On the Capacity-Performance Trade-off of Online Policy in Delayed Mobile Offloading

Transcription

1 On th Capacty-Prformanc Trad-off of Onln Polcy n Dlayd Mobl Offloadng Han Dng and I-Hong Hou Abstract WF offloadng, whr mobl usrs opportunstcally obtan data through WF rathr than cllular ntworks, s a promsng tchnqu to gratly mprov spctrum ffcncy and rduc cllular ntwork congston. W consdr a systm whr th srvc provdr dploys multpl WF hotspots to offload mobl traffc, and study th schdulng polcy to maxmz th amount of offloadd data. Snc usrs movmnts ar unprdctabl, w focus on onln schdulng polcy whr APs hav no knowldg about usrs moblty pattrns. W study prformanc of onln polcs by comparng aganst th optmal offln polcy. W prov that any workconsrvng polcy s abl to offload at last half as much data as th offln polcy, and thn propos an onln polcy such that whn th rqustd data by ach usr s vry larg, th polcy can offload ( )/ as much data as th offln polcy, whr s Eulr s constant. W furthr study th cas whr th srvc provdr can ncras th capacty of WF so as to provd som guarants on th amount of offloadd data. W drv a lowr-bound on th trad-off btwn capacty and th amount of offloadd data, and propos a smpl onln polcy that achvs ths lowr-bound. In addton, w show that our polcy only nds half as much capacty as currnt mchansms to provd th sam prformanc guarant. I. INTODUCTION Wth th ncrasng numbr of smart phon usrs subscrbng to 3G/4G ntworks, th mobl data traffc grows rapdly n rcnt yars. Th global mobl data traffc growth rat n 203 xcds8%, and s xpctd to grow at a 6% compound annual growth rat (CAG) from 203 to 208 []. Cllular ntworks fac th challng of srv ths grat ncras n data consumpton. Snc ntrfrnc btwn lnks s th major obstacl to dramatcally ncrasng th capacty of wrlss ntworks, many studs hav bn proposd to mgrat traffc from th hgh-powr and hgh-ntrfrnc macro bas statons to ntworks wth smallr transmsson powr and ntrfrnc, such as fmtoclls [2], WF [3], and moblto-mobl opportunstc ntworks [4]. Offloadng traffc through WF has bn shown to b an ffctv way to rduc th mobl traffc [5] [3]. WF s fastr and uss lss nrgy to transmt data whn thr s a conncton Han Dng s wth Dpartmnt of ECE, Txas A&M Unvrsty, Collg Staton, Txas, , USA. Emal: hdng@tamu.du I-Hong Hou s wth Dpartmnt of ECE, Txas A&M Unvrsty, Collg Staton, Txas, , USA. Emal: hou@tamu.du Ths matral s basd upon work supportd n part by th U. S. Army sarch Laboratory and th U. S. Army sarch Offc undr contract/grant numbr W9NF and NPP Grant of Qatar Natonal sarch Fund (a mmbr of Qatar Foundaton). [5]. Thus WF can sgnfcantly rduc th mobl traffc through macro bas statons n th nxt svral yars. For nstanc, 45% of th global mobl traffc s offloadd usng WF n 203, and th rat s stmatd to ras to 52% n 208 []. In ths papr, w study th problm of usng WF for dlayd mobl offloadng [6] [8]. In dlayd mobl offloadng, a larg amount of mobl usrs nd to obtan dlay-tolrant data,such as Dropbox synchronzaton and App updats, from srvc provdrs. Each mobl usr sts a dadln for ts data, and opportunstcally obtans ths data through WF whnvr t s connctd to WF accss ponts (APs), so as to rduc traffc on cllular ntworks. Du to ts own moblty pattrns, a mobl usr may only hav ntrmttnt WF connctons. If a usr fals to obtan all ts data by th dadln, t downloads th rmanng data through cllular ntworks. Whn multpl usrs ar connctd to on WF AP, th AP maks dcson on whch usr to srv. W am to dsgn schdulng polcs for WF APs that maxmz th amount of data offloadd to WF. W focus on WF offloadng bcaus t has bn xtnsvly dployd. Howvr, all our rsults can b drctly appld to othr mans of mobl offloadng, such as offloadng through fmto cll ntworks. W show that th problm of maxmzng th amount of offloadd data can b formulatd as a lnar programmng problm, and an offln polcy can solv t wth standard lnar programmng tchnqus. Howvr, such a formulaton rqurs th knowldg of moblty pattrns of all mobl usrs n advanc. Instad, w study th prformanc of onln schdulng polcs that mak schdulng dcsons only basd on systm hstory and th currnt locatons of usrs. Whn all APs us th sam transmsson rats for any connctd usrs, w show that any workconsrvng schdulng polcy s abl to offload at last 50% as much data as th optmal offln polcy. On th othr hand, whn APs may us dffrnt transmsson rats for dffrnt usrs basd on thr ndvdual channl qualts, w propos a smpl onln algorthm that guarants to dlvr at last as much data as th optmal offln polcy whn th rqustd data amount s larg, whr s Eulr s constanc. A fracton of of data offloadd to WF may not b suffcnt to rduc th congston. Hnc, w furthr nvstgat th cas whn wrlss srvc provdrs hav a hard rqurