Towards Stability Analysis of Data Transport Mechanisms: a Fluid Model and an Application

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1 Towards Stability Analysis of Data Transport Mechanisms: a Fluid Model and an Application Gayane Vardoyan *, C. V. Hollot, Don Towsley* * College of Information and Computer Sciences, Department of Electrical and Computer Engineering

2 Background n Stability well-understood: TCP Reno, STCP. n Less so: CUBIC, H-TCP. n Reason: lack of a suitable modeling framework. 2

3 Contributions of this Work n New modeling framework applicable to wide variety of loss-based protocols. n Application to TCP CUBIC. n Result: CUBIC is locally asymptotically stable. n Simulation framework to test and validate the models. 3

4 Brief Introduction to Stability unstable equilibrium global stable equilibrium two local stable equilibria and one unstable equilibrium 4

5 Stability: Difference in Responses x(t) stable, monotonic response with convergence time 5

6 Stability: Difference in Responses x(t) stable, oscillatory response with convergence time 6

7 Stability: Difference in Responses x(t) oscillatory response without convergence time 7

8 Stability: Difference in Responses Undamped oscillation is undesirable behavior. Convergence to a fixed point is desirable behavior. 8

9 Instability with TCP CUBIC Cwnd (segments) Fluid model simulation. Link capacity of 1 Gbps, delay of 100 ms Time (s) This choice of initial conditions leads to unstable behavior. 9

10 Stability with TCP CUBIC Cwnd (segments) Fluid model simulation. Link capacity of 1 Gbps, delay of 100 ms Time (s) This choice of initial conditions leads to stable behavior. 10

11 The Lesson TCP CUBIC can be ill-behaved. In general, more deviation from fixed-point à more instability. Impact on performance metrics like bandwidth utilization. Effective modeling à conditions for stability à ensure efficient operation of the protocol. 11

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sha1_base64="xlchz3hyeu/ck4jc9xcf0etzao0=">aaab53icbvbns8naej3ur1q/qh69lbbbu0leug9flx5bmlbqhrlzttu1m03y3ygl9bd48adi1b/kzx/jts1bwx8mpn6bywzemaiujet+o4wv1bx1jejmawt7z3evvh9wr+numfrzlglvcqlgwsx6hhubruqhjukbzxb0m/wbj6g0j+wdgscyrhqgez8zaqzueoqwk27vnyesey8nfchr75a/or2yprfkwwtvuu25iqkyqgxnaieltqoxowxeb9i2vniidzdndp2qe6v0sd9wtqqhm/x3reyjrcdradsjaoz60zuk/3nt1pqvg4zljduo2xxrpxxexgt6nelxhcyisswukw5vjwxifwxgzloyixilly8t/6x6vfua55xadz5gey7gge7bgwuows3uwqcgcm/wcm/og/pivdsf89ack88cwh84nz9un4zq</latexit> Definitions Term Definition per-flow capacity W max (t) s(t) W (t) p(t) link delay the size of the cwnd immediately before loss the time elapsed since last loss the cwnd as a function of time a loss probability function fixed-point value of 13

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<latexit sha1_base64="dh/5ouazgwbhns4b529alentnsk=">aaab53icbvbns8naej34wetx1aoxxsj4koki6q3oxwmlxhbaudbbsbt2swm7g6ge/givhls8+pe8+w/ctjlo64obx3szzmwlu8g1cd1vz2v1bx1js7rv3t7z3duvhbw+6crtdh2wies1q6prcim+4uzgo1vi41bgkxzdtv3weyrne3lvxikgmr1ihnfgjzwarfepujv3brjmvijuoucjv/nq9howxsgne1trjuemjsipmpwjnjs7mcaushedymdsswpuqt47dejordinuajssunm6u+jnmzaj+pqdsbudpwinxx/8zqzia6cnms0myjzffgucwismv2a9llczstyesout7csnqskmmozkdsqvmwxl4l/xruuec2lav2mskmex3acz+dbjdthdhrgawoez3ifn+frexheny9564ptzbzbhzifpztgjls=</latexit> TCP CUBIC: Definition Congestion window function is given by W (t) =c s(t) r! 3 3 Wmax (t)b + W max(t) c - scaling factor, - multiplicative decrease constant, s(t) W max (t) last loss. - elapsed time since last loss, - size of cwnd immediately before 14

