Stochastic Power Control for Time-Varying Long-Term Fading Wireless Networks

Transcription

1 Sochasc Power Corol for Tme-Varyg Log-Term Fadg Wreless Neworks ohammed Olama Deparme of Elecrcal ad Compuer Egeerg, Uversy of Teessee, Koxvlle, TN 37996, USA. Emal: Seddk. Djouad Deparme of Elecrcal ad Compuer Egeerg, Uversy of Teessee, Koxvlle, TN 37996, USA. Emal: Charalambos D. Charalambous Deparme of Elecrcal ad Compuer Egeerg, Uversy of Cyprus, 678 Ncosa, Cyprus. Emal: The power corol (PC) of wreless eworks s formulaed usg a sochasc opmal corol framework. The performace of sochasc opmal power corol for me-varyg log erm fadg chaels, whch he evoluo of he dyamcal chael s descrbed by a sochasc dffereal equaos (SDE) s deermed. Ulke he commo radom or free space sac models usually ecouered he leraure, he SDE esseally capure he spaal-emporal varaos of logormal fadg wreless chaels as well as he radomess. The soluo of he sochasc opmal corol s obaed hrough pah-wse opmzao, whch s solved by lear programmg usg predcable power corol sraeges (PPCS). The algorhm ca be mplemeed usg a erave umercal scheme. The performace measure of he algorhm s erferece or ouage probably. Smulao resuls show ha he performace of PPCS usg sochasc models ouperforms he performace of PC based o sac models. The PPCS algorhm ca be used as log as he chael model does o chage sgfcaly. If predcable corol sraeges do o hold, s show ha he proposed power corol problem reduces o parcular covex opmzaos. Keywords ad phrases: wreless commucaos, power corol, sochasc corol, sochasc dffereal equaos, logormal shadowg, radom processes, radom varables. I. INTRODUCTION Power corol s mpora o mprove performace of wreless commucao sysems. os of he research ha has bee doe hs feld deals maly wh sac wreless chael models. Bu realy, wreless chaels are dyamc due o he relave moo bewee rasmers ad recevers ad emporal varaos of he propagao evrome, for e.g., due o movg scaerers []. Therefore dyamcal chael models are more realsc ha sac oes.

2 The power allocao problem has bee suded exesvely as a egevalue problem for o-egave marces [-3], resulg erave power corol algorhms (PCA s) ha coverge each user s power o he mmum power [4-7], ad as opmzao-based approaches [8]. uch of hs prevous work deals wh sac chael models. Ay power corol (PC) scheme ha aemps o follow fas fades would eed o be hghly effce o be mplemeed pracce, or cur a power pealy due o ese sgal processg ad may requre freque commucao wh s assged base sao. The scheme roduced by [8], whereby he sascs of he receved sgal o erferece rao (SIR) are used o allocae power, raher ha a saaeous SIR. The allocao decsos ca he be made o a much slower me scale. Prevous aemps a capacy deermaos CDA sysems have bee based o a load balacg vew of he PC problem [9]. Ths reflecs a esseally sac or a bes quas-sac vew of he PC problem, whch largely gores he dyamcs of chael fadg as well as user mobly. I hs formulao PC a successve samplg me pos s vewed as a powse opmzao problem wh oal sascal depedece assumed bewee he varables (corol or sgal) a dsc me pos. I a sochasc framework, aemps a recogzg he me correlaed aure of sgals are made [], where blockg s defed va he sojour me of global erferece above a gve level, whch f suffcely log, duces blockg. Dowlk PC for fadg chaels s suded [] by a heavy raffc lm where averagg mehods are used. Sochasc corol approach for uplk logormal fadg chaels s suded [], whch a bouded rae power adjusme model s proposed. Rece work o dyamc PC wh sochasc chael varao ca be foud [3-5]. I coras o hose papers, he modellg ad aalyss of PC sraeges vesgaed here employ wreless models whch are me-varyg ad subjec o fadg. The radom varables characerzg he saaeous power sac chael models are geeralzed o dyamcal models cludg radom processes wh me-varyg sascs. Sce wreless chaels have radom ad me varyg (TV) properes, hs paper suggess usg dyamcal (me varyg) chael models. The dyamcs of he chael s capured by a sochasc dffereal equao (SDE). A sochasc PCA s appled o deerme he opmal rasmed powers. The correc usage of ay PCA ad hereby he power opmzao of he chael models, requre he use of such chael models ha capure boh emporal ad spaal varaos he chael, whch exhb more realsc behavour of wreless sysems ha he sac models usually ecouered he leraure. Sce few emporal or eve spaal-emporal dyamcal models have so far bee vesgaed wh he applcao of ay PCA, he suggesed dyamcal models ad he PCA wll hus provde a far more realsc ad effce opmum corol for wreless chaels. The proposed PCA s based o predcable power corol sraegy (PPCS) ha was frs roduced [6]. The PPCS algorhm s prove o be effecvely applcable o such dyamcal models for a opmal PC. The PPCS algorhm s used hs paper o mmze he oal rasmed power, whch akes o accou he presece of ose he chael ad makes use of me varyg dyamcal lk for a effce PC of rasmers. Also he erave PCA preseed [5, 7] ca be used o deerme he opmal powers eravely. Ths helps allowg auoomous execuo a he ode or lk level, requrg mmal usage of ework commucao resources for corol sgallg. The ouage probably s used as a performace measure for he proposed algorhm. Smulao resuls are provded comparg he performace of he proposed mehod (PPCS) wh he performace of PC based o sac models. The resuls show ha he former ouperforms he laer as expeced. Regardg wreless chael modellg, s oed ha rado chaels experece boh large-scale fadg (log-erm fadg (LTF)) ad small-scale fadg (shor-erm fadg (STF)). LTF s modelled by logormal dsrbuos ad STF ca be modelled by Raylegh or Rcea dsrbuos [7]. I geeral, LTF ad STF are cosdered as supermposed ad may be reaed separaely [7, 8]. I hs paper, we cosder dyamc modellg ad power corol for LTF whch predoma suburba areas. The STF case has bee cosdered [6].

