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Components are exposed to a combination of extreme shocks and\n                    <jats:italic>\u03b4<\/jats:italic>\n                    \u2010shock. The reliability of each subsystem's components is modeled using a phase\u2010type distribution and Markov chain approach, with component reliability being incorporated into subsystem reliability calculations. Disjoint cases are classified based on the number of failed components in shared regions to derive the shared component failure probability matrix and the initial probability distribution. Using the transfer matrix and transition rate matrix, the overall system reliability is determined through the Markov chain embedding approach. For this model, the total cost function of the system was determined by calculating the cost functions of different scenarios and different subsystems, and the optimal maintenance time for the system was obtained. The effectiveness of the method is verified by taking the unmanned aerial vehicle swarm working in a harsh environment as an example.\n                  <\/jats:p>","DOI":"10.1002\/qre.70143","type":"journal-article","created":{"date-parts":[[2025,12,28]],"date-time":"2025-12-28T10:19:08Z","timestamp":1766917148000},"page":"1302-1320","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Reliability Assessment and Maintenance Policy of Polygonal\n                    <i>k<\/i>\n                    \u2010Out\u2010of\u2010\n                    <i>n<\/i>\n                    Systems Under Mixed Shock Sources"],"prefix":"10.1002","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-7977-6158","authenticated-orcid":false,"given":"Ruida","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mechanical Engineering and Automation Northeastern University  Shenyang 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