Mean Field Convergence of a model of multiple TCP connections through a buffer implementing RED

HAL Id: inria-00071733

https://hal.inria.fr/inria-00071733

Submitted on 23 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,

Mean Field Convergence of a model of multiple TCP connections through a buffer implementing RED

David R. Mcdonald, Julien Reynier

To cite this version:

David R. Mcdonald, Julien Reynier. Mean Field Convergence of a model of multiple TCP connections through a buffer implementing RED. [Research Report] RR-4850, INRIA. 2003. �inria-00071733�

a p p o r t

d e r e c h e r c h e

THÈME 1

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Mean Field Convergence of a model of multiple TCP connections through a buffer implementing RED

D. R. McDonald and J. Reynier

N° 4850

June 2003

0"1'2 34')56%"57 8"%9:; ,<*=> ?') A@CBED

FHGJIKGJLNMOF"PRQTSRUWV

∗ SRQVYX)GTI6Z\[]Q^WZ`_

†

acbedgfKhji?kmlnpohrqtsvuwhgxo+yvo+xdgfKhpo

z{|O}(hgxa~l?€

lqt‚ƒ‚)|O{+x„ƒh!{hp…,behg{…,beh‡†)ˆ0‰‹ŠŒŽkmsƒ†eh6ŽŽ‘"k’‰ei“‚]qt”•hpo

–:—0˜r™rš`›ƒœ™ l?>ž Ÿ¡l?qt†e„ƒ|Of¢£qt{+¤¥y¦ž!hgxhp…gx+§¨|O†2©6b]qrª‹hj«)hphg†¬o+sƒ””•hpo+xhp„N­~behg†Yfsƒ¤¥x+§¥‚ƒ¤¨h

a€®zNohpo,o+§¨|O†eo£qt{h~fsƒ¤¥x+§¥‚ƒ¤¨h¯uƒhp„Kx+bƒ{|Osƒ”b#q!«)|Ox+x+¤¨hg†ehp…,°=«ƒsv±Thg{r²acbeh§¨„ƒhrq!§¨o>x|³„ƒhgxhp…gx…p|O†v´

”•hpo+x+§¨|O†µ«)h¯¶¡|O{h®x+behc«ƒsv±hg{|\ª‹hg{(·]|\­o«¸y„v{|O‚ƒ‚ƒ§¥†ƒ”6|O{fjqt{°¸§¥†ƒ”“‚]qO…,°•hgxo­~§¥x+bjq‚ƒ{|O«]qt«ƒ§¥¤¥§¥x¹y

x+b]qtx§¥†e…g{hrqOo,hpo­~§¥x+b:x+b]h?º‹sehgseh¤¨hg†ƒ”x+bJ²>acbeh?|O«v}+hp…gx+§¥ª•hpocqt{hqt†Hhpº¸sƒ§¥x,qt«ƒ¤¨h„v§¨o+x+{+§¥«ƒsƒx+§¨|O†#|t¶

‚]qO…,°‹hgx6¤¨|oor»1{hp„vse…php„¦oy¸†e…,bƒ{|O†ƒ§¥¼\qtx+§¨|O†Nx|O”•hgx+b]h¯{­~§¥x+b¦{,hg„ƒse…phg„¦‚]qO…,°‹hgx6¤¨|oor»0„ƒhg¤½qry•»qt†e„

„ƒhg¤½qryHªqt{+§½qtx+§¨|O†J²

¾~qO…p…phg¤¥¤¥§¿»•À…gž!|O†]qt¤¨„µqt†e„"lhgyR†ƒ§¨hg{®Á¥igÂRb]qrª‹h‚ƒ{|O‚)|ohp„µqfK|¸„ƒhg¤R¶¡|O{fsƒ¤¥x+§¥‚ƒ¤¨hca€®z$…p|O†v´

†ehp…gx+§¨|O†eo®§¥†:x+beh?…p|O†ƒ”•hpo+x+§¨|O†wqrª‹|O§¨„eqt†e…ph{hg”§¥fKh?fsƒ¤¥x+§¥‚ƒ¤¨h¯uƒhp„:x+bƒ{|Osƒ”bwq³«)|Ox+x+¤¨hg†ehp…,°K«ƒsv±Thg{

§¥fµ‚ƒ¤¨hgfKhg†¸x+§¥†ƒ”Ãl>ž=²2acbeh6­~§¥†e„ƒ|\­Äo§¥¼rhpo?|t¶£hrqO…,bÅa€®zÆohpoo+§¨|O†hgª•|O¤¥ª‹h6¤¥§¥°•h6§¥†e„ƒhg‚)hg†e„ƒhg†¸x

„vyR†]qtfµ§¨…rqt¤o+yvo+xhgfKo³…p|Osƒ‚ƒ¤¨hp„¦«¸yx+behjº‹sehgsehK¤¨hg†ƒ”x+bYqtx³x+behµ«ƒsƒ±hg{r²:acbehK°•hgy§¨„ƒhrqw§¥†Á¥i¯Â

§¨ox|K…g|O†]o+§¨„ƒhg{x+beh³bƒ§¨o+x|O”{,qtfÇ|t¶­~§¥†e„ƒ|\­;o+§¥¼rhpoqOo?q={,qt†e„ƒ|OfÈfKhrqOo+sƒ{h³…p|Osƒ‚ƒ¤¨hp„É­~§¥x+bÃx+beh

º¸sehgseh²Ê?hg{h“­®h³‚ƒ{|\ª‹h“x+beh³…p|O†O}+hp…gx+sƒ{h³fjqO„ƒh³§¥†¦Á¥igÂ1x+b]qtxqOo~xbeh“†¸sƒf«)hg{?|t¶>…p|O†ƒ†ehp…gx+§¨|O†eo

xhg†e„ƒo£x|“§¥†vËe†e§ÌxÍy=x+bƒ§¨oo+yRoxhgf‡…p|O†¸ª‹hg{+”•hpo£x|q“„ƒhgxhg{+fµ§¥†ƒ§¨o+x+§¨…fKhrqt†v´ÎË]hg¤¨„:¤¥§¥fµ§¥x…p|Ofµ‚ƒ{+§¨o+§¥†ƒ”

x+beh“­~§¥†e„ƒ|\­;o+§¥¼rh“„ƒhg†eo+§¥x¹yw…g|Ose‚ƒ¤¨hg„í~§¥x+bÏqµ„ƒhgxhg{+fµ§¥†ƒ§¨o+x+§¨…6º¸sehgseh²

ÐwыÒÓÔÕ š`֘ a€®z»el?>ž=»vfKhrqt†v´ÎË]hg¤¨„T»]„vyR†]qtfµ§¨…rqt¤0o+yvo+xhgfKor²

∗ ×Ø(ÙOÚÛÎÜ¿Ý~Ø(ÞpÜ~ß,àá!ÚÜ¿âtØ(ÝÚÜ¿ãåä(æ(çJè0ÞtãåégØÍÛ¿æÎãêÜìëjßíà£î0ÜÎÜïÚ(ðÚ\ç1ñ\Ýñ\æÎòpóÝ~Ú,Ü¿â`æ¡ÜïÚ,Ü+ôõtß,ÜÎÜïÚ(ðTÚ\ôä+ÚrçJö1Ø(æÎØ+ÚÛ¿ä¹â æÎõtÙ`ٍßíÛÎÜ¿Ø+ñã÷ÞÙtÚ,ÛÎÜ1øpëùú`ûeö1üHòíÛïÚ,ÞpÜ0ý0þpÿíÿ

† ûeä(ß÷ØcùßíÛ¿ÝÚåØúrõtÙÍÛ¿ãåØ(õ\ۿدç¯õ åã÷ØÍÞ¸ôö0عë`Þ`ãåØÍÛ¹óØ(Þtæ(ôàۃö0ØÍæÎØ+Ú,Û¿äïâ"ãåÞ`ãêÜ¿ãÚÜ¿Ø+ñ"ñ\õ`Û¿ãåÞtòÚ,Þ"ãåÞpÜ¿ØÍÛ¿Þ`æÎâtãåÙ"ÚÜ Ü¿âtØè0ÞtãåégØÍÛ¿æÎãêÜìëßíà]î0ÜÎÜïÚ(ðTÚ

*=~2'2' %""¿~'e'2?'2&%" @;B¦D

˜ qt¤¥”•|O{+§¥x+bƒfKh“l>ž;q6ngxno+se””•n¯{,n?¤¥|O{,oº‹seh?‚ƒ¤¥seo+§¨hgsƒ{ocohpoo+§¨|O†eo®a€®z o|O†¸x®hg†