3 The paper s orgazed as follows. I Seco II, me-varyg LTF chael models whch he evoluo of he chael s descrbed by a SDE as roduced. I Seco III, he sochasc opmal PC model s proposed. I subseco III-A, he soluo of he proposed PC algorhm s obaed hrough lear programmg usg PPCS. A erave PC scheme s ouled subseco III-B o smplfy he mplemeao of he proposed PCA. ore geeral power corol cases are preseed subseco III-C. I Seco IV, smulao resuls are preseed ad he performace of PPCS usg me varyg (dyamcal) models s compared wh oe of me vara (sac) models. Fally Seco V gves he cocluso of he work developed hs paper. II. TIE VARYING LOGNORAL FADING CHANNEL ODEL Wreless commucao eworks are subjec o me-spread (mul-pah), Doppler spread (mevaraos), pah loss, ad erferece serously degradg her performace. I addo o expoeal power pah loss, wreless chaels suffer from sochasc shor-erm fadg due o mulpah, ad log erm fadg due o shadowg depedg o he geographcal area. If he moble happes o be some less populaed area wh few buldgs, vehcles, mouas ec., s sgal wll udergo log erm (log ormal) fadg [7]. I hs case, he proposed he me varyg model ha capures boh space ad me varaos s frs roduced. The power loss (PL) db for a gve pah s gve by [7]: d PL( d)[ db]: = PL( d)[ db] + α log + X, d d () d where PL( d ) s average power loss db a a referece dsace d from he rasmer, α s he pahloss expoe whch depeds o he propagao medum, ad X s a zero-mea Gaussa dsrbued radom varable, whch represes he varably of power loss due o umerous reflecos occurrg alog he pah ad possbly ay oher uceray of he propagao evrome from oe observao sa o he ex. I Dyamcal TV model, he average pah loss becomes a radom process deoed { X (, τ )}, whch s a fuco of boh me ad locao represeed by τ, where τ = d/c, d s he, τ τ pah legh, c s he speed of lgh, τ o = d o /c ad d o s he referece dsace. The process { X (, τ )} represes how much power he sgal loses a a parcular dsace as a fuco of me., τ τ kx (, τ ) The sgal aeuao s defed by S (, τ ) e, where k c/ c = l /. The process = ad ( ) X (, τ ) s geeraed by a mea-reverg verso of a geeral lear SDE gve by [9]: ( ) dx (, τ) = β(, τ) ( (, τ) X (, τ) d + δ(, τ) dw ( ), X (, τ) = NPLd ( ( )[ db]; σ ) () { } depede of (, ) where W () κ, ad ( )[ ] s he sadard Browa moo (zero drf, u varace) whch s assumed o be X τ, ad N ( µ ; κ ) deoes a Gaussa radom varable wh mea µ ad varace PL d db s he average pah loss db. The parameer (, ) τ models he average mevaryg power pah-loss a dsace d from rasmer, whch correspods o PL( d)[ db ] a d dexed by. Ths model racks ad coverges o hs value as me progresses. The saaeous drf β (, τ ) (, τ ) X (, τ ), β, τ ( ) represes he effec of pullg he process owards ( τ ), whle ( ) 3

