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Our method is near optimal among all approximation methods using discrete distributions. We apply the developed method to compute the stationary distribution of virtual waiting time and associated performance measures for a GI\/GI\/1+GI queue in which the large market assumption may not hold and the patience time may follow any bounded distribution. Numerical experiments show that our method outperforms steady-state simulation, phase-type approximation, diffusion approximation, and fluid approximation, particularly for medium or small arrival intensities in overloaded or balanced loaded queues.<\/jats:p>\n                  <jats:p>Funding: S. Li and S. Mehrotra were supported by the National Science Foundation grant CMMI-1763035.<\/jats:p>","DOI":"10.1287\/moor.2024.0468","type":"journal-article","created":{"date-parts":[[2025,6,12]],"date-time":"2025-06-12T12:01:28Z","timestamp":1749729688000},"page":"1626-1658","source":"Crossref","is-referenced-by-count":0,"title":["A New Finite Approximation Method for Evaluating Steady-State Performance of a Continuous-State Markov Chain with an Application to Queues with Customer Abandonment"],"prefix":"10.1287","volume":"51","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-1406-5803","authenticated-orcid":false,"given":"Shukai","family":"Li","sequence":"first","affiliation":[{"name":"Northwestern University, Evanston, Illinois 60208"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1106-1901","authenticated-orcid":false,"given":"Sanjay","family":"Mehrotra","sequence":"additional","affiliation":[{"name":"Northwestern University, Evanston, Illinois 60208"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"109","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1937-5956.2007.tb00288.x"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.2013.1730"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1287\/opre.2016.1542"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1093962237"},{"key":"B5","volume-title":"Collectively Compact Operator Approximation Theory and Applications to Integral Equations","author":"Anselone PM","year":"1971"},{"key":"B6","unstructured":"Baccelli F, Hebuterne G (1981) On queues with impatient customers.\n                      Performance\u201981 Proc. 8th Internat. 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