…p|O†e…gsƒ{+{hg†e…ph?„eqt†eo>sƒ†j{|Osƒxhgsƒ{(´¿”•|Osƒ¤¨hgx®„

ngx+{,qt†ƒ”¤¨hgfKhp†¸xr²

§¨„ƒnphhpo+x£„ƒh„ƒngxhp…gxhg{¤¨q³…p|O†ƒ”•hpo(´

x+§¨|O†Åqrªqt†¸x„ qrª•|O§¥{{hgfµ‚ƒ¤¥§0¤½qµËe¤¨h„ qtx+xhg†¸xh"hg†„ƒngx+{+sƒ§¨o,qt†¸x!|Oshg†Ïfjqt{º‹s]qt†¸x¤¨hpo‚]qOº¸sehgxo

qrª‹hp…wsƒ†ehɂƒ{|O«]qt«ƒ§¥¤¥§¥xnϺ¸sƒ§…g{|¨x#qrª•hp…ä½q$¤¨|O†ƒ”sehgsƒ{H„ƒhä½qÅËe¤¨hɄ qtx+xhg†¸xh² hpo:|O«v}+hp…gx+§Ì¶¡o

o|O†¸xc„ƒh?„v§¨o+x+{+§¥«ƒsehg{~npº‹sƒ§¥x,qt«ƒ¤¨hgfKhg†¸x~¤¨hgo®‚)hg{+xhpo®„eh‚]qOº¸sehgxor»R„ƒh{np„vsƒ§¥{h¤½q6o+yR†e…,bƒ{|O†ƒ§¨o,qtx+§¨|O†

hg†¸x+{h¤¨hpoo|Osƒ{,…ghpo®a€®zYhgx®„ƒh{np„vsƒ§¥{h?¤¥hpo®„ƒng¤½qt§¨ohgx¤¨hgsƒ{o£ªqt{+§½qtx+§¨|O†eor²¾~qO…p…phg¤¥¤¥§¿»RÀ…pž!|O†]qt¤¨„

qt†e„El?h¯yR†ƒ§¨hg{KÁ¥ig£|O†¸x³‚ƒ{|O‚)|onKsƒ†$fK|¸„ƒdg¤¨hj„ƒhpo6…p|O†ƒ†eh¯uR§¨|O†eoa€®z „eqt†eo³¤¨hµ{ng”§¥fKhK„v§¥x6„ƒh

€®|O†ƒ”•hpo+x+§¨|O†1ª‹|O§¨„eqt†e…ph Ÿ¡ngªR§¥xhgfKhg†¸x„ƒh“…p|O†ƒ”•hpo+x+§¨|O†2©cº‹sƒ§Jo|O†¸xhg†w…p|O†e…gsƒ{+{hg†e…ph6„eqt†eo~sƒ†

{|Osƒxhgsƒ{“º¸s hg¤Ì¤¨hpo!…p|O†ƒ”•hpox+§¨|O†ƒ†ehg†¸xr²6€®h{|Osƒxhgsƒ{!sƒx+§¥¤¥§¨oh"l>ž=² hpox,qt§¥¤¥¤¨hpo!„ƒhpo?¶¡hg†gx+{hpo!„ƒh

…,b]qOº¸sehc…p|O†ƒ†]híuv§¨|O†Ka€®z¦o|O†¸x„ƒhpooyRo+xdgfKhpo„vyR†]qtfµ§¨º‹sehpo>º‹se§vngª‹|O¤¥sehg†¸x§¥†e„ƒng‚)hg†e„eqtfµfjhg†¸x

qrª‹hp……p|OfµfKh?…p|Osƒ‚ƒ¤½qt”•h¤½q³x,qt§¥¤¥¤¨h„ƒh¤½q!Ëe¤¨h„

qtx+xhg†¸xh²

§¨„ƒnph…phg†¸x+{,qt¤¨h?„eqt†eo?Á¥igÂ)hgox£„eh…p|O†v´

o+§¨„ƒng{hg{£¤bƒ§¨o+x|O”{,qtfµfKh?„ƒhpo>x,qt§¥¤¥¤¨hpo„ƒhc¶¡hg†gx+{h…p|OfµfKhsƒ†eh~fKhpo+sƒ{,hqt¤¨nrqtx|O§¥{h…p|Osƒ‚e¤¥nphqrª•hp…

¤½qÃËe¤¨h:„ qtx+xhg†¸xh²Åž“qt†eo"¤¨hj‚ƒ{npohg†¸xµqt{+x+§¨…g¤¨hH†e|Oseo"„ƒngfK|O†¸x+{|O†eo=¤½qυp|O†O}(hp…gx+sƒ{h:¶¿qt§¥xh:„eqt†eo

Á¥i¯Âº‹sƒ§>„v§¥x6º‹sehµ¤¥|O{,oº‹sehµ¤¨h=†e|Of«ƒ{hK„ƒhK…p|O†ƒ†eh¯uv§¥|O†]o6a€®z xhg†e„$ª‹hg{o!¤ §¥†vËe†ƒ§¿»0…phKo+yvo+xdgfKh

…p|O†¸ª‹hg{+”•hª•hg{o?sƒ†eh6¤¥§¥fµ§¥xh"„vs…,b]qtfµ‚fK|\y•hg†Åº‹sƒ§hpo+x„ƒngxhg{+fµ§¥†ƒ§¨o+xh²“€®hgx+xh6¤¥§¥fµ§¥xh"hpo+x?sƒ†

o+yvo+xdgfKh…p|Ofµ‚ƒ{hg†]qt†¸x“sƒ†eh"„ƒhg†eo+§¥xn=„ƒhx,qt§¥¤¥¤¨hpo!„ƒh6¶¡hg†gx+{hpo!hgxsƒ†ehx,qt§¥¤¥¤¨h=„ƒh6Ëe¤¨h"„ qtx+xhg†¸xh

„ƒngxhg{+fµ§¥†ƒ§¨o+xh²

CÕ ™r˜ Ó œ ˜Ã a€®z»el?>ž=»]€b]qtfµ‚ÉfK|\y•hg†J»eoyRo+xdgfKhpo„vyR†]qtfµ§¨º‹sehpo\²

'])“"%1'2

¹fjqt”§¥†eh£x+beho…phg†]qt{+§¨|?­~behg{h

N ­®|O{°³o+x,qtx+§¨|O†eo0§¥†=qsƒ†ƒ§¥ª•hg{o+§¥x¹y6„ƒhg‚]qt{+x+fKhg†¸xqt{,h£…p|O†ƒ†]hg…gxhp„

«¸y³q~o­~§¥x…,behp„6hgx+behg{+†ehgx0x|?q„ƒhg‚]qt{+x+fKhg†¸x,qt¤R{|Osƒxhg{r²¿¶ƒhgª‹hg{+y!­®|O{+°“o+x,qtx+§¨|O†o§Ìf"sƒ¤¥x,qt†ehp|Oseo+¤¥y

a~z o!q=Ë]¤¥h6x|:o|OfKh6„ƒ§¥ox,qt†¸x?fjqO…,bƒ§¥†eh6x+b]h¯†Ïxbeh6|Osƒx+‚ƒsƒx«ƒsv±hg{?§¥†Ïx+beh6{|Osƒxhg{?­~§¥¤¥¤0«)hq

«)|Ox+x+¤¨hg†ehp…,°²! ¦hKox+se„vyÅx+behK§¥†¸xhg{,qO…gx+§¨|O†Y|t¶

N a€®z#"$¹zC…p|O†ƒ†ehp…gx+§¨|O†eo"§¥†¦x+behj…p|O†ƒ”•hpo+x+§¨|O†

qrª‹|O§¨„eqt†e…ph“‚ƒb]qOo,h“|t¶a€®zl?hg†e|={|Osƒxhp„Ãx+bƒ{|Osƒ”bÏq=«)|Ox+x+¤¨hg†ehp…,°wº¸sehgseh²

%?‚)|O†w{hp…phg§¥ªR§¥†ƒ”:q"‚]qO…,°•hgxx+beh!{hp…g§¥‚ƒ§¨hg†¸x?ohg†e„ƒo~«]qO…,°#qt†ÉqO…,°R†e|\­~¤¨hp„v”•hgfKhg†¸x?‚]qO…,°•hgxo|

x+behg{hç¥oj|O†ehÃl|Osƒ†e„ a1{+§¥‚ac§¥fKh¦Ÿ¡la~a©"«)h¯xÍ­®hphg†x+behwx+§¥fKhÉq‚]qO…,°‹hgxK§¨oKohg†¸x:qt†e„ x+beh

qO…,°R†e|\­~¤¨hp„v”•hgfKhg†¸x“§¨o{hp…phg§¥ª•hp„T²“acbehµqO…,°¸†e|\­~¤¨hp„v”•hgfKhg†Rx³…p|O†¸x,qt§¥†eo!x+beh"ohpº¸sehg†e…ph†¸sƒf«)hg{

|t¶Jx+behbƒ§¥”behpo+x®ªOqt¤¥seh§¥†:x+beh«¸yRxh?o+x+{hrqtf ose…p…ghpo,o(¶Wsƒ¤¥¤¥yµ{hp…phg§¥ª‹hp„jsƒ‚:x|6x+bƒ§¨o‚)|O§¥†¸x®§¥†:x+§¥fKh²

.behg†E‚]qO…,°•hgxo"qt{+{+§¥ª•hK|Osƒx|t¶c|O{„ƒhg{|O{qt{hµ¤¨|o+x³x+behg†N„vsƒ‚ƒ¤¥§¨…rqtxh:qO…,°R†e|\­~¤¨hp„v”•hgfKhg†¸xo=qt{h

”•hg†ehg{,qtxhp„T²£qO…,bÅo|Osƒ{…ph³§¥fµ‚ƒ¤¨hgfKhg†¸xo“qµ­~§¥†e„ƒ|\­ ·]|\­C…p|O†¸x+{|O¤0­~bƒ§¨…,bϤ¥§¥fµ§¥xo?x+beh³†¸sƒf«)hg{

|t¶!‚]qO…,°‹hgxoµ¶¡{|Ofx+bƒ§¨oj…p|O†ƒ†ehp…gx+§¨|O†Æqt¤¥¤¨|\­®hp„ §¥†¸x|Ex+behw†ehgx¹­®|O{+° „vsƒ{+§¥†ƒ”N|O†ehÃl?|Osƒ†e„ a1{+§¥‚

ac§¥fKhjŸ¡la~a©¯²

acb]hK¤¥§¥†ƒ°${,qtxh:|t¶~x+behK{|Osƒxhg{"§¥o

N L ‚]qO…,°•hgxo‚)hg{o,hp…g|O†]„T²& Eh:qOoosƒfKhKx+behK‚]qO…,°‹hgxo

¶¡{|Of qt¤¥¤)…p|O†ƒ†ehp…gx+§¨|O†eo}(|O§¥†:x+beh?º‹sehgsehqtxx+beh«)|Ox+x+¤¨hg†ehp…,°K«esv±hg{cqt†e„j­®h?„ƒhg†e|Oxh?«¸y

Q_N(t)

x+beh6qrª‹hg{,qt”•h³º‹sehgseh“‚)hg{~·]|\­³²' ¦h³qOoo+sƒfKh“x+beh³o…,behp„vsƒ¤¥§¥†ƒ”jx|=«)h()*+"²

¦h£§¥fjqt”§¥†ehx+beh®o|Osƒ{…ph­~{+§¥xhpo0§¥xo…gsƒ{+{hg†¸x­~§¥†e„ƒ|`­¬o+§¥¼rh®qt†e„x+beh…gsƒ{+{hg†¸xla~a¦§¥†"hrqO…,b

‚]qO…,°‹hgx§¥xo,hg†e„ƒo­~behg{h®«¸y³x+beh®…gsƒ{+{hg†Rxla~aN­®hfKhrqt†=x+behcla~aY|t¶]x+beh®¤½qOo+xqO…,°¸†e|\­~¤¨hp„v”•hp„

‚]qO…,°‹hgxr²

, W^N_n (t) §¨o~„ƒh¯Ëe†ehp„wx|"«)h!x+beh!­~§¥†e„ƒ|\­ o+§¥¼rh!­~{+§¥x+xhg†w§¥†Éq‚]qO…,°‹hgxc¶W{|Of&…p|O†ƒ†ehp…gx+§¨|O†

n

qt{+{+§¥ª¸§¥†ƒ”Hqtx~x+beh“{|Osƒxhg{qtx~x+§¥fKh

t^-

, R^N_n(t)§¨o„ƒh¯Ëe†ehp„:x|6«)hx+behla~a­~{+§¥x+xhg†H§¥†#q³‚]qO…,°•hgx£¶¡{|Of …p|O†ƒ†ehp…gx+§¨|O†

nqt{+{+§¥ª¸§¥†ƒ”

qtxcx+beh{|Osƒxhg{qtxcx+§¥fKh

t ŸWx+bƒ§¨o~la~a §¨ocx+beh!osƒf|t¶0x+b]h‚ƒ{|O‚]qt”‹qtx+§¨|O†É„ƒhg¤½qry:‚ƒ¤¥seocx+beh º‹sehgsehg§¥†ƒ”:„ƒhg¤½qryH§¥†Ãx+beh“{|Osƒxhg{í©¯²

,

Ehco+b]qt¤¥¤ƒqOoosƒfKhx+b]qtx…p|O†ƒ†ehp…gx+§¨|O†eo>…rqt†=«)hc„v§¥ª¸§¨„ƒhp„"§¥†

d…g¤½qOoohpo­~behg{h~…p|O†ƒ†ehp…gx+§¨|O†eo

§¥†Ã…g¤½qOo,o

c b]qrª‹h“…p|OfµfK|O†Éx+{,qt†eo+fµ§¨o,o+§¨|O†Ãx+§¥fKh

T_c^²

%?†e„ƒhg{a€®zClhg†e|ƒ»hpo+x,qt«ƒ¤¥§¨o+behp„N…p|O†ƒ†ehp…gx+§¨|O†eo=h¯uvhp…gsƒxh:…p|O†ƒ”•hpo+x+§¨|O†¬qrª‹|O§¨„eqt†e…phK­~behg{h

x+beh~­~§¥†e„ƒ|\­ o+§¥¼rh~|t¶ThrqO…,bj…p|O†ƒ†ehp…gx+§¨|O†j§¥†e…g{hrqOohpo£«¸y"|O†eh~‚]qO…,°‹hgx£hrqO…,bKx+§¥fKhq‚]qO…,°‹hgx>fjqt°•hpo

q"{|Osƒ†e„#x+{+§¥‚J»v§¿²êh²>hrqO…,b

R^N_n qOo~¤¨|O†ƒ”jqOoc†e|=¤¨|oohpo~|O{cx+§¥fKhp|Osƒxo|¸…p…gsƒ{r²>žsƒ{+§¥†ƒ”µx+bƒ§¨o~‚ƒb]qOoh x+behK{,qtxhKx+behK­~§¥†e„ƒ|\­ |t¶…p|O†ƒ†ehp…gx+§¨|O†

n §¥†e…g{hrqOo,hpo6§¨oqt‚ƒ‚ƒ{|ruv§¥fjqtxhg¤¥y

1/R^N_n ‚2qO…,°•hgxo6‚)hg{

ohp…p|O†e„T²;acbehÏ|O†ƒ¤¥y x+bƒ§¥†ƒ”Y{hpo+x+{,qt§¥†ƒ§¥†ƒ”Yx+behϔ{|\­~x+bÆ|t¶³x+{,qt†eo+fµ§¨oo+§¨|O†.{,qtxhpo:§¨o#qE¤¨|oo:|O{

x+§¥fKhp|Osƒxr² .behg†Éq=¤¨|oo|¸…p…gsƒ{ox+beh“­~§¥†e„ƒ|\­ §¨o~{hp„vse…php„w«Ry:b]qt¤Ì¶+²

acb]h:o,|Osƒ{…ph:„eh¯x,hg…gxoKqä¨|oo­~behg†Yx+be{hphH„vsƒ‚ƒ¤¥§¨…rqtxhwqO…,°¸†e|\­~¤¨hp„v”•hgfKhp†¸xoKqt{+{+§¥ª‹h²acbeh

o|Osƒ{…ph“x+behg†Ãox,qt{+xoq6¶¿qOo+x~{hgx+{,qt†eo+fµ§¥x."\¶ÎqOox{hp…p|\ª•hg{+yH«¸y:§¥fµfKhp„v§½qtxhg¤¥yw{hpohg†e„v§¥†ƒ”Kx+beh!¤¨|o+x

