ow-rte Utiity-Bsed Resource Aoctio i Wireess Networks Peiju Liu, Rd Berry, Miche L. Hoig ECE Deprtmet, Northwester Uiversity herid Rod, Evsto, IL 68 UA peiju,rberry,mh @ece.wu.edu cott Jord ECE Deprtmet, Uiversity of Cifori, Irvie D Egg. Tower, Irvie, CA 9697-6 sjord@uci.edu Abstrct We cosider forwrd-ik power octio i wireess etwork with stochsticy vryig dt requests. We ssume user s service prefereces re specified vi utiity fuctio tht depeds o the received dt rte. The octio of power cross users is studied, where this octio my deped o both user s che d utiity. The objective is to mximize the time-verged utiity rte subject to stochstic tot power costrit t the trsmitter. For rge, heviy oded etwork, we itroduce Gussi pproximtio for the tot trsmitted power, which is used to decompose the power costrit ito three more trctbe costrits. We preset soutio to this probem tht is combitio of dmissio cotro d pricig of power. The optim trde-off betwee these pproches is chrcterized. Numeric exmpes re give to iustrte these ides. I. INTRODUCTION The efficiet octio of rdio resources, such s trsmissio power, is esseti for supportig diverse ppictios over wireess etworks. This pper ivestigtes resource octio for the forwrd ik i wireess etwork usig utiity-bsed pproch, where user s service preferece is specified by sige quity idictor or utiity fuctio. Oe dvtge of such pproch is tht differet utiity fuctios c be used to ccommodte wide rge of trffic fows uder sige frmework. Aso, utiity fuctios c be used to cpture my commo defiitios of firess withi etwork [6]. Utiity-bsed resource octio hs recety received ttetio both for wire-ie [], [], [] d wireess etworks[], [8], []. Reted work ddressig the forwrd ik i CDMA etwork c be foud i [], where the probem of mximizig ggregte utiity subject to costrits o vibe trsmissio power d spredig codes ws studied. The soutio to this probem c be iterpreted i pricig frmework, where prices per uit power d per code re ouced, d users mximize their surpus (utiity mius cost). The optim octio of resources c be foud by choosig the correct resource prices. I [] d much of the other prior work, the focus is o octig resources for sttic situtio, where the umber of users is fixed d ech user wi fuy utiize whtever resources it is octed. I this pper, we cosider situtio where trffic is dymicy chgig over the time period i which resources re octed. I this cse, rdom trffic vritios must be tke ito ccout whe octig resources. We cosider mode where the bse sttio s trsmissio power is octed mog the users. Trsmissio requests rdomy rrive t the bse sttio; ech request cotis fixed This work ws supported by the Motoro-Northwester Ceter for Teecommuictios, d by NF uder grt CCR-99. mout of dt. The rte t which request is trsmitted, d hece the time to serve request, depeds oy o the power octio d the user s che. The utiity user derives is bsed o how fst the request is served. We study the probem of octig trsmissio power to mximize the time verge utiity rte, give costrit o the tot power trsmitted by the bse sttio. ice the trffic is rdomy vryig, the tot power trsmitted by the bse-sttio is rdom process. We cosider stochstic costrit o this process, which imits the tot power to be ess th give vue with high probbiity. We chrcterize the soutio to this probem for system with rge umber of users. This soutio c be viewed i pricig frmework s i []; however, there re sever fudmet differeces. First, i dditio to pricig, expicit dmissio cotro is so eeded. ecod, the price tht is used is ot fixed price for the costried resource, the