15 <latexit sha1_base64="y4c6lzfnezmegionan7rovwsi08=">aaab6nicbvbns8naej34wetx1aoxxslus0leug9flx4rgftoq9lsn+3s3ststors+he8efdx6i/y5r9x0+agrq8ghu/nmdmvtkuw6lrfzsrq2vrgzmmrvl2zu7dfoth8nemmgfdzihpddqnhustcr4gst1pnqqolb4wj29xvpxftrbi/4djlgakdwescucylvg3pepwqw3dnimvek0gvcjr7la9up2gz4jeysy3peg6kwyrqfezyabmbgz5snqid3re0poqbydk7dupordinuajtxuhm6u+jcvxgjfvooxxfovn0cve/r5nhdbvmrjxmygm2xxrlkmbc8sdjx2joui4touwleythq6opqxtp2ybglb68tpzz+nxdu7+onm6knepwdcdqaw8uoqf30aqfgazhgv7hzvhoi/pufmxbv5xi5gj+wpn8ab0sjzi=</latexit> <latexit 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sha1_base64="y4c6lzfnezmegionan7rovwsi08=">aaab6nicbvbns8naej34wetx1aoxxslus0leug9flx4rgftoq9lsn+3s3ststors+he8efdx6i/y5r9x0+agrq8ghu/nmdmvtkuw6lrfzsrq2vrgzmmrvl2zu7dfoth8nemmgfdzihpddqnhustcr4gst1pnqqolb4wj29xvpxftrbi/4djlgakdwescucylvg3pepwqw3dnimvek0gvcjr7la9up2gz4jeysy3peg6kwyrqfezyabmbgz5snqid3re0poqbydk7dupordinuajtxuhm6u+jcvxgjfvooxxfovn0cve/r5nhdbvmrjxmygm2xxrlkmbc8sdjx2joui4touwleythq6opqxtp2ybglb68tpzz+nxdu7+onm6knepwdcdqaw8uoqf30aqfgazhgv7hzvhoi/pufmxbv5xi5gj+wpn8ab0sjzi=</latexit> ŝ<latexit 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sha1_base64="8vi6x9d9ju+x45l07vtcqzsbqm8=">aaab9hicbvbns8naej3ur1q/qh69lbbbu0leug9flx4rgftoy9lsn+3s3u3y3agl5h948adi1r/jzx/jts1bwx8mpn6bywzemhcmjet+o6wl5zxvtfj6zwnza3unurt3p+nueeqtmmeqhwjnozpun8xw2k4uxslktbworiz+64eqzwj5a8yjdqqesbyxgo2v7rtdbljw3su6aj/lvwrnrbttoexifaqgbzq96le3h5nuugkix1p3pdcxqyavyyttvnjnnu0wgeeb7vgqsaa6ykzx5+jikn0uxcqwngiq/p7isnb6lelbkbaz6nlviv7ndvitnqczk0lqqcszrvhkkynrjaluz4osw8ewykkyvrwrivaygbtuxybgzb+8spyt+kxduzmtns6lnmpwaidwdb6cqqouoqk+efdwdk/w5jw6l8678zfrltnfzd78gfp5a4hakrs=</latexit> ŝ<latexit sha1_base64="z4aywzrrmb7bdvbo7ljyyh19v2c=">aaab7xicbvbns8naej3ur1q/qh69lbbbu0leqn6kxjxwmlbqhrlzbtqlm03ynqgl9ed48adi1f/jzx/jts1bwx8mpn6bywzemeph0hw/ndla+sbmvnm7sro7t39qptx6nemmgfdzihpdcanhuijuo0djo6nmna4lb4fj25nffulaieq94ctlquyhskscubrsuzeimjtpv1pz6+4czjv4balbgva/+tubjcyluuimqtfdz00xyklgwssfvnqz4sllyzrkxusvjbkj8vm5u3jmlqgjem1lizmrvydyghsziupbgvmcmwvvjv7ndtomrojcqdrdrthiuzrjggmz/u4gqnogcmijzvrywwkbuu0z2oqqngrv+evv4l/ur+ve/wwtevokuyytoivz8kabtbidfvjayazp8apvtuq8oo/ox6k15bqzx/ahzucpfhepma==</latexit> <latexit