{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:14:46Z","timestamp":1760242486827,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2017,9,15]],"date-time":"2017-09-15T00:00:00Z","timestamp":1505433600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In wireless powered communication networks (WPCNs), it is essential to research energy efficiency fairness in order to evaluate the balance of nodes for receiving information and harvesting energy. In this paper, we propose an efficient iterative algorithm for optimal energy efficiency proportional fairness in WPCN. The main idea is to use stochastic geometry to derive the mean proportionally fairness utility function with respect to user association probability and receive threshold. Subsequently, we prove that the relaxed proportionally fairness utility function is a concave function for user association probability and receive threshold, respectively. At the same time, a sub-optimal algorithm by exploiting alternating optimization approach is proposed. Through numerical simulations, we demonstrate that our sub-optimal algorithm can obtain a result close to optimal energy efficiency proportional fairness with significant reduction of computational complexity.<\/jats:p>","DOI":"10.3390\/s17092125","type":"journal-article","created":{"date-parts":[[2017,9,15]],"date-time":"2017-09-15T10:22:10Z","timestamp":1505470930000},"page":"2125","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Optimal Energy Efficiency Fairness of Nodes in Wireless Powered Communication Networks"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4463-0632","authenticated-orcid":false,"given":"Jing","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qingjie","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Derrick","family":"Ng","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney 2052, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Minho","family":"Jo","sequence":"additional","affiliation":[{"name":"Department of Computer Convergence Software, Korea University, Sejong Metropolitan 30019, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,9,15]]},"reference":[{"key":"ref_1","unstructured":"CISCO (2017). Cisco Visual Networking Index: Global Mobile Data Traffic Forecast Update, 2016\u20132021 White Paper, Cisco Mobile VNI."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1936","DOI":"10.1109\/JSAC.2017.2720898","article-title":"Energy efficient user association and power allocation in millimeter wave based Ultra Dense Networks with energy harvesting base stations","volume":"35","author":"Zhang","year":"2017","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1109\/MWC.2016.7462488","article-title":"User-centric ultra-dense networks for 5G: Challenges, methodologies, and directions","volume":"23","author":"Chen","year":"2016","journal-title":"IEEE Wirel. Commun."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Wong, V.W.S., Schober, R., Ng, D.W.K., and Wang, L.-C. (2017). Key Technologies for 5G Wireless Systems, Cambridge University Press.","DOI":"10.1017\/9781316771655"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1109\/JSAC.2016.2545539","article-title":"Ultra dense small cell networks: Turning density into energy efficiency","volume":"34","author":"Samarakoon","year":"2016","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"706","DOI":"10.1109\/JSYST.2015.2475377","article-title":"Energy-saving transmission for green macrocell-small cell systems: A system-level perspective","volume":"11","author":"Chung","year":"2017","journal-title":"IEEE Syst. J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1810","DOI":"10.1109\/TVT.2016.2565539","article-title":"Energy and spectrum efficient user association in millimeter wave backhaul small cell networks","volume":"66","author":"Mesodiakaki","year":"2017","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"7540","DOI":"10.1109\/TVT.2017.2673245","article-title":"Energy efficient noncooperative power control in small-cell networks","volume":"66","author":"Jiang","year":"2017","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Zhang, J., Xiang, L., Ng, D.W.K., Jo, M., and Chen, M. (2017). Energy efficiency evaluation of Multi-tier cellular uplink transmission under maximum power constraint. IEEE Trans. Wirel. Commun.","DOI":"10.1109\/TWC.2017.2739142"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2099","DOI":"10.1007\/s11277-014-1626-1","article-title":"An energy efficient barrier coverage algorithm for wireless sensor networks","volume":"77","author":"Habib","year":"2014","journal-title":"Wirel. Pers. Commun."},{"key":"ref_11","unstructured":"Mohammad, S., Nicola, C., and Enzo, B. (2016). Energy-efficient adaptive resource management for real-time vehicular cloud services. IEEE Trans. Cloud Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1007\/s11227-016-1785-9","article-title":"P-SEP: A prolong stable election routing algorithm for energy-limited heterogeneous fog-supported wireless sensor networks","volume":"73","author":"Paola","year":"2017","journal-title":"J. Supercomput."