‚]qO…,°‹hgxr² ¦hqOoosƒfKh~x+beh¤¨|oohpoqt{h|O†ƒ¤¥y=”•hg†ehg{,qtxhp„:«¸y=x+behl?>ž‡Ÿ¡lqt†e„ƒ|Of £qt{+¤¥yKž!hgxhp…¯´

x+§¨|O†2©“qO…gx+§¥ª‹hµ«ƒsv±Thg{“fjqt†]qt”•hgfKhg†Rxo,…,behgfKhK|O{“«¸yx,qt§¥¤Ì´ï„v{|O‚J² Ehµ†ehg”¤¨hg…gx6x+behµ‚)|oo§¥«ƒ§¥¤¥§¥x¹y

x+b]qtx„vsƒ‚ƒ¤¥§¨…rqtxh!qO…,°¸†e|\­~¤¨hp„v”•hgfKhp†¸xocqt{,h~”•hg†ehg{,qtxhp„:«¸y=‚]qO…,°•hgxo®qt{+{+§¥ªR§¥†ƒ”|Osƒx|t¶J|O{„ƒhg{r²) ¦h

­~§¥¤¥¤qOoo+sƒfKh³x+beh“«ƒsv±Thg{be|O¤¨„ƒo

B ‚]qO…,°•hgxoqt†e„Ãx+b]qtx|O†]…gh³x+be§¥o«ƒsv±Thg{o+‚]qO…ph³§¨oh¯uRb2qtseo+xhp„

qt{+{+§¥ªR§¥†ƒ”j‚2qO…,°•hgxoqt{h„v{|O‚ƒ‚)hp„T² ¸se…,bÏx,qt§¥¤Ì´ï„v{|O‚eo…p|OfKh6§¥†qO„ƒ„v§¥x+§¨|O†x|Kx+beh6l£žÄfKhp…,b]q`´

†ƒ§¨o+fòÊhg{hc­®h~x,qt°‹h~x+beh„v{|O‚K‚ƒ{|O«]qt«ƒ§¥¤¥§¥x¹yK|t¶l?>ž Ÿ¡|t¶Jqt†K§¥†e…p|Ofµ§¥†ƒ”6‚]qO…,°‹hgx>«)h¯¶¡|O{h~«)hg§¥†ƒ”

‚ƒ{|¸…phpoo,hg„2©®x|=«)h6q¶¡sƒ†e…gx+§¨|O†É|t¶x+beh³º‹sehgseh³o+§¥¼rh“­~bƒ§¨…,bç¨o~¼rhg{|=¶Î|O{qµº‹sehgseh“¤¨hg†ƒ”x+bë)hg¤¨|\­

Q_min «ƒsƒx~{§¥o,hpo~¤¥§¥†ehrqt{+¤¥y#x|

p_max ^qtx Q_max qt†e„Ãhpº¸s]qt¤¨o~x|

1 ^qt«)|\ª•h Q_max^²

ï¶qt¤¥¤

N …p|O†ƒ†ehp…gx+§¨|O†eojqt{h:§¥† …p|O†ƒ”•hpo+x+§¨|O†qrª•|O§¨„eqt†e…ph#­®hH…rqt†¬{h¯¶¡|O{+fsƒ¤½qtxh#x+bƒ§¨oµ„v{|O‚

‚ƒ{|O«]qt«ƒ§¥¤¥§¥x¹y:§¥†#xhg{+fKo~|t¶

Q^N»Rx+beh!º‹sehgseh!o+§¥¼rh„v§¥ªR§¨„ƒhp„H«¸y

N^»ƒqOo F(Q^N(t))»¸­~b]h¯{,h

F ^§¨o~q

„v§¨o+x+{+§¥«ƒsƒx+§¨|O†Ï¶Wsƒ†]…¯x§¥|O†Ï­~bƒ§¨…,bϧ¨o¼rhg{|K«)hg¤¨|\­

q_min =Q_min/N «ƒsex{+§¨ohpo¤¥§¥†ehrqt{+¤¥ywx|

p_max

qtx q_max = Q_max/N qt†e„Ã}(sƒfµ‚eox|

1 ^­~behg† Q^N(t) ≥ q_max^² +¶…p|Osƒ{ohjx+b]hjx,qt§¥¤Ì´ï„v{|O‚

o…,behgfKh®…rqt†"«)h…p|O†eo+§¨„ƒhg{hp„=qOo0x+beh¤¥§¥fµ§¥x+§¥†ƒ”!…rqOo,h£­~behg†

q_min = 0^» q_max =b^qt†e„ p_max= 0^²

acb]hH”•|•qt¤|t¶?x+bƒ§¨o=‚]qt‚)hg{µ§¨o=x|‚ƒ{|\ª‹hHx+behw…p|O†O}(hp…gx+sƒ{hw§¥†4Á¥i¯Â~­~bƒ§¨…,b…g¤¨qt§¥fKoµx+b]qtxµx+beh

bƒ§¨o+x|O”{,qtfÈ|t¶x+beh!­~§¥†e„ƒ|\­ o§¥¼rhpoc§¥†Éqt†¸yH…g¤½qOoo

c …g|O†Rª•hg{+”•hpox|Kq"„ƒhgxhg{+fµ§¥†ƒ§¨o+x§¥…³fKhrqt†wË]hg¤¨„

¤¥§¥fµ§¥x­~§¥x+bÏfKhrqOosƒ{h

M_c(t, dw) qtx?x+§¥fKh

t qt†e„ÉfK|O{hp|\ª‹hg{x+beh6{hg¤½qtx+§¥ª‹hº¸sehgseho+§¥¼rh

Q^N(t)

…p|O†¸ª‹hg{+”•hpox|Kqµ„ƒhgxhg{+fµ§¥†ƒ§¨o+x+§¨…³¤¥§¥fµ§¥x

Q(t)^²

Ñ¸Õ š Ñ !"#$ &%0 ('$ *) 0,+.-/

N → ∞^{-0"1 2}

/ &"1 &3' 234. 65

7 & 1

/8 c ^{'9% : (;}

/

M_c(t, dw) ^{3<1 1 ("1 9% (; >=;} M_c(s−R_c(s), dv;s, dw)-<"1

: (;

@?

A (; 9= B"1 B3' 23C D5 E "

t ^{E "} t−Rc(t)

F

G ={g∈C_b¹(IR⁺) :g(0) = 0} ^3<1 C_b¹(IR⁺) ^{("1 G B=2 ;}

3' H"1*=2 & . "' E "'$I '; 7

g_c ∈ G, c= 1, . . . d hg_c, M_c(t)i − hg_c, M_c(0)i

= Z _t

0

h 1

R_c(s)hdg_c(w)

dw , M_c(s, dw)i ^{Ÿ(i²¥i\©}

+ h(g_c(w/2)−g(w))v, M_c(s−R_c(s), dv;s, dw)i (1−R˙_c(s))

Rc(s−Rc(s))K(s−R_c(s))i ds.

= Z t

0

h 1

R_c(s)hdgc(w)

dw , Mc(s, dw)i

+ h(g_c(w/2)−g(w)), e(s, s−R_c(s), w)M_c(s, dw)i (1−R˙_c(s))

R_c(s−R_c(s))K(s−R_c(s))i ds.

= Z _t

0

h 1

R_c(s)hdg_c(w)

dw , M_c(s, dw)i+ 1

R_c(s−R_c(s))K(s−R_c(s))(1−R˙_c(s))^{Ÿ(i²å•©}

·hg_c(w), e(s, s−R_c(s),2w)·M_c(s,2dw)−e(s, s−R_c(s), w)·M_c(s, dw)ii ds.

3<1

hg_c, M_c(t)i=R∞

w=0g_c(w)M_c(t, dw)^{- 3<1} R_c(t) =T_c+Q(t−R_c(t))/L

3<1

e(s, s−R_c(s), w) =hv,M_c(s−R_c(s), dv;s, dw) Mc(t, dw) i

(B"1

H 0 '

. B"1 3' 23 "1 /; %0 ('$ "1 3' 23

( 23

w

2'$"1 : ; D5

'$>% !! ; (;

" "

Q(t) ^{. ; %}

dQ(t) dt =

d

X

c=1

Z

w

wM_c(t, dw)(1−K(t))

Rc(t) −L ^{Ÿ(i²å‘•©}

− Ã _d

X

c=1

hw, M_c(t, dw)i(1−K(t)) R_c(t) −L

!+

χ{Q(t) =q_max} ^{Ÿ+i²‰¸©}

− Ã _d

X

c=1

hw, M_c(t, dw)i(1−K(t)) R_c(t) −L

!−

χ{Q(t) = 0}). ^{Ÿ(i²åŒ•©}

0< Q(t)< q_max^- K(t) =F(Q(t)) ^3<1 Q(t) =q_max ^- K(t) ^{(( ; =;}

d

X

c=1

hw, M_c(t, dw)i(1−K(t)) R_c(t) =L.

oo+sefµ§Ì†e”

gc(0) = 0 qt†e„wx+b]qtx

gc(w)→0 ^qOo w→ ∞ ­®h“…rqt†Ã{hg­~{+§¥xh:Ÿ(i²å•©~qOo

hg_c, M_c(t)i − hg_c, M_c(0)i

= Z t

0

h− 1

Rc(s)hg_c(w),M_c(s, dw)

dw i+ 1

Rc(s−Rc(s))K(s−R_c(s))(1−R˙_c(s))

·hg_c(w), e(s, s−R_c(s),2w)·M_c(s,2dw)−e(s, s−R_c(s), w)·M_c(s, dw)ii ds.

