Optimized Caching in Systems with Heterogeneous Client Populations *

Transcription

1 Ome chg Ssems h Heeogeeous e ouos * Dee L. Ege Mche. Fes M. Veo Uves o Ssche Uves o scos - Mso Ssoo S S7N 5A9 Mso I Absc A mo queso ovg o-em ccess o ge e she es such s ou veo es s ho o eecve use ego (ox seves h c soe some o he cose o he ces. he ox cchg obem s moe comex he coex o couous me es becuse o he ee o cose bh s e s soge coss he ox seves becuse o he bh shg ossbes ove b ece oose mucs eve echques. hs e eveos e hgh ece c moes o eemg om ox cche coe such evomes. Secc he e moes o heeogeeous ssems hch he ox seves hve ee ce oos seve cbes. esus om he moes ove gee sghs o cchg seges o such ssems sugges h m be useu o emo ece cos moes cu ssems o eeme h coe shou be cche esose o he mesue ce oo. Iouco I ssems h suo o-em ccess o e-she eve cos c be ge euce hough he use o ego (o ox seves h soe some o he cose o he equesg ces. hs e esses he queso o hch ge ou e she shou be soe he ego/ox seves. he coex s o ssem cog mue heeogeeous ox seves oe o moe she emoe seves (.e. he omo souces hch e sc om he ox seves mucs eve o coseve seve eo bh. e ssume he ssem uses he ece oose oe mc ssce eve echque [] evee Seco. As exe Seco. ou esus e so cbe o ohe o-em mucs eve echques. Ou go s o cheve bo sghs o ox cchg seges o such ssems he h o comue he ecse cche coes o cu ssem. Hoeve he moes e hgh ece c souo me s h e secos. hus he moes seve s "ooes" o cos moes h cou be emoe cu ssems o comue he om cche coe o gve mesue ce oo. Mos o o o o-e eb cchg s e s sbue veo-o-em (VOD checues hs ocuse o eemg o hch seve(s ech ee e shou be oce so s o ome ssem cos/eomce [ 4 8 6]. Ohe ee o hs cocee seges o mc cchg evs o om couous me es so s o ss oe o moe equess h ve cose me o evous eques [4]. hs o o o hoe e ceme ev cchg hs o cosee he mc o she eve o ou es h s ebe b mucs eve echques. I cu hee s e e-o beee cchg he es h e equese mos eque cchg ess ou es h hve ess cos shg he evee om he emoe seve. Fuhemoe o segmee mucs eve echques such s oe mc ssce he ee segmes o e e sme hve moe eque mucss (.e. hghe bh equeme e be h ee ces e mucs (.e. ess cos shg o emoe eve h he e segmes o he e. hs es o ohe e cos e-o beee cchg he ee o cu ou e o cchg he segmes o m moe es. * hs esech s suoe b he Nu Sceces Egeeg esech ouc o ue O-64 b he A Foce Oce o Scec esech ue F b he No Scece Fouo ue ece o hs esbshe h cchg us he segmes o obecs hs umbe o ohe vges cug hg he ec o commuco h emoe seve hg he ec o bocs echques [] cg ohe smoohg o vbe-b-e veo [3]. o ee eomce Evuo Sec Issue o Iee eomce Moeg Se..

2 Heeogee ox oos seve cbes s mo co becuse occus cc ssems becuse ouces e eso beee ( og he soe ech ox ccog o he oc ce oo (b mxmg uom ox cche coes so s o cheve he gees ossbe shg o emove seve mucss o ucche ems. I ohe os heeogee m cuse vegece beee he gob om cche coguos h s vu om o ech ego se. A e go o hs o s o cee moes h ove sgh o ho hs eso s esove o vous s o heeogee. e cose he bove e-os he coex o eemg hch shou be cche o he ss o ego ox seve. e cose h he ou es o gve mume seve such s he ou ecues o sce euco seve o he ou shos o eeme seve mgh be exece o em hgh em o ehs s og s goo co o o oge. equess o o ou coe o he seve be esese h equess o he ou coe. Due o s bh coseos he ox seve e beeve h mes sese o sem-sc cche he ou es (.e. ove eos o eve sbe mesue ccess es he h mc cchg he es esose o vu ce equess. he queso esse hs e s o gve se o es h sece eques es hch es o oos o he es shou be sem-sc cche oe o mme he eve cos o he es? Moes h cue ce cos-shg s e s seve bh equemes o he e mucs eve echques e eee o evue he sem-sc cche esg e-os. A evous omo moe [] evee Seco. eemes h shou be soe homogeeous ox seves. he moe comues he cche coe h mmes sece eve cos obecve ue sece bh soge coss he ox ssumg oe mc ssce mucs eve he cb o cche oe o sece umbe o segmes o ech e. e mo hs evous moe o su ssems hch he ox seves e h esec o he bh soge cces h esec o he ce oos. A e chege s o cee moes h e ece o sove h o us o g he ese sghs o heeogeeous ssems. e hve eveoe o such moes hch e escbe Seco 3. he moes e e Seco 4 o su ox seve cchg seges he mc o heeogee. he sghs esg ces eve om he esus cue: Eve ssems h que sog heeogeeous eues s oe moe cos eecve o soe he sme o he ox seves he h o cose o cche coe ech ox ccog o he oc ce eeeces seve chcescs. I s oe moe cos-eecve o soe us segmes he h ee es he ox seves. Heeogee ceses hs eec s come h he homogeeous ssem. he mmum o eve cos s he obecve ego ox seve h sc ce oo c somemes uece he cche b he ohe ego ox seves uexece s. he mmum eve cos o he ces o gve ox seve s he obecve he om se o o soe h seve m be que ee h he "soc om" se h mmes o eve cos o ces. ses hee he e o obsevos move he use o ece cos moes o gue cche coe cu ssems. Seco 5 oves he cocusos o hs e suggess uu ocs o uue esech. Bcgou Seco. eves segmee mucs eve echques he oe mc ssce eve echque h s ssume he eve cos moes eveoe hs e. Seco. eves he evous cos moe o eemg om cche coe oe mc ssce ssems h homogeeous ox seves. he e moes h be eveoe Seco 3 c be e o homogeeous ssems s sec cse o he u mees; hus he ee equos o he evous homogeeous ssem moe e o eee hs e. o ee eomce Evuo Sec Issue o Iee eomce Moeg Se..

