Stochastic Game Formulation for Cognitive Radio Networks- Short Paper

Transcription

1 Sochasc Game Formulaon for Cognve Rado Neworks- Shor Paper Fangwen Fu and haela van der Schaar Elecrcal Engneerng Deparmen Unversy of Calforna Los Angeles (UCLA) Los Angeles Calforna USA {fwfu Absrac In hs paper we model he varous wreless users n a cognve rado nework as a collecon of selfsh auonomous agens ha sraegcally nerac n order o compee for he dynamcally avalable specrum opporunes. We propose a sochasc game framework o model how he compeon among users for specrum opporunes evolves over me. A each sage of he dynamc resource allocaon a specrum moderaor aucons he avalable resources and he users sraegcally bd for he requred resources. Based on he observed resource allocaon and correspondng rewards from prevous allocaons we propose a bes response learnng algorhm ha can be deployed by wreless users o mprove her bddng polcy a each sage. The smulaon resuls show ha by deployng he proposed bes response learnng algorhm he wreless users can sgnfcanly mprove her own performance n erms of boh he packe loss rae and he ncurred cos for he used resources. Keywords- ul-user Resource anagemen; Ineracve Learnng Cognve Rado Neworks Sochasc game I. INTRODUCTION One vson for emergng cognve rado neworks assumes ha ceran porons of he specrum wll be opened up for secondary users (SUs) whch can auonomously and opporunscally share he specrum once prmary users (PUs) are no acve [3][4][5]. Imporanly n cognve rado neworks heerogeneous wreless users wh dfferen ules delay olerances raffc characerscs nerference avodance knowledge and ably o adap wll need o coexs n he same band. Curren soluons do no provde far or effcen resource managemen for delay-sensve applcaons such as mulmeda sreamng n he cognve rado neworks. Thus o enable he prolferaon of mulmeda applcaons over cognve rado neworks wreless soluons for dynamc specrum access and resource managemen wll need o consder he sysem dynamcs as well as he heerogeney of wreless users. oreover SUs wll need o possess learnng ables o be able o sraegcally nfluence and adap o he dynamc specrum dvson. Usng her knowledge SU can proacvely harves resources based on her dynamc resource requremens as well as opmally adap her cross-layer ransmsson sraeges o he envronmen dynamcs and mevaryng gahered resources. Such dynamc and compeve soluons for specrum access and proocol desgn lead o more effcen and far wreless neworks han curren soluons whch requre SUs o blndly follow predeermned or sac proocol rules []. In our consdered cognve rado neworks he SUs are modeled as raonal and sraegc. We model he specrum managemen as a sochasc game [6] n whch he SUs smulaneously and repeaedly compee for he avalable Ths work was suppored n par by NSF CAREER Award CCF and n par by grans from ONR. nework resources. The compeon for he dynamc resources s asssed by a cenral coordnaor (smlar o ha n exsng wreless LAN sandards such as 802.e HCF [7]). We refer o hs coordnaor as he cenral specrum moderaor (CS). The role of he CS s o allocae resources o he SUs based on pre-deermned uly maxmzaon rule. In hs paper o explcly consder he sraegc behavor of he auonomous SUs and he nformaonally-decenralzed naure of he compeon for wreless resources we assume ha he CS deploys an aucon mechansm for dynamcally allocang resources. In order o capure he nework dynamcs we allow he CS o repeaedly aucon he avalable specrum opporunes based on he PUs behavors. eanwhle each SU s allowed o sraegcally adap s bddng sraegy based on nformaon abou he avalable specrum opporunes s source and channel characerscs and he mpac of he oher SUs bddng acons. Usng hs sochasc wreless allocaon framework we can develop a learnng mehodology for SUs o mprove her polces for playng he aucon game.e. he polces for generang he bds o compee for he avalable resources. The dealed learnng algorhm s presened n [2]. Specfcally durng he repeaed mul-user neracon he SUs can observe paral hsorc nformaon of he oucome of he aucon game hrough whch he SUs can esmae he mpac on her fuure rewards and hen adop her bes response n order o