­~behg{h Mc(s,dw) dw

§¨ox+beh(e{hp…,behgx?„ƒhg{+§¥ªOqtx+§¥ª‹h³|t¶x+beh³fKhrqOo+sƒ{h

M_c ­~§¥x+bÃ{hpo+‚)hp…gxx|

w^{²®€®|O†v´}

ohpº¸sehg†¸x+¤¥y‹»

dM_c(t)

dt =− 1 R_c(t)

M_c(t, dw)

dw + 1

R_c(t−R_c(t))K(t−R_c(t))(1−R˙_c(s)) ^{Ÿ(i² ©}

·(e(t, t−R_c(t),2w)(M_c(t, d(2w))−e(t, t−R_c(t), w)M_c(t, dw))i .

¹† Rhp…gx+§¨|O†:‘v²å‘v²å!§¥†ÃÁ¥i¯Âe­®hcfjqO„ƒhq!o+fK|¸|Ox+bjqt‚e‚ƒ{|ruR§¥fjqtx+§¨|O†jx|

e(t, t−R_c(t), w) ^«]qOohp„

|O†Ex+beh=¶ÎqO…gx“xb]qtx6|O†ehµla~aħ¥†$xbeh=‚]qOo+x³x+behµ­~§¥†e„ƒ|\­ o§¥¼rh=­cqOo“fK|ox“¤¥§¥°‹hg¤¥yÏx+behK…gsƒ{+{hg†¸x

­~§¥†e„ƒ|\­4o§¥¼rhfµ§¥†¸seo~|O†eh|O{®x¹­~§¨…phx+beh…gsƒ{+{,h¯†Rxc­~§¥†e„ƒ|\­4o+§¥¼rh?§Ì¶q6¤¨|oo­~qOo®„eh¯x,hg…gxhp„#§¥†Hx+beh

§¥†¸xhg{+§¥fò .§¥x+bÏx+bƒ§¨o!qt‚ƒ‚ƒ{|ruv§¥fjqtx+§¨|O†Å­®h6seohp„YŸ(i² ©x|:hgª‹|O¤¥ª‹h"qj„v§¨o…g{hgxh=qt‚ƒ‚ƒ{|ruR§¥fjqtx+§¨|O†

|t¶6x+behÏfKhrqOo+sƒ{h

Mc

Ÿ¡h¯uv…phg‚ƒxÅÁ¥igÂ!|O†ƒ¤¥y x+{hrqtxo#|O†eh…g¤½qOoo¯©¯²;acbehφ¸sƒfKhg{+§¨…rqt¤!{hpo+sƒ¤¥xowqt{h

h¯uƒ…phg¤¥¤¨hg†¸x"q`¶¡xhg{6|O†ehK…p|O{+{,hg…gxo6¶¡|O{³x+behµ¶ÎqO…gx6x+b]qtx"q#‚ƒ{|O‚)|O{+x+§¨|O†Y|t¶cx+behj…p|O†ƒ†ehp…gx+§¨|O†eo§¥†Nqt†

+?‚ƒ†ehgxjo+§¥fsƒ¤½qtx+§¨|O† qt{h#§¥† x+§¥fKhp|Osƒxϟ¡|Osƒ{µfK|¸„ƒhg¤?qOoo+sƒfKhpoK…p|O†ƒ†ehp…gx+§¨|O†eoK§¥†eo+x,qt†¸x,qt†ehp|Oseo¤¥y

{hpo+sƒfKh³…p|O†ƒ”•hpox+§¨|O†Ïqrª‹|O§¨„eqt†e…ph³§¶x+behgy:¶¿qt¤¥¤T§¥†¸x|µx+§¥fKhp|Osƒxí©¯²

a0|#§¥¤¥¤¥seo+x+{,qtxh=x+beh"fKhrqt†ÅË]hg¤¨„¤¥§¥fµ§¥x“­®h"‚)hg{(¶¡|O{fKhg„¦qt† +?‚ƒ†ehgx³o+§¥fsƒ¤½qtx+§¨|O†$­~§¥x+b

N = 200^» N = 400 ^qt†e„ N = 800 o,|Osƒ{…phpor²cqO…,bo|Osƒ{…ph6o,hg†e„ƒo‚]qO…,°•hgxo|t¶>o+§¥¼rh

536 ^«¸yRxhpox|

qwa‘#{|Osƒxhg{³­~§¥x+bYqHx+{,qt†eo+fµ§¨o,o+§¨|O†E{,qtxhK|t¶c‰‰e²O‘ Àhg”‹qt«ƒ§¥xo³‚)hg{6o,hp…g|O†]„E|O{

L = 10433

‚]qO…,°‹hgxoK‚)hg{:ohp…p|O†e„T² ¦hÉqOoo+sƒfKhwx+behÉo|Osƒ{…phpo:qt¤¥¤b]qrª•hÉqÅx+{íqt†eo+fµ§¨oo+§¨|O† „ƒhg¤½qry |t¶µirŽŽ

fµ§¥¤¥¤¥§¨ohp…p|O†e„ƒor²?acb]h6{|Osƒxhg{§¥fµ‚ƒ¤¨hgfKhg†¸xo!l?>ž ­~§¥x+b

p_max = 0.05 ¶Î|O{qt¤¥¤0x+beh"o+§¥fsƒ¤½qtx+§¨|O†eo

«ƒsƒx“­®h"{hpo…rqt¤¨h

Qmax

x|H«)h

1000 ^­~§¥x+b 200 o|Osƒ{…phpor»

2000 ^­§Ìxb 400 o|Osƒ{…phpo³qt†e„

4000

­~§¥x+b

800 o|Osƒ{…phpo\² ¸§¥†e…ph

Q_max o…rqt¤¨hpo"­§Ìxb

N»x+behwqrª•hg{,qt”•h#º¸sehgsehHo+§¥¼rhH„ƒ|¸hpoKqOo"­®hg¤¥¤

­~bƒ§¥¤¨h!b]|O¤¥„ƒ§Ì†e”µx+beh!¤¨|ooc‚ƒ{,|O«]qt«ƒ§¥¤¥§ÌxÍy:˃uƒhp„H­~bƒ§¨…,b祆wx+sƒ{+†wbe|O¤¨„ƒo~x+beh³qrª•hg{,qt”•h!­§Ì†]„ƒ|\­ o+§¥¼rh

˃uƒhp„T²?o

N §¥†e…g{hrqOohpo­®h®o,hph£x+beh·ese…gx+s]qtx+§¨|O†eo§¥†"x+beh®{hg¤½qtx+§¥ª•hcº¸sehgseh®o+§¥¼rh!ŸW§¥†=‚]qO…,°‹hgxo‚)hg{

…p|O†ƒ†ehp…gx+§¨|O†2©!„ƒhp…g{hrqOohpor² Eh=qt¤¨o,|Hohphx+beh"{hg¤½qtx+§¥ª•h=º¸sehgseh=o+§¥¼rh"|t¶£x+beh=ÀÅqtx+¤½qt«Å†¸sƒfKhg{+§¨…rqt¤

o+§¥fsƒ¤½qtx+§¨|O†É§¨oq=«ƒ§¥x~bƒ§¥”bJ²£acbƒ§¨o~§¨o~«)hp…rqtseo,h³|t¶x+§¥fKhp|OsƒxoqOo„v§¨o…gseo,ohp„#§¥†¦Á¥i¯Âï²

hg§¥x+behg{

Mc(t, dw) ^|O{ Mc(s−Rc(s), dv;s, dw) §¨o³qHox,qtxh"«ƒsƒx“x+behKqt«)|\ª‹h=hpº¸s]qtx+§¨|O†

„ƒ|¸hpo~‚ƒ{|\ªR§¨„ƒh“hg†e|Osƒ”bw§¥†v¶Î|O{+fjqtx+§¨|O†Ãx|µhgª•|O¤¥ª‹hx+beh“oyRo+xhgfò hgx