3 houghou he eme o he e (ox o emoe seve s ssume o hve eough s bh memo bh eo /o bh o suo gve umbe o smueous sems sece e eve e. Fo smc es e ssume o hve he sme eve e hough he cchg moes c es be exee o he mue e semg es. he xe-e sems be eee o s "ches" houghou he e seve bh be mesue us o he (xe-e ches h he seve c suo.. Segmee Mucs Deve he coveo och o o-em eve o veo ohe couous me es s o oce e sem (o che o ech ce eques. ece oose segmee mucs eve echques [ ] cheve sgc bh svgs b vg ech e o cesg-se segmes (s s s 3 s. he seve he emos mucs smsso scheue hch he sme segmes e mucs moe eque h he ge segmes such s he scheue use Fgue. Ech ce eceves oe smsso o ech segme such s he g she sequece o smssos Fgue. B eque mucsg he smes s segme (s he me h ce mus o beg ecevg he e s sm. Sce he ge segmes e mucs ess eque ces mus be ee o eceve hose segmes he o he he e eee o bc bchg ogehe h ohe ces h m be ee os. Fo exme ce ho eceves he se s smsso ee o eceve segme s cocue bueg he seco segme u s eee o bc. he sucue o he smsso scheue esues h o s mucs ce mgh eceve he ce c eceve ech ohe e segme beoe o he me s eee o bc. (he ee c ve hs Fgue. Mucsg he ge segmes ess eque ge euces seve bh usge o hgh ce eques es. eg segme se s : s : s 3 : 3 g segme se s 4 : s 5 : s 6 : s 7 : s 8 : me Fgue : Exme Segmee Mucs Achecue oe Dmc Ssce (8 8 3 Mos o he segmeo-bse eve echques emo sc smsso scheues hch xe mou o seve bh (e.g. equ o mes he e semg e s couous ssge o mucsg he segmes o secc ou e. A exceo s he oe mc ssce echque [ ] hch mc ssgs seve bh o e segme smssos esose o he comg ce equess. he oe mc ssce echque os he e segmes o o ses: "eg segme se" (s s "g segme se" (s s. Seve bh equ o smsso cuse use Fgue b he she se smsso eos o ech segme se s scheue s eee o eve he esecve e segme se esose o ce equess. he eg segme smsso cuse uses bh equ o ches hs see uo equ o he egh o segme ( o ech che; he g segme smsso cuse uses bh equ o ches hs see uo o ech che. Scheug o he eg g cuses mus be cooe so h o ech ce e beee bc o ech se s voe. h hs cos eg segme cuses e scheue s e s ossbe o ee eomce Evuo Sec Issue o Iee eomce Moeg Se.. 3

4 g segme cuses e scheue s e s ossbe so h he e c be she b he mxmum umbe o uue ce equess. A e ce eques ehe eceves he eg (g e segmes om cuse h s scheue o evous eques o he sme e o scheues e eg (o g segme smsso cuse. Noe h he cuse hs she eveg he s segme he se he bh oce o he cuse (equ o o - mes he e semg e c be eoce o cuse o segme smssos o ee e eeg o ce equess. he moes hs e eeme ho much o seve bh mus be oce o eg segme smsso cuses g segme smsso cuses esecve so h he oe cuse(s c be oce he ech e ce eques ves. Dmc scheug c ove ue mmee sevce o ce equess [] s e s segmee mucs eve o ess ou es coexs hee e ou ves ove me. oe mc scheug moves he ecec o segmee mucs smssos becuse he eg segme smsso cuses e sme e use moe ece. oe mucss so move ox cche eomce becuse o ech e ego ox seve m cche us he eg segme se boh ses o segmes o ehe se. ece o [] shos h he vege seve bh eque e e he usg oe mc ssce mucss o mmee o-em e eve gos smoc o oghmc he ce eques e. Ohe o-em mucs eve echques such s he echque o s sem g gce chg o cooe mucs [7957] he ece oose hechc mucs sem megg bh smmg echques [34] so sm oos o ou couous me e moe eque h e oos o he e. Ssems h use hese eve echques ou hve sm e-os h esec o om ox cche coe sm om cchg seges ( he eve o e sue Seco 4 s o ssems h use he oe mc ssce mucs echque.. Moe o omug Ome che oe Homogeeous Ssems A evous moe [] eemes he ox cche coe h mmes eve cos o ssems h ( ec ox seve bh soge cbes (b ssc homogeeous ego ce oos (c oe mc ssce eve ( es o equ se eve e. Fo ech e he ox seves c soe ehe oe o us he eg segme se o he e. I hs seco e ove oveve o he moe he esus h ee evous obe om he moe hch oves he sg o o eveog he e moes o ssems h heeogeeous ox seve cbes oos. he homogeeous moe uses sme c esmes o he me vege o he umbe o emoe seve ches ( emoe ox seve ches ( ego use ech ce s seve mmee ue gve se o e eques es s uco o he e segme ses h e soe he ox. esus gve [] sho h hese sme bh esmes e ve cose o he ee o he cuve o me ce g me vesus he vese o he umbe o vbe seve ches h he ee o he cuve s c que sh. Sce he ssem c ove mmee sevce o ce equess he vege ce g me s c ve cose o eo e he (sh ee o he cuve. be gves he moe u ouu mees. Noe h he s ou u mees ( sec he coguo o he oe mc ssce ssem. Noe so h he mee β c cue coseo o oxme eo bh coss s e s he seve esouce coss o ech e o che o sem. he e ssem coss comug he cche coe h mmes eve cos e he mxmum ox seve bh ( N ches soge cc ( N segmes. he e moe ouus e he vues h sec o ech e hehe he e shou be u cche cche o o cche he ox seves. ve h β s he cos o o ox seve che emoe seve ches gve h s he umbe o ox seves he ssem he evous homogeeous ssem moe o om ox cche coe s ee s oos: o ee eomce Evuo Sec Issue o Iee eomce Moeg Se.. 4