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ng, D.W.K., and Schober, R. (2012, January 3\u20137). Energy-efficient power allocation for M2M communications with energy harvesting transmitter. Proceedings of the IEEE Globecom Workshops, Anaheim, CA, USA.","DOI":"10.1109\/GLOCOMW.2012.6477832"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1109\/MCOM.2015.7081080","article-title":"Application of smart antenna technologies in simultaneous wireless information and power transfer","volume":"53","author":"Ding","year":"2015","journal-title":"IEEE Commun. Mag."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1109\/TCOMM.2015.2498171","article-title":"Secure D2D communication in large-scale cognitive cellular networks: A wireless power transfer model","volume":"64","author":"Liu","year":"2016","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"7405","DOI":"10.1109\/ACCESS.2016.2600242","article-title":"Energy harvesting aided device-to-device communication underlaying the cellular downlink","volume":"5","author":"Gupta","year":"2016","journal-title":"IEEE Access"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1406","DOI":"10.1109\/TWC.2015.2489651","article-title":"Heterogeneous cellular network with energy harvesting-based D2D communication","volume":"15","author":"Yang","year":"2016","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Zhang, Y.X., Zhang, J., Sun, Y., and Ng, D.W.K. (2017, January 21\u201325). Energy-efficient transmission for wireless powerec D2D communication networks. Proceedings of the IEEE International Conference on Communications (ICC), Paris, France.","DOI":"10.1109\/ICC.2017.7996666"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4939","DOI":"10.1109\/TWC.2017.2704084","article-title":"Dilemma at RF energy harvesting relay: Downlink energy relaying or uplink information transfer?","volume":"16","author":"Mishra","year":"2017","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"6132","DOI":"10.1109\/TWC.2016.2578925","article-title":"Optimum transmission policies for energy harvesting sensor networks powered by a mobile control center","volume":"15","author":"Li","year":"2016","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5743","DOI":"10.1109\/TVT.2015.2453206","article-title":"Throughput maximization for multiuser MIMO wireless powered communication networks","volume":"65","author":"Hwang","year":"2016","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1518","DOI":"10.1109\/JSAC.2016.2551538","article-title":"Self-sustainable communications with RF energy harvesting: Ginibre point process modeling and analysis","volume":"34","author":"Lu","year":"2016","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2291","DOI":"10.1109\/TCOMM.2017.2676815","article-title":"Performance analysis and optimization for SWIPT wireless sensor networks","volume":"65","author":"Pan","year":"2017","journal-title":"IEEE Trans. Commun."},{"key":"ref_24","unstructured":"Docomo, N. (2010, January 12\u201316). Performance of eICIC with control channel coverage limitation. Proceedings of the 3GPP TSG RAN WG1 Meeting 61, R1-103264, Montreal, QC, Canada."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1835","DOI":"10.1109\/TWC.2015.2496942","article-title":"Optimal user-cell association for massive MIMO wireless networks","volume":"15","author":"Bethanabhotla","year":"2016","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1109\/WCL.2013.111713.130707","article-title":"On association cells in random heterogeneous networks","volume":"3","author":"Singh","year":"2014","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Haenggi, M. (2012). Stochastic Geometry for Wireless Networks, Cambridge University Press.","DOI":"10.1017\/CBO9781139043816"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3484","DOI":"10.1109\/TWC.2012.081612.111361","article-title":"Heterogeneous cellular networks with flexible cell association: A comprehensive downlink SINR analysis","volume":"11","author":"Jo","year":"2012","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3122","DOI":"10.1109\/TCOMM.2011.100411.100541","article-title":"A tractable approach to coverage and rate in cellular networks","volume":"59","author":"Andrews","year":"2011","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"6898","DOI":"10.1109\/TWC.2016.2593440","article-title":"User-centric energy efficiency maximization for wireless powered communications","volume":"15","author":"Wu","year":"2016","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_31","unstructured":"Liu, D., Chen, Y., Chai, K.K., and Zhang, T.K. (2014, January 10\u201313). Nash bargaining solution based user association optimization in HetNets. Proceedings of the IEEE 11th Consumer Communications and Networking Conference (CCNC), Las Vegas, NV, USA."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1109\/LCOMM.2014.012314.140090","article-title":"Opportunistic user association for multi-service HetNets using Nash bargaining solution","volume":"18","author":"Liu","year":"2014","journal-title":"IEEE Commun. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Tse, D., and Viswanath, P. (2005). Fundamentals of Wireless Communications, Cambridge University Press.","DOI":"10.1017\/CBO9780511807213"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1016\/j.physa.2007.07.063","article-title":"On the size distribution of Poisson Voronoi cells","volume":"385","author":"Ferenc","year":"2007","journal-title":"Phys. A Stat. Mech. 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