sha1_base64="z4aywzrrmb7bdvbo7ljyyh19v2c=">aaab7xicbvbns8naej3ur1q/qh69lbbbu0leqn6kxjxwmlbqhrlzbtqlm03ynqgl9ed48adi1f/jzx/jts1bwx8mpn6bywzemeph0hw/ndla+sbmvnm7sro7t39qptx6nemmgfdzihpdcanhuijuo0djo6nmna4lb4fj25nffulaieq94ctlquyhskscubrsuzeimjtpv1pz6+4czjv4balbgva/+tubjcyluuimqtfdz00xyklgwssfvnqz4sllyzrkxusvjbkj8vm5u3jmlqgjem1lizmrvydyghsziupbgvmcmwvvjv7ndtomrojcqdrdrthiuzrjggmz/u4gqnogcmijzvrywwkbuu0z2oqqngrv+evv4l/ur+ve/wwtevokuyytoivz8kabtbidfvjayazp8apvtuq8oo/ox6k15bqzx/ahzucpfhepma==</latexit> <latexit sha1_base64="z4aywzrrmb7bdvbo7ljyyh19v2c=">aaab7xicbvbns8naej3ur1q/qh69lbbbu0leqn6kxjxwmlbqhrlzbtqlm03ynqgl9ed48adi1f/jzx/jts1bwx8mpn6bywzemeph0hw/ndla+sbmvnm7sro7t39qptx6nemmgfdzihpdcanhuijuo0djo6nmna4lb4fj25nffulaieq94ctlquyhskscubrsuzeimjtpv1pz6+4czjv4balbgva/+tubjcyluuimqtfdz00xyklgwssfvnqz4sllyzrkxusvjbkj8vm5u3jmlqgjem1lizmrvydyghsziupbgvmcmwvvjv7ndtomrojcqdrdrthiuzrjggmz/u4gqnogcmijzvrywwkbuu0z2oqqngrv+evv4l/ur+ve/wwtevokuyytoivz8kabtbidfvjayazp8apvtuq8oo/ox6k15bqzx/ahzucpfhepma==</latexit> <latexit sha1_base64="z4aywzrrmb7bdvbo7ljyyh19v2c=">aaab7xicbvbns8naej3ur1q/qh69lbbbu0leqn6kxjxwmlbqhrlzbtqlm03ynqgl9ed48adi1f/jzx/jts1bwx8mpn6bywzemeph0hw/ndla+sbmvnm7sro7t39qptx6nemmgfdzihpdcanhuijuo0djo6nmna4lb4fj25nffulaieq94ctlquyhskscubrsuzeimjtpv1pz6+4czjv4balbgva/+tubjcyluuimqtfdz00xyklgwssfvnqz4sllyzrkxusvjbkj8vm5u3jmlqgjem1lizmrvydyghsziupbgvmcmwvvjv7ndtomrojcqdrdrthiuzrjggmz/u4gqnogcmijzvrywwkbuu0z2oqqngrv+evv4l/ur+ve/wwtevokuyytoivz8kabtbidfvjayazp8apvtuq8oo/ox6k15bqzx/ahzucpfhepma==</latexit> t<latexit