A Theoretical Framework for Selecting the Cost Function for Source Routing

Transcription

1 A Theoreal Framework for Seleg he Cos Fuo for Soure Roug Gag Cheg ad Nrwa Asar Seor ember IEEE Absra Fdg a feasble pah sube o mulple osras a ework s a NP-omplee problem ad has bee exesvely suded ay proposed soure roug algorhms akle hs problem by rasformg o he shores pah seleo problem whh s P-omplee wh a egraed os fuo ha maps he mul-osras of eah lk o a sgle os However how o sele a approprae os fuo s a mpora ssue ha has rarely bee addressed leraure I hs paper we provde a heoreal framework for pkg a os fuo ha a mprove he performae of soure roug erms of omplexy overgee ad probably of fdg a feasble pah Idex Terms ulple addvely osraed QoS roug os fuo NP-omplee I INTRODUCTION he remedous growh of he global Iere has gve rse T o a varey of applaos ha requre qualy-of-serve (QoS beyod wha s provded by he urre bes-effor IP pake delvery serve Oe of he hallegg ssues s o sele feasble pahs ha sasfy dffere qualy-of-serve requremes Ths problem s kow as QoS roug I geeral sae dsrbuo ad roug sraegy [] are he wo ssues relaed o QoS roug Sae dsrbuo addresses he ssue of exhagg he sae formao hroughou he ework [] Roug sraegy s used o fd a feasble pah ha mees he QoS requremes I hs paper we fous o he laer ask ad assume ha aurae ework sae formao s avalable o eah ode A umber of researh works have also addressed aurae formao [3 4] whh s however beyod he sope of hs paper QoS osras a be aegorzed o hree ypes: oave addve ad mulplave Se oave parameers se he upper lms of all he lks alog a pah suh as badwdh we a smply prue all he lks ad odes ha do o sasfy he QoS osras We a also over mulplave parameers o addve parameers by usg he logarhm fuo For sae we a ake -log(- p as he replaeme for loss rae p Thus we fous oly o addve osras hs paper I has bee proved ha mulple Auhors are wh he Advaed Neworkg Laboraory Newark NJ 7 USA (orrespodg auhor o provde phoe: ; fax: ; e-mal: asar@edu Ths work has bee suppored par by he New Jersey Commsso o Hgher Eduao va he NJ I-TOWER proe ad he New Jersey Commsso o See ad Tehology va he NJ Ceer for Wreless Teleommuaos addvely osraed QoS roug s NP-omplee [5] Hee aklg hs problem requres heurss I [6] a heurs algorhm was proposed based o a lear os fuo for wo addve osras; hs s a CP (ulple Cosraed Pah Seleo [] problem wh wo addve osras A bary searh sraegy for fdg he approprae value of k he lear os fuo w + kw or kw + w where w ( p ( are wo respeve weghs of he pah p was proposed ad a herarhal Dksra algorhm was rodued o fd he pah I was show ha he wors-ase omplexy of he algorhm s Ο (log B( m+ log where B s he upper boud of he parameer k m s he umber of lks ad s he umber of odes Smlar o [6] Lagrage Relaxao Based Aggregaed Cos (LARAC was proposed [7] for he Delay Cosraed Leas Cos pah problem (DCLC Ths algorhm s based o a lear os fuo λ + λd where deoes he os d he delay ad λ a adusable parameer I dffers from [6] o how λ s defed: λ s ompued by Lagrage Relaxao sead of he bary searh I was show ha he ompuaoal omplexy 4 of hs algorhm was Ο ( m log m However [8] for he same problem (DCLC a o-lear os fuo was proposed afer osderg he shoromg of he lear os fuo ay proposed soure roug algorhms rasform he mulple osraed QoS roug problem o a shores pah seleo problem wh a egraed os fuo ha map he mul-osras of eah lk