5 m subec o ( β ( D emoe ego ( N ( ego ches ego N segmes I he bove moe he smbo eeses he veco h comoes.... Noe h he exesso o be mme s he o eve cos o es o ces egos. Hoeve sce he ego ce ouos e ssc he sme vg he o eve cos b gves he cos o eve o vu ego hose ces coecve o / o he emoe eve cos. hus s uo suggess he cche coe h mmes o eve cos s he sme s he coe h mmes vu ego s cos he homogeeous ssem. Noe h β he moe comues he ox cche coe h mmes he use o emoe seve bh. Equos o comug emoe ego D ego o heeogeeous ssem moes e gve he Aex scusse he ex seco. Fo he es o he homogeeous moe equos hch e sec cses o he heeogeeous ssem equos he ee s eee o []. 3 Moes o Ome che oe Heeogeeous Ssems I hs seco e eveo o e omo moes h em su o heeogeeous ssems hch egos m hve eg ce eques es ox seve cbes o e seeco equeces. he go s o cee ece moes h c be use o eve useu sghs esg ces o cche coe such heeogeeous ssems. oo Ssem N ches Nsegmes β Ouu emoe ego be : mees o he Homogeeous Ssem Moe mee Deo umbe o es o eques e o e (om egos mee Deo mxmum umbe o ches ech ox seve soge cos ech ox seve (mesue us o he se o segme s umbe o ox seves he cos o o ox seve che emoe seve che umbe o segmes he eg segme se o umbe o segmes he eg g segme ses se o he h segme (eve o he se o he s he ges segme se ( he g segme se mee Deo umbe o ches eee he emoe seve umbe o ches eee he ox seve Dego soge eee ech ox seve (mesue us o he se o segme s equs e s soe o he emoe seve; ohese equs o he eg segme se o e s cche ego; ohese equs he ee e s cche ego; ohese o ee eomce Evuo Sec Issue o Iee eomce Moeg Se.. 5

6 he moe Seco 3. ocuses o he mc o heeogeeous ego ce eques es seve cbes. he moe Seco 3. s esge o su he mc o heeogeeous e seeco equeces. I he ccuos o vu ox seve cos e ssume h he ox seve oes o cche gve segme se shes equ he cos o emoe mucss o he segme se. hs cg oc s bse o he oo h ox seve h oes o cche he segme se s o "subsco" o emoe eve o he e h ee ooo o he vege umbe o emoe seve ches o sems eque o eve he segme ve b he umbe o subscos. hs oc smes he ccoug he emoe seve m so sm he commuco beee he ox he emoe seve. he oc oves ce be o ego h hghe eques e o gve segme se h s cche. Hoeve he ce be s ve sm sce seve bh usge gos o oghmc he eques e. I s so sgho o mo he moes so h he cos o ech emoe mucs s she equ b he ox seves h se o he mucs he memeo eo o he moe ee ccoug s eeme ohhe. 3. Heeogeeous e eques es Seve bes o cee ece useu moe oe ox seve s ssume o hve hghe o oe ce eques e ossb so ee bh soge cc h ohe ox seves hch hve he sme eques e seve cbes. Leg "" eoe he sc seve h hs hghe o oe ce eques e "" eoe o he ohe (o-sc ox seves be ees he e u ouu mees o hs heeogeeous moe. I o o hese e mees he u ouu mees o he ox seves ( N ches N segmes ego ech hve suesc ( o o eoe he e o seve (sc o o-sc h he D ego mee es o. As beoe he e ouus o he moe e he mees h sec hehe ech e s u o cche ech e o ego ox seve. Noe h hee e e such mees o ech e ue o ossbe s o suescs o he vues. Fe ocos h mme o eve cos e obe b sovg he oog omo moe: m subec o D D emoe ego ego ego ego ego ego ( β ( ( ( ( ( N ( N ( N ( N ches ches segmes segmes be : Ne mees o he eques e Seve Heeogee Omo Moe Iu mee Deo co o he o equess h e om ces beogg o he sc ego Ouu mee Deo emoe he comoe o he emoe eve cos (s mesue umbes o ches ooe o emoe he sc ox seve o ech ohe ox seve esecve equs e s cche he sc o-sc ox seves ccog o he suescs esecve ( he e s o cche ego; o (he eg segme se s cche ego; he e s u cche he esecve ego seve; equs ohese. 6