sha1_base64="l+mdhfflejvcyqvuvkqqvzbwxvy=">aaab53icbvbns8naen34wetx1aoxxsj4koki6q3oxwmlxhbaudbbsbt2swm7e6ge/givhls8+pe8+w/ctjlo64obx3szzmwluykmuu63s7k6tr6xwdoqb+/s7u1xdg4ftjjpdj5pzklbitmghqifbupopxpyhepohapbqd96am1eou5xneiqs4eskeamrdtexqxq1twz6dlxclilbrq9yle3n/asbovcmmm6nptikdongkuylluzgztxertax1lfyjbbpjt0qk+t0qdrom0ppdp190toympgcwg7y4zds+hnxf+8tobrvzallwyiis8xrzmkmndp17qvnhcuy0sy18lesvmqacbrzlo2ixilly8t/7x2xfoaf9x6tzfgiryte3jgphjj6usonihpoahytf7jm/povdjvzse8dcupzo7ihzifp06tjmw=</latexit> <latexit sha1_base64="l+mdhfflejvcyqvuvkqqvzbwxvy=">aaab53icbvbns8naen34wetx1aoxxsj4koki6q3oxwmlxhbaudbbsbt2swm7e6ge/givhls8+pe8+w/ctjlo64obx3szzmwluykmuu63s7k6tr6xwdoqb+/s7u1xdg4ftjjpdj5pzklbitmghqifbupopxpyhepohapbqd96am1eou5xneiqs4eskeamrdtexqxq1twz6dlxclilbrq9yle3n/asbovcmmm6nptikdongkuylluzgztxertax1lfyjbbpjt0qk+t0qdrom0ppdp190toympgcwg7y4zds+hnxf+8tobrvzallwyiis8xrzmkmndp17qvnhcuy0sy18lesvmqacbrzlo2ixilly8t/7x2xfoaf9x6tzfgiryte3jgphjj6usonihpoahytf7jm/povdjvzse8dcupzo7ihzifp06tjmw=</latexit> <latexit sha1_base64="l+mdhfflejvcyqvuvkqqvzbwxvy=">aaab53icbvbns8naen34wetx1aoxxsj4koki6q3oxwmlxhbaudbbsbt2swm7e6ge/givhls8+pe8+w/ctjlo64obx3szzmwluykmuu63s7k6tr6xwdoqb+/s7u1xdg4ftjjpdj5pzklbitmghqifbupopxpyhepohapbqd96am1eou5xneiqs4eskeamrdtexqxq1twz6dlxclilbrq9yle3n/asbovcmmm6nptikdongkuylluzgztxertax1lfyjbbpjt0qk+t0qdrom0ppdp190toympgcwg7y4zds+hnxf+8tobrvzallwyiis8xrzmkmndp17qvnhcuy0sy18lesvmqacbrzlo2ixilly8t/7x2xfoaf9x6tzfgiryte3jgphjj6usonihpoahytf7jm/povdjvzse8dcupzo7ihzifp06tjmw=</latexit> <latexit sha1_base64="l+mdhfflejvcyqvuvkqqvzbwxvy=">aaab53icbvbns8naen34wetx1aoxxsj4koki6q3oxwmlxhbaudbbsbt2swm7e6ge/givhls8+pe8+w/ctjlo64obx3szzmwluykmuu63s7k6tr6xwdoqb+/s7u1xdg4ftjjpdj5pzklbitmghqifbupopxpyhepohapbqd96am1eou5xneiqs4eskeamrdtexqxq1twz6dlxclilbrq9yle3n/asbovcmmm6nptikdongkuylluzgztxertax1lfyjbbpjt0qk+t0qdrom0ppdp190toympgcwg7y4zds+hnxf+8tobrvzallwyiis8xrzmkmndp17qvnhcuy0sy18lesvmqacbrzlo2ixilly8t/7x2xfoaf9x6tzfgiryte3jgphjj6usonihpoahytf7jm/povdjvzse8dcupzo7ihzifp06tjmw=</latexit> TCP CUBIC in Steady State W (t) Ŵ max 15