µ_c(t) „ƒhg†e|Oxh“x+beh!‚ƒ{|¸…phpoo

{M_c(s, dw);t−1 ≤ s ≤ t} ŸÎqt¤¥¤la~a oqt{hK¤¨hpoo³x+b]qt†

1^©¯² %o+§¥†ƒ”NŸ(i²¥i\©!­®hK…rqt†Nhgª‹|O¤¥ª‹h

Mc(t, dw) ^¶¡{|Of t ^x| t+δt ^{­~bƒ§¥¤¨h} Mc(t−s+δt, dw) §¨oH|O«ƒx,qt§¥†ehp„.«¸y qEx+§¥fKhobƒ§Ì¶Wxr²

%?†v¶¡|O{+x+sƒ†2qtxhg¤Ìy

µ_c §¨oj†e|Ox:‚ƒ{,qO…gx+§¨…rqt¤!o+x,qtx,hO² ª•hg† §Ì¶³­®hτv§¨o…g{hgx+§¥¼rhÅqt†e„ |O†ƒ¤¥y °•hphg‚ x+beh x+{,q`}+hp…gx|O{+y=|t¶x+behc‚ƒ{|¸…phpo,o>|O†:q‚]qt{+x+§¥x+§¨|O†j”§¥ª¸§¥†ƒ”

{M_c(s_i, dw);t−1 =s₀< s₁. . . s_n=t}

§¥xo+x+§¥¤¥¤T{hpº¸sƒ§¥{hpox|¸|µfse…,bɅp|Ofµ‚ƒsƒxhg{fKhgfK|O{+y#x|µo|O¤¥ª‹h!†¸sƒfKhg{+§¨…rqt¤¥¤¥y‹²

¦h…rqt† qrª‹|O§¨„.x+bƒ§¨oH‚ƒ{|O«ƒ¤¨hgf «¸y „ƒh¯Ëe†ƒ§¥†ƒ”qNo,hpº‹sehg†e…phÅ|t¶x+§¥fKhpo

t^c_k ¶Î|O{#hrqO…,b4…g¤½qOo,o o+se…,bx+b]qtx

t^c_k+n+1−R_c(t^c_k+n+1) = t^c_k^² ï¶>­®h6‚ƒ§¨…,°

n o+sj…g§¨hg†¸x+¤¥yɤ½qt{+”•hx+bƒ§¨o”§¥ª•hpo!qKËe†eh

‚]qt{+x+§¥x+§¨|O†J²>ž!h¯Ëe†eh

Φ^c(t) =x+beh!Ëe{o+x°Ho+se…,bÃx+b]qtx

t^c_k > t^² ¦h­~§¥¤¥¤2…p|O†eox+{+se…gx®|Osƒ{o|O¤¥sƒx+§¨|O†

«¸y¬{hp…gsƒ{+{hg†e…phö¡{|Ofx+§¥fKh

t_i ^x| t_i+1 ^«¸y „ƒh¯Ëe†ƒ§¥†ƒ”

t_i+1 = min_c(t^c_Φc(ti)) o+x,qt{x+§¥†ƒ”E¶W{|Of x+§¥fKh

t₀ = 0²h¯uRx"qOo,o+sƒfKhjx+b]qtx6¶Î|O{hrqO…,b¬…g¤½qOoo­®hjb]qrª•hj«)hphg† qt«ƒ¤¨hjx|υrqt¤¨…gsƒ¤½qtxhwqt†e„

2 4 6 8 10 12 14 0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time

Queue size / N

OPNET N=200

§¥”sƒ{h4iÄlhg¤½qtx+§¥ª‹h¬º‹s]h¯s]h¬o+§¥¼rh¬­~§¥x+b ŽŽ

o|Osƒ{…phpo

0 2 4 6 8 10 12 14

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time

Queue size / N

OPNET N=400

0§¥”sƒ{h Älhg¤½qtx+§¥ª‹h¬º¸sehgseh o+§¥¼rhY­~§¥x+b ‰‹ŽŽ

o|Osƒ{…phpo

0 2 4 6 8 10 12 14

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time

Queue size / N

OPNET N=800

§¥”sƒ{h ‘Älhg¤½qtx+§¥ª‹h¬º‹s]h¯s]h¬o+§¥¼rh¬­~§¥x+b ŠŽŽ

o|Osƒ{…phpo

0 2 4 6 8 10 12 14

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time

Q limit

"N=infinity" Matlab

0§¥”sƒ{hN‰?sehgseh¬o+§¥¼rhN|t¶jx+behNfKhrqt†;Ë]hg¤¨„

¤Ì§¥fµ§¥x

o,qrª‹h=x+behKª•hp…gx|O{

V_c^M(t)»q#„v§¨o,…g{hgx+§¥¼rhp„Eª•hg{o§¨|O†¦|t¶

M_c(t^c_k) ^¶Î|O{ k = m−n, . . . m ^­~behg{h m= Φ^c(t)−1²!ŸWx+behpohqt{h³fjqt{+”§¥†]qt¤¨o†e|Oxx+beh6hg†¸x+§¥{h}(|O§¥†¸x?„v§¨o+x+{+§¥«ƒsƒx+§¨|O†2©¯² ¤¨o|:qOoo+sƒfKh

­®hÉoíqrª•hÃx+behê‹hp…gx|O{:|t¶³°•hg{+†ehg¤¨o

V_c^T(t) ”§¥ª•hg†«Ry

T^c(t^c_k) ^¶¡|O{ k = m−n, . . . m ^­~behg{h

M_c(t^c_k+1) =M_c(t^c_k)◦T^c(t^c_k+1) ^qt†e„ m §¨o“qOo“qt«)|\ª•h²(§¥†]qt¤¥¤¥yqOoo+sƒfKh"x+b]qtx!­®h"o,q\ª•hx+beh

°‹hg{+†ehg¤¨o

S_m^c =Qm

k=m−nT^c(t^c_k)^²

¦hc…rqt†µ†e|\­ hgª•|O¤¥ª‹hc|Osƒ{>o+yvo+xhgfÄx|

t_i+1²cxhrqO…,bKoxhg‚=­®hc|O†ƒ¤¥y"hgª‹|O¤¥ª‹h®x+beh~º‹sehgsehqt†e„

x+beh!|O†eh!…g¤½qOo,o

d^­~behg{h t_i+1=t^d_Φd(ti)

²acbeh§¥†¸ª•hg{o,h?°‹hg{+†ehg¤

(S_m^d)⁻¹ ”§¥ª‹hpo®x+beh!…p|O†e„v§¥x+§¨|O†]qt¤

„v§¨o+x+{+§¥«ƒsƒx+§¨|O†Æ|t¶!x+behÉ­~§¥†e„ƒ|\­o:|t¶³…g¤½qOoo

d |O†ehÉla~a&«)h¯¶¡|O{hÉx+§¥fKh

t_i ”§Ìª‹hg†x+behí~§¥†e„ƒ|\­

qtx:x+§¥fKh

ti

²;€cqt¤¨…gsƒ¤½qtxhx+behυp|O†e„v§¥x+§¨|O†]qt¤“h¯uv‚)hp…gx,qtx+§¨|O†

e(ti, ti −R_d(ti), w)^{² .§¥x+b x+bƒ§¨o}

­®hɅrqt† seohNŸ(i²å•©µx|¦…rqt¤¨…gsƒ¤½qtxh

T_m+1^{d ²Æž{|O‚} T_m−n^{d ²} ¦hÏqt¤¨o|Esƒ‚)„eqtxh

S^d(m+ 1) = (T_m−n^d )⁻¹S^d(m)T^d(m+ 1)²'§¥†]qt¤¥¤¥yH­®h³…rqt¤¨…gsƒ¤½qtxh

Q(t_i+1) seo+§¥†ƒ”Kx+beh

M_c(Φ^c(t_i)−1)^²

¸§¥†e…ph6­®hqt{h“§¥†Rxhg{hpo+xhp„祆q=fjqtx+behgfjqtx+§¨…rqO¤0‚ƒ{|¸|t¶£|t¶x+beh6…p|O†¸ª‹hg{+”•hg†e…ph6x|Kx+beh³fKhpqt†

Ë]hg¤¨„í®h!­~§¥¤¥¤1†e|Oxx+{+yHx|!}Íseox+§Ì¶Wy:x+beh“†¸sƒfKhg{+§¨…rqt¤0‚ƒ{|¸…php„vsƒ{h³¤¨hrqO„v§¥†ƒ”jx|µx+beh6qt«)|\ª‹h“{hpo+sƒ¤¥xor²

