Decentralized Algorithms for Sequential Network Time Synchronization

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1 Decetalzed Algothms fo Sequetal etwok me Sychozato Maxme Cohe Depatmet of Electcal Egeeg echo Hafa 32, Isael Abstact Accuate clock sychozato s mpotat may dstbuted applcatos. Stadad algothms, such as the etwok me Potocol (P, essetally ely o pawse offset estmato betwee adacet odes. Some ecet wok toduced moe elaboate algothms fo clock offset estmato, whch take to accout the algebac costats mposed o the sum of offsets ove etwok cycles, usg a least-squaes famewok. hese algothms ae teatve ad decetalzed atue, equg seveal cycles of local commucato amog eghbos fo covegece. I ths pape, we exted ths appoach towads a sequetal estmato famewok, whch allows to copoate tal tme estmates alog wth the ucetaty, as well as multple ouds of pawse measuemets. We popose a decetalzed mplemetato of the estmato algothm that employs oly local boadcasts ad establsh ts covegece to the optmal cetalzed soluto. We also peset some smulato esults to llustate the pefomace beefts of the suggested algothms. Keywods: etwok clock sychozato; decetalzed algothms; Kalma flteg; ecusve estmato. I. IRODUCIO Accuate clock sychozato has bee extesve studed ad appled the cotext of commucato etwoks, fom the Iteet [] to seso etwoks [7]. he task of sychozg clocks dstbuted systems s usually accomplshed va the exchage of tme-stamped messages (pobe packets betwee the dstbuted ettes ode to coodate the tme. hee s a lage lteatue o how to sychoze clocks tadtoal etwoked systems; amog these, the etwok me Potocol (P s the most wdely accepted stadad fo sychozg clocks ove the Iteet [], []. hs potocol essetally uses a so-called heachcal appoach by sedg pobe messages alog a layeed spag tee of the etwok. Moe ecetly, a ovel appoach fo tme sychozato temed CP Classless me Potocol [4] was poposed. hs o-heachcal appoach explots covex optmzato theoy ode to mmze a quadatc obectve fucto of clock offsets. It was show that CP substatally outpefoms heachcal schemes such as P tems of clock accuacy wthout ceasg complexty. A elated appoach that eles o Least-Squaes ft was poposed [3], [5]. he accuacy of ahum Shmk Depatmet of Electcal Egeeg echo Hafa 32, Isael shmk@ee.techo.ac.l clock sychozato was mpoved by explotg global etwok-wde costats (.e., the elatve offsets must sum up to zeo ove etwok loops. he cetal chaactestc of these methods s the use of a dstbuted algothm that eques oly local boadcasts amog eghbog odes. he algothms ae teatve, ad typcally covege wth a small umbe o ouds. he wok [] exteds the same LS appoach to a Weghted Least-Squaes (WLS famewok, whee each measuemet may be assged a dffeet weght. It s teestg to ote, followg [], that the tme sychozato poblem s mathematcally equvalet to a geeal class of dstbuted estmato poblem of addtve quattes ove a etwok, a class that cludes seso localzato ove a seso etwok ( Catesa coodates. he basc Least-Squaes famewok of [3] cosdeed the offset estmato poblem usg oly a sgle set of measuemets. Ou goal hee s to exted ths famewok to a sequetal estmato oe, whch hadles po estmates of clock offsets as well as multple measuemet sets. hese goals ca be cast the famewok of Kalma Flteg, whch deed eadly povdes a optmal cetalzed soluto. Howeve, as the obtaed equatos ae ot eadly ameable to a decetalzed mplemetato, we esot to the equvalet least-squaes fomulato, ad employ localzed least squaes teatos to obta a decetalzed algothm. hs algothm employs oly local boadcasts betwee eghbos whe the tal covaace matx s dagoal. A ch lteatue exsts o dstbuted mplemetato of the Kalma Flte (KF, datg back to [5], [4]. Moe ecet wok, such as [2], focused o etwok stuctues