{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:31:36Z","timestamp":1787340696432,"version":"3.56.0"},"reference-count":27,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Math."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>Stability analysis of a single-lane microscopic car-following model is studied analytically from the perspective of delayed reactions of human drivers. In the literature, the delayed reactions of the drivers are modeled with discrete delays, which assume that drivers make their control decisions based on the stimuli they receive from a point of time in the history. We improve this model by introducing a distribution of delays, which assumes that the control actions are based on information distributed over an interval of time in history. Such an assumption is more realistic, as it takes into consideration the memory capabilities of the drivers and the inevitable heterogeneity of their delay times. We calculate exact stability regions in the parameter space of some realistic delay distributions. Case studies are provided demonstrating the application of the results.<\/jats:p>","DOI":"10.1137\/060673813","type":"journal-article","created":{"date-parts":[[2007,12,20]],"date-time":"2007-12-20T18:32:01Z","timestamp":1198175521000},"page":"738-759","source":"Crossref","is-referenced-by-count":172,"title":["Stability of Traffic Flow Behavior with Distributed Delays Modeling the Memory Effects of the Drivers"],"prefix":"10.1137","volume":"68","author":[{"given":"Rifat","family":"Sipahi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fatihcan M.","family":"Atay","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Silviu-Iulian","family":"Niculescu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,12,20]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"F. M. Atay,\n                      Distributed delays facilitate amplitude death of coupled oscillators\n                      , Phys. Rev. Lett., 91 (2003), paper 094101.","DOI":"10.1103\/PhysRevLett.91.094101"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.58.5429"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141000376086"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1287\/opre.6.2.165"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/508772"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/S0378-4371(02)01457-7"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1207\/STHF0203_1"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1287\/opre.7.1.79"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2005.02.034"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"K. Gu, V. L. Kharitonov, and J. Chen,\n                      Stability of Time-Delay Systems\n                      , Birkh\u00e4user Boston, Cambridge, MA, 2003.","DOI":"10.1007\/978-1-4612-0039-0"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1993.1312"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.73.1067"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"A. R. Horn and C. R. Johnson,\n                      Matrix Analysis\n                      , Cambridge University Press, Cambridge, UK, 1985.","DOI":"10.1017\/CBO9780511810817"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"G. Iooss and D. D. Joseph,\n                      Elementary Stability and Bifurcation Theory\n                      , Undergrad. Texts in Math., Springer-Verlag, New York, 1980.","DOI":"10.1007\/978-1-4684-9336-8"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"W. Michiels and S.I. Niculescu,\n                      Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach\n                      , Adv. Des. Control 12, SIAM, Philadelphia, 2007.","DOI":"10.1137\/1.9780898718645"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"R. M. Murray, ed.\n                      Control in an Information Rich World: Report of the Panel on Future Directions in Control, Dynamics, and Systems\n                      , SIAM, Philadelphia, 2003.","DOI":"10.1137\/1.9780898718010"},{"key":"R17","unstructured":"S.I. Niculescu,\n                      Delay Effects on Stability: A Robust Control Approach\n                      , Lecture Notes in Control and Inform. Sci. 269, Springer-Verlag, Heidelberg, 2001."},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-004-0625-4"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"G. Orosz, R. E. Wilson, and B. Krauskopf,\n                      Global bifurcation investigation of an optimal velocity traffic model with driver reaction time\n                      , Phys. Rev. E, 70 (2004), paper 026207.","DOI":"10.1103\/PhysRevE.70.026207"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2005.09.004"},{"key":"R21","unstructured":"R. W. Rothery, in Traffic Flow Theory, 2nd ed., TRB Special Report Volume 165, N. H. Gartner, C. J. Messner, and A. J. Rathi, eds., Transportation Research Board (TRB), Washington, DC, 1992, Chapter 4."},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2005.03.022"},{"key":"R23","doi-asserted-by":"crossref","unstructured":"R. Sipahi and S.I. Niculescu,\n                      Analytical stability study of a deterministic car following model under multiple delay interactions\n                      , in Proceedings of the 6th IFAC Time Delay Systems Workshop, L'Aquila, Italy, 2006, International Federation of Automatic Control\/Elsevier, Oxford, UK, 2006.","DOI":"10.3182\/20060710-3-IT-4901.00031"},{"key":"R24","unstructured":"G. Stepan,\n                      Retarded Dynamical Systems: Stability and Characteristic Function\n                      , Longman Scientific & Technical\/John Wiley & Sons, New York, 1989."},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1007\/BF01211858"},{"key":"R26","doi-asserted-by":"crossref","unstructured":"M. Treiber and D. Helbing,\n                      Memory effects in microscopic traffic models and wide scattering in flow-density data\n                      , Phys. Rev. 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