effecvely compee for he channel opporunes. The paper s organzed as follows. In Secon II we nroduce a sochasc game formulaon for mul-user neracon n he cognve rado neworks. In Secon III we specfy he deals of he sochasc game n our cognve rado neworks and characerze he bes response o play hs game. In Secon IV we presen he smulaon resuls followed by he conclusons n Secon V. II. DYNAIC ULTI-USER WIRELESS RESOURCE GAE FORULATION As menoned n he nroducon we focus n hs paper on developng wreless resource markes for secondary neworks (SN). In SN he SUs can opporunscally ulze he nework resources ha are vacaed by he PUs. For llusraon purposes we assume ha he SN consss of SUs whch are ndexed by { }. The SUs compee for he dynamcally avalable ransmsson opporunes based on her own prvae nformaon and knowledge abou oher SUs and avalable resources (and/or PUs behavors). In each me slo Δ T he SUs compee wh each oher for specrum access and gven he allocaed ransmsson opporunes hey deploy opmzed cross-layer sraeges o ransm her delay-sensve bsreams /08/$ IEEE

2 Durng each me slo a sae of nework resources can be defned o represen he avalable ransmsson opporunes n a SN whch s denoed by w W where W s he se of possble resource saes. We can also defne saes for he SUs. For nsance he saes may represen her prvae nformaon whch ncludes he raffc and channel characerscs. The curren sae of a SU s denoed by s S where S s he se of possble saes of SU. A each me slo SU wll deploy an acon o compee for he nework resources. Ths acon s denoed by a A where A s he se of possble acons. An example of he acons n wreless neworks s he seleced ransm power n nerference channels or he declared resource reques lke he TSPEC n 802.e WLANs. In hs paper we formulae he mul-user wreless cognve resource compeon as a sochasc game. Formally he sochasc game s defned as a uple ( I SWA P s P w R) where I s he se of agens (SUs).e. I ={... } S s he se of sae profles of all SUs.e. S = S S wh S beng he sae se of SU W s he se of nework resource sae. A s he jon acon space A = A A wh A beng he acon se avalable for SU o play he resource sharng game P s s a ranson probably funcon defned as a mappng from he curren sae profle s S correspondng jon acons a A and he nex sae profle s ' S o a real number beween 0 and.e. P s: S A S [0]. P w s a ranson probably funcon defned as a mappng from he curren resource sae w W and he nex sae w W o a real number beween 0 and.e. P w: W W [0]. R s a reward vecor funcon defned as a mappng from he avalable resource w he curren sae profle s S and correspondng jon acons a A o an -dmensonal real vecor wh each elemen beng he reward o a parcular agen.e. RW : S A. The sae ranson P w for he nework resource sae s deermned by he PUs and no by he SUs. In oher words he SUs acons wll no affec he nework resource sae ranson. Ths srucure acually opens an opporuny o allow he PUs o be agens wh hgher prores n hs sochasc game. III. SPECIFICATION OF STOCHASTIC GAE FOR COGINITVE RADIO NETWORKS As an llusraon example we consder ha he SN can be formed across N channels each ndexed by j {... N}. A each me slo each channel s assumed o be n one of he followng wo saes: ON (hs channel s currenly used by he PUs) or OFF (hs channel s no used by he PUs and hence s an opporuny for he SUs o use). Whn each me slo he channel s only OFF or ON [8]. A me slo he avalably of each channel j s denoed by w j { 0} wh w j beng 0 f he channel s n ON sae and beng f s n OFF sae. The channel avalably profle for he N channels s represened by w = [ w... w ] whch s he N sae of he nework resource a me slo. A. Saes for each SU We assume ha he SU deploys a delay-sensve applcaon. The daa of he applcaon layer s packezed wh an average packe lengh. In hs paper we consder mulmeda applcaons where he applcaon packes have a hard delay deadlne.e. he packes wll expre n J sages afer hey are ready for ransmsson. Then we can defne he T sae of he buffer as v = [ v vj ] where v j ( j J ) s he number of packes wang for ransmsson ha have a remanng lfe me of j me slos. The condon of channel j experenced by SU s represened by he Sgnal-o-Nose Rao (SNR) and s denoed as c j n db. The channel condon profle s gven by c = [ c cn ]. To model he dynamcs experenced by SU a me n he cognve rado nework we defne a sae s = ( v c ) S whch encapsulaes he curren buffer sae as well as he sae of each channel. The envronmen experenced by each SU s characerzed by he packe arrvals from he (mulmeda) source (.e. source/raffc characerzaon) conneced wh he ransmer he specrum opporunes released by PUs and he channel condons. Dfferen models can be used by a SU o characerze he envronmen. However he accuracy of he deployed models wll only affec he performance of he proposed soluon and no he general framework for muluser sochasc game model presened here. B. Sae ranson and sage reward Snce he nework resource sae s no affeced by he acons performed by he SUs he ranson of w can be modeled as a fne sae arkov chan [8]. The ranson probably s denoed by ( p w w w ). In hs example we assume ha he ranson probably ( p w w w ) s known by all he SUs and CS. However more complcaed models for he nework resource sae ranson can also be nvolved n our sochasc game framework [9]. When SU receves he resource allocaon z can ransm n packes durng me slo whch s denoed as Ψ( ) T n( s ) = c z z () where Ψ( ) s he effecve rae funcon he form of whch depends on he proocols mplemened a he SU. Then he buffer sae can be updaed as v2 max ( n v 0) v j v j = v( j ) max n vm 0 (2) m= v J Y where Y s he random varable represenng he number of packes arrvng a me slo havng lfe me J. The dsrbuon of Y s denoed by p Y ( l ). Hence he ranson probable s gven by f v.( 2) sasfes eq P () Y l p( v v z ) = & Y = l (3) 0 ow.. The channel condon c depends on he channel gan and he power level for ransmsson. The channel gan s generally modeled as a FSC [0]. In hs example we also consder ha he power allocaon s consan durng he daa ransmsson and hence he channel condon c can be formulaed as a FSC wh ranson probably c c. ( ) p

3 The sae ranson probably for SU s gven by ps( s s z ) = p( v v z ) p( c c ). (4) Here we assume ha he ranson of he channel condon s ndependen of he ranson of buffer sae. The uly for he delay-sensve applcaon a me slo s defned here as J u( s z ) = mn n v j λg max { v n0} (5) j = where λ g s he parameer o rade-off he receved and los packes. ore sophscaed uly formulaons for mulmeda ransmsson whch consder he explc mpac on he mulmeda qualy (e.g. PSNR) can be found n []. C. Resource allocaon rule We model he mul-user wreless resource allocaon as an aucon for specrum opporunes held by he CS durng each me slo. The SUs calculae he acon a based on he nformaon abou he nework resources and her own prvae nformaon abou he envronmen hey experence. In hs aucon game he acon s he compeon bd.e. a s he amoun of bd submed o he CS. We use he acon and bd nerchangeably. Subsequenly each SU subms he bd a o he CS. Afer recevng he bd vecors from he SUs he CS compues he channel allocaon z for each SU based on he submed bds. To compel he SUs o declare her bds ruhfully [] he CS also compues he paymen τ ha he SUs have o pay for he use of resources durng he curren sage of he game. The negave value of he paymen means he absolue value ha SU has o pay he CS for he used resources. The aucon resul s hen ransmed back o he SUs whch can deploy her ransmsson sraeges n dfferen layers and send daa over he assgned channel. Afer he daa ransmsson anoher aucon sars a he nex me slo. The compuaon of he allocaon z and paymen τ s descrbed as follows. Afer each SU subms he bd vecor he CS performs wo compuaons: () channel allocaon and () paymen compuaon. Durng he frs phase he CS allocaes he resources o SUs based on s adoped farness rule e.g. maxmzng he oal socal welfare : op z = arg max ( ) h a z w (6) z = where h ( ) s he uly funcon of SU seen by he CS. Noe ha hs uly can be represened by eher he effecve rae or me on he nework allocaed o each user or can be deermned n he uly doman by consderng he resulng uly-rae funcons of he deployed mulmeda coders []. Unlke he aucons n [4][5] We wll consder n hs paper a second prce aucon mechansm [2] for deermnng he ax ha needs o be pad by SU based on he above op opmal