ad developed cosesus-based algothms that employ oly local commucato. hs le of wok allows measuemet to be pocessed locally, howeve each ode keeps a copy of the ete state vecto. I ou famewok, each ode keeps oly data elated to ts ow offset. Iteestg wok ths le s peseted [9], whee decetalzed appoxmato schemes ae peseted. Ou goal hee s to develop exact algothms, fo the moe lmted poblem that we cosde. Fally, we peset smulato esults ove seveal etwok topologes fo evaluatg ad compag the accuacy of the poposed tme sychozato schemes. We povde seveal teestg compasos, whee the Kalma Flte appoach outpefoms the exstg algothms. Submtted to etcoop 2. Coespodg autho:. Shmk

2 he pape s ogazed as follows. I secto II, we descbe the model ad fomulate the estmato poblem state-space fom. I secto III, we peset a teatve, decetalzed algothm that combes po offset estmates wth a sgle set of measuemets, ad establsh ts covegece. Whle the developmet s caed out fo a geeal tal state covaace matx, the local commucato stuctue s mataed oly f ths matx s dagoal. Hece, a ecusve exteso of ths algothm to the case of multple measuemet sets s ot appaet. he case of multple measuemet sets s take up Secto IV, whee a exact algothm s developed. We also cosde fo compaso puposes a smple sub-optmal algothm that eglects the off-dagoal tems of the vese covaace matx. Secto V pesets some smulato esults that evaluate ad compae the pefomace of the poposed algothms. Fally, the coclusos ad some otes o futue dectos ae epoted secto VI. II. MODEL AD PROBLEM DEFIIIOS We model the etwok as a dected gaph G= ( V, Ε wth = V odes deoted { Λ Λ Λ }, ad m =Ε edges., 2,..., Each edge epesets the ablty to tasmt ad eceve packets betwee the coespodg pa of odes. he edge coectg odes Λ ad Λ s deoted by e. We assume that all the edges ae bdectoal, so that e Ε mples e Ε, ad that the etwok gaph s coected, amely thee exsts a path betwee ay pa of odes the etwok. Deote by the set of odes whch ae the eghbos of Λ,.e., oe edge away fom ode Λ, ad let be the umbe of such eghbos. We assume that each ode the etwok keeps a local clock, ad ou goal s to estmate the tme offset of each clock wth espect to some global efeece. Wthout loss of geealty, we may assg ode Λ as the efeece tme ode, so that all clock offsets ae estmated wth espect to ths ode's clock. A. Clock Model A stadad model fo the clock dft at a ode follows the lea fom: ( t = αt+ τ, whee α ad τ ae the skew (ate devato ad the offset paametes espectvely, t s the eal tme (o efeece tme ad ( t s the local tme at ode Λ. hs model s kow as the two paametes lea model (see [5] ad the efeeces thee. As ode seves as the efeece, we have by default τ = ad α =. We focus hee o the smplfed model whee all clocks u at the same speed, so that thee s o skew ( α = fo all. hs assumpto s appopate whe the measuemet tme spa s shot so that the ate devato s small elatve to the skew. B. Measuemet Model We cosde a two-way offset measuemet scheme. Each etwok ode ( Λ, =, 2,..., seds pobe packets to each of ts eghbos. Upo sedg a packet k m the sede Λ stamps the packet wth ts local tme ( k m, ad the eceve Λ stamps the packet upo ecevg t wth ts local tme R ( k m. he, ode Λ etasmts the packet back to the souce wth tme stamp ( k m, ad the souce stamps ts local tme R( k whe ecevg the packet back. We thus obta m R ( k ( k = D ( k τ + τ + ε. m m m Hee D ( k m s the popagato delay ove lk e, ε s a addtve ose that epesets the adom queug delay (ad the othe ukow flueces ad τ τ s the dffeece betwee the two clock offsets. Assumg that D ( k = D ( k (symmetc popagato delay we obta: m m Oˆ ( = τ τ + v ( 2 whee v = ( ε ε /2 s the effectve measuemet ose. We shall assume that v s zeo mea, wth covaace >. Remak: I pactce, the effectve measuemet O ˆ may be obtaed by pocessg seveal subsequet packet exchages. Oe way to combe these successve measuemets s to cosde the aveage. Moe efed methods such as a mmum flte may favo measuemets that eflect smalle delays, to accout fo chagg cogesto codtos (cf. []. I ay case, the combed measuemet s take to be of the above fom. C. State-Space Equatos Ou obectve s to sychoze all the clocks the etwok wth the efeece tme. hs s equvalet to estmatg the offset τ at each etwok ode. We stat by fomulatg the statstcal estmato poblem state space fom. Defe the state as the colum vecto (,... x τ2 τ (ecall that τ = by defto. As x s assumed to be costat fo the tme fame of teest, we ca wte x( + = x(, whee s the tme (o step dex. he tal state x ( s assumed to have Ex( = x, kow fst ad secod ode statstcs: [ ] cov [ x( ] P =. hs allows to take to accout po fomato about the accuacy of the tal clock offsets dffeet odes. Fo example, the etwok may cota seveal efeece odes whch keep accuate tme, whch taslates to small covaace etes P. he effectve measuemet fo each pa of eghbog odes s gve by equato (. hus, O ˆ measues the offset dffeece fo these odes, plus a addtve ose v. he ose v s assumed to have zeo mea, wth fte covaace >, ad s depedet acoss ode pas. We collect the dvdual measuemets a colum vecto y = ( Oˆ, Ε, 2

3 whch epesets a sgle measuemet set. We assume that a ew measuemet set y ( s obtaed at each step. We ote that eed ot efe to actual tme, but athe coespods to the epoch whe the -th measuemet set y ( become avalable. o expess the measuemet equato vecto fom, defe the gaph cdece matx A whose dmesos ae (odes m (edges, ad whee the ow coespodg to ode Λ, we have a ety + fo all edges e ogatg Λ, a ety - fo all edges e tematg Λ, ad othewse. Fo a coected gaph, the ak of the cdece matx s. hus, deletg ay ow fom the cdece matx yelds a full ow ak matx, whch s called the educed cdece matx. Hee, we wll wok wth the ( m matx A obtaed by deletg the ow coespodg to the efeece ode Λ. he model may ow be summazed state space fom: x ( = x ( y ( = Ax ( + v ( ; wth tal codtos E[ x( ] = x, cov [ x( ] = P. he measuemet ose sequece { v ( } s assumed to be a whte ose sequece wth zeo mea ad covaace R ( = R>. We futhe assume that thee s o coelato betwee the measuemet ose ove dffeet lks, so that R = dag{, E}, as a dagoal matx wth elemets > alog the dagoal. Fally, the measuemet ose ad tal state x ( ae ucoelated. Ou goal s to estmate the offset vecto x( = x based o the measuemet sets y(,, y( ad the po fomato x, P. hs s of couse a classcal poblem sequetal estmato theoy. As s well kow, the Kalma flte povdes the optmal lea soluto the MMSE (Mmal Mea Squae Eo sese, whch futhe cocdes wth the optmal (codtoal expectato soluto ude the Gaussa assumpto. Howeve, whle the cetalzed KF equatos ae easly wtte, t s ot eadly see how they may dstbuted. We thus poceed to develop dstbuted algothms that covege to the optmal KF soluto. III. SIGLE MEASUREME SE We stat by cosdeg a sgle measuemet update, amely the poblem of estmatg x( = x based o the measuemet y( = y ad the po fomato x, P. As s well kow (e.g., [7], Secto 5.3, the KF equatos ae equvaletly obtaed as a soluto to a Least-Squaes detemstc poblem, whch ou case educes to the mmum of the followg obectve fucto: J( x = ( x x P ( x x + ( y A x R ( y A x (2 he fst tem s elated to the tal fomato egadg the clock offsets, wheeas the secod tem s assocated wth the sgle set of measuemets ad ts coespodg covaace matx R. I the developmet of a dstbuted algothm, we wll fd t moe coveet to mapulate the above detemstc LS poblem athe tha statg wth the KF equatos. A. Basele Algothm We fst evew the exstg algothm toduced [3], [4], [5]. I ths case, the obectve fucto s gve by:, ( ˆ 2 J( x = ( y A x ( y A x = O τ + τ he fst ode optmalty codtos ae J = = 2 τ + = Fom ths, we obta: τ ( ˆ = O + τ (3 ( AA x ( A y ( Oˆ τ τ he above equato must be satsfed by the optmal soluto of the offset estmato poblem. Whle ths s a set of lea equatos, a dect soluto caot be caed out a decetalzed mae. Istead, a decetalzed teatve algothm was suggested ad show to covegece to the optmal cetalzed soluto. hs algothm smply teates the above equato, whch ca be smply tepeted as follows. Each ode obtas fom ts eghbos the cuet estmates fo the ow clock offsets, ad computes ts offset estmate as the aveage of all ts eghbos' estmates plus the coespodg elatve measuemets. hs pocedue s the same as [3], [5] ad oe ca easly show that ths s equvalet to the algothm [4]. Ou obectve s to exted the pevous esult to a wde famewok ad we wll obta ths pocedue as a specal case of a moe geeal algothm. ext, we wll cosde the moe geeal famewok that cludes the tal covaace matx the obectve fucto addto to a weghtg matx R. he aalyss s dvded two cases: o-dagoal ad dagoal tal covaace matx. B. Addg Ital Codtos ad Measuemet Weghts We ow cosde the dstbuted soluto of equato (2, that copoates the tal estmate x wth covaace P, as well as possbly dffeet weghts fo the measuemet as expessed by the covaace matx R. We develop a teatve algothm, ad establsh ts covegece to the optmal soluto of (2. Howeve, t wll be see that the algothm s ot tuly dstbuted, the sese that each ode must eceve data fom o-eghbog odes, uless the tal covaace matx s dagoal. he latte case wll be cosdeed the ext subsecto. 3

4 Poceedg smlaly to above, the fst-ode optmalty codtos fo (2 ae J τ ( ˆ O ( P ( k k( τ = + + τ τ τ = k k= o, equvaletly, τ ( ˆ O ( P ( ( ( ( P m m m I τ τ = + + τ τ (4 m= m whee I = + ( P hese equatos motvate the followg sychoous teatve algothm fo the soluto: k ( O ( P τ = ˆ + τ + τ [ ( k+ ( ˆ ˆ ( I. ( k ( P ( ˆ τm τm( ] m= m ( wth talzato ˆ τ = τ(, = 2,3,.... Hee, k s the teato umbe. hs algothm ca be tepeted as a the classcal Jacob teato fo the soluto of the lea equatos (5, o, equvaletly, as a local least-squaes algothm whee each ode mmzes at each teato the obectve fucto (2 ove ts ow offset, gve the cuet offset estmates of the othe odes. he followg covegece esult s ext establshed. heoem. Suppose that: ( he matx R s dagoal ad Postve Sem-Defte, that s: ( <,. (2 he tal covaace matx P s a M-matx, amely: ( P ( P ad ( P ( (3 he clock adustmet opeato (5 s appled sychoously by all odes ( = 2,3,... all teatos. ( he, the teated estmatos ˆ τ k ( = 2,3,... covege (as k to the optmal offsets that mmze the obectve fucto (2. he poof s povded the Appedx. he ma poblem wth the last teatve algothm s that, geeal, each ode eeds to commucate wth all the othe odes ad ot oly wth ts eghbos. hus, each ode s equed to be awae of the global topology of the etwok, ad the algothm does ot satsfy the equemet of local fomato exchages oly. Fotuately, ths poblem does ot exst whe tal covaace matx P s dagoal. m (5 C. Dagoal P We hecefoth specalze the dscusso to the case whee the tal covaace matx P (hece ts vese P s dagoal, wth dagoal elemets ( p. hs wll be the case whe the tal estmates of clock skews ae obtaed by the dffeet odes depedetly. Fo example, some odes may have a GPS eceves whch allows the to obta a accuate estmate of the tme. O, statg fom a tally accuate estmate, each ode has afte some tme a added ucetaty due to the estmated dft of ts clock. he decetalzed teatve pocedue (5 ow educes to: ˆ τ ( O ˆ τ ( k+ ˆ ( k τ ( = + + p + p I wods, τ s computed at each teato as a weghted aveage betwee the modfed estmates obtaed fom adacet odes, ad ts po estmate. Obseve that