trsmissio power; the price isted depeds o the product of the trsmissio power d eergy, resutig i o-ier price for the required power. Our focus is o the situtio where trffic vritios occur o much fster time-sce th tht over which resource octio is doe. pecificy, we ssume power is octed bsed o the users che gi d utiity, d this ssigmet is fixed over the time period of iterest. I prticur, the power octio does ot deped o the istteous system stte (e.g., the umber of ctive requests), but oy o the og term sttistics of the system. A terte pproch woud be to tke ito ccout the curret system stte d reocte resources t every rriv d deprture. This type of pproch hs bee studied i [], vi dymic progrmmig techiques. Cery, octig resources o fster time-sce my improve the resutig utiity rte. However, such pproch my ot be fesibe, due to vrious system costrits, d wi require more compicted octio poicy. Aso, sice the octio cosidered here is ot stte depedet, user receives fixed utiity upo dmissio. I cotrst, with stte depedet reoctios, the utiity user receives c vry depedig o future evets. The rest of the pper is orgized s foows. I ect. II, we itroduce mode for the forwrd ik of sige ce. I ect. III, we formute costried optimiztio probem where the objective is to mximize the time-verged utiity rte subject to stochstic costrit o the tot power. I ect. IV, soutio to this probem bsed o decomposig the power costrit ito three more trctbe costrits is preseted. We the idetify the system behvior uder optimity. I ect. V, we preset umeric resuts iustrtig these ides.
D Fig.. utiity Exmpe utiity fuctio for dt trffic. II. YTEM MODEL We describe mode for the forwrd ik withi sige ce, where the bse sttio trsmits simuteousy to ctive users, d trsmissios to differet users re ssumed to be orthogo. For exmpe, this modes CDMA system with orthogo spredig codes. uppose tht user is octed trsmissio power. rte The received ig-to-iterferece Pus Noise Rtio (INR) for this user is give by, where is the che gi for tht user d is the tot oise pus iterferece eve. We ssume tht the received dt rte for user is fuctio of the received power, or equivety received INR; this retioship is give by, where is icresig fuctio. User requests re modeed s rrivig t the bse sttio ccordig to Poisso process with over rte. Ech request cosists of uit of dt with fixed egth. Ech dt uit is referred to s pcket, however, this coud so be sequece of pckets or fie depedig o the situtio. We cosider system with rge umber of users, d ssume ech request correspods to ew user. The che gi of ech user is ssumed to be distributed o the iterv!#"$ &%('), where!+*-, d &%('/., with cotiuous desity fuctio, /. This desity c be used to mode the users geogrphic distributio withi the ce, d so propgtio effects such s rdom shdowig. The che gi correspodig to ech rriv is chose idepedety ccordig to this distributio d stys fixed durig the etire trsmissio of the pcket. A utiity fuctio is ssocited witch request; this refects user s desired Quity of ervice (Qo). We ssume tht utiity depeds oy o the trsmissio rte. ice ech pcket hs fixed egth, this is equivet to defiig utiity s fuctio of pcket trsmissio time. I this pper, users re ssumed to hve the sme utiity fuctio, 6 ; however, this formutio c tury be exteded to cses with mutipe utiity csses. We ssume tht,789, d tht 6 is icresig, cocve d cotiuousy differetibe with respect to, for :*,. These re commo ssumptios for so-ced estic trffic, which describes my dt ppictios [7]. A exmpe utiity fuctio is depicted i Fig.. ice users hve the sme utiity fuctio, the power octed to user depeds oy o the che gi. For ech <;=, it wi be usefu to defie the fuctio >?