7 Noe h e hve use he oo o eese he veco hose comoes e.... As he homogeeous ssem moe he soge equemes o ech e o ox seve e comue b summg ove he segmes soe h e o seve. he ccuos o eque emoe seve bh eque bh o ech e o ox seve use he sme och s he homogeeous moe bu e moe comex o o esos. Fs ech e o ox seve m om cche ee segme ses. Seco hee e comex ecos h esec o ho mucs smsso cuses e she b ces om he ee es o ox seves. he ee equos h cue hese eecs e ove Seco A. o Aex A. e use he em "soc om" o eoe he ox seve cche coe h mmes o eve cos s exesse he bove omo moe. I heeogeeous ssems h hve smmec ce oos o seve cbes he "soc om" cche coe m e om he "vu om" cche coe h s obe cu (comeve ego sevce ove ems o mme s o eve cos. o ssess hehe hese o es o souos e o gve ssem e eve vu om cche coe o he sc ox seve ue xe cche coe (such s he soc om coe he o-sc ox seves. Sm e eve he vu om cche coe o he o-sc ox seves ue xe cche coe o he sc ox seve. he oog omo obem mmes he eve cos o he sc ego ox seve h he O suesc O eog he xe ocos ssume o he o-sc egos: m subec o emoe ego ( β ( D ego ego O ( N ( N O ches segmes he eve cos exesso h s mme he bove moe cues o he cos o emoe seve mucss h s ooe o he sc ego. hs cos s he sum o ( / o he ches eque o mucs segmes h e o cche b o he seves (b o he bh eque o mucs segmes h e o soe he sc ox seve bu e soe he ohe ox seves. he ee equos o hs he ogous moe o mmg he eve cos o he o-sc ox seves e gve Seco A. o Aex A. 3. Heeogeeous Fe Seeco Fequeces I hs moe he es he ox seves e ech oe o equ-se gous ech gou o ox seves hs eeece (.e. ge co o he egos ce equess o sc gou o es. Ech ox seve hs he sme ce eques e o ech o s (- o-eee gous o es. Aso he eve seeco equeces o he es h gou e he sme o gous o ox seves. he e moe oo o he ssem h hese heeogeeous e seeco equeces s gve be 3. Ahough ech ox seve m om cche ee e segmes he smme he ego ce oos mes h ech ox seve cche he sme segmes om s esecve eee gou o es. Sm ech ox seve cche he sme segmes om s esecve o-eee gous. hs ge smes he moe oo es o ece moe h g hs e cche ceme vbes ( e e. 7

8 be 3: Ne mees o he Fe Seeco Fequec Heeogee Omo Moe Iu Ouu mee Deo co o he o equess om ego h e o s eee (o gou o es umbe o gous o es mee Deo equs e s cche ech ox seve o hch he e s gou s he ego s eee gou ech ox seve o hch he e s gou s o he eee gou ccog o he suescs esecve ( he e s o cche ego; o he eg segme se s cche eeg/o-eeg ego seve; he e s u cche eeg/o-eeg ego seve; equs ohese. he smme he ego ce oos so mes h he soc om ox cche coe s he vu om coe. hs om cche coe s comue b sovg he oog omo obem: m subec o ( β ( D emoe ego ( N ( ego ches ego N segmes he seve bhs soge equemes e comue usg he equos o heeogeeous e seeco equeces gve Seco A. o Aex A. 3.3 Souo o he Heeogeeous Omo Moes Ahough he ee equos o he heeogeeous moes ee bove e someh comex boh he obecve ucos coss ech moe e e ucos o he b vbes. hs s e go ou esg o he moes s hese obems e cbe mxe ege ogms (MIs []. Fo he uoses o obg souos o he moes e omue he omo obems he AMS moeg guge [3] sove hem usg e ogmmg [9] bse bch bou souo seg. Des o seve comuo ehcemes o he bsc moe e gve [5]. 4 esus Isghs evous esus [] shoe h homogeeous ssems s oe moe cos-eecve o cche he segmes o m es he h he comee o ee es. he esus so shoe h he om cche coe ees o e ssem mees cug β he eve coss o s bh soge cc he ox seves. I cu he esus shoe h s β ceses o he he ego seve bh cos s cve o he om cche coe he ox seves shou cche (segmes o ess ou es. hs seco eses esus sghs o he oog hee es o heeogeeous ssems:. ssems h - homogeeous seves oe seve h sgc ghe ce eques e. ssems h - homogeeous seves oe seve h sgc heve eques e 3. ssems h ou gous o ego seves ech gou o seves hvg hghe co o equess o cu eee gou o es. 8