f the matx P s equal to zeo ad R = I, we obta the equato (3 as the basc LS case descbed above. hs algothm eques oly local boadcasts betwee adacet odes. Evdetly, the covegece esult heoem apples hee as a specal case. IV. MUILPLE MEASUREME SES I the pevous secto we have developed a decetalzed algothm fo the case of a sgle measuemet set, ude the assumpto that P s dagoal. We ext cosde the case whe multple measuemet sets become avalable sequetally, wth the goal of pesetg a ecusve veso of the pevous decetalzed algothms. As ts tus out, a smple ecusve exteso of the sgle-measuemet case, the style of the Kalma Flte, wll ot wok hee. he easo s that eve f the tal covaace matx P s dagoal, whch we assume, afte the fst teato the state covaace matx P wll ot be dagoal ay moe (as ca be easly vefed, the estmates wll become coelated. hus, although the teatve pocedue (5 ca be fomally epeated, t wll ot be dstbuted ay moe, as each ode has to commucate ad stoe fomato elated to all the othe odes ove the etwok ad ot oly wth the oe-hop eghbos. Ou obectve s theefoe to deve a alteatve ecuso that s sutable fo decetalzed mplemetato. hs s doe the ext subsecto. Fo compaso puposes we also meto, the subsequet subsecto, a appoxmate algothm that smply eglects the off-dagoal elemets of the covaace matx afte each measuemet update. A. Optmal Decetalzed Algothm Fo the multple measuemet update case, the equvalet Least Squaes obectve fucto s gve by: k = J ( = ( xx P ( x x + ( y( k A x R ( y( k A x (7 (6 4

5 Fo otatoal smplcty assume that the matx R s detcal fo each set of measuemets. We popose the followg teatve algothm: ˆ τ ( = ˆ τ ( + I ( [ Oˆ ˆ τ ( ˆ τ ( ] { ( ( k+ ( ( k ( k + ( [ ( ˆ τ ( ˆ τ ( ]}, I( = I( + = ( P +, ( fo = 2,3,...,, talzed wth ˆ τ = τ (, I ( = ( P. he above set of equatos s a decetalzed, sychoous ad ecusve algothm that computes at each step the estmated offsets ad the coespodg eo vaaces. he ma advatage of ths algothm s ts local atue; each etwok ode eeds to commucate oly wth ts eghbos. We ow descbe wods the teatve pocedue (8. At tme, we assume that the estmate of ˆ τ ( s gve. ( k he, ˆ τ ( k =,2,... s computed based o ˆ τ ( ad the last measuemet set y (. We assume that a suffcet umbe of teatos s pefomed at each tme, so that the estmate ˆ τ ( s accuate. Remaks:. We pot out that the suggested ecuso slghtly devates fom a stadad ecusve estmato scheme due to the pesece of (tme o measuemet cout as a facto the ecuso. 2. It may be show that the elemets of I ( ae the dagoal etes of the vese covaace matx the Kalma flte equatos, amely I( = ( P. I fact, the above teato fo I ( s the same as the teato ove the dagoal elemets the fomato fom of the Kalma covaace update: ( ( P + = P + AR A (9 Obseve howeve that we do ot compute the o-dagoal elemets of the vese covaace matx. he poposed algothm may be deved s by dffeetatg J( ad J ( wth espect to the offsets vecto x ad set the patal devatves to zeo. he algebac detals (whch ca be foud [2] ae omtted sce the pocedue s smla to the pevous case. A alteatve devato, also peseted [2], may stat wth the KF equatos fomato fom. Howeve, the ecuso (8 s ot equvalet to the KF ecuso, ad s ot eadly see fom these equatos. We ext addess the covegece the set of equatos (8 to the optmal cetalzed soluto. heoem 2. Suppose that: (8 ( Assumptos (a ad (b fom heoem hold. (2 he clock adustmet opeato (8 s appled sychoously by all odes ( = 2,3,... all teatos, ecusvely fo sets of measuemets. (3 A suffcet umbe of teatos s pefomed afte each ( k measuemet set, so that ˆ τ ( coveges to ˆ τ (. he, fo each, the teated estmatos ( k ˆ τ ( = 2,3,... covege (as k to the optmal offsets that mmze the obectve fucto (7. he poof s povded the Appedx. ext, we