@, which retes the utiity received by user with che gi to the trsmitted power?. This fuctio is give by >?@ABC?@@(D () E The foowig c be exteded to the cse where the egth of ech request is rdom, but we wi ot ddress this here. Notice tht >?@ wi be differet for users with differet che gis eve though these users hve the sme 6. III. PROBLEM FORMULATION Our objective is to octe trsmissio power to mximize the utiity rte give costrit o the tot trsmissio power. A power octio is specified by fuctio GF IHJLKNM tht idictes the power used to trsmit pcket to user with che gi B;. If??O,, the correspodig requests re cosidered bocked d ot trsmitted. If?QPR,, the correspodig pckets re trsmitted with trsmissio time give by T?@ For UVXW)"$YZ"[D[D[D, et \? deote the che gi of the U th rriv, d et ]^`_@ deote the umber of rrivs i the iterv,b"@_@. For give power octio, the time verge utiity rte is give by c d e fgh W i j fk m / >?\o@bzpqrrs>?\ttu@vs" where the expecttio is tke with respect to )/. Let wx`_@ deote the set of ctive trsmissios t time _. The tot power trsmitted t time _ c the be writte s Nyoz{`_@A ` )} j ~`k?\(d This is stochstic process with sttistics depedet o the rriv process, power octio d che distributio. We ssume tht uder y power octio,yoz{`_@ajlnyoz{ i distributio s _ J, wheretyoz{ is rdom vribe with the stedy-stte distributio. We cosider stochstic costrit o the tot power. pecificy, yoz{op ƒ/ G ˆ, where ˆ 6P, is sm costt. The resource octio probem c be formy stted s Probem MAXU: mximize Œ g Ž Zp A>?tu () subject to Tyoz{GP ƒ - ˆ )D () IV. UTILITY BAED POWER ALLOCATION A. Power Costrit Decompositio Let be sm costt such tht &%('s! šœ, for some iteger š. For UT,b"[D[D[D#" š, defie!qžsuo. For UT,b"[D[D[D#" š: œw, et ]^`U be rdom vribe represetig the umber of ctive users i stedy-stte with che gi i "$ M [. The stedy-stte tot power,yoz{, c the be pproximted s: Nyoz{RŸV m#? ]^`U(D ice rrivs re Poisso with over rte, the ]^`U is pproximtey the occupcy of s ) queue with r- Therefore, riv rte / t d service time. ]^`U is pproximtey Poisso distributed, d so pa]^`u@?ÿ Vr]^`U@/ŸI / / ƒ ]^, where ]^ ts ƒ
h ' e / t we hve (D Tkig expected vues, d ettig J,, px Nyoz{6? ]^ ƒ D () Likewise, ssumig ]^`U("@UV:,b"[D[D[D#" š W re idepedet, the vrice oftyoz{ is give by: VrNyoz{6 ƒ ]^ D () For rge umber of ctive users, we pproximteayoz{ by Gussi rdom vribe, yiedig, Nyoz{ P ƒ ƒb /? ]^ ƒ ]^ ƒ ~ ˆ )" where? )_ is the compemetry cumutive distributio fuctio (c.d.f.) of the stdrd Gussi rdom vribe. This c be simpified to? ]^ ƒ /ž where & ˆ. ]^B / d ice ƒ, we hve: ƒ ]^ = ƒ" (6)? ]^ ƒ BZpq="!x@ (7) ƒ ]^ BZpq=?#!x@ (8) where!xa? is the eergy octed to user with che gi. A ictive user uses zero eergy. ubstitutig (7) d (8) ito (6), costrit () c be pproximted by: Zpq="!x@ ž %$ Zpq=?#!x@ ƒ (9) Fiy, this c be further decomposed ito three prts: & ' ( Zp ="!x@& ) verge eergy p =?#!x@ ) ž,+ : ƒ trdeoff betwee ) d verge power * eergy () We wi refer to Probem MAXU whe () is repced with () s Probem MAXUA. A soutio to Probem MAXUA is provided ext. We proceed i two steps. First, the utiity mximizig power ssigmet is foud subject to the first two costrits i () for give vues of ) d. Next, the combitio of ) d tht yieds the highest utiity rte is derived. B. outio with Fixed ) d Give vues of ) d, cosider the foowig probem: Probem P: mximize Œ g Ž Zpq A>?tu subject to Zp ="!x@ ) () p =?