16 <latexit sha1_base64="ilacqpywm/xx7b1jqwsh1vfl10m=">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</latexit> <latexit sha1_base64="ilacqpywm/xx7b1jqwsh1vfl10m=">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</latexit> <latexit sha1_base64="ilacqpywm/xx7b1jqwsh1vfl10m=">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</latexit> <latexit sha1_base64="ilacqpywm/xx7b1jqwsh1vfl10m=">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</latexit> <latexit sha1_base64="ombsvi/lx8alw+grlh7hdyieinu=">aaacj3icbvdlsgmxfm34rpvvdekmwir20tjtbhuhft24rgbtovnkjpnpqzmpkjtcgezz3pgrbkrudomfmol0oa0haifn3hute5xicawm+wusla+srq0xnoqbw9s7u6w9/xsvxpkyng1fklsouuzwglwbg2ddsdlio4j1npf15ncemfq8do5gerg+t4yb9zglokvb6dl2jkgj26lanu1cspefziurtwwgcvrlr3lbi7wfhu6sctqytzoolc26oqvejnamlnemruhp1xzdgvssacqiuj3ljkcfeamccpyw7vixinaxgbkepghxmeon00vtfkwvf3uh1ccapfv/dytev2rio7rsjzbs814m/uf1yvdo+gkpohhyqpohvfhgchgwgna5zbterbncjdd/xxredc6gsy3qekz5lrdju1e/r1u3j+xm1synajper6iclhskmuggtvabufsintebejeejbfjw/jms5emwc8b+gpj+wchn6yp</latexit> <latexit sha1_base64="ombsvi/lx8alw+grlh7hdyieinu=">aaacj3icbvdlsgmxfm34rpvvdekmwir20tjtbhuhft24rgbtovnkjpnpqzmpkjtcgezz3pgrbkrudomfmol0oa0haifn3hute5xicawm+wusla+srq0xnoqbw9s7u6w9/xsvxpkyng1fklsouuzwglwbg2ddsdlio4j1npf15ncemfq8do5gerg+t4yb9zglokvb6dl2jkgj26lanu1cspefziurtwwgcvrlr3lbi7wfhu6sctqytzoolc26oqvejnamlnemruhp1xzdgvssacqiuj3ljkcfeamccpyw7vixinaxgbkepghxmeon00vtfkwvf3uh1ccapfv/dytev2rio7rsjzbs814m/uf1yvdo+gkpohhyqpohvfhgchgwgna5zbterbncjdd/xxredc6gsy3qekz5lrdju1e/r1u3j+xm1synajper6iclhskmuggtvabufsintebejeejbfjw/jms5emwc8b+gpj+wchn6yp</latexit> <latexit sha1_base64="ombsvi/lx8alw+grlh7hdyieinu=">aaacj3icbvdlsgmxfm34rpvvdekmwir20tjtbhuhft24rgbtovnkjpnpqzmpkjtcgezz3pgrbkrudomfmol0oa0haifn3hute5xicawm+wusla+srq0xnoqbw9s7u6w9/xsvxpkyng1fklsouuzwglwbg2ddsdlio4j1npf15ncemfq8do5gerg+t4yb9zglokvb6dl2jkgj26lanu1cspefziurtwwgcvrlr3lbi7wfhu6sctqytzoolc26oqvejnamlnemruhp1xzdgvssacqiuj3ljkcfeamccpyw7vixinaxgbkepghxmeon00vtfkwvf3uh1ccapfv/dytev2rio7rsjzbs814m/uf1yvdo+gkpohhyqpohvfhgchgwgna5zbterbncjdd/xxredc6gsy3qekz5lrdju1e/r1u3j+xm1synajper6iclhskmuggtvabufsintebejeejbfjw/jms5emwc8b+gpj+wchn6yp</latexit> <latexit sha1_base64="ombsvi/lx8alw+grlh7hdyieinu=">aaacj3icbvdlsgmxfm34rpvvdekmwir20tjtbhuhft24rgbtovnkjpnpqzmpkjtcgezz3pgrbkrudomfmol0oa0haifn3hute5xicawm+wusla+srq0xnoqbw9s7u6w9/xsvxpkyng1fklsouuzwglwbg2ddsdlio4j1npf15ncemfq8do5gerg+t4yb9zglokvb6dl2jkgj26lanu1cspefziurtwwgcvrlr3lbi7wfhu6sctqytzoolc26oqvejnamlnemruhp1xzdgvssacqiuj3ljkcfeamccpyw7vixinaxgbkepghxmeon00vtfkwvf3uh1ccapfv/dytev2rio7rsjzbs814m/uf1yvdo+gkpohhyqpohvfhgchgwgna5zbterbncjdd/xxredc6gsy3qekz5lrdju1e/r1u3j+xm1synajper6iclhskmuggtvabufsintebejeejbfjw/jms5emwc8b+gpj+wchn6yp</latexit> Modeling CUBIC n Reno: AIMD dw (t) dt = 1 W (t) 2 (t ) n Scalable TCP: MIMD n CUBIC: W (t) =c s(t) r! 3 3 Wmax (t)b + W max(t) c 16