popose a smple sub-optmal algothm fo the case whee multple sets of measuemets ae avalable. B. A Sub-Optmal Decetalzed Algothm Fo the case whee P s o-dagoal, we obtaed (5 that the estmated offset of ode Λ depeds o all the othe offsets ad ot oly o those of ts eghbos. Oe ca cosde the aïve sub-optmal algothm that eglects the off-dagoal tems of the vese covaace matx: τ ˆ τ τ k ( O ( P ( k ( ˆ + ˆ = + + ( + ( P he decetalzed sub-optmal ecusve algothm fo the multple measuemet sceao s gve by: ˆ τ ( k+ ( = ˆ τ( ˆ ( ( + ˆ ˆ O ( P + k ( τ ( τ ( Hee, we may eglect the off-dagoal tems befoe vetg the fomato matx ( P, ode to educe the algothm complexty. I ths case, we wll vet a dagoal matx ad hece the tme computato wll sgfcatly decease. V. UMERICAL RESULS I ths secto, we mplemet some of the algothms that we pevously developed fo typcal poblems ad we compae the esults wth the exstg algothms. Moe extesve compasos ca be foud [2]. he covegece ates of the decetalzed algothm ae ot peseted hee as covegece s acheved afte a elatvely small umbe of teatos ad the esults ae vey smla tha [4]. Cosde two dffeet etwok topologes: - etwok : a 4 ode etwok wth 997 edges. - etwok 2: a 7 ode etwok wth 2 edges. he fst case we aalyze s the oe whee % of the odes ae pefectly sychozed to the global tme (though a GPS satellte eceve fo example, ad the emade ae ot sychozed at all. amely, fo these abtay 4 odes we take the tal vaaces to be vey small (. ad the offsets equal to zeo, ad fo the est of the odes, the vaaces ted 5

6 to fty ad the offsets ae adomly chose accodg to a ufom dstbuto. he gaphcal compaso betwee the decetalzed CP algothm (equato (3 ad the Decetalzed Kalma Flte (DKF (equato (6 s peseted Fg. 2. As expected, the DKF algothm outpefoms the decetalzed CP method tems of clock accuacy. Fg. 2 shows the facto of odes wth clock offset wth espect to the efeece tme ode that s ot gate tha t fo the dffeet algothms. I othe wods, the y-axs epesets the facto of odes wth clock offset, elatg to the global tme, ot geate tha the value descbed by the x-axs. DKF Vs. Decetalzed CP - etwok Facto of odes CKF ad CLS Vs. SOA (= - etwok 2 CKF SOA CLS Absolute value clock offset CKF ad CLS Vs. SOA (=5 - etwok 2 Facto of odes Decetalzed CP DKF Facto of odes CKF SOA CLS Absolute value clock offset Fgue. Compaso betwee the decetalzed CP ad DKF algothms (wth % of odes sychozed va GPS etwok. he secod pat of ths secto s devoted to the compaso of the ecusve Cetalzed Kalma Flte (CKF algothm to the Sub-Optmal Algothm (SOA that eglects the offdagoal tems of the vese covaace matx (see secto V. B. We cosde the topology of etwok 2 ad we check seveal values of (the umbe of measuemets. he queug delay s adomzed accodace wth the Kalma Flte assumptos, amely omally dstbuted wth zeo mea ad covaace matx R : [.,2] (, R U Q R delay I addto, we cosde that % of the odes ae pefectly sychozed to the global tme ad the emade ae ot sychozed at all (smla to the case Fg. 2. I ths aalyss, we also compae the esults to the Cetalzed Least- Squaes (CLS algothm. Fg. 3 pesets the esults fo the offsets obtaed by applyg the optmal CKF method, the SOA ad the CLS algothms fo two dffeet values of. As expected, the optmal algothm gves the best esults. he suboptmal algothm gves elatvely poo esults but educes the complexty ad s ot dvegg. Moeove, we obtaed that the sub-optmal algothm s eve wose ( tems of clock accuacy tha the basc cetalzed Least-Squaes method (that does ot take to accout the tal covaace matx Absolute value clock offset Fgue 2. Compaso betwee CKF, SOA ad CLS (wth R U[.,2 ] ad P I etwok 2 fo =, 5. VI. COCLUSIO I ths pape, we have developed seveal decetalzed algothms fo estmatg the offset at each etwok ode