{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:34:26Z","timestamp":1787337266679,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>In this paper tools are developed to analyse a recently proposed random matrix model of communication networks that employ additive\u2010increase multiplicative\u2010decrease (AIMD) congestion control algorithms. We investigate properties of the Markov process describing the evolution of the window sizes of network users. Using paracontractivity properties of the matrices involved in the model, it is shown that the process has a unique invariant probability, and the support of this probability is characterized. Based on these results we obtain a weak law of large numbers for the average distribution of resources between the users of a network. This shows that under reasonable assumptions such networks have a well\u2010defined stochastic equilibrium. ns2 simulation results are discussed to validate the obtained formulae. (The simulation program ns2, or network simulator, is an industry standard for the simulation of Internet dynamics.)<\/jats:p>","DOI":"10.1137\/040620953","type":"journal-article","created":{"date-parts":[[2006,10,30]],"date-time":"2006-10-30T12:46:56Z","timestamp":1162212416000},"page":"703-723","source":"Crossref","is-referenced-by-count":21,"title":["Stochastic Equilibria of AIMD Communication Networks"],"prefix":"10.1137","volume":"28","author":[{"given":"F.","family":"Wirth","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R.","family":"Stanojevi\u0107","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R.","family":"Shorten","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D.","family":"Leith","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,9,19]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-5316(02)00106-2"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"F. Baccelli and D. Hong,\n                      AIMD, fairness and fractal scaling of TCP traffic\n                      , in Proceedings of the IEEE INFOCOM 2002, New York, 2002, pp. 229\u2013238.","DOI":"10.1109\/INFCOM.2002.1019264"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"F. Baccelli and D. Hong,\n                      Interaction of TCP flows as billiards\n                      , in Proceedings of the IEEE INFOCOM 2003, San Francisco, 2003, pp. 841\u2013853.","DOI":"10.1109\/TNET.2005.852883"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"A. Berman and R. J. Plemmons,\n                      Nonnegative Matrices in the Mathematical Sciences\n                      , SIAM, Philadelphia, 1994.","DOI":"10.1137\/1.9781611971262"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"P. Brown,\n                      Resource sharing of TCP connections with different round trip times\n                      , in Proceedings of the IEEE INFOCOM 2000, Tel Aviv, Israel, 2000, pp. 1734\u20131741.","DOI":"10.1109\/INFCOM.2000.832573"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"Y. Chait, C. V. Hollot, V. Misra, H. Han, and Y. Halevi,\n                      Dynamic analysis of congested TCP networks\n                      , in Proceedings of American Control Conference, San Diego, 1999, pp. 2430\u20132435.","DOI":"10.1109\/ACC.2002.1024007"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(90)90206-R"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700004168"},{"key":"R9","first-page":"115","volume":"3","author":"Floyd S.","year":"1992","journal-title":"J. Internetworking: Practice and Experience"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"D. Hartfiel,\n                      Nonhomogeneous Matrix Products\n                      , World Scientific, Singapore, 2002.","DOI":"10.1142\/4707"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"J. Hespanha,\n                      Stochastic hybrid systems: Application to communication networks\n                      , in Hybrid Systems, Computation, and Control, R. Alur and G. J. Pappas, eds., Proceedings of HSCC 2004, Lecture Notes in Comput. Sci. 2993, Springer, Heidelberg, 2004, pp. 387\u2013401.","DOI":"10.1007\/978-3-540-24743-2_26"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"J. Hespanha, S. Bohacek, K. Obraczka, and J. Lee,\n                      Hybrid modeling of TCP congestion control\n                      , in Hybrid Systems, Computation, and Control, M. D. Di Benedetto and A. L. Sangiovanni\u2010Vicentelli, eds., Proceedings of the HSCC 2001, Lecture Notes in Comput. Sci. 2034, Springer, Heidelberg, 2001, pp. 291\u2013304.","DOI":"10.1007\/3-540-45351-2_25"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"C. V. Hollot and Y. Chait,\n                      Nonlinear stability analysis of a class of TCP\/AQM networks\n                      , in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, 2001, pp. 2309\u20132314.","DOI":"10.1109\/CDC.2001.980604"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"C. V. Hollot, V. Misra, D. Towsley, and W. Gong,\n                      A control theoretic analysis of RED\n                      , in Proceedings of the IEEE INFOCOM, Anchorage, AK, 2001, pp. 1510\u20131519.","DOI":"10.1109\/INFCOM.2001.916647"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2002.1008360"},{"key":"R16","unstructured":"D. Hong and D. Lebedev,\n                      Many TCP User Asymptotic Analysis of the AIMD Model\n                      , Tech. report INRIA 4229, INRIA Rocquencourt, 2001."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1109\/90.974534"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"F. P. Kelly,\n                      Mathematical modelling of the internet\n                      , in Proceedings of the ICIAM 99 (Edinburgh), 4th International Congress of Industrial Applied Mathematics, Oxford University Press, Oxford, UK, 2000, pp. 105\u2013116.","DOI":"10.1093\/oso\/9780198505143.003.0010"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2003.819295"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1109\/37.980245"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/S0005-1098(99)00128-4"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2002.1008356"},{"key":"R23","doi-asserted-by":"crossref","unstructured":"S. P. Meyn and R. L. Tweedie,\n                      Markov Chains and Stochastic Stability\n                      , Comm. Control Engrg. Ser., Springer, London, 1993.","DOI":"10.1007\/978-1-4471-3267-7"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1109\/TNET.2006.876178"},{"key":"R25","unstructured":"R. Shorten, C. King, F. Wirth, and D. Leith,\n                      Modelling TCP congestion control dynamics in droptail environments\n                      , Automatica, to appear."},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2004.09.017"},{"key":"R27","unstructured":"R. Srikant,\n                      Internet Congestion Control\n                      , Control Theory 14, Birkh\u00e4user Boston, Boston, MA, 2004."},{"key":"R28","doi-asserted-by":"crossref","unstructured":"G. Vinnicombe,\n                      On the stability of networks operating TCP\u2010like congestion control\n                      , in Proceedings of the 15th IFAC World Congress on Automatic Control, Barcelona, Spain, 2002.","DOI":"10.3182\/20020721-6-ES-1901.00370"},{"key":"R29","unstructured":"L. Xu, K. Harfoush, and I. Rhee,\n                      Binary increase congestion control (BIC) for fast long\u2010distance networks\n                      , in Proceedings of IEEE INFOCOM 2004, Hong Kong, 2004, pp. 2514\u20132524."}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/040620953","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:53:28Z","timestamp":1787334808000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/040620953"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1]]},"references-count":29,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["10.1137\/040620953"],"URL":"https:\/\/doi.org\/10.1137\/040620953","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,1]]}}}