¦h=­~§¥¤¥¤£qt¤¨o,|#§¥”†e|O{hµx+§¥fKhp|Osƒxo"qt†e„$o¤¨|\­c´ïo+x,qt{+xqOo“­®hg¤¥¤£qOo³o‚)hp…g§½qt¤>„ƒhgx,qt§¥¤¨o6|t¶~…p|O†ƒ”•hpo+x+§¨|O†

qrª‹|O§¨„eqt†e…phµ­~bƒ§¨…,bE­®|Osƒ¤¨„E|O†ƒ¤¥yo,hg{+ª•h=x|w|O«eo…gsƒ{hµx+beh=fjqt§¥†E§¨„ƒhrqOor² ¦h"­~§¥¤¥¤>†ehgª‹h¯{x+behg¤¨hgo,o

o+°‹hgx…,bwbe|\­ x+behpoh³h¯uvxhg†eo+§¨|O†eo…p|Osƒ¤¨„w«)h“b]qt†e„v¤¨hp„T²

acb]hµo+x+{+se…gx+sƒ{hK|t¶®x+beh=‚]qt‚)hg{“§¨o6qOo¶Î|O¤¥¤¨|\­or² ï† vhg…gx+§¨|O†NH­®h=fK|¸„ƒhg¤>x+beh

N ‚]qt{+x+§¨…g¤¨h o+yvo+xhgf |t¶­~§¥†e„e|\­o…p|Osƒ‚ƒ¤¨hp„j­~§¥x+bjx+behº‹sehgseh?qt†e„Kx+behg†K¶Î|O{+fsƒ¤½qtxhx+bƒ§¨o®qOoq!bƒ§¨ox|O”{,qtf |t¶

­~§¥†e„ƒ|\­4o+§¥¼rhpoc…p|Osƒ‚ƒ¤¨hp„H­~§¥x+bÃq6º¸sehgseh²'¹† Rhp…gx+§¨|O†Ã‘6­®ho+sƒfµfjqt{+§¥¼rhx+beh?fKhrqt†HË]hg¤¨„:¤¥§¥fµ§¥xr²

acbeh‚ƒ{|¸|t¶1|t¶x+behh¯uv§¨o+xhg†e…ph|t¶x+bƒ§¨o£¤¥§¥fµ§¥x>¶Î|O¤¥¤¨|\­o£§¥† Rhp…gx+§¨|O†:‰e²)¹† Rhp…gx+§¨|O†HŒ“­®hhpo+x,qt«ƒ¤¥§¨o+b

x+beh³…p|O†¸ª‹hg{+”•hg†e…ph³x|Kq=sƒ†ƒ§¨º‹seh“¤¥§¥fµ§¥xr²0§¥†]qt¤¥¤¥y#§¥† Rhp…gx+§¨|O† ­®h³hpo+x,qt«ƒ¤¥§¨o+bÉacbehp|O{hgf i²

- 5"

N *"]'2 ,0&3“3ƒ'2>

", , +'

N ^{! "#$%} !&'&

+?sƒ{®fK|¸„ƒhg¤x,qt°•hpo®§¥†¸x|=qO…p…p|Osƒ†¸xcx+beh„ƒhg¤½qryj|t¶0|O†eh?{|Osƒ†e„:x+{+§¥‚Hx§ÌfKh«)hgx¹­®hphg†#x+beh?x+§¥fKh?x+beh

‚]qO…,°‹hgx§¨o°R§¥¤¥¤¨hp„Ïqt†e„Ãx+beh6x+§¥fKh³­~behg†Ïx+beh³«ƒsv±hg{?{hp…ghg§¥ª‹hpox+beh³{,hg„ƒse…phg„Ï{,qtxh²# EhqOoo+sƒfKh

­~§¥†e„ƒ|\­ {hp„vse…gx+§¨|O†eoµqtx…p|O†ƒ†ehp…gx+§¨|O†

n |¸…p…gsƒ{6«)hp…rqtseo,hj|t¶~¤¨|o,ohpo6|O†ehK{|Osƒ†e„¦x+{+§¥‚Nx+§¥fKhK§¥†

x+beh"‚]qOo+xr²!a0|:Ëe{o+x!|O{„ƒhg{r»x+beh‚ƒ{|O«]qt«ƒ§¥¤¥§¥x¹y|t¶®qj­~§¥†e„e|\­Ä{hp„vse…gx+§¨|O†$«)h¯xÍ­®hphg†Åx+§¥fKh

t ^qt†e„

t+h ^§¨o

Z _t+h−R^N_n(t+h)

t−R^N_n(t)

W_n^N(s)

RT T_n^N(s)F(Q^N(s))ds∼[1−d

dtR^N_n(t)]W(t−R^N_n(t))

R^N_n(t−R^N_n(t))F(Q^N(t−R^N_n(t)))h

o+§¥†e…ph=x+beh"‚ƒ{|O«]qt«ƒ§¥¤¥§¥x¹y$qH‚]qO…,°•hgx“§¨o“„v{|O‚ƒ‚)hp„ŧ¨o!‚ƒ{|O‚)|O{+x+§¨|O†]qt¤£x|

W(t−R^N_n(t))/R^N_n(t− R^N_n(t))»)x+behx{,qt†eo+fµ§¨oo+§¨|O†Å{,qtxh=|O†ehx+§¥fKh"§¥†x+beh"‚]qOo+xr»)x+§¥fKhpo

F(Q^N(s))»)x+beh=„v{|O‚Å{,qtxh

|O†eh{|Osƒ†e„:x{+§¥‚H§¥†Hx+beh‚]qOo+xr²) Eh?x+behg{h¯¶¡|O{,h?fK|¸„ƒhg¤x+beh‚ƒ{|¸…phpooc|t¶1­~§¥†e„ƒ|\­4{hp„vse…gx+§¨|O†eoc«¸y

qµz|O§¨oo|O†Ã‚ƒ{|¸…phpoo­~§¥x+bÉo+x|¸…,b]qOo+x+§¨…“§¥†¸xhg†eo§¥x¹y

λ^N_n(t) := [1− d

dtR^N_n(t)]W_n^N(t−R_n^N(t))

R^N_n(t−R^N_n(t))F(Q^N(t−R_n^N(t)))

ŸW­®hÅrqt† qOoosƒfKh

W_n^N(t) = w_n(0) ^¶Î|O{ t < 0^©¯² acbehwxhg{+f

[1− _dt^dR^N_n(t)] ^§¥† λ^N_n ^­cqOo

|\ª‹hg{+¤¨|¸|O°•hp„w§¥†¦Á¥i¯Âï²

hgx

{N_n(t);n = 1, . . .} «)h:§¥†e„ƒhg‚)hg†e„ƒhg†¸xjz|O§¨oo|O†¬‚ƒ{|¸…phpoo,hpo"­~§¥x+b¬§¥†¸xhg†eo+§¥x¹y

1 ^qt†e„

¤¨hgx

Λ^N_n(t) =R_t

0λ^N_n(s)ds «)h"x+beh=o+x|¸…,b]qOox+§¨…"§¥†¸xhg†eo+§¥x¹yɶÎ|O{x+beh=z|O§¨o,o|O†Å‚)|O§¥†¸x“‚ƒ{|¸…phpoo“|t¶

¤¨|oo,hgo~|t¶0…p|O†ƒ†ehp…gx+§¨|O†

n²Êhg†e…phx+beh¤¨|oohpoc|t¶…p|O†ƒ†ehp…gx+§¨|O†

n|¸…p…gsƒ{qO…p…p|O{„v§¥†ƒ”Kx|xbehx+§¥fKh …,b]qt†ƒ”•hp„Éz|O§¨oo|O†Ã‚ƒ{|¸…phpo,o

N_n(Λ^N_n(t))^²

¾chg¤¨|\­.­®h?­~§¥¤¥¤)fjqt°‹hq“·ese§¥„wqt‚ƒ‚ƒ{|ruR§¥fjqtx+§¨|O†Ã|t¶Jx+beh?x+behº‹sehgseh!qt†e„:x+beh?­~§¥†e„ƒ|\­Æo§¥¼rhpo

x+b¸seo§¥”†e|O{+§¥†ƒ”“·ese…gx+s]qtx+§¨|O†eo®qtxx+behc‚]qO…,°•hgx>¤¨hgª•hg¤¿²acbƒ§¨o£qt¤¥¤¨|\­oseox,|!„ƒhpo,…g{+§¥«)hcx+behc­~§¥†e„ƒ|\­o

qOo~§¥†e„ƒhg‚)hg†e„ƒhg†¸x„v§Ì±Thg{hg†¸x+§½qt«ƒ¤¨h6„vyR†]qtfµ§¨…rqt¤1o+yvo+xhgfKo…p|Osƒ‚ƒ¤¨hp„í~§¥x+bÏq·esƒ§¨„w«ƒsv±Thg{r²