wth espect to the efeece tme, utlzg a sequetal estmato famewok. he essetal chaactestc of these algothms s the decetalzed atue; each ode ca estmate ts clock offset by exchagg packets wth ts oe-hop eghbos oly. We exted the exstg Least-Squaes based algothms so that we may assg dffeet weghts to the measuemets accodg to the accuacy, clude a-po fomato, ad povde a ecusve estmato scheme. he ma algothm s both decetalzed (eques oly local boadcasts, ecusve (woks o-le applcatos ad coveges to the optmal cetalzed soluto. Fally, some umecal esults wee peseted to show that, as expected, the poposed algothm outpefoms the exstg methods. We close the pape by metog seveal extesos of teest. A dscout facto s easly copoated to the obectve fucto (7 ode to gve a hghe weght to the moe ecet measuemets, ad leads to smla algothms. hs wll be useful whe the offsets ae tme-vayg. he poposed algothms may also be exteded to hadle dyamc chages the commucato topology by cosdeg tempoay lk falues, followg the teatmet []. Futhe detals elated to these two ssues may be foud [2]. Oe may also cosde moe elaboate state dyamcs to model possble tme vaatos the clock offsets. he smplest s addg a whte system ose the state space model. Iteestgly, the esults of ths pape ae ot easly extedable to ths model. Aothe mao ssue s the copoato of the clock skew paamete to the clock model (see secto II. A. 6

7 hese sceaos wee patally vestgated [2] ad may be cosdeed as dectos fo futue eseach. APPEDIX Poof of heoem : Let us ecall that the geeal obectve fucto s gve by: J = ( xx P ( x x + ( y Ax R ( y Ax Let us aalyze the covegece popetes of the geeal case, whee P s ot ecessaly assumed to be a dagoal matx. We ecall that teato (5 caot be easly decetalzed whe P s ot dagoal as we pevously explaed. Howeve, the teato s stll well defed mathematcally. he sychoous teato ca be wtte vecto fom: ( D P ( AR A AR y P x P ˆ τ = ˆ τ + ˆ τ + ˆ τ ( k+ ( k ( k ( k Hee: = ( ( ( P = D = ( P = othewse othewse he optmal soluto (equvalet to pefomg the cetalzed potocol s gve by: ( AR A P ( AR y P x * τ = + + Let us defe: ( k ( k τ ˆ τ τ *. he we obta afte some mapulatos: ( k+ ( k τ = Mτ whee: M I ( D + P ( AR A + P ( hus, the covegece of the sequece ˆ τ k * to τ s equvalet to the covegece of τ ( k to the zeo vecto, whch s detemed by the matx M. he ecessay ad suffcet codto fo ths covegece s that the spectal adus of M s stctly smalle tha. he followg esult s well kow (see, e.g., [6], chapte 6. Poposto. Cosde a o-egatve squae matx A wth the followg popetes: a All the ow sums of A ae smalle o equal tha. b At least oe ow ths sum s stctly smalle tha. c he matx A s educble (.e., thee exsts a path fom ay ode to ay othe ode the etwok. he, ρ ( A <. ( k Accodg to Poposto, τ f the suffcet codtos apply to the matx M. I ode to show that the spectal adus of M s stctly smalle tha, we wll eque that the matx M s both o-egatve ad sub-stochastc (the ow sums ae smalle tha oe. he elemets of the matx M ca be detemed by specto as the followg: M =, ad ( P ( ( + ( P ( P ( ( P + ad, ae eghbos M = othewse + Let us fd the codtos fo the ow sums of the matx M to be smalle tha : ( P ( ( + P M = + P = ( ( ( ( P ( + ( P f ad oly f: ( It follows that M P. I othe wods, we obtaed that the ecessay codto s that fo each ode Λ, the ow sum of the matx P has to be o-egatve. Requg that all the etes of the matx M ae oegatve leads to: P ; P ( ( ( Hece, we ca wte: ( P ( P he above equemet ca be see as a dagoal domace codto ove the matx P. I the case that the ode Λ s adacet to the efeece ode, the coespodg ow sum of the M matx s gve by: ( ( ( ( ( P P + P ( + ( P ( ( ( ( P P ( + ( P = < I the case that Λ s ot adacet to the efeece ode, the coespodg ow sum of the M matx s gve by: ( ( P ( P < + P ( ( Hece, we have show that at least oe ow, the ow sum of M s stctly smalle tha. Actually, we poved that the teato matx vefes all the suffcet codtos fo covegece. amely, the ow sums of the matx M ae less 7