9 I ech cse e e eese uesg he mc o he gve e o heeogee o he om ox cche esg ues. Fo he s o es o ssems e e so eese he exe o hch hee s vegece beee he "soc om" ox cche coe h mmes o eve cos he "vu om" coe h mmes he cos o he ces oc o gve ox seve. he vu om cchg seges e mo comeve ogos e ovg he ox cche sevces. he e esus hs seco e summe Seco 5. I ech exeme scusse beo he ssem mees e s gve be 4 uess ohese se. he eu coss o ox soge (N segmes bh (N ches be 4 be ce he bsee ssem cbes. hese bsee ques ee eve ssumg he seve hs sx ss ech h soge cc equ o 3. hoe es bh o smg 48 smueous sems. Noe h N segmes 3654 mes soge o oxme 9% o he o o he es h ces c eques. he eu mees o he ssce eve checue ( ee seece o be eeseve o eve checue h mgh be eoe cce. hs checue hs sum o segme eghs equ o 89 he sum o eg se segme ses equ o 9; hus he eg segme se s 9/89 o 5% o he e. Fo he exemes eoe hs e he om cchg seg s ve sm o ohe vues o bu 7 cheves he oes ome eve cos mos eve cse. hus e eo he esus o hs vue o. he smsso me o he s segme s s equ o he e egh ( bes ve b he sum o he segme eghs ve b he e semg e. I s es o see om he equos he Aex h he e egh semg e ec ech em he eque seve bh ( emoe o ego o hough he ouc o he e eques e. I ech exeme beo o ce eques v e s exesse us o he vege umbe o ce equess h ve e u o me equ o he smsso me o segme s (.e.. hus he esus ho o e egh semg e h e ooo o he sece coss o ox s cc bh. Fe seeco equeces ove o h gou o es he moe ee Seco 3. e ssume o hve he Z( sbuo. I he ghs he hegh o he bc b o ech e (o obec oe o ecesg e ou ces he ecege o he e h s soe gve ox seve gve mmum cos souo. be 4: mees Use he Exemes mee Deu Vue mee Deo umbe o es ce eques e mesue equess e u o me equ o he smsso me o segme s N ches 88 mxmum umbe o ches ech ox seve N segmes 3654 soge cc ech ox seve umbe o ox seves 7 umbe o segmes he eg segme se o umbe o segmes he eg g segme ses 4488 se o he h segme (eve o he se o he s segme he ges segme se ( he g segme se 4. Oe ox Seve h Lgh eques-e he ssem cosee hee hs e ego seves o ce eques e ( equ o. Oe ox seve hs (gh ce eques e equ o.5; he ohe e ox seves ech hve ec (hev ce eques e e equ o. Fgue eses he soc om vu om cche coe ech e o ox seve he seves hve he sme soge cc bh equ o he bsee vues gve be 4. he esus sho h he soc om cche coe s he sme o boh he gh hev ox seves hs s so 9

10 he coe h s vu om o ech ego. hus hs heeogeeous ssem comeve ox sevce oves c vu ome he soe he esecve seves ve he soc om cche coguo. Fuhemoe he om cche coe o ech vue o β s e ec o he om cche coe o homogeeous ssem h e ox seves h ech hve eques e equ o he hev seve eques e o e (o sho. hus he ox seves hve he sme bh soge cc he vu om coe s he sme s he soc om coe o boh gh hev ox seves he om coe o he hev ox seves s he sme he gh seve s o ese he om coe o he gh ox seve s he sme s he om coe o he hev seves. ecege che ego ecege che ego ecege che ego Obec ou Obec ou Obec ou ( β gh & hev (b β. gh & hev (c β. gh & hev soc & vu soc & vu soc & vu Fgue : Bsee Ssem h Oe Lgh Seve Om che oe ( hev gh.5 ecege che ego Obec ou ecege che ego Obec ou ecege che ego Obec ou ( β. gh ox (b β. hev ox (c β hev ox soc & vu soc om coe vu om Fgue 3: Oe Lgh Seve h euce Soge Om che oe ( hev gh.5

11 Fgue 3 eses he om cche coe he he ox seve h he ghe eques e hs euce soge bh ech equ o 5% o he bsee ques. I hs cse: he soc om cche coe es o ech e o ox seve s sho Fgues 3 3b. Fo he ox seves h gee soge bh (.e. he seves h hev eques e he soc om cche coe es ge om he vu om cche coe s sho Fgues 3b 3c. I he gh eques-e seve cches o he 5% o he mos ou es he soc om cchg seg o he hev eques e ox seves (sho Fgue 3b s o u cche he es ou es he h he hgh ou es h he soe Fgue b. hs uses ve sog uece o he gh seve o he cchg seg hev seves ehs soge h oe mgh ce. he exo o hs mc s h he emoe seve mus mucs he g segme se o he mos ou es eque o he gh seve. hus he gees gob vge s o he hev seves o she he cos o hose mucss se cche he ess ou es h he gh seve ees ve eque. e so vesge cses hee he ox seve h gh ce o hs cese soge bh equ o ce he soge bh he ox seves h hev ce o. I some cses o exme he β equs. o mme o eve cos he gh seve cches exc he sme segmes s he hev seves; s ex soge bh cbes e o use. Hoeve he vu om souo o he gh seve h he cche segmes he hev seves xe s he e o mmg o eve cos shos h comeve gh seve ou soe o emoe. hus g he vu soc om cchg seges e mc o he seve h hs gee soge bh. 4. Oe ox Seve h Hev eques-e he s ssem cosee hee hs eeve ox seves ech h he bsee (.e. eu bh soge cc. o ce eques e s equ o. Oe ox seve s ce o s equ o equess e ; he ohe e ox seves hve ce eques e o. he β equs eo he soc om vu om cche coe o o he ox seves s e ec o he coe sho Fgue. hs coe s u he om cche coe homogeeous ssem h β o ce eques e gee h eo he sme ox soge cc bh. he β equ o. he soc om cche coe gve Fgue 4b s g he sme o o he ox seves. Hoeve he soc om cche coe o he gh seves s ee h he he hev seve s o ese (Fgue 4. Fuhemoe s eesg o oe h es u cche he heeogeeous ssem (Fgue 4b e ess ou h he es u cche Fgue 4 bu e moe ou h he es u cche he homogeeous ssem sho Fgue 4c. Noe h Fgue 4c s he homogeeous ssem h o eques e equ o he o eques e he heeogeeous ssem. Mmg he gh seve eve cos he xg segmes cche he hev seve s Fgue 4b esus vu om cchg seg h u cches sgh moe ou se o es. Fuhemoe he gh seve cche coguos e xe hese vu om vues he hev seve's vu om seg s o cche he sme segmes s he gh seve. I hs cse s ossbe h coug o comeve ome vu ego seve eve coss m e o usbe souo. Oe vue o he omo moe s o scove such suos hch oe mgh o evse e cg seges. e ex cose ssem hch he hev eques e ego ox seve hs ce he bsee seve soge bh he ohe seves hve he bsee soge bh. Ove ce eques e s equ o. he hev ego's ce eques e s equ o ; ech o he ohe e egos'ce eques e oxme equ o 9. he esus e esee Fgue 5. I e mme he vu eve cos o he gh seve he xg he cche coe he hev seves s Fgue 3c he gh seve cchg seg ems s Fgue 3.