4 represes he speed of adjusme owards hs value. Fally, δ (, τ ) corols he saaeous varace or volaly of he process for he saaeous drf. The al codo of X (, τ ) ca be obaed from a geomerc Browa moo model whch calculaes X ( τ ) for a fxed = as a fuco of τ. Le { θ (, τ) } { β (, τ), (, τ), δ (, τ) }. If he radom processes { θ(, τ) } bouded, he () has a uque soluo for every X (, τ ) gve by []:, are measurable ad X = e X + e u u du+ u dw u (3) β([, ], τ) β([ u, ], τ) (, τ) (, τ) [ β(, τ) (, τ) δ(, τ) ( )] where β([, ], τ) β( u, τ) du. oreover, usg Io s sochasc dffereal rule o wh respec o τ we oba, S (, τ ) = e kx(, τ ) ds(, τ) = S(, τ) kβ(, τ)[ (, τ) X (, τ)] + k δ (, τ) d + kδ(, τ) dw( ) S (, τ ) = e kx (, τ ) (4) Ths model capures he emporal ad spaal varaos of he propagao evrome as he radom θ, τ ca be used o model he me ad space varyg characerscs of he chael. I parameers { ( )} s requred ha he mea of power-loss process EX [ (, τ )] should rack me ad space varaos of he average power-loss. For example, le / T π (, τ) = m +.5e s T (5) where m s he average power loss ad T s he observao erval, δ ( ) = 4 ad () β = 5 (hese parameers are deermed from expermeal measuremes as wll be show a he ed of hs X, τ as a fuco of dsace ad me are represeed Fgure (). The seco). The varaos of ( ) emporal varaos of he evrome are capured by a TV (, ) τ whch was chose such as o flucuae aroud dffere average power losses m s, hus each curve correspods o a dffere locao. I spaal-emporal logormal model, besdes al dsaces, he moo of mobles,.e., her veloces ad drecos of moo wh respec o her base saos are mpora facors o evaluae me varyg power pah losses for he lks volved. Ths ca be llusraed a smple way for he case of oe rasmer ad oe recever. Cosder a recever a a dsace d from a rasmer ha moves wh a cera cosa velocy υ m a dreco defed by a arbrary cosa agle θ, where θ s he 4

5 agle bewee he dreco of moo of he moble ad he dsace vecor ha sars from he rasmer owards he recever as show Fgure (). Fgure (): ea-reverg power pah loss as a fuco of ad τ, for a gve me varyg (, τ ). Tx d Rx d() υm θ Fgure (): A recever (moble) a a dsace d from a rasmer (base sao), moves wh velocy υ m ad dreco gve by θ wh respec o he rasmer-recever axs. A me, he dsace from he rasmer o he recever d () s gve by: d = d+ + = d + + d (6) ( ) ( υmcos θ) ( υms θ) ( υm ) υmcosθ Therefore, he average power-loss a ha ew locao s gve by: d () (, τ) = PLd ( ( ))[ db] = PLd ( )[ db] + αlog + ξ( ) (7) d where d() d ad ( ) d s defed (6), α s he power loss coeffce ad ξ () s a arbrary fuco of me represeg he emporal varaos he propagao evrome lke he appearace ad dsappearace of addoal scaers. PL d s he average power loss db a referece dsace d, () I ca be show ha he spaal correlao of he logormal mea-reverg model () agrees wh ha foud he leraure [-3]. I parcular, was repored ha he spaal correlao for shadow fadg 5