17 New Approach Observation: loss-based protocols have in common: n max cwnd before loss, W max (t) n time since last loss, s(t) Derive DEs for instead of for W(t)! 17

18 Example: TCP CUBIC 18

19 Example: TCP Reno 19

20 <latexit sha1_base64="ukefaldpmq3x3x67a/ir8hysi4u=">aaab6xicbvbns8naej34wetx1aoxxslus0mkon6kxjxwnlbqhrlzbtqlm03ynqil9cd48adi1x/kzx/jts1bwx8mpn6bywzemeph0hw/nzxvtfwnzcjwcxtnd2+/dhd4ajjmm+6zrca6fvldpvdcr4gst1lnarxk3gyhn1o/+cs1eyl6wfhkg5j2lygeo2il+0rtrfsqu1v3brjmvjyuiuejw/rq9bkwxvwhk9sytuemgiyprseknxq7meepzupa521lfy25ccazuyfk1co9eixalkiyu39pjglszcgobwdmcwawvan4n9fomlomxkklgxlf5ouitbjmyprv0hoam5qjsyjtwt5k2ibqytcmu7qheisvlxo/vr2qenfn5fp1nkybjueekudbbdthfhrga4m+pmmrvdnsexheny9564qtzxzbhzifp6+kjo8=</latexit> <latexit 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sha1_base64="vbrc/bxx+k9ccqoeoefymkvf/li=">aaab6xicbvbns8naej34wetx1aoxxslus0luug9flx4rgltoq9lst+3szsbstoqs+ho8efdx6j/y5r9x2+agrq8ghu/nmdmvtkqw6lrfztlyyuraemgjulm1vbnb2tt/nhgqgfdzlgpddknhuijuo0djm4nmnaolb4tdm4nfeolaifg94cjhqut7svqeo2il+8rzsadudqvufgsredkpq456p/tv7sysjbhcjqkxlc9nmmiorsekhxfbqeejzupa5y1lfy24cblpqwnybjuu6cxalkiyvx9pzdqyzhsftjoioddz3kt8z2ul2lsmmqgsfllis0w9vbkmyerv0hwam5qjsyjtwt5k2ibqytcmu7qhepmvlxl/thpv9e7oy7xrpi0chmirvmcdc6jbldtbbwz9eizxehok8+k8ox+z1iunnzmap3a+fwcxdozw</latexit> New Model System of differential equations (1) (2) dw max (t) dt dt ds(t) dt = (W max (t) W (t)) =1 s(t) W (t ) W (t ) p(t ) p(t ) (3) Loss probability function C p(t) = max 1 W (t), 0 20