12 ecege che ego Obec ou ecege che ego Obec ou ecege che ego Obec ou ( β. homogeeous (b β. hev & gh (c β. homogeeous soc om coe Fgue 4: Bsee Ssem h Oe Hev Seve Om che oe ( hev gh ecege che ego Obec ou ecege che ego Obec ou ecege che ego Obec ou ( β gh (b β hev (c β hev soc & vu soc om vu om ecege che ego Obec ou ecege che ego Obec ou ecege che ego Obec ou ( β. hev (e β. hev & gh ( β. hev soc om soc om vu om Fgue 5: Oe Hev Seve h Ex Soge Om che oe ( hev gh 9

13 Fo β. he soc om cche coe o he ox seve h gee soge bh (e.g. Fgues 5b 5e es ge om he vu om cche coe o h seve (e.g. Fgues 5c 5 esecve. o mme o eve cos he β he hev ox seve uses s ex soge o cche he es ou es (Fgue 5b he h he mos ou es (Fgue 5c. o mme o eve cos he β. he hev ox seve soes exc h he gh ox seve soes (Fgue 5e; s moe ecoomc o he hev e seve o she he emoe sems h e evee o he gh egos h o use ox seve bh o eve he ex segmes cou soe (Fgue 5. he esus o hs he evous seco ce h ox seves h gh ce eques es c ge uece he soc om cche coe o ox seves h heve ce eques es he soge bh he gh ox seves s sgc ess h he coesog cbes he hev seves. 4.3 Heeogeeous Fe oues he ex se o exemes coses ssems h eve ego seves oe o ou gous o hee seves o hue es oe o ou gous o es. Ech gou o ox seves hs ee eee gou o es. he co o ego ce equess o he eee gou s gve b he mee ; ech o-eee gou o es s seece h equec ( /3. Seeco equec h ech gou (eee o o s moee b Z( sbuo. he o ce eques e s equ o equess e. Fgue 6 oves esus o he cse h ox seves hve he bsee soge cc bh; Fgue 7 oves esus o he cse h ox seves hve hee mes he bsee soge cb. Fgues 6 7 sho he ge o eee gou ccess equeces ue hch oxes soe he sme cche coe. Fgues 6bc o 7bc sho he ee om cche coe o he eee gou he ohe gous he he eee gou ccess equeces e hghe. e cocusos om hese esus e: he eee gou ccess equec s hgh eough moe segmes e cche o he eee gou h o he ohe gous (Fgue 6b come h 6c o Fgue 7b come h 7c. he hesho vue o bove hch he oxes soe moe o he ces eee gou s hghe he he ox seves hve gee bh soge cc. he he oxes soe moe o he eee gou he cche s ge eough o u cche some o he es (.e. Fgue 7b he es ou eee es e u cche he h he mos ou eee es. hs s g ue o he vges o shg he cos o emoe mucss o he e segmes o he mos ou es. he ox bh soge e me (Fgue 6 cchg o segmes o subse o he es ox seves es o be esbe comomse he ee egos hve ee eee es Sm ox cche coe mog he heeogeeous seves s mos cc o eve moe ou es. 5 ocusos I hs e e hve eveoe ece c moes o eemg om cchg seges o couous me es ssems hee he ox seves hve heeogeeous ce ouos /o seve cbes. Heeogee s cuc ssue o ox cchg s mes ume coc beee he obecves o cchg he mos useu oc o mxmg commo cche coes ( hus emoe e equess coss ego seves so s o cheve he gees ossbe shg o mucss om emoe seve. e hve e he moes o ob sghs o ho hs coc s esove o secc es o heeogee. he esus o hose exemes ce h: 3

