{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:35:52Z","timestamp":1760236552095,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"23","license":[{"start":{"date-parts":[[2021,12,2]],"date-time":"2021-12-02T00:00:00Z","timestamp":1638403200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Chilean agencies ANID-fondecyt de iniciaci\u00f3n grant 11201049","award":["11201049"],"award-info":[{"award-number":["11201049"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper proposes a new theoretical stochastic model based on an abstraction of the opportunistic model for opportunistic networks. The model is capable of systematically computing the network parameters, such as the number of possible routes, the probability of successful transmission, the expected number of broadcast transmissions, and the expected number of receptions. The usual theoretical stochastic model explored in the methodologies available in the literature is based on Markov chains, and the main novelty of this paper is the employment of a percolation stochastic model, whose main benefit is to obtain the network parameters directly. Additionally, the proposed approach is capable to deal with values of probability specified by bounded intervals or by a density function. The model is validated via Monte Carlo simulations, and a computational toolbox (R-packet) is provided to make the reproduction of the results presented in the paper easier. The technique is illustrated through a numerical example where the proposed model is applied to compute the energy consumption when transmitting a packet via an opportunistic network.<\/jats:p>","DOI":"10.3390\/s21238058","type":"journal-article","created":{"date-parts":[[2021,12,6]],"date-time":"2021-12-06T03:10:38Z","timestamp":1638760238000},"page":"8058","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Novel Theoretical Probabilistic Model for Opportunistic Routing with Applications in Energy Consumption for WSNs"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4818-6006","authenticated-orcid":false,"given":"Christian E.","family":"Galarza","sequence":"first","affiliation":[{"name":"Escuela Superior Polit\u00e9cnica del Litoral\u2014ESPOL, Facultad de Ciencias Naturales y Matem\u00e1ticas, V\u00eda Perimetral 5, Guayaquil 090150, Ecuador"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3924-1907","authenticated-orcid":false,"given":"Jonathan\u00a0M.","family":"Palma","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Faculty of Engineering, University of Talca, Curic\u00f3 3344158, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9994-0339","authenticated-orcid":false,"given":"Cecilia\u00a0F.","family":"Morais","sequence":"additional","affiliation":[{"name":"Center of Exact, Environmental and Technological Sciences, Pontifical Catholic University of Campinas, Campinas 13086-900, SP, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8024-5080","authenticated-orcid":false,"given":"Jaime","family":"Utria","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Statistics, Fluminense Federal University\u2014UFF, Niter\u00f3i 24210-201, RJ, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2910-5140","authenticated-orcid":false,"given":"Leonardo\u00a0P.","family":"Carvalho","sequence":"additional","affiliation":[{"name":"Discrete Technology and Production Automation (DTPA), Rijksuniversiteit Groningen, 9712 CP Groningen, The Netherlands"},{"name":"Polytechnic School, University of S\u00e3o Paulo, S\u00e3o Paulo 05508-900, SP, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2136-2305","authenticated-orcid":false,"given":"Daniel","family":"Bustos","sequence":"additional","affiliation":[{"name":"Centro de Investigaci\u00f3n de Estudios Avanzados del Maule (CIEAM), Vicerrector\u00eda de Investigaci\u00f3n y Postgrado, Universidad Cat\u00f3lica del Maule, Talca 3460000, Chile"},{"name":"Laboratorio de Bioinform\u00e1tica y Qu\u00edmica Computacional (LBQC), Facultad de Medicina, Universidad Cat\u00f3lica del Maule, Talca 3460000, Chile"},{"name":"Escuela de Bioingenier\u00eda M\u00e9dica, Facultad de Medicina, Universidad Cat\u00f3lica del Maule, Talca 3460000, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8225-7058","authenticated-orcid":false,"given":"Ricardo C. 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Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1145\/1090191.1080108","article-title":"ExOR: Opportunistic Multi-hop Routing for Wireless Networks","volume":"35","author":"Biswas","year":"2005","journal-title":"ACM Sigcomm Comput. Commun. Rev."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1109\/MWC.001.2000189","article-title":"Social-Aware Routing in Mobile Opportunistic Networks","volume":"28","author":"Zhang","year":"2021","journal-title":"IEEE Wirel. Commun."