21 New Model: First DE change in maximum congestion window size dw max (t) dt = (W max (t) W (t)) W (t ) p(t ) delayed by one round trip time, since loss occurs at a congestion point, not at the source packet loss rate 21

22 New Model: Second DE time since last loss grows by one time unit, and is reset to zero upon new loss dt ds(t) dt =1 s(t) W (t ) p(t ) delayed by one round trip time, since loss occurs at a congestion point, not at the source packet loss rate 22

23 l to TCP Reno and CUBIC, which are both cwnd -based, we use (1) in the interest of Figure 1: W Wmax (t), TCPReno. Reno. Figure 1: (t), W (t), Wmax (t),and ands(t) s(t) for for TCP. 1 illustrates Wmax (t), s(t), and W (t) for TCP Reno. To adapt model (1) to TCP s(t) the amount of time sincelast loss, which which variables all all lossthe amount ofcwnd time elapsed since are variables common to define Reno s as aelapsed function oflast Wloss, and are s(t). At thecommon time oftoloss, Wloss(t) = max (t) drithms. Since W (t), a function Wmax(t) (t) and and s(t), s(t), ititisiscompletely determined by by Wbecomes (t), is aisfunction ofof W completely determined s algorithms. halved.since This the initial value of W (t) in the new congestion epoch. W (t) max 2: DEs. The result is the following model 2: The by result the following model eases oneissegment for every round-trip time, so the total increase is s(t)/ after (t) loss. Hence, W (t ) maxlast has elapsed(1) sincedwthe = (W (t) W (t)) max dwmax (t) W (t )p(t ) System of differential dt = (Wmax (t) W (t)) p(t ) (1) W s(t) dt ds(t) Wmax (t (t) ) equations (1) =W (t ) p(t + ). (2) ds(t)dt =W1(t) s(t) 2 (2) = 1 s(t) p(t ) dt ve, p(t model ) is afor loss probability function. The expression W (t )p(t )/ describes the fluid Reno is (1) combined with (2). ime-delayed, Razumikhin s Theorem toloss show that the candidate ispoint, suitable and of which is delayed because occurs not at ) is aloss, lossuse probability The expression Wat )p(t characteristics )/ describes the ss packet probability function canfunction. bebycustomized according to(tthe the congestion specific of a ty holds in a neighborhood of the fixed point of the system. A consequence of the ource. The first DE describes the behavior of W, which takes the value of W (t) right cket is delayed because occurs at the point, function not at em, loss, such which as queue size andby AQM policy. loss Formax simplicity, we congestion use the following rization will exponential for CUBIC, butthe wew can e a loss. isatthat the we time of not loss, prove Wmax (t) > W (t),stability Wmax decreases by amount The first DE describes theifbehavior of Wmax,then which takes the value of (t) still right sequent models: W (t);lyapunov otherwise,stability. it increases by thewe same amount. The second DEon describes the ptotic and Finally, derive convergence results the system s x (t) oss. At the time of loss, if W (t) > W (t), then by the amount Wmax isdecreases Loss ution of the time since last loss s(t),max which growsc by one unit and reset to zero uponprobability loss. W (t); otherwise, it increases by the same amount. The second DE describes = max scheme 1, 0.of maximum rate and time (3) (3)to p(t) system can be adapted a rate-based in terms since the function W (t) of the time since