channel assgnmen z. Ths aucon mechansm enables each SU o ruhfully declare her resource requremen a each sage. Ths ax equals: τ op j j j j z j= j j= j = h ( a z w) max h ( a z w). (7) For smplcy we can denoe he oupu of he resource allocaon game as r = ( z τ ) = Ω( a w ). D. Selecng he Polcy for Playng he Resource anagemen Game In he cognve rado nework we assume ha he sochasc game s played by all SUs for an nfne number of sages. Ths assumpon s reasonable for applcaons havng a long duraon such as vdeo sreamng vdeoconferencng ec. In our nework seng we defne a hsory of he sochasc game up o me as h = { s w a b z τ... s w a b z τ s } H whch summarzes all prevous saes and he acons aken by he SUs as well as he oucomes a each sage of he aucon game and H s he se of all possble hsory up o me. However durng he sochasc game each SU canno observe he enre hsory bu raher par of he hsory h. The observaon of SU s denoed as o O and o h. Noe ha he curren sae s can be always observed.e. s o. Then a bddng polcy π : O A for SU a he me s defned as a mappng from he observaons up o he me no he specfc acon.e. a = π( o ). Furhermore a polcy profle π for SU aggregaes he bddng polces abou how o play he game over he enre course of he sochasc game.e. 0 π = ( π... π...). The polcy profle for all he SUs a me slo s denoed as π = ( π... π ) = ( π π ). The reward for SU a he me slo s R( s r ) = u( s z ) τ. Snce he resource allocaon also depends on oher SUs saes and acons he reward s furher expressed by R( s Ω( a a w )). We defne he bes response β for SU o oher SUs polces π as β ( π ) arg max Q (( π π ) ) (8) = s w - π where Q( π π - s w ) s he oal dscouned sum of rewards whch s defned as k k k k k - k k= The facor α ( 0 α ) ( ( )) Q (( π π ) s w ) = ( α ) R s Ω a a w (9) < s he dscouned facor deermned by a specfc applcaon (for nsance for vdeo sreamng applcaons hs facor can be se based on he olerable delay). The oal dscouned sum of rewards n Eq. (9) consss of wo pars: () he curren sage reward and () he expeced fuure reward dscouned by α. Noe ha SU canno ndependenly deermne he above value whou explcly knowng he polces and saes of oher SUs. The SU maxmzes he oal dscouned sum of fuure rewards n order o selec he bddng polcy whch explcly consders he mpac of he curren bd vecor on he expeced fuure rewards. The cenral ssue n he sochasc game s how he bes response polces can be deermned by he SUs. Ths wll be dscussed n Secon III.E. E. Characerzng he bes response polcy Recall ha durng each me slo he CS announces an aucon based on he avalable resources and hen SUs bd for he resources. To enable he successful deploymen of hs resource aucon mechansm we can prove smlarly o our pror work n [] ha SUs have no ncenve o msrepresen her nformaon.e. hey adhere o he ruh ellng polcy. We assume ha a each me slo SU has preference U j over he channel j whch capure he benef derved when usng ha channel. The preference U j s nerpreed as he benef obaned by SU when usng channel j compared o he benef when hs channel s no used. Noe ha hs benef

4 ( ) also ncludes he expeced fuure rewards. The opmal bd op a j ha SU can ake on he channel j a me s he bd maxmzng he ne benef U j τ. In he aucon dscussed n Secon III.C he opmal bd ha SU can make s op a j = U j.e. he opmal bd for SU s o announces s rue preference o he CS []. The proof s omed here due o space lmaons snce s smlar o ha n []. The paymen made by SU s compued by he CS based on he nconvenence ncurred by oher SUs due o SU durng ha me slo []. Nex we defne he preference U j n he conex of he sochasc game model. Usng he channel j when s avalable SU obans he mmedae gan u( s e j ) by ransmng he packes n s buffer where e j ndcaes ha channel j s allocaed o SU durng he curren me slo. SU hen moves no nex sae s from whch may oban he fuure reward Q ( π s w ). On he oher hand f no channel s assgned o SU receves he mmedae gan u( s 0 ) and hen moves no he nex sae s from whch may oban he fuure reward Q ( π s w ). We defne a feasble se of channel assgnmens o SU s opponens gven SU s channel allocaon z as Z -( z ) wh Z -( z ) = { Z N z {0}} kj = y k k j z j z j kj k z = = kj. The preference over he curren sae can be hen compued as U ( s w ) = j u ( s ej ) α p ( ) ( ) s s s ej pw w w (0) p ( ) ( ) s sk sk k Q s S z π s w w W Z ( ) Z ej k = u ( ) s 0 α p ( ) ( ) s s s 0 pw w w s S p ( ) s sk sk zk Q w W Z Z 0 k = ( π s w ) From hs equaon s clear ha he rue value U j depends on s own curren sae s bu also he oher SUs saes s he channel allocaons Z ( e j ) o he oher users when channel j s assgned o SU Z ( 0) when SU s no assgned o any channel and he sae ranson models ( ps sk sk z k ) k. However he oher SUs saes he channel allocaons and he sae ranson models of oher SUs are no known o SU and s hus mpossble for each SU o deermne s preference Uj ( s w ). Whou knowng he oher SUs saes and sae ranson models SU canno derve s opmal bddng sraegy op a j = Uj ( s w ). However f SU chooses he bd vecor by only maxmzng he mmedae reward u ( s z ) τ.e. he oal dscouned sum of reward degeneraes n Q( s w π ) = u ( s z ) τ by seng α = 0. Then he preference over channel j becomes Uj ( s y ) = u ( s e j ) u ( s 0). Snce now U j only depends on he sae s SU can compue boh he opmal bd vecor as well as he opmal bddng polcy. We refer o hs opmal bddng polcy as he myopc polcy snce only akes he mmedae reward no consderaon and gnores he fuure mpac. The myopc polcy s referred o as myopc π. To solve he dffcul problem of opmal bddng polcy selecon when α 0 an SU needs o forecas he mpac of s curren bddng acons on he expeced fuure rewards dscouned by α. The forecas can be performed usng learnng from s pas experences. F. Learnng for playng he game A key queson s wha needs o be learned whn a wreless sochasc game n order o mprove he polcy of an SU. Recall ha he opmal bddng polcy for SU s o generae a bd vecor ha represens s preferences U j j for usng dfferen channels. From III.E we can see ha SU needs o learn: () he sae space of oher SUs.e. S ; () he curren sae of oher SUs.e. s ; () he ranson probably of oher SUs.e. p ( s sk sk z k ); (v) he k resource allocaon Z-( e j ) j and Z -() 0 ; and (v) he dscouned sum of rewards Q ( π ( s s ) w ). However SU can only observes he nformaon ο = { s w a b z τ... s w a b z τ s } from whch SU canno accuraely nfer he oher SUs sae space and ranson probably. oreover capurng he exac nformaon abou oher SUs requres heavy compuaonal and sorage complexy. Insead we allow SU o classfy he space S no H classes each of whch s represened by a represenave sae s h h {... H}. By dvdng he sae space S he ranson probably p ( s sk sk z k ) s approxmaed k by ( ps s s z ) where s and s are he represenave saes of he classes ha s and s belong o. Ths approxmaon s performed by aggregang all oher SUs saes no one represenave sae and assumng ha he ranson depends on he resource allocaon z. Noe ha he classfcaon on he sae space S and approxmaon of he ranson probably and dscouned sum of rewards affecs he learnng performance. Hence a user should radeoff an ncreased learnng complexy for an ncreased learnng performance. The ranson probably ( p s s ) s z can be approxmaed usng occurrence frequency and he average rewards V (( s s )) can be learned usng he algorhm smlar o he Q-learnng [3]. The dealed learnng algorhm s presened n [2]. IV. SIULATION RESULTS In hs secon we am a quanfyng he performance of our proposed sochasc neracon and learnng framework. We assume ha he SUs compee for he avalable specrum opporunes n order o ransm delay-sensve mulmeda daa. In hs smulaon we consder fve SUs compeng for he avalable channel opporunes n he WLAN-lke cognve rado nework. The packe arrvals of all he fve SUs are modeled usng a Posson process wh he same average arrval rae of 2bps. The number of channels s 3 and he channel condon of all he fve SUs on each channel akes only hree values ( K = 3 ) whch are 8dB 23dB and 26dB. The ranson probables are l l 2 pj = pj = 0.4 pj = 0.2 pj = pj = 0.4 l 3 p j = 0.2 j l. The parameers of he model of he avalably of he channels o he SUs are NF FN pj = 0.7 pj = 0.3. The lengh of he me slo Δ T s 2 also 0 s. Smlar parameers are used for he fve SUs n order o clearly llusrae he performance dfferences obaned based on he dfferen sraeges.

