-title":"IEEE Trans. Wirel. Commun."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/14\/5279\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:50:35Z","timestamp":1760140235000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/14\/5279"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,14]]},"references-count":57,"journal-issue":{"issue":"14","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["s22145279"],"URL":"https:\/\/doi.org\/10.3390\/s22145279","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2022,7,14]]}}}