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2452","DOI":"10.1109\/TWC.2009.071388","article-title":"AsOR: An energy efficient multi-hop opportunistic routing protocol for wireless sensor networks over Rayleigh fading channels","volume":"8","author":"Wei","year":"2009","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1934","DOI":"10.1109\/TPDS.2011.70","article-title":"Energy-Efficient Opportunistic Routing in Wireless Sensor Networks","volume":"22","author":"Mao","year":"2011","journal-title":"IEEE Trans. Parallel Distrib. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Landsiedel, O., Ghadimi, E., Duquennoy, S., and Johansson, M. (2012, January 16\u201320). Low power, low delay: Opportunistic routing meets duty cycling. Proceedings of the ACM\/IEEE 11th International Conference on Information Processing in Sensor Networks (IPSN), Beijing, China.","DOI":"10.1145\/2185677.2185731"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Spachos, P., Chatzimisios, P., and Hatzinakos, D. (2012). Energy Aware Opportunistic Routing in Wireless Sensor Networks. IEEE Globecom Workshops, IEEE.","DOI":"10.1109\/GLOCOMW.2012.6477606"},{"key":"ref_10","unstructured":"Roedig, U., and Sreenan, C.J. (2009, January 11\u201313). Wireless Sensor Networks. Proceedings of the Wireless Sensor Networks: 6th European Conference, EWSN 2009, Cork, Ireland. Chapter Potentials of Opportunistic Routing in Energy-Constrained Wireless Sensor Networks."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"e4368","DOI":"10.1002\/dac.4368","article-title":"Neural network-based routing protocol for opportunistic networks with intelligent water drop optimization","volume":"33","author":"Kumaram","year":"2020","journal-title":"Int. J. Commun. Syst."},{"key":"ref_12","unstructured":"Jadhav, P., and Satao, R. (2016, January 26\u201327). A Survey on Opportunistic Routing Protocols for Wireless Sensor Networks. Proceedings of the International Conference on Communication, Computing and Virtualization (ICCCV) 2016, Amsterdam, The Netherlands."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Ramanathan, R., Hansen, R., Basu, P., Rosales-Hain, R., and Krishnan, R. (2007, January 11). Prioritized Epidemic Routing for Opportunistic Networks. Proceedings of the 1st International MobiSys Workshop on Mobile Opportunistic Networking (MobiOpp\u201907), New York, NY, USA.","DOI":"10.1145\/1247694.1247707"},{"key":"ref_14","first-page":"22:1","article-title":"Opportunistic Routing in Wireless Networks: Models, Algorithms, and Classifications","volume":"47","author":"Boukerche","year":"2014","journal-title":"ACM Comput. Surv."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1109\/TNET.2011.2159844","article-title":"Adaptive Opportunistic Routing for Wireless Ad Hoc Networks","volume":"20","author":"Bhorkar","year":"2012","journal-title":"IEEE\/ACM Trans. Netw."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"e4364","DOI":"10.1002\/dac.4364","article-title":"Message trust-based secure multipath routing protocol for opportunistic networks","volume":"33","author":"Dhurandher","year":"2020","journal-title":"Int. J. Commun. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.comcom.2020.07.010","article-title":"Energy-efficient unmanned aerial vehicle scanning approach with node clustering in opportunistic networks","volume":"161","author":"Bacanli","year":"2020","journal-title":"Comput. Commun."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"107327","DOI":"10.1016\/j.comnet.2020.107327","article-title":"Topology optimization against cascading failures on wireless sensor networks using a memetic algorithm","volume":"177","author":"Fu","year":"2020","journal-title":"Comput. Netw."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Khalid, K., Woungang, I., Dhurandher, S.K., Singh, J., and JPC Rodrigues, J. (2020). Energy-efficient check-and-spray geocast routing protocol for opportunistic networks. Information, 11.","DOI":"10.3390\/info11110504"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Dhurandher, S.K., Singh, J., Obaidat, M.S., Woungang, I., Srivastava, S., and Rodrigues, J.J. Reinforcement learning-based routing protocol for opportunistic networks. Proceedings of the ICC 2020-2020 IEEE International Conference on Communications (ICC).","DOI":"10.1109\/ICC40277.2020.9149039"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Rahmani, A.M., Tehrani, Z.H., and Souri, A. (2021). Evaluation of energy consumption in routing protocols for opportunistic networks. Telecommun. Syst., 1\u201333.","DOI":"10.1007\/s11235-021-00803-z"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"100384","DOI":"10.1016\/j.iot.2021.100384","article-title":"Reinforcement learning-based fuzzy geocast routing protocol for opportunistic networks","volume":"14","author":"Khalid","year":"2021","journal-title":"Internet