o a sgle os However how o sele a approprae os fuo s rarely addressed leraure; hereby we wll provde a heoreal framework for hs ssue II PROBLE FORULATION We wll provde a geeral framework for seleg a os fuo whh here s o lmao o he umber of QoS osras Se oave osras a be easly addressed by prug ad mulplave osras a be geerally overed o addve osras whou loss of geeraly we oly osder addve osras ad formulae he problem as follows: Defo : ulple Addvely Cosraed Pah Seleo (ACP: Assume a ework s modeled as a dreed graph GNE ( where N s he se of all odes ad E s he se of all lks Eah lk oeed from ode u o v deoed by E s assoaed wh addve parameers: /3/$7 3 IEEE 63

2 w Gve a se of osras ( ad a par of odes s ad ACP s o fd a pah p from s o sube o W ( p w ( u v < e p for all uv Defo : Ay pah seleed by ACP s a feasble pah; ha s ay pah p from s o ha mees he requreme W ( p w ( u v e p for all s a feasble uv pah Noaos: f ( x : Cos fuo where x ( x x x C : The veor represeao of he QoS osras ( 3 W : The wegh veor of pah p e W ( W W W where W ( p w ( u v e p uv 4 C( p : The os of pah p C f( w ( e w ( e w ( e where f ( s he os fuo Noe ha C f( W for uv uv uv f( W( p f( w( uv w( uv w ( uv ( e uv However f f ( x s lear e f ( x x f ( W f( w w w ( w e uv p w ( w f ( w ( w ( w( C ( III A FRAEWOR FOR SELECTING THE COST FUNCTION ay proposed soure roug algorhms are assoaed wh os fuos whh are esseal for solvg he problem Thus desgg a approprae os fuo s a key ssue hs kd of approahes I hs paper by assumg he os fuo s ouous we prese some bas feaures of he os fuo ad he mpa of hese feaures o he performae of roug algorhms Whou loss of geeraly he os fuo for raversg a lk from ode u o v s desrable o have he followg properes: f ( ; eah varable f ( orrespods o a QoS parameer suh as delay ad er I s uve ha he os for raversg a lk wh QoS value (eg he os of raversg a lk whh does o ause ay delay should be > f x > ad f x ; e he os fuo s reasg wh respe o eah addve parameer f( x x x 3 ; e he os fuo s oave mplyg ha f w ( u v e uv he C f( W f( C I fa mos soure roug algorhms proposed he leraure possess hs propery Noe ha some oher o-ouous fuos proposed for QoS roug a also be haraerzed by fuos whh sasfy he above properes For example / f ( x x max{ x x } equals o ( x + x whh lm sasfes he above properes The followg heorems ad lemmas are derved by assumg ha he os fuo f ( x sasfes Properes -3 [Theorem ] No feasble pah exss f he leas os pah has he os larger ha f ( C Proof: By orado Assume pah p sasfes he osra C ad he leas os amog all pahs s larger ha f ( C ; ha s C > f( C p C( p > f( C (3 Also f( x x x f ( W( p C( p (4 Thus from (3 ad (4 f( W( p > f( C (5 However se ad pah p sasfes he osra C W( { } ( p f W f( C (6 whh orads (5 ad hus Theorem s proved The followg lemma wll falae he proof of Theorem [Lemma ] Le g f( xe where e s he h u veor he m-dmesoal spae ad g( ym y The: lm[ max ( ym] g ( g ( m ym dg where g( dx x Proof: Le Thus f g ( else Ly ( y y λ y λ[ g( y ] m m (7 63