£Ñ š Ñ ™ › Ñ 1›v™ WÕ¹Õ š Ô Ö ÕRÔ ˜ acbehg{hcqt{hx+bƒ{hph®ohg‚]qt{,qtxh‚eb]qOohpo 1…p|O†ƒ”•hpo+x+§¨|O†

qrª‹|O§¨„eqt†e…ph»x+§¥fKhp|Osƒx£qt†e„=o+¤¨|\­ o+x,qt{+xr²ž?sƒ{+§¥†ƒ”³…p|O†ƒ”•hpo+x+§¨|O†jqrª‹|O§¨„eqt†e…ph»­~behg†µ†e|?¤¨|o,o|¸…p…gsƒ{o

x+beh³­~§¥†e„ƒ|\­;o+§¥¼rh“§¥†e…g{hrqOohpo¤¥§¥†ehrqt{+¤¥yÃqtx{,qtxh

1/R^N_n(t) «ƒsƒx§Ì¶x+beh6o|Osƒ{…ph³„ƒhgxhp…gxox+beh³¤¨|oo

|t¶q"‚]qO…,°‹hgx?qtx~x+§¥fKh

t−R^N_n(t) «)hp…rqtseo,h“x+bƒ{hph³„vsƒ‚ƒ¤¥§¨…rqtxh6qO…,°R†e|\­~¤¨hp„v”•hgfKhg†¸x,o?qt{+{§Ìª‹h»ƒx+beh o|Osƒ{…ph~…gsƒxox+behc…gsƒ{+{hg†¸x

W_n^N(t⁻)o+§¥¼rh®«¸y6b]qt¤Ì¶2x|

W_n^N(t⁻)/2²acbehco+¤¨|\­ ox,qt{+xx+bƒ{hpo+bƒbe|O¤¨„

Ÿ¡oox+bƒ{hpo+b2©§¨oohgxx|

H_n^N(t) =W_n^N(t⁻)/2^²

acb]hÏo|Osƒ{…phÉx+behg† «)hg”§¥†eo:¶ÎqOoxj{hgx+{,qt†eo+fµ§¥xwqt†e„ ¶¿qOo+xj{hp…p|\ª‹hg{+y•² acbehɤ¨|o+x:‚]qO…,°‹hgx:§¨o

{hgx+{,qt†eo+fµ§¥x+xhp„ qt†e„¬x+bƒ{|Osƒ”b ­§Ì†]„ƒ|\­&§¥†v·2qtx+§¨|O† ‚]qO…,°‹h¯x,oµ…p|O†Rx+§¥†¸sehwx|Å«)hwohg†¸xjqOo=§Ì¶x+beh

­~§¥†e„ƒ|\­ o+§¥¼rh~§¨o>…p|O†eo+x,qt†¸xŸ¡|O{qtx¤¨hrqOoxx+behqrª‹hg{,qt”•h~x+{,qt†eofµ§¨oo+§¨|O†K{,qtxh~§¨o>…p|O†eo§¨o+xhg†¸x>­~§¥x+b:q

…p|O†eo+x,qt†¸x­~§¥†e„ƒ|\­Äo+§¥¼rh

W_n^N(t⁻)/2^©¯² behg†Ïx+beh6{,h¯x{,qt†eo+fµ§¥x+xhp„Ï‚]qO…,°‹hgx?§¨o!qO…,°¸†e|\­~¤¨hp„v”•hp„

…p|O†ƒ”•hpo+x+§¨|O†#qrª•|O§¨„eqt†e…ph{hpo+sƒfKhpor²Êhg†e…ph~x+behhgª•|O¤¥sƒx+§¨|O†:|t¶Tx+beh~­~§¥†e„ƒ|\­.o+§¥¼rh~§¥†Kx+beh…p|O†ƒ”•hpo(´

x+§¨|O†Ïqrª‹|O§¨„eqt†e…ph“‚ƒb]qOo,h!§¨o„ƒhpo,…g{+§¥«)hp„w«¸yHx+beh!¶¡|O¤¥¤¨|\­~§¥†ƒ”:o+x|¸…,b]qOo+x+§¨…³„v§Ì±Thg{hg†¸x+§½qt¤0hpº‹s]qtx+§¨|O†

dW_n^N(t) = 1

R_n^N(t)(1−S_n^N(t))dt−W_n^N(t⁻)

2 dN_n(Λ^N_n(t)),

­~§¥x+b

W_n^N(0) = w_n(0)^» n = 1, . . . , N o‚)hp…¯§ÌË]hp„T²³Êhg{h

S_n^N(t) = 1 §Ì¶£x+bƒ§¨o!…p|O†ƒ†ehp…gx+§¨|O†$§¨o o+x+§¥¤¥¤J§¥†#¶¿qOo+x~{hp…p|\ª•hg{+yÃqtx~x+§¥fKh

t qt†e„w§¨o~¼rhg{|K|Ox+behg{+­~§¨oh²

ï¶ W_n^N(t⁻)<4 x+behg†Ïxbeho|Osƒ{…ph"…rqt† x{hp…phg§¥ª‹h6x+bƒ{hph"„vsƒ‚ƒ¤¥§¨…rqtxh ?€“oo|j§¥x?­~§¥¤¥¤”•|

§¥†¸x|?x+beh£x+§¥fKhp|Osƒx‚ƒb]qOoh£¶¡|O{qt‚ƒ‚ƒ{|ruv§¥fjqtxhg¤¥y|O†eho,hp…p|O†e„T²'+?x+behg{hgª•hg†¸x,o0¤¥§¥°•h£x+beh¤¨|oo|t¶ex+beh

{hgx+{,qt†eo+fµ§¥x+xhp„:‚]qO…,°•hgx£­~§¥¤¥¤)qt¤¨o|“‚ƒ{|\ª‹|O°‹hq!x+§¥fKhp|Osƒxr²¿¶­®h­~§¨o+behp„Kx|“fK|¸„ƒhg¤2x+§¥fKhp|Osƒxo£­®h

…p|Osƒ¤¨„:„ƒh¯Ëe†ehq“¶Wsƒ†e…gx+§¨|O†

U(W_n^N(t⁻))hpº¸s]qt¤2x|6|O†eh§Ì¶Tx+beh?…p|O†ƒ†ehp…gx+§¨|O†eo¶Îqt¤¥¤¨o£§¥†¸x|6x+§¥fKhp|Osƒx

„vsƒ{+§¥†ƒ”K¶ÎqOox~{hp…p|\ª•hg{+yÃqt†e„w¼rhg{|µ§Ì¶†e|Oxr²£Ê?hg†e…ph“x+beh³‚)|O§¥†¸x‚ƒ{|¸…phpoo|t¶¶¿qt¤¥¤¥§¥†ƒ”K§¥†¸x|µx+§¥fKhp|Osƒx

§¨o6”§¥ª•hg†¦«¸y

U(W_n^N(t⁻))dN_n(Λ^N_n(t))²#ž?sƒ{+§¥†ƒ”Ïx+behKx+§¥fKhp|Osƒx"‚ƒb]qOohKx+behK­~§¥†e„ƒ|\­ o§¥¼rhµ§¨o

¼rhg{|ƒ²>acbeh³…p|O†ƒ†ehp…gx+§¨|O†Ï§¨o„ƒhpo…g{+§¥«)hp„ë¸yHoox+bƒ{hpo+bÉqt†e„wx+beh“{hgfjqt§¥†ƒ§¥†ƒ”jx+§¥fKh“§¥†Ãx+§¥fKhp|Osƒxr²

¶¡xhg{x+beh?x+§¥fKhp|Osƒxc‚ƒb]qOohhg¤½qt‚eohpo®x+beho|Ose{…phhg†¸xhg{oco+¤¨|\­c´ïo+x,qt{+x~qt†]„H„ƒ|Osƒ«ƒ¤¨hpo®§¥xo®­~§¥†v´

„ƒ|\­ o+§¥¼rhÏox,qt{+x+§¥†ƒ”N¶W{|Of |O†ehÏhgª‹hg{+y la~aÈsƒ†¸x+§¥¤x+behÏ­~§¥†e„ƒ|\­ o+§¥¼rhÏ{hrqO…,behpo#oo+x+bƒ{hpo+bÆqtx

­~bƒ§¨…,b#x+§¥fKh…p|O†ƒ”•hpo+x+§¨|O†Ãqrª‹|O§¨„eqt†e…ph{hpo+x,qt{+xor²'ï¶0qt†e|Ox+behg{c¤¨|oo§¨oc„ƒhgxhp…gxhp„H«)h¯¶Î|O{h?{hrqO…,bƒ§¥†ƒ”