14 ecege che ego Obec ou ecege che ego Obec ou ecege che ego Obec ou (.5.4 gous (b.5.9 eee gou (c.5.9 ohe gous Fgue 6: eee Fe ou e ego Bsee Ssem ( β 4 ecege che ego Obec ou ecege che ego Obec ou ecege che ego Obec ou (.5.75 gous (b.9 eee gou (c.9 ohe gous Fgue 7: eee Fe ou e ego hee mes Bsee ox Seve bes ( β 4. he he ox cche co ho o he ou es s mos s om o cche he segmes o m es he h segmes o ee es s s sho Fgues -6. I he cse o heeogeeous e seeco equeces hee s cese eec o cche segmes he h u es ue o he move cos-shg o emoe eve he seves cche he sme segmes.. Eve ssems h que sog heeogeeous eues (e.g. Fgues 4b 5e 6 7 s oe moe cos eecve o soe he sme o he ox seves he h og cche coe ech ox o he oc ce oo. he exceos o hs ue e he he ox seves hve ee soge cc bh (e.g. Fgues 3c o Fgues 5b o he ces ee ox seves hve ve sog eeeces o ee es (e.g. Fgues 6bc o Fgues 7bc. 3. he mmg he o cos o eve ego ox seve h sc ce oo c somemes uece he se o segmes cche b ohe ox seves uexece s. I cu seve h oe soge cc o ee eee es c uece ohe seves o u cche he es ou es he h he mos ou es (e.g. Fgue 3b come o 3c Fgue 5b come o 5c Fgue 5e come o 5 o Fgue 7b come o 7c. 4. he mmg he eve cos o gve ox seve he esug om se o segmes o be cche h seve m be que ee h he "soc om" se h ou mme ove cos o eve. hs occue cu o ox seves h h gee soge cc bh. Noe h such cses m be he emoe ove s bes ees o ecouge he ego ox seves h ess soge bh o cese he ox seve soge bh. 4

15 Due o he comex ueces o heeogeeous ce ouos o he soc vu om cchg seges he ox seves m be useu o cose eog moes cu ssems o comue om cche coe o he mesue ce oos. O-gog esech cues: ( exeg he moes hs e o cue moe gee heeogee ego ce oos ( vesgg om cchg seges o ee couous me es o mucs eve echques h em moe exbe og o he beee he ox he emoe seve [3] (3 evsg o-e cchg gohms h cheve e-om cche coe. eeeces [].. Agg J. L. o. S. Yu A emuo Bse m Bocsg Scheme o Veo-o-Dem Ssems oc. IEEE I. o. o Mume omug Ssems (IMS 96 Hoshm J Jue 996. [].. Bs B. V. e os-bse ogm Aoco o Dsbue Mume-o-Dem Ssems IEEE MuMe 3 3 (F [3] A. Booe D. ec A. Meeus AMS: A Use s ue he Scec ess Souh S Fcsco A 988. [4] D.. Bubec L.. oe Hechc Soge Mgeme Dsbue VOD Ssem IEEE MuMe 3 3 (F [5] Y.. A. Hu. Vu Omg chg eomce oc. IS&/SIE o. o Mume omug Neog 999 (MMN 99 S Jose A Ju [6]. o S. I os-ae ox chg Agohms oc. USENI Smosum o Iee echooges Ssems (USIS Moee A Decembe [7] S.. e D. D. E. Log Imovg Veo-o-Dem Seve Ecec hough Sem g oc. 6 h I. o. o omue ommucos Neos (IN 97 Ls Vegs Nev Seembe [8] A. hhuho. B. Dg. Neees M. F. Sch. J. oe A Hechc Iee Obec che oc. 996 USENI echc o. Ju 996. [9]. B. Dg Le ogmmg Exesos ceo Uves ess ceo NJ 963. [] D. L. Ege M.. Veo Dmc Ssce Bocss o Veo-o-Dem oc. 4h I. osho o Mume Iomo Ssems (MIS 98 Isbu ue Seembe [] D. L. Ege M.. Fes M.. Veo Ome ego chg o O-Dem D Deve oc. IS&/SIE o. o Mume omug Neog 999 (MMN 99 S Jose A Ju [] D. L. Ege M.. Veo J. Zho Mmg Bh equemes o O-Dem D Deve oc. 5 h I. osho o Mume Iomo Ssems (MIS 99 I es A Ocobe [3] D. L. Ege M.. Veo J. Zho Om Ece Megg Scheues o Veo-o-Dem Seves oc. 7 h AM I. Mume o. (AM Mume 99 Oo FL Novembe [4] D. L. Ege M.. Veo J. Zho Bh Smmg: A echque o os-eecve Veo-o- Dem oc. IS&/SIE o. o Mume omug Neog (MMN S Jose A Ju. [5] M.. Fes.. Mee Moes Souo o O-Dem D Deve obems. M. os (e. Nocovex Omo s Acos Vo. 4 ue (://.cs.sc.eu/mh-og/ech-eos/99-4.s. [6] L. o J. uose D. ose Ece Schemes o Bocsg ou Veos oc. 8h I. osho o Neo Oeg Ssems Suo o Dg Auo Veo (NOSSDAV 98 mbge U Ju 998. [7] L. o D. ose Sug Iseous Veo-o-Dem Sevces Usg ooe Mucs oc. 999 IEEE I. o. o Mume omug Ssems (IMS 99 Foece I Jue 999. [8]. A. Hu S. Sheu Ssce Bocsg: A Ne Bocsg Scheme o Meoo Veo-o-Dem Ssems oc. AM SIOMM 97 o. es Fce Seembe [9]. A. Hu Y. S. Sheu chg: A Mucs echque o ue Veo-o-Dem Sevces oc. 6 h AM I. Mume o. (AM Mume 98 Bso U.. Seembe []. L. Nemhuse L. A. ose Iege ombo Omo e Ne Yo NY 988. [] J.-. Nussbume B. V. e F. Sch J... Sebe Neog equemes o Iecve Veo o Dem IEEE Jou o Seece Aes ommucos 3 5 (Jue [] J.-F. s D. D. E. Log. E. Me Zeo-De Bocsg oocos o Veo-o-Dem oc. 7 h AM I. Mume o. (AM Mume 99 Oo FL Novembe [3] S. Se J. exo D. ose ox ex chg o Mume Sems oc. IEEE Iocom 99 Ne Yo NY Mch 999. [4]. e H. M. V A. D D. Sm esouce-bse chg o eb Seves oc. IS&/SIE o. o Mume omug Neog 998 (MMN 98 S Jose o Ju