Things"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"012801","DOI":"10.1103\/PhysRevE.89.012801","article-title":"Resiliently evolving supply-demand networks","volume":"89","author":"Rubido","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Li, Y., and Zhang, Z.L. (2010, January 14\u201319). Random Walks on Digraphs: A Theoretical Framework for Estimating Transmission Costs in Wireless Routing. Proceedings of the IEEE INFOCOM, San Diego, CA, USA.","DOI":"10.1109\/INFCOM.2010.5462109"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Darehshoorzadeh, A., Sanchez, M.I., and Boukerche, A. (2014, January 9\u201311). Modeling and Analysis of Opportunistic Routing in Multi-hop Wireless Networks. Proceedings of the IEEE 22nd International Symposium on Modelling, Analysis & Simulation of Computer and Telecommunication Systems, Paris, France.","DOI":"10.1109\/MASCOTS.2014.49"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Darehshoorzadeh, A. (2012). Opportunistic Routing in Wireless Mesh Networks. [Ph.D. Thesis, Universitat Politecnica de Catalunya].","DOI":"10.1109\/IWCMC.2012.6314199"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Cerd\u00e0-Alabern, L., Pla, V., and Darehshoorzadeh, A. (2010, January 22\u201323). On the Performance Modeling of Opportunistic Routing. Proceedings of the Second International Workshop on Mobile Opportunistic Networking, Pisa, Italy.","DOI":"10.1145\/1755743.1755747"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1109\/TII.2014.2374071","article-title":"Opportunistic Routing Algorithm for Relay Node Selection in Wireless Sensor Networks","volume":"11","author":"Luo","year":"2015","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_29","first-page":"23","article-title":"The R project in statistical computing","volume":"1","author":"Ripley","year":"2001","journal-title":"MSOR Connect. Newsl. Ltsn Maths Stats Netw."},{"key":"ref_30","unstructured":"Galarza, C.E., and Palma, J.M. (2017). Package\u2018Opportunistic\u2019:Routing Distribution, Broadcasts, Transmissions and Receptions in an Opportunistic Network, [2nd ed.]. Available online: https:\/\/cran.r-project.org\/web\/packages\/Opportunistic\/index.html."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Heimlicher, S., Karaliopoulos, M., Levy, H., and May, M. (2007, January 28\u201330). End-to-end vs. Hop-by-hop Transport Under Intermittent Connectivity. Proceedings of the 1st International Conference on Autonomic Computing and Communication Systems, Rome, Italy. Autonomics \u201907.","DOI":"10.4108\/ICST.AUTONOMICS2007.2225"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Palma, J.M., Carvalho, L.D., Goncalves, A.P., Galarza, C.E., and De Oliveira, A.M. (2015, January 14\u201316). Application of Control Theory Markov Systems to Minimize the Number of Transmissions in a Multi-hop Network. Proceedings of the 2015 Asia-Pacific Conference on Computer Aided System Engineering, Quito, Ecuador.","DOI":"10.1109\/APCASE.2015.59"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Grimmett, G.R. (1999). Percolation, Springer Science & Business Media. [1st ed.].","DOI":"10.1007\/978-3-662-03981-6"},{"key":"ref_34","unstructured":"Weiss, N., Holmes, P., and Hardy, M. (2006). A Course in Probability, Pearson Addison Wesley."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Cacciapuoti, A.S., Caleffi, M., and Paura, L. (2009, January 12\u201314). A theoretical model for opportunistic routing in ad hoc networks. Proceedings of the 2009 International Conference on Ultra Modern Telecommunications Workshops, St. Petersburg, Russia.","DOI":"10.1109\/ICUMT.2009.5345582"},{"key":"ref_36","unstructured":"Duran-Faundez, C., Palma, J.M., Orellana-Romero, E., and Melin, P. (2014, January 5\u20137). Effect of packet sampling time on a colony of mobile routing robots for communication link maintenance using IEEE 802.15. 4 devices. Proceedings of the 10th International Conference on Modeling, Optimization and Simulation, Nancy, France."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Palma, J.M., Duran-Faundez, C., De P. Carvalho, L., Morais, C., Oliveira, R.C.L.F., Rubio, E., and Herman, K. (2018). Protocol for Energy-Efficiency in Networked Control Systems Based on WSN. Sensors, 18.","DOI":"10.3390\/s18082590"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/23\/8058\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:38:50Z","timestamp":1760168330000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/23\/8058"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,2]]},"references-count":37,"journal-issue":{"issue":"23","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["s21238058"],"URL":"https:\/\/doi.org\/10.3390\/s21238058","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2021,12,2]]}}}