{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T08:05:13Z","timestamp":1776240313944,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,20]],"date-time":"2022-05-20T00:00:00Z","timestamp":1653004800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper studies a Markov model of a queuing-inventory system with primary, retrial, and feedback customers. Primary customers form a Poisson flow, and if an inventory level is positive upon their arrival, they instantly receive the items. If the inventory level is equal to zero upon arrival of a primary customer, then this customer, according to the Bernoulli scheme, either leaves the system or goes into an infinite buffer to repeat their request in the future. The rate of retrial customers is constant, and if the inventory level is zero upon arrival of a retrial customer, then this customer, according to the Bernoulli scheme, either leaves orbit or remains in orbit to repeat its request in the future. According to the Bernoulli scheme, each served primary or retrial customer either leaves the system or feedbacks into orbit to repeat their request. Destructive customers that form a Poisson flow cause damage to items. Unlike primary, retrial, and feedback customers, destructive customers do not require items, since, upon arrival of such customers, the inventory level instantly decreases by one. The system adopted one of two replenishment policies: (s, Q) or (s, S). In both policies, the lead time is a random variable that has an exponential distribution. It is shown that the mathematical model of the system under study was a two-dimensional Markov chain with an infinite state space. Algorithms for calculating the elements of the generating matrices of the constructed chains were developed, and the ergodicity conditions for both policies were found. To calculate the steady-state probabilities, a matrix-geometric method was used. Formulas were found for calculating the main performance measures of the system. The results of the numerical experiments, including the minimization of the total cost, are demonstrated.<\/jats:p>","DOI":"10.3390\/axioms11050241","type":"journal-article","created":{"date-parts":[[2022,5,20]],"date-time":"2022-05-20T13:56:12Z","timestamp":1653054972000},"page":"241","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Retrial Queuing-Inventory Systems with Delayed Feedback and Instantaneous Damaging of Items"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3721-7820","authenticated-orcid":false,"given":"Agassi","family":"Melikov","sequence":"first","affiliation":[{"name":"Institute of Control Systems, National Academy of Science, Baku AZ 1141, Azerbaijan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5073-9124","authenticated-orcid":false,"given":"Sevinj","family":"Aliyeva","sequence":"additional","affiliation":[{"name":"Faculty of Engineering, Baku Engineering University, Baku AZ 0101, Azerbaijan"}]},{"given":"Sajeev S.","family":"Nair","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government Engineering College, Thrissur 680009, India"}]},{"given":"B. Krishna","family":"Kumar","sequence":"additional","affiliation":[{"name":"Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai 600127, India"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Artalejo, J.R., and Gomez-Corral, A. (2008). Retrial Queueing Systems: A Computational Approach, Springer.","DOI":"10.1007\/978-3-540-78725-9"},{"key":"ref_2","unstructured":"Phung-Duc, T. (2019). Retrial queueing models: A survey on theory and applications. arXiv."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.apm.2017.10.025","article-title":"On an unreliable-server retrial queue with customer feedback and impatience","volume":"55","author":"Chang","year":"2018","journal-title":"Appl. Math. 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