#!x@& - D () To gi isight ito this probem, we first cosider ech of the costrits seprtey. First, we exmie the probem with oy the eergy costrit, i.e., Probem P: mximize Œ g Ž Zpq A>?tu Zp ="!x@ -)D subject to From () d (7), ptyoz?bzp ="!x@, so tht Probem P is equivet to costriig the verge sum power. To cotiue, we ssume tht the trsmissio rte is proportio to the received power, i.e.,?@ {?(" () where is give costt.. It foows directy from () tht the eergy cosumed by user depeds oy o whether user s trsmissio power is ozero, d ot o the specific power eve, i.e.,!xa/? { " for? P-, (),b" for?a,bd ice utiity is stricty icresig i received power, it foows from () tht the soutio to Probem P is for ech pcket to be either deied trsmissio (bocked) or trsmitted with ifiite power. If o users re bocked d the eergy costrit () is vioted, the dmissio cotro is required to bock some users. The choice of which users re bocked depeds o whether 6 is bouded s 9J. If 6 is bouded, the users requirig the highest eergy shoud be bocked uti () is stisfied. I this wy the fewest users re bocked, d therefore () is mximized. If 6 is ubouded, the mxim utiity rte is so ubouded, d which users re bocked is rbitrry s og s () is stisfied. I either cse,ayoz{`_@&r, with probbiity W dtyoz{`_@v wheever ew request rrives. Of course, this is ot reistic. This behvior is eimited by ddig costrit (). Next cosider Probem P with oy costrit (): Probem P: mximize Œ g Ž Zp A>?tu pq=?#!x@ D subject to This is mthemticy equivet to the probem studied i []; s i [], the soutio c be ttied vi pricig scheme. Theorem : Cosider the foowig pricig scheme: che depedet price per uit trsmit power of the form!x is ouced; users respod by requestig power to mximize their surpus (utiity mius cost), i.e., A 6 j87 k A>?tu!x?$ 7D () If is set such tht () is stisfied witquity, this pricig scheme provides power octio tht soves Probem P. This theorem c be esiy prove usig the Kuh-Tucker optimity coditios. The set of ctive users d the ssiged 9 A ier retioship betwee rte d power is resobe pproximtio for my prctic systems. For rge eough rtes, cpcity cosidertios impy tht this is optimistic.
W power eves re determied by, which c be iterpreted s fixed uit price o the product of power d eergy. For ech ctive user, the mrgi utiity qj with respect to power equs the price per uit power: j87 k. Ictive users hve ower mrgi utiity t zero th the price, qj j87 k j87 k m#.. ice >?@ is cocve, qj j87 k i.e., is decresig with?. I other words, for ictive users, opertig t y positive power gives utiity tht is ess th the cost (egtive surpus). We c those ictive users itimidted due to combitio of high price d sm iiti sope of >. Assumig users hve the sme 6 d tht () hods, the set of users tht re itimidted c be chrcterized s foows: Theorem : There exists threshod P, such tht?p, if d oy if =P. The threshod stisfies: Z 6 7 m# t { qj k The theorem foows esiy from the fct tht qj k d tht is decresig i. This theorem impies tht user with ow che gi is peized twice. First, this user requires more trsmissio power to chieve the sme 7] ; secod, the user is chrged higher uit price per power. Notice tht s icreses, becomes smer d? icreses for ctive users. This i tur icreses the utiity for ech ctive user d hece resuts i higher utiity rte. Aso otice tht the costrit i Probem P does ot deped o the trffic itesity, but oy o the che distributio, /. It foows tht chges i the rriv rte, for fixed /, wi ot effect the optim price i Theorem. Now we retur to Probem P. The soutio to this probem wi be combitio of dmissio cotro, s i Probem P, d the pricig pproch from Theorem. The optim combitio of these pproches c be foud i the foowig sequece: ) Assume?