6 moble commucaos, whch compares successfully wh expermeal daa, could be modeled usg a expoeally decreasg fuco mulpled by he varace of he power-loss process as follows: c c Cov ( ) σ e = σ e (8) d/ X ( v/ X ) X X X where σ X s he covarace of he power-loss process, d s he dsace bewee wo cosecuve samples, v s he velocy of he moble ad X c s he effecve correlao dsace whch s proporoal o he desy of he propagao evrome, correspodg o he dsace whe he ormalzed correlao falls o e [3]. I he remag par of hs seco, we show ha our overall spaal dyamcal model capures deed hese correlao properes. Cosder he space-me mea-reverg logormal model (). Whou loss of geeraly cosder he θ, τ = β τ,, τ, δ τ ad he space-me model parcular case where he parameers { ( )} { ( ) ( ) ( )} gve by (), wh (, τ ) beg gve by (7). Le X (, τ ) X (, τ) EX [ (, τ) ], he we have: dx (, τ) = β( τ) X (, τ) d + δ( τ) dw (), X (, τ) = N(; σ ) (9) The soluo of (9) s gve by: ( )( ) ( )( u ) X (, ) e βτ βτ τ = X (, τ) + e δ( τ) dw( u) () The mea of he process X (, τ ) s gve by: βτ ( )( E X (, τ) = e ) E X (, τ) = () sce E X (, τ ) =. The covarace of X (, τ ) s gve by: ( τ τ )( τ τ ) Cov (, v) = E X (, ) E X (, ) X ( v, ) E X ( v, ) X βτ ( )( + v) βτ ( ) δ ( τ) ( )( v ) = e e σ + e βτ ( ) βτ ( ) () where v m(, v). Leg v= + he, δ ( τ) ( )( ) (, ) β( τ)( ) β( τ) σ ( X + = + ) (3) Cov e e e βτ βτ ( ) 6

7 The covarace of he overall dyamcal model dcaes wha proporo of he evrome remas he same from oe observao sa or locao o he ex, separaed by he samplg erval. Sce he moble s moo, mples ha hs correspods o a spaal covarace. Now f we choose he δ ( τ ) varace of he al codo such ha σ =, he, β ( τ ) Cov ( ) ( ) ( ) (, δ τ ) ( ) X X ( ) e βτ e βτ + = = σ βτ Cov (4) Expresso (4) dcaes ha he spaal covarace of our overall dyamcal model correspods o he repored expermeal spaal covarace gve by (8). The comparso furher dcaes ha β(τ) s a characersc of boh he propagao evrome ad he separao dsace of wo cosecuve samples,.e., β(τ) s versely proporoal o he desy of he propagao evrome, ad drecly proporoal o he sample separao dsace. Fally, we oe ha he spaal covarace s a mpora characersc for our dyamcal mea-reverg shadow fadg model sce ca be clearly used order o defy he radom parameers {β(τ), δ(τ)}. Ths could be accomplshed by usg expermeal daa of Cov ( ) X order o defy β(τ) whch we ca use furher cojuco wh σ order o defy δ(τ). Noe ha he varace of he al codo of he power loss process, σ, should evably crease wh dsace, or equvalely δ(τ) should crease ad/or β(τ) should decrease. Fgure (3) shows he mea reverg power pah loss X (,τ), velocy ad dsace as a fuco of me for a moble wh parameers v= 8Km/hr, θ = 35 degrees,ad d= 5 meers ( ξ ( ) = ). I ca be see Fgure (3) ha he mea of he sochasc power pah loss X (,τ) cocde wh he average power pah loss (, τ ). oreover, he varao of X (,τ) s due o uceraes he wreless chael such as movemes of objecs or obsacles bewee rasmer ad recever. Fgure (3): ea-reverg power pah loss X (,τ) for a moble he LTF umercal example. The moble moves closer sarg from po 5 meers wh a agle of 35 degrees ad susodal speed wh average 8 km/hr (. m/s). 7