16 6 [5] S. Vsh. Imes m Bocsg o Veo-o-Dem Sevce oc. IS&/SIE o. o Mume omug Neog 995 (MMN 95 S Jose A Febu [6] S. Vsh. Imes Meoo Ae Veo-o-Dem Sevce usg m Bocsg Mume Ssems 4 4 (Augus Aex A Hee e ove equos o comug he eque emoe seve ox seve bhs ( emoe ego ox seve soge cces (D ego s ucos o he e eques es he segmes cche he heeogeeous ox seves. A. Heeogeeous e eques es Seve bes I he heeogeeous ssem moe escbe Seco 3. o he e ox seves e ssume o be ec exce o sge ox seve h hs sc (hghe o oe ce eques e ossb so ee seve bh soge cc. I o o he oo ee bes he ee equos o hs moe so use ques x x ( x hch e ee s he mxmum es hch smsso cuses c be oce o mucss o he eg/g (/ segme se o e sgushe b he seve ( emoe sc o - ohe ox b hehe equess om sc ( o o-sc ( ego eceve o he e om he emoe se ( (o he g segme se om he emoe se ( o eceve o he e om he ego se (. Fuhe s he bo o he e eoes he ges segme se he eg segme se eoes he uo o smsso o segme s. Sce he eg g segme ses e o uo cue - ches esecve he vege umbe o ches eque he emoe seve he o es o ox seves he soge eque he ox seves e gve b he oog equos: ( ( emoe ego ( ( ego ( ( ( ego D ( ( ego D ( he emoe seve che cos h s ooe o (ces o he sc ox seve s he sum o ( / o he ches eque o mucs segmes h e o cche b o he seves (b o he bh eque o mucs segmes h e o soe he sc ox seve bu e soe he ohe ox seves:

17 7 ( ( emoe Sm he emoe seve che cos ooe o (ces o o-sc ox seve s gve b: ( ( emoe he mxmum es hch smsso cuses c be oce e eeme b he v e o equess he seve o ees b he se o he smsso cuse cchu o (ee s he eo o me om he begg o smsso cuse ug hch e vg eques c be seve b h cuse. smsso cuses o eg g segme ses hve cchu os o eg eghs. Fuhe he egh o he cchu o o eg segme se smsso cuse ees o he he cuse begs eosh o he e o he cchu o o he coesog g segme se smsso cuse. Fo ech sc cse he e m be comue s he vese o he mmum vege me beee he os o e cuses o e. Fo exme cose he cse hch ces eceve he g segme se o e om he emoe se. Sce he se o he cchu o o g segme se smsso cuse s sce he vege me om he e o he cchu o u e ce eques o e s / he mxmum oco e o e smsso cuses o he g segme se o e s comue s: I hs equo hose gve beo he suesc oos * ( -c - ( o' ce ( o e emoe h he oog megs: equo hch * suesc es hos o cosse subsuo houghou he equo ( h suesc oso o he * h o ; equo hch - suesc es hos o (o ecess cosse subsuo ( h suesc oso o he - h o ; equo hch suesc x es hee x hos ehe x o s cosse subsue houghou he equo ( h suesc oso. Noe h he bove exesso gves he oco e o smsso cuses he seve h ume bh. Sce he seve hs me bh he bove exesso s he mxmum oco e s hee m be queueg o ce equess bchg o hese equess he queue o ce bg. he emg equos o he moe h heeogeeous ce eques es seve cbes e s oos: ( (

18 8 ( ( ( ( ( ( ( ( ( * * * ( ( ( ( ( * * * ( ( ( A. Heeogeeous Fe Seeco Fequeces I he heeogeeous ssem escbe Seco 3. o he e he es he ox seves e ech oe o equ-se gous ech gou o ox seves hs eeece (.e. ge co o he egos ce equess o sc gou o es. Ech ox seve hs he sme eques e o ech o s (- o-eee gous o es. Aso he eve seeco equeces o he es h gou e he sme o gous o ox seves. I o o he oo ee bes 3 he ee equos o hs moe so use ques x x ( x hch e ee s he mxmum es hch smsso cuses c be oce o mucss o he eg/g (/ segme se o e sgushe b he seve ( emoe ox seve h hs eeece o o he e s gou o ox seve h oes o hve eeece o he e s gou b hehe equess om eeg ( o o-eeg ( ego eceve o he e om he emoe se ( (o he g segme se om he emoe se ( o eceve o he e om he ego se (. Noe h hough ech ox seve m om cche ee e segmes ue o he smme he ego ce oos ech ox seve cche he sme segmes om s esecve eee o-eee gous. hs ge smes he moe. he umbe o ches eque he emoe seve ech ox seve he soge eque ech ox seve e gve b he equos beo hee he ex us o ove he es h sge gou sce he gous o es e smmec: ( ( emoe ( ( ( ego

19 9 ( ( ego D ( ( ( he mxmum es hch smsso cuses c be oce e comue sm sho s he evous moes. Usg he c o' ce o oo use evous o he eques e seve heeogee moe e hve: ( ( ( ( ( ( ( ( ( ( ( * * * ( ( ( ( ( * * * ( ( (