lastbyloss s(t), each which oneweunit is reset to zero uponofloss. rate decrease, simply dividing DEgrows by. by Since will and be describing applications P CUBIC FluidtoModel m can be adapted acubic, rate-based incwnd terms maximum and since model TCP whichscheme areexample both we use (1)rate ingiven the interest of at W (t)to must bereno eitherand derived explicitly, for as in-based, (2)offor TCP Reno or in time the definition paper. ecrease, simply dividing each DE by. Since we will be describing oller, congestion as in (7) for by CUBIC. IC s window function is defined in terms of the time since applications last loss s(t)of igure 1 illustrates Wimmediately (t), s(t), and W forlast TCP Reno. To(t) adapt TCP of to TCP Reno and CUBIC, which are(t)the both cwnd -based, we use model (1) in (1) the to interest max um value of cwnd before loss Wmax [2]:, we define Reno s cwnd as a function of Wmax (t) and s(t). At the time of loss, W (t) = Congestion!3 r 4 (t)illustrates is halved. W This becomes the initial (t) value oftcp W (t) Reno. in the new congestion epoch. W x1 (t)b for To adapt model (1) to(t) TCP max window max (t), s(t), and 3WW W (t) = c s(t) + W (t) increases by one segment for every round-trip time, so the max total increase is s(t)/ after(7) define Reno s cwnd as a function of Wcmax (t) and s(t). At the time of loss, W (t) = function time has elapsed since the last loss. Hence, halved. This becomes the initial value of W (t) in the new congestion epoch. W (t) a multiplicative decrease factorround-trip and c is(t)atime, scaling factor. Figure 2 illustrates the Wmax s(t) ases by one segment forofevery so the total increase is s(t)/ after 23 UNIVERSITY MW ASSACHUSETTS, A MHERST College of Information and Computer Sciences (t) = +. (2) f CUBIC s cwndthe over time. opaque red curves represent behavior in steady 2 has elapsed since last loss. 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sha1_base64="1iqgaietf4jr3uotfm2vgwdbj+q=">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</latexit> <latexit sha1_base64="1iqgaietf4jr3uotfm2vgwdbj+q=">aaaclhicbvdlsgmxfm34rpvvdekmwardwgzeubdcurcuk1grnkvk0kwbmskmyr1pgcypcuoviociilu/w3taha8lurzouyebe/xycgouo3jmzufmfxyls8xlldw19dlg5q2jes14nuuy0nc+nvwkxesgqpk7whma+pi3/p7fwg/cc21epg5ggpnwsltkbijrsfs7demadyclqpm9ecncdgoeiijuamomsgufscekryedzaedpdgnohfyfuzw2qwyw3hzwn+bnwvlnk1au/rcohflqq6aswpm03njakvug2csz0wsgb5t1qdd3rrq0zcbvppfm+fdy3rwegn7focc/e5iawjmmpttzeihz35ry/i/rzlacnjkhyot4ipnfgwjxbdhcxs4izrniicwukaf/stmpaopaxtw0ybg/t75l6gfvk4r3vvruxo+taoatteo2kmeokzvdivqqi4yektpaitencfn1xl3piajm87us4v+lpp5befcqn0=</latexit> Asymptotic Stability Given > 0, for any t>0, 9 > 0 s.t. ˆx x 0 < =) ˆx x(t) < and lim ˆx x(t) =0. t!1 <latexit sha1_base64="pvc3trcjt5oyczrz2ipaohrbju4=">aaab7xicbvbns8naej3ur1q/qh69lbbbu0leug9flx4rgftoq9lsn+3szsbstsqs+io8efdx6v/x5r9x2+agrq8ghu/nmdmvtkuw6lrftmlldw19o7xz2dre2d2r7h88mcttjpsskyluh9rwkrt3uadk7vrzgoest8lrzdrvpxjtrkluczzyikydjslbkfqp1r1szj8mvwrnrbszkgxifaqgbzq96le3n7as5gqzpmz0pdffikcabzn8uulmhqeujeiadyxvnoymygfntsijvfoksrqthwsm/p7iawzmoa5tz0xxaba9qfif18kwugxyodimuwlzrvemcszk+jvpc80zyrellglhbyvssdvlaboq2bc8xzexix9wv6p7d+e1xnwrrhmo4bhowymlamatnmehbin4hld4c1lnxxl3puatjaeyoyq/cd5/abwlj50=</latexit> x 0 convergence to fixed point Lyapunov stability trajectory x(t) =W (t) 24