8 Now cosder a cellular ework wh rasmers ad recevers. The receved sgal a he h base sao ca be expressed as: y () = u () s () S () + d () (5) j j j j= where u j s he corol pu of rasmer j, whch acs as scalg o he formao sgal s j, d s he chael dsurbace or ose a recever, ad S j () s he sgal aeuao coeffce bewee rasmer j ad he recever assged o rasmer. Therefore, he spaal-emporal model () for rasmers ad recevers ca be descrbed by: ( ) dx (, τ) = β (, τ) ( (, τ) X (, τ) d + δ (, τ) dw ( ), j j j j j j X (, τ) = N( PL( d)[ db] ; σ ),, j j j Ths model s used o geerae he lk gas for he proposed PCA for log erm fadg commucao eworks. (6) III. POWER CONTROL ODEL The corol of he rasmed power provdes for a effce ad opmal performace of wreless commucao sysems. I ca crease sysem capacy ad qualy of commucaos, ad reduce moble baery power cosumpo. A. Power corol scheme Cosder a wreless ework of rasmers ad recevers. The measure of qualy of servce QoS ca be defed by he sgal o erferece rao (SIR) as [6]: m p subjec o ( p,... p ) = j pg pg + η j j (7) whch s equvale o m p subjec o ( p,... p ) = j= pg pg + η j j (8) where,< <. Here p deoes he power of rasmer, g j > deoes he + chael ga of rasmer j o he recever assged o rasmer, > s he requred SIR hreshold ad η > s he ose power level a he h recever,, j. The cosra equao (8) dyamc case s gve usg he pah-wse QoS of each user wh respec o he power sgals over a me erval [,T] as: 8

9 T p () S () d T T j= j j p () S () d + d () d, =,, (9) where S () s he sgal aeuao coeffces from rasmer j o recever assged o rasmer j a me, d () s he chael dsurbace a he h recever a me, ad. s he Eucldea orm. The sgal aeuao coeffces Sj ( ) kx (, τ ) S (, τ ) = e, where k = c/ ad c = ( ) QoS (8) wh respec o he sysems (6) s ow defed by: are geeraed usg he SDE () ad he relao l /. Cosequely, a aural geeralzao of he T m p ( ) d, subjec o = ( p,... p ) T T T p () s S () d p () s S () d+ d () d, =,, () j j j j= Now, we prese a soluo o () by frs roducg he commucao meag of predcable power corol sraeges (PPCS). I wreless cellular eworks, s praccal o observe ad esmae chaels a base saos ad he commucae he formao o he rasmers o adjus her corol pu sgals { u () }. Sce chael expereces delays, ad corol are o feasble couously me bu oly a = dscree me sas, he cocep of predcable sraeges s roduced [6]. Le he chael formao a ay me be deoed by { S (), s ()}, ad le he corol pu sgal for a rasmer a dscree me be { u ();,,, = T}. A me k, he base sao observes he chael formao { S s } ( ), ( ) k k. Usg he cocep of predcable sraegy, he base sao deermes he corol = sraegy { u ( )} k = for he ex me sa k. The laer s commucaed back o he rasmers whch hold hese values durg he me erval [, k k). A me k, a ew se of chael formao { S ( ), ( )} k s k = s observed a he base sao ad he me k + corol sraeges { u ( ) } k + are = compued ad he commucaed o he rasmers ad held cosa durg he me erval [ k, k + ). Such decso sraeges are called predcable sraeges. Usg he cocep of PPCS over ay me, +, equao () s equvale o: erval [ ] k k m p ( ) p( k+ ) = k+ subjec o p ( ) Γ G (, ) ( G (, ) p ( ) + η( )) k+ I k k+ k k+ k+ k+ () 9