œP, for y. Give s d (), check the eergy costrit (). If it is vioted, bock users with che gis : L where is seected such tht costrit () is tight. Otherwise, dmit users. By covetio, if () is stisfied witquity, we set?x!. If () is oose, we set equ to y rbitrry vue ess th!. ) Fid so tht () is bidig for the set of ctive users. The optim power octio is give by () for the ctive users. Bocked users re ssiged zero power. The reso users with the owest che gis re bocked is tht with the sme 6, these users wys derive the owest utiity for y give. Therefore, there exists eergy iduced cutoff threshod such tht oy users with < B re bocked vi dmissio cotro. We ote tht t the optimum, () is wys bidig, wheres () my ot be bidig. By formutig the probem with both costrits, the resource octio is ccompished i two steps. The combitio of the eergy costrit d the rriv rte my require dmissio cotro, i.e., some users my be bocked to stisfy the eergy costrit. Amog the remiig users, power eves re determied vi pricig. ome users my so be itimidted depedig o the price. Admissio cotro d itimidtio re chrcterized by the che gi threshods, d, respectivey. We distiguish the foowig cses. C: V*B! d Q*B. (Active users re determied by.) C:! P d! *. (A users re ctive.) C: AP d AP!. (Active users re determied by.) C. Optim Admissio Cotro/Pricig Trde-off Give ) d, we hve show tht the optim soutio to Probem P cosists of combitio of dmissio cotro d pricig. Returig to Probem MAXUA, otice tht y pir of vues ) d tht stisfy )+ž + : ƒ (6) resuts i soutio to Probem P tht is so fesibe power octio for Probem MAXUA. To sove Probem MAXUA, we wt to fid the combitio tht mximizes the utiity rte. It is sufficiet to cosider vues of ) d such tht (6) is tight. ice if )<ž-z,+ X.ƒ, we c wys icrese to the poit where (6) is bidig. A rger ows ower, d therefore higher utiity rte. Theorem : Cosider P with costrits )". As ) icreses from, to ƒ, the optim soutio trsitios through the cses C, C, C i oe of the foowig sequeces: C J C J C or C J C. Proof: Let A deote the set of vues of ) for which the optim soutio to P is i C. Defie A d A simiry. At ) X,, xi &%(' d QX, ; therefore, ; A. As ) icreses, decreses; this resuts i decresig with ) d icresig. This impies tht if ) ; A the ) ; A for )A ) d ikewise, if ); A the ); A for )* ). Whe ) ƒ, X:,, i which cse, thus, ƒ ; A. Therefore, the oy possibe sequeces re C J C J C or C J C, which of these occurs depeds o!. As oted previousy, the costrit () is tight uder optim power octio. If the eergy costrit () is oose, the ) c be decresed, owig for rger without viotig (6). This resuts i higher utiity rte. Therefore we hve the foowig: Theorem : The power octio which soves Probem MAXUA stisfies both () d () witquity. Corory: The optim ) ; A, d. A is the oy regio where both costrits re tight. I A or A, the eergy costrit is wys oose. V. NUMERICAL REULT I this sectio, we preset umeric resuts to iustrte the ides from the previous sectio. Throughout this sectio, users re ssumed to hve the sme utiity fuctio: 69W& " (7) d the che distributio is ssumed to be give by: / " " for =; W)"( (D (8)!