3 dl( y y y λ dy λg ( y g ( y (8 λ By leg y y y g( ym (9 Also g Ly ( y y λ y Thus ym s maxmzed by (9 ( Defe I ( x as he verse fuo of g ( x So Se g max( y I ( I ( ( m g ad g ( we have di d I ad Hee max( ym reases dx dx wh Thus I ( lm[max( ] lm[ ( ] lm[ y I ] m I ( f ( x g lm g ( ( x x else [Theorem ] A pah p s a feasble pah f C m{ } Proof: Se wm( eu v m g ( w( e f( W ( e (3 So for ay pah p uv uv g( w( eu v ( The for a pah p wh C C p (4 m{ } x g( w( eu v C m{ } (5 Cosder he ase ha f ( x Lemma w ( e u v by m{ } (6 Cosder he oher ase ha { } Thus m{ } C w( e u v p(7 Thus W ( p Tha s pah p s a feasble pah w( eu v { } (8 Noe ha a pah p wh C f( C may o be a feasble pah For example osder f ( x x x+ x wh C ( ad a pah p wh W( p ( I s obvous ha C f( W < f( C bu pah p s o a feasble pah The followg wo defos a be used as he soppg rera searhg for feasble pahs Defo 3: Gve a ework G wh os fuo f ad QoS osra C o feasble pah exss from ode s o f he leas os pah p from ode s o has C > Bu ( f G C Bu ( f G C s kow as a upper boud of he os of a feasble pah Defo 4: Gve a ework G wh os fuo f ad QoS osra C f ay pah p from ode s o wh C B ( f G C s a feasble pah B ( f G C s kow as he gh boud of he os of a feasble pah From Defos 3 ad 4 he os of a feasble pah ao be larger ha Bu ( f G C ad ha of a feasble pah s larger ha B ( f G C I also follows from Theorems ad ha B ( u f G C f ( C ad B( f G C [Theorem 3] If λ pah p Proof: Se λ The m{ } w m{ } he ( uv N w λ W p for ay W w m{ } we have uv N w λ w w (9 w λ w W ( uv p ( uv p λ ( W w w ( uv p ( uv p w [Lemma ] If λ m{ } ad < λ for uv N w ay pah p wh W also mples ha W < Proof: From Theorem 3 W W λ W W λ λ < ( Lemma mples ha f < λ he h osra s me as log as he h osra s me Thus he h osra 633

4 a be omed for he ACP problem; e he -osraed For example le C ( λ λ ad he os problem a be redued o a (--osraed problem fuo f ( xy x + y k I hs ase Au ( f w [Lemma 3] Defe λ m{ } B ( f G C ad Fg shows he area overed by he gh boud reases uv N w wh k Thus r( f as k ; e a feasble wll m{ } where h ( x f ( λ x λ x λ x o be seleed h Proof: The proof s smlar o ha of Theorem However oe ha m{ h ( } does o hage wh f w [Lemma 4] If λ m{ } ad here are N odes uv N w he os fuo s lear whle m{ h ( } dereases wh f( x he ework B ( f G C m{( N h ( } where x x f > Thus we oeure N h ( x f ( λ x λ x λ x ha for ay olear os fuo here mus exs a lear Proof: The proof s smlar o ha of Theorem os fuo wh a larger r( f whe s large eough [Lemma 5] Gve a os fuo f ad QoS osra C Wh C ( λ λ Fg ad Fg 3 show he a -hop pah p sasfyg C m{ h ( } s a feasble overage raos for boh f ( x y x + y ad ( k x f ( xy ( e ( k y + e /( k k whe pah Proof: The proof s smlar o ha of Theorem ad 3 Smlar o he defo of he gh boud m{ h ( } s defed as he -hop gh boud; ha s he os of ay -hop feasble pah ao be less ha hs boud From Theorem ad Lemma 5 s kow ha a -hop pah wh os bewee he upper boud ad -hop gh boud may be a feasble pah Iuvely s desrable o have he upper boud ad gh boud lose o eah oher o mmze he possbly of seleg a wrog pah Coaeae W W W of lks of a -hop pah o a veor ( W W W Thus he area overed by he -hop gh boud s A( f dwdw dw ( f ( W m{ h ( } W C Smlarly he area overed by he upper boud s A ( f dwdw dw (3 u f ( W f ( W C Based o he above uo he overage rao defed below A ( f r( f Au ( f (4 a be used as a fgure of mer o how robus he os fuo s avodg a wrog pah seleo The larger he rao he beer he os fuo Fgure A ( f for x + y 634 Fgure Coverage Rao for Fgure 3 Coverage Rao for 3 Whe 3 he lear os fuo (e x+ y whe k yelds he larges overage rao as ompared o f ( ad f ( Furhermore he lear os fuo possesses a dsgushed propery C f( W ha makes he desg of QoS roug algorhm (seleo of os fuo parameers suh as [6] sraghforward ad easy Thus he lear os fuo s preferred for soure QoS roug algorhms Ths oluso assumes ha he os of a pah s he sum of he oss of all dvdual lks omprsg he pah; may o be feasble for algorhms whh he assumpo does o hold For example a good approah proposed [] s ha he os of pah p s f ( W e C f( W