10 where k+ k k k g (, + ): = S ( ) d,,, G (, ) dag( g (, ),, g (, )) I k k+ k k+ k k+ k+ k k k η (, + ): = d ( ) d,,, =, { } G (, ) = ( g (, ),,, k k+ k k+ η(, ) ( (, ),, (, )) T k k+ = η k k+ η k k+, p ( ) ( ( ),, ( )) T k+ = p k+ p k+, Γ= dag (,, ), dag() deoes a dagoal marx wh s argume as dagoal eres, ad T sads for marx or vecor raspose. The opmzao () s a lear programmg problem vecor of ukows ( ), + s a me erval such ha he chael model does o chage sgfcaly. p +. Here [ ] k k k The performace measure s erferece or ouage probably. I s defed as he probably ha a radomly chose lk wll fal due o excessve erferece []. Therefore, smaller ouage probably mples larger capacy of he wreless ework. A lk wh receved SIR rcvd less ha or equal o hreshold SIR h s cosdered a commucao falure. The Ouage Probably F ( rcvd ) s expressed as F( h ) = Pr{ rcvd h}, where F ( ) s he dsrbuo of rcvd. rcvd I he ex seco, we show ha he proposed PCA ca be mplemeed umercally usg a erave scheme. Ths s covee for o-le mplemeao sce helps allowg auoomous execuo a he ode or lk level, requrg mmal usage of ework commucao resources for corol sgalg. B. Ierave power corol scheme By usg PPCS he PC occurs oly a dscree me sas ad s assumed ha he chael does o chage sgfcaly he erval [ k, k + ], he he erave algorhm descrbed [5, 7] ca be used o deerme he opmal rasmed powers. Defe F ( k, k+ ) Γ* GI ( k, k+ )* G ( k, k+ ) ad u (, ) Γ* G (, )* η ( ), he k k+ I k k+ k+ T η ( k ) η ( k ) η( ) + + k+ k k+ = G( k, k+ ) G( k, k+ ) G( k, k+ ) u (, ),,, ad, j. The cosra () ca be rewre as: k k+ k+ k k+ Gj k k+ (, ) Fj ( k, k+ ) = where G (, ) k k+ ( I F (, )) P ( ) u (, ) () The marx F ( k, k + ) has oegave elemes ad s rreducble. The exsece of a feasble power vecor P ( k + ) > sasfyg () s equvale o ρ F(, ) k <, where ρ k+ F( k, k+ ) s he maxmum modulus egevalue of F ( k, k + ). The power vecor * P ( k+ ) = ( I F( k, k+ )) u( k, k+ ) s he opmal power vecor sasfyg (), ad he erao, P ( ) = F (, ) P ( ) + u (, ) (3) k+ k k+ k k k+ coverges o * P ( k + ) whe ρ F(, ) k k+ <. Equao (3) ca be wre as follows:

11 P ( k+ ) = Gj( k, k+ ) Pj( k) + η( k, k+ ) (4) G ( k, k+ ) j= ad also as: P ( ) = P ( ) (5) k+ k (, ) rcvd k k+ where = +, rcvd = rcvd +, =,,. I ca be show ha he erave PC (4) ad (5) rcvd coverges o he opmal (mmal) power vecor. The umercal mplemeao of he erave scheme, +. ca be carred ou durg processg he ervals [ ] C. Geeralzaos of he power corol scheme Whou predcable power corol sraeges, wo formulaos erms of covex opmzao usg lear programmg echques ad sochasc corol wh egral or expoeal-of-egral cosras are roduced. Boh problems are formulaed ex. The frs problem s formulaed erms of covex opmzao ad lear programmg as: k k k+ m p ( ) d, subjec o = k ( p,... p ) k+ k+ k+ p () s S () d p () s S () d+ d () d, =,, (6) j j j j= k k k Accordg o he above formulao usg predcable sraeges hs s a covex opmzao problem. Also, ay erval [, T] ca be cosdered as = < <... < k < k+... < T = T ad solvg he problem over a sequece of ervals. I should be oed ha f p() s couous almos everywhere k+ he erval [ k, k + ) he he egral p () d ca be approxmaed by Rema sums as close as k desred, ad he above problem (6) reduces o a lear programmg problem aga. The secod problem s formulaed erms of sochasc corol wh egral or expoeal-of-egral cosras as: T m E p ( ) (,... ) d p p, subjec o = k k + + k+ J, T ( p) E pj() sjsj() d p() ss() d d() d,,, j= + = k k k (7)