5 In hs smulaon we consder only wo scenaros. In scenaro () all SUs deploy a myopc bddng sraegy myopc π = whle n scenaro (2) SU 5 deploys he mul-user learnng-based bddng sraegy π L 5 wh he dsc and he oher SUs deploy he myopc bddng myopc sraegy π = The packe loss rae and cos per me slo ncurred by he SUs are presened n Table. The accumulaed packe loss and cos of SU 5 for he fve scenaros are ploed n Fgure (a) and (b) respecvely. The average ax and cos s agan compued whn a me wndow of T = 000 slos. From Table we noe ha SU 5 sgnfcanly reduces he packe loss rae by 4.6% and average cos by 6.% by adopng he bes response learnng-based bddng sraegy. Fgure (a) and (b) furher verfy he mprovemen of he performance for SU 5. However oher SUs performances are decreased as hey need now o compee agans a learnng SU (.e. SU 5) whch s able o make beer bds for he avalable resources. V. CONCLUSION In hs paper we model he cognve rado resource allocaon problem as a sochasc game played among sraegc SUs. A each sage of he game he CS deploys a generalzed second prce aucon mechansm o allocae he avalable specrum resource. The SUs are allowed o smulaneously and ndependenly make bd decson on ha resource by consderng her curren saes experenced envronmen as well as he esmaed fuure reward. To mprove he bd decson a each sage we propose a bes response learnng algorhm o predc he possble fuure reward a each sae. The smulaon resuls show ha our proposed learnng algorhm can sgnfcanly mprove he SUs performance. Our fuure work wll focus on analyzng he performance of cognve rado neworks where mulple SUs are deployng varous learnng sraeges and proocols. Table. Performance of SU ~5 n he fve SUs nework SU SU 2 Packe Loss Average Packe Loss Average Rae (%) cos Rae (%) cos SU3 SU SU Accumulaed packe loss Learnng-based bddng sraegy yopc bddng sraegy Tme slo (a) Accumulaed cos Learnng-based bddng sraegy yopc bddng sraegy Tme slo (b) Fgure. The accumulaed packe loss and cos of SU 5 n he wo scenaros (a) accumulaed packe loss over he me slo; (b) accumulaed cos over he me slo REFERENCES [] F. Fu and. van der Schaar Non-collaborave resource managemen for wreless mulmeda applcaons usng mechansm desgn IEEE Transacon on ulmeda vol. 9 no. 4 pp Jun [2] F. Fu and. van der Schaar Learnng o Compee for Resources n Wreless Sochasc Games IEEE Transacons on Vehcular Technology o appear. [3] Federal Communcaons Commsson Specrum Polcy Task Force Rep. ET Docke No Nov [4] S. Haykn Cognve rado: Bran-empowered wreless communcaons IEEE J. Sel. Areas Commun. vol. 23 no. 2 Feb [5] F. A. Ian W.Y. Lee.C. Vuran and S. ohany NeX generaon/dynamc specrum access/cognve rado wreless nework: a survey Compuer Neworks vol 50 no. 3 Sep [6] D. Fudenberg and D. K. Levne The heory of learnng n games Cambrdge A: IT Press 999. [7] IEEE 802.e/D5.0 wreless medum access conrol (AC) and physcal layer (PHY) specfcaons: edum access conrol (AC) enhancemens for Qualy of Servce (QoS) draf supplemen June [8] S. Shankar C.T. Chou K. Challapal and S. angold Specrum agle rado: capacy and QoS mplcaons of dynamc specrum assgnmen Global Telecommuncaons Conference Nov [9] L. Kaelblng. Lman and A. Cassandra. Plannng and acng n parally observable sochasc domans. Arfcal Inellgence Volume 0 pp [0] Q. Zhang and S.A. Kassam Fne-sae arkov model for Raylegh fadng channels IEEE Transacon on Communcaons vol. 47 no. Nov []. Jackson echansm heory In he Encyclopeda of Lfe Suppor Sysems [2] P. Klemperer Aucon heory: A gude o he leraure J. Economcs Surveys vol. 3 no. 3 pp Jul [3] C. Wakns and P. Dayan Q-learnng. Techncal Noe achne Learnng vol [4] C. Kloeck H. Jaekel and F. Jondral Dynamc and Local Combned Prcng Allocaon and Bllng Sysem wh Cognve Rados IEEE DySpan [5] N. Amay and L. J. Greensen Resource aucon mulple access (RAA) n he cellular envronmen IEEE Transacons on Vehcular Technology Nov. 994.