{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,11]],"date-time":"2026-08-11T17:00:38Z","timestamp":1786467638707,"version":"build-2736575974"},"reference-count":34,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2026,8,11]],"date-time":"2026-08-11T00:00:00Z","timestamp":1786406400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["J. ACM"],"published-print":{"date-parts":[[2026,8,31]]},"abstract":"<jats:p>We consider a dynamic model of traffic that has received a lot of attention in the past few years. Infinitesimally small agents aim to travel from a source to a destination as quickly as possible. Flow patterns vary over time, and congestion effects are modeled via queues, which form based on the deterministic queueing model whenever the inflow into a link exceeds its capacity.<\/jats:p>\n                  <jats:p>Are equilibria in this model meaningful as a prediction of traffic behavior? For this to be the case, a certain notion of stability under ongoing perturbations is needed. Real traffic consists of discrete, atomic \u201cpackets\u201d, rather than being a continuous flow of non-atomic agents. Users may not choose an absolutely quickest route available, if there are multiple routes with very similar travel times. We would hope that in both these situations \u2014 a discrete packet model, with packet size going to 0, and \u03b5-equilibria, with \u03b5 going to 0 \u2014 equilibria converge to continuous and exact equilibria in the flow over time model. No such convergence results were known.<\/jats:p>\n                  <jats:p>We show that such a convergence result does hold in single-commodity instances for both of these settings, in a unified way. More precisely, we introduce a notion of \u201cstrict\u201d \u03b5-equilibria, and show that these must converge to the exact dynamic equilibrium in the limit as \u03b5 \u2192 0. We then show that results for the two settings mentioned can be deduced from this with only moderate further technical effort.<\/jats:p>","DOI":"10.1145\/3815167","type":"journal-article","created":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T11:38:08Z","timestamp":1777981088000},"page":"1-40","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Convergence of Approximate and Packet Routing Equilibria to Nash Flows Over Time"],"prefix":"10.1145","volume":"73","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8897-5459","authenticated-orcid":false,"given":"Neil","family":"Olver","sequence":"first","affiliation":[{"name":"Department of Mathematics, The London School of Economics and Political Science","place":["London, United Kingdom of Great Britain and Northern Ireland"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2953-1115","authenticated-orcid":false,"given":"Leon","family":"Sering","sequence":"additional","affiliation":[{"name":"ETH Zurich","place":["Zurich, Switzerland"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7499-5958","authenticated-orcid":false,"given":"Laura","family":"Vargas Koch","sequence":"additional","affiliation":[{"name":"School of Business and Economics, RWTH Aachen University","place":["Aachen, Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2026,8,11]]},"reference":[{"key":"e_1_3_4_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-16170-4_8"},{"key":"e_1_3_4_3_2","doi-asserted-by":"publisher","DOI":"10.4086\/toc.2017.v013a013"},{"key":"e_1_3_4_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.geb.2013.09.004"},{"key":"e_1_3_4_5_2","doi-asserted-by":"publisher","DOI":"10.1145\/3033274.3085101"},{"key":"e_1_3_4_6_2","doi-asserted-by":"publisher","DOI":"10.5555\/3202838.3202841"},{"key":"e_1_3_4_7_2","doi-asserted-by":"publisher","DOI":"10.1287\/opre.2020.2081"},{"key":"e_1_3_4_8_2","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2021.1173"},{"key":"e_1_3_4_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.trb.2018.08.015"},{"key":"e_1_3_4_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-022-01772-0"},{"key":"e_1_3_4_11_2","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2022.1336"},{"issue":"310","key":"e_1_3_4_12_2","first-page":"1","article-title":"Prediction equilibrium for dynamic network flows","volume":"24","author":"Graf L.","year":"2023","unstructured":"L. Graf, T. Harks, K. Kollias, and M. Markl. 2023. Prediction equilibrium for dynamic network flows. Journal of Machine Learning Research 24, 310 (2023), 1\u201333. 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Sering. 2020. Nash Flows Over Time. Ph. D. Dissertation. Technische Universit\u00e4t Berlin."},{"key":"e_1_3_4_29_2","first-page":"12:1","volume-title":"Proceedings of the 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems","author":"Sering L.","year":"2018","unstructured":"L. Sering and M. Skutella. 2018. Multi-source multi-sink Nash flows over time. In Proceedings of the 18th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems. 12:1\u201312:20."},{"key":"e_1_3_4_30_2","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611975482.57"},{"key":"e_1_3_4_31_2","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2022.1318"},{"key":"e_1_3_4_32_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-92702-8_11"},{"issue":"2","key":"e_1_3_4_33_2","first-page":"251","article-title":"Congestion theory and transport investment","volume":"59","author":"Vickrey W. S.","year":"1969","unstructured":"W. S. Vickrey. 1969. Congestion theory and transport investment. 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