12 = If here exss a se of { } each J ( ), T p we ca roduce L λ ( u λ ) such ha he QoS are feasble, by employg Lagrage mulplers λ for k+ T E p () () () d+ λ pj sjsj d j= k, = m ( p,... p ) k + k+ = p() ss () d d() d + k k λ λ ad he solvg he problem l( λ, u ) = sup L ( u, λ ). Furher, ca be show ha L ( u, λ ) sasfes a dyamc programmg equao of he Hamlo-Jacob- Bellma ype [4]. Smlarly, he QoS ca be cosdered as powse cosras ad pursue he problem λ (8) T m E p ( ) (,... ) d p p, subjec o = p j () sjsj () p() ss() + d(), [, T], =,, (9) j= Noe ha f p (), for =,,, are covex he erval [, T], ad k+ k+ k+ p j() sjsj() d p() ss() d d() d j + s covex p, he equaos (7) = k k k ad (9) are covex opmzao problems, sce her objecve fucos ad cosras are covex. To llusrae he effcecy of he proposed PCA usg PPCS, smulao resuls are preseed he ex seco. IV. NUERICAL RESULTS I hs example, he performace of he proposed PCA for dyamcal LTF chael s compared wh he performace of he power corol algorhm proposed [] ha s based o he well-kow logormal model ecouered he leraure [7]. I s assumed ha oly mobles (rasmers) are movable whle base saos (recevers) are fxed hroughou he me of smulao. The cellular model has he s = for followg feaures: Number of rasmers ad recevers s = 4, he formao sgal () =,...,, al dsaces of all mobles wh respec o her ow base saos d are geeraed as uformly depede decally dsrbued (..d.) radom varables (r.v. s) [ ] meers, cross al dsaces of all mobles wh respec o oher base saos dj, j, are geeraed as uformly..d. r.v. s [5-55] meers, he agle θ j bewee he dreco of moo of moble j ad he dsace vecor passes hrough base sao ad he moble j are geeraed as uformly..d. r.v. s [ 8] degrees, he average veloces of mobles are geeraed as uformly..d. r.v. s [4 ] km/hr, all mobles move a susodal varable veloces aroud her average veloces, power pah loss expoe s 3.5, al referece dsace from each of he rasmers s meers, power pah loss a he al

13 referece dsace s 67 db, δ () = 4 ad ( ) r.v. s wh zero mea ad varace = 4* -8. β = 5 for he SDE s, η s are..d. Gaussa The SIR hreshold h s vared from fve o hry fve seps of fve, ad for each value of h he ouage probably s compued every 5 mllsecod for 5 secods. The ouage probably s compued usg oe-carlo smulaos. The ouage probably graphs of hs example for boh PC usg PPCS based o he proposed model ad PC based o he logormal model are show Fgure (4a) ad (4b) respecvely. Fgure (4) shows how he ouage probably chages wh respec o SIR hreshold ad me. As he SIR hreshold creases he ouage probably crease. Ths s obvous sce we expec more users o fal as SIR hreshold creases. The ouage probably s also chagg wh respec o me. Ths s because he mobles are movg dffere drecos wh dffere veloces. A ay me, some mobles are movg owards her ow base saos ad ohers are movg away from her base saos. The average ouage probably over all me ervals s show Fgure (5). The ouage probably s ploed versus hreshold SIR h, whch vares from 5 o db. The performace of PPCS usg he sochasc models s compared wh he performace of PC usg he sac models. From Fgure (5), he performace of PPCS usg he sochasc models s much beer ha he oe for PC usg he sac models. For example, a db SIR hreshold, he ouage probably s reduced from.6 for PC sac case o.8 for PPCS sochasc case, hs represes a mproveme of over 3%. The PPCS algorhm ouperforms he referece algorhm by a order of magude. I ca be see ha as hreshold SIR h creases he performace gap bewee PPCS usg sochasc models ad PC usg sac models decreases. Ths s because he effec of hreshold SIR h (requred QoS) s doma. Fgure (6) δ, τ = 8). I shows he average ouage probably over all me ervals for hgher ose varace ( ( ) hs case he sochasc power pah loss X (, τ ) have hgher varaos or flucuaos aroud he average pah loss (, τ ), sce hs parameer corols he saaeous varace of he sochasc power pah loss. Therefore, PC based o sac models wh hgh varace gves hgher ouage probably ha he oe wh low varace as oced Fgure (5) ad (6). Ths s because he acual (sochasc) chael s o close o he average (sac) oe. For example, a db SIR hreshold he ouage probably he sac case s abou.3 whle he dyamcal s abou., a mproveme of over 37%. Therefore, he opmal rasmed power for he sac model s o loger opmal whe s used for he sochasc model. Thus, sochasc models provde a far more realsc ad effce opmum corol ha sac oes. 3