8 o equal tha (ad at least oe ow ths sum s stctly smalle tha, the matx M s educble ad all ts etes ae o-egatve. As a esult, we poved the covegece of the decetalzed algothm to the optmal soluto pefomed by the cetalzed Kalma Flte fo the most geeal case. o sum up, the covegece codtos ae gve by: P ; P ; P ( ( ( ( Poof of heoem 2: he case of = was teated heoem. Ou poof eles o the followg lemma. Lemma. Suppose that P satsfes the covegece codtos of heoem, amely P s a M-matx. Let P be computed usg (9, the P s a M-matx fo all. Poof Equato (9 coespods to the measuemet update equato of the vese covaace matx of the KF. Recallg that the matx R s assumed to be dagoal, let us aalyze the popetes of the matx AR A. Fo the educed cdece matx, we have: AR A = v Hee, v s a vecto wth o-egatve compoets. he stuctue of the matx AR A s as follows: AR A > ; AR A ( ( he ow sums ae ( AR A = fo each ode Λ that s ot adacet to the efeece ode. Moeove, f Λ s adacet to the efeece, ths sum s a stctly postve umbe. We coclude that f the a-po vese covaace matx ( P vefes the covegece codtos, the the a-posteo vese covaace matx ( P wll vefy them too. hs lemma mmedately mples the covegece of the ecusve exteso (fo seveal measuemet sets of equato (5 to the optmal soluto, whee at each step, the ew covaace matx s computed accodg to (9. Sce the teatos (8 ae equvalet to the pocedue (5, we obta the clamed covegece heoem 2. [3] A. Gdha ad P. R. Kuma, Dstbuted clock sychozato ove weless etwoks: algothms ad aalyss, Poc. 45 th IEEE REFERECES [] P. Baooah, ad J. P. Hespaha, Dstbuted estmato fom elatve measuemets seso etwoks, Poc. 3 d It. Cof. o Itellget Sesg ad Ifomato Pocessg, 25. [2] M. Cohe, etwok me Sychozato Usg Decetalzed Kalma Flteg, M. Sc. Reseach hess, echo, Aug. 29, avalable at: Cofeece o Decso ad Cotol, Sa Dego, Dec. 26. [4] O. Guewtz, I. Cdo ad M. Sd, etwok tme sychozato usg clock offset optmzato, IEEE Iteatoal Cofeece o etwok Potocols, pp , Atlata, GA, ovembe 23. [5] H. R. Hashempou, S. Roy ad A. J. Laub, "Decetalzed stuctues fo paallel Kalma flteg", IEEE as. Automatc Cotol, vol. 33, o., pp , Ja [6] R. A. Ho ad C. R. Johso, Matx aalyss, Cambdge Uvesty Pess, Campdge, UK, 99 (pp [7] A. H. Jazwsk, Stochastc pocesses ad flteg theoy, Academc Pess, Y, 97. Repted by Dove, 27. [8] C.. Kelley, Iteatve methods fo lea ad olea equatos, SIAM, Phladelpha, PA, 995. [9] U. A. Kha ad J. M. F. Moua, "Dstbutg the Kalma Flte fo lage-scale systems", IEEE as. o Sgal Pocessg, Vol. 56, o., Octobe 28, pp [] D. L. Mlls, Iteet tme sychozato: the etwok tme potocol, IEEE as. Commu., vol. COM-39, o., pp , Oct. 99. [] D. L. Mlls, etwok me Potocol (veso 3 specfcato, mplemetato ad aalyss, etwok Wokg Goup Repot. Uv. of Delawae, RFC-35, pp. 3, 992. [2] R. Olfat-Sabe, Dstbuted Kalma flteg fo seso etwoks, Poc. 46 th IEEE Cofeece o Decso ad Cotol, ew Oleas, Dec. 27. [3] K. Plae ad P. R. Kuma, "Obect tackg by scatteed dectoal sesos", Poc. 44 th IEEE Cofeece o Decso ad Cotol, pp , Sevlle, Spa, Dec. 25. [4] B. S. Rao ad H. F. Duat-Whyte, "Fully decetalzed algothm fo multseso Kalma flteg", IEEE Poceedgs-D, Vol. 38, o. 5, Septembe 99. [5] R. Sols, V. Boka, ad P. R. Kuma, A ew dstbuted tme sychozato potocol fo multhop weless etwoks, Poc. 45 th IEEE Cofeece o Decso ad Cotol, pp , Sa Dego, Dec. 26. [6] J. Stoe ad R. Bulsch, Itoducto to umecal aalyss, 3 d Ed. Spge Velag, ew Yok, 22. [7] F. Svkaya ad B. Yee, "me sychozato seso etwoks: a suvey", IEEE etwok, July 24, pp