Averge utiity per user.8.6. λ = λ = λ = λ = λ =6 Averge utiity per user vs. ε for differet λ.. A A A 6 7 8 9 Eergy costrit ε Fig.. Averge utiity per user vs. for differet rriv rtes,. Fig.. Averge utiity per user.98.96.9.9.9.88.86 Utiity vs. threshod q λ = ~ λ = λ = λ = λ =.8...6.8....6.8. q! "!# Mximum $&%&' vs. threshod () for differet s. Fig.. & h mi..... & h mi vs. ε with λ = h i h mi Eergy Costrit ε, vs. for differet s. & h mi..... & h mi vs. ε, with λ = h i h mi Eergy Costrit ε This is bsed o distce-bsed tteutio formu q / where users re distributed uiformy og the rdius withi ce, i.e., x W)" ^;:,b"[w{. We ssume the fie egth is ormized so tht 7, which is the ier scig fctor betwee trsmissio rte d received power. I other words, oe uit of received power resuts i oe uit competio time. Fig. shows how the verge utiity per user vries with d ) (d therefore ) whe ƒ LW, d ˆ,bD,bW. The cssifictio of the resutig octio is idicted o the figure. Notice tht the mximum poit is wys i A s predicted. Aso observe tht s ) icreses, the soutio trsitios from C J C J C whe is sm ( 9W)"[W,b"$Y, ). As icreses, the octio trsitios directy from C to C (,b"), ). Fig. shows how d vry with ) uder differet rriv rtes,. The miimum che gi! W is so 7 show. For.!, we choose to stisfy 7 " 7 ) so tht the curve is exteded cotiuousy from where *!. Observe tht, s expected, decreses with ) d icreses. Whe XW,, the system trsitios from C to C whe fs beow!, d from C to C whe icreses bove!. Whe,, the itersectio poit of d is rger th! ; i this cse the soutio trsitios directy from C to C. Fig. shows the mximum verge utiity per user, p A>?tu versus the threshod [ for differet rriv rtes. As icreses, the verge utiity per user decreses; however, the over utiity rte Zpq A>?tu icreses. A smer [, or equivety tighter power costrit, resuts i ower utiity per user. Notice the verge utiity per user is isesitive to ˆ whe is sm (. W, ). This is becuse the utiity fuctio we use ( 6T9W ) is retivey ft (cose to ) whe becomes rge. Whe is sm, the optim power octio ets users operte i the rge of where the utiity fuctio is retivey ft. VI. CONCLUION We hve studied forwrd ik power octio for stochsticy vryig trffic withi sige ce. We use stochstic tot power costrit i order to octe resources t sow rte retive to the dymics of the trffic. pecificy, power ssigmets deped oy o the users che stte, utiity fuctio d the og-rge trffic sttistics. We give pproximtio for the power costrit tht resuts i three trctbe costrits, d show tht combitio of dmissio cotro d pricig of power mximizes time-verged utiity rte. We ctegorize the trdeoff betwee dmissio cotro d pricig iduced itimidtio ito three cses d show tht the soutio is wys i cse. Numeric resuts iustrte the trdeoff d show tht for sm, the derived utiity rte is isesitive to the choice of ) over wide regio. REFERENCE [] N. Feg, N. Mdym, d D. J. Goodm, Joit power d rte optimiztio for wireess dt services bsed o utiity fuctios, I Proc. CI 999, Johs Hopkis Uiversity, Mrch 999. []. Kysudrm d M. Needhm d R. Agrw, Utiity fuctiobsed optim resource octio with mixture of reoctio-toert d reoctio-itoert users, I Proc. of PIE, vo., Aug.. []. Kuiyur d R. rikt, Ed-to-Ed Cogestio Cotro chemes: Utiity fuctios, Rdom Losses d Mrks, I Proc. INFOCOM, pp. -,. [] F. Key, Chrgig d rte cotro for estic trffic, Europe Trs. o Teecommuictios, 8: 7, 997. [] P. Liu, M. Hoig d. Jord, Forwrd-Lik CDMA Resource Aoctio Bsed o Pricig, Proceedigs IEEE Wireess Commuictios d Networkig Coferece,. [6] J. Mo d J. Wrd, Fir ed-to-ed widow-bsed cogestio cotro, I Proc. PIE 98 Itertio ymposium o Voice, Video d Dt Commuictio, Bosto, MA, Oct. 998. [7]. heker, Fudmet desig issues for the future iteret, IEEE Jour o eected Ares i Commuictios, vo., pp. 76-88, 99. [8] L. og d N. B. Mdym, Hierrchic IR d rte cotro o the forwrd ik for CDMA dt users uder dey d error costrits, IEEE Jour o eected Ares i Commuictios, 9():87 88, October.