14 (a) Fgure (4): Ouage probably for dyamcal log erm fadg model. (a) Usg PPCS algorhm o sochasc models. (b) Usg PC o sac models. (b) 4

15 Fgure (5): Average ouage probably for dyamcal log erm fadg chael model wh δ () = 4. Performace comparso. Fgure (6): Average ouage probably for dyamcal log erm fadg chael model wh δ () = 8. Performace comparso. 5

16 V. CONCLUSIONS I hs paper, a opmal PCA based o a dyamcal model for log-erm fadg wreless chael s proposed. ore realsc me-varyg logormal wreless chael models are used. The dyamcs of he chael s depced by a SDE, whch esseally capure he spaal-emporal varaos of wreless fadg commucao eworks. The opmal PCA s show o reduce o a smple lear programmg problem f predcable power corol sraeges (PPCS) are used. Ierave algorhms ca be used o solve for he opmzao problem. Geeralzaos o he power corol problem based o covex opmzao echques are provded f PPCS are o assumed. The performace measure s erferece or ouage probably. Numercal resuls preseed for hs algorhm dcae ha he performace of PPCS usg sochasc models ouperforms he performace of PC based o sac models by a order of magude. Resuls show ha here are poeally large gas o be acheved by usg sochasc models. The hgher he varace of he sochasc power pah loss process, he worse he performace of he PCA based o sac models. I hs case, he mproveme provded by he PCA based o TV models s more sgfca. The PPCS algorhm ca be used as log as he chael model does o chage sgfcaly. VI. REFERENCES [] Huag,., P. E. Caes, C. D. Charalambous, R. alhame. Power corol wreless sysems: A sochasc corol formulao, Proceedgs of he Amerca Corol Coferece, pp , Arlgo, VA, USA,. [] J. Zader, Performace of opmum rasmer power corol cellular rado sysems, IEEE Tras. o Vehcular Tech., vol. 4, o., Feb. 99. [3] J. Ae, Power balacg sysems employg frequecy reuse, COSAT Techcal Revew, vol. 3, 973. [4] S. A. Gradh, J. Zader ad R. Yaes, Cosraed power corol, Wreless Persoal Commucaos, vol., o. 3, Aug [5] N. Bambos ad S. Kadukur, Power-corolled mulple access schemes for ex-geerao wreless packe eworks, IEEE Wreless Commucaos, vol. 9, ssue 3, Jue. [6] X. L ad Z. Gajc, A mproves SIR-based power corol for CDA sysems usg Seffese eraos, Proceedg of he Coferece o Iformao Scece ad Sysems, ar.. [7] G. J. Fosch ad Z. ljac, A smple dsrbued auoomous power corol algorhm ad s covergece, IEEE Tras. o Vehcular Tech., vol. 4, o.4, Nov [8] S. Kadukur ad S. Boyd, Opmal power corol erferece-lmed fadg wreless chaels wh ouage-probably specfcaos, IEEE Trasacos o Wreless Commucaos, vol., o., pp ,. [9] A.. Verb ad A. J. Verb, Erlag capacy of a power-corolled CDA sysem, IEEE J. Selec. Areas Commu., vol., pp. 89-9, Sep [] N. B. adayam, P. C. Che, ad J.. Holzma, mum durao ouage for cellular sysems: A level crossg aalyss, Proc. 46 h IEEE Cof. Vehcular Techol., Alaa, GA, Apr. 996, pp [] R. Buche ad H. J. Kusher, Corol of moble commucaos wh me-varyg chaels heavy raffc, IEEE Tras. Auoma. Cor., vol. 47, pp. 99-3, Jue. []. Huag, P. E. Caes, ad R. P. alhame, Uplk power adjusme wreless commucao sysems: A sochasc corol aalyss, IEEE Tras. Auoma. Cor., vol. 49, o., pp , Oc. 4. [3] J. F. Chamberlad ad V. V. Veeravall, Deceralzed dyamc power corol for cellular CDA sysems, IEEE Tras. Wreless Commu., vol., pp , ay 3. [4] L. Sog, N. B. adayam, ad Z. Gajc, Aalyss of a up/dow power corol algorhm for he CDA reverse lk uder fadg, IEEE J. Selec. Areas Commu., vol. 9, pp , Feb.. 6

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