5 where f ( s he os fuo I hs ase pah p a always be vewed as a lk oeg he soure ad he desao wh wegh W Hee f here are oly wo osras ad f ( xy x + y k he suess rao of a fdg feasble pah usg hs algorhm always reases wh k regardless he umber of he hop ou from he soure o he desao as show Fg So hs ase he lear os fuo s o opmal ad our oluso s o applable aymore IV SIULATIONS We odu our smulaos wo ework opologes: oe s he ework opology preseed [6] [] ad he oher s a 7 7 mesh ework The os fuos adoped for omparsos are f ( x y x+ y f ( x y x + y ad x y f3( x y ( e + e The QoS roug algorhms used smulaos are Dksra algorhm I boh smulaos he lk weghs are depede ad uformly dsrbued from o wo QoS osras are se o be equal ad rease from 5 o 5 wh a reme of All daa are obaed by rug requess To ruly refle algorhms apably fdg a feasble pah we propose he followg more approprae suess rao defo as our performae dex: Toal umber of suess reques of he algorhm SR (6 Toal umber of suess reques of he opmal algorhm The algorhm ha a always loae a feasble pah as log as exss s refereed o as he opmal algorhm Here s aheved smply by floodg whh s raher exhausve The smulao resuls are show Fgures 4 ad 5 SR f(xy f(xy f3(xy Cosra Fgure 4 SR he 3-ode ework I a be observed ha boh smulaos he QoS roug algorhm wh he lear os fuo aheves he hghes suess rao fdg a feasble pah whh oforms o our oeure V CONCLUSIONS Alhough soure roug has bee exesvely suded he pas mos proposed mulple osraed roug algorhms are heurs aure I hs paper we have provded a heoreal framework for seleg he os fuo for mulple addvely osraed QoS roug By deployg our proposed overage rao as he mer for evaluag he os fuo s advsable o use a lear os fuo for he approah whh he os of a pah equals he sum of he oss of all dvdual lks omprsg he pah REFERENCES [] S Che ad Nahsed A overvew of qualy of serve roug for ex-geerao hgh-speed ework: problems ad soluos IEEE Nework vol o 6 pp Deember 998 [] A Shakh J Rexford ad G Sh Evaluag he mpa of sale lk sae o qualy-of-serve roug IEEE/AC Trasaos o Neworkg vol 9 o pp 6-76 Aprl [3] R Guer ad A Orda QoS based roug eworks wh aurae formao: heory ad algorhms Proeedgs of he INFOCO 97 pp [4] J Wag W Wag J Che ad S Che A radomzed QoS roug algorhm o eworks wh aurae lk-sae formao Proeedg of WCC-ICCT vol pp 67-6 [5] Z Wag ad J Crowrof Qualy of Serve roug for supporg mulmeda applaos IEEE Joural o Seleed Areas o Commuaos vol 4 o 7 pp 8-34 Sepember 996 [6] T orkmaz ruz ad S Tragoudas A effe algorhms for fdg a pah sube o wo addve osras Proeedgs of he AC SIGETRICS pp Jue [7] A Juer B Szyaovszk I es ad Rako Lagrage releaxao based mehod for he QoS roug problem Proeedgs of IEEE INFOCO vol pp [8] L Guo ad I aa Searh spae reduo QoS roug Proeedgs of 9 h IEEE Ieraoal Coferee o Dsrbued Compug Sysems pp [9] H De Neve ad P va eghem A mulple qualy of serve roug algorhm for PNNI Proeedgs of 998 IEEE AT workshop pp [] T orkmaz ad ruz "ul-osraed opmal pah seleo" Proeedgs of he IEEE INFOCO Coferee pp Aprl 99 SR f(xy f(xy f3(xy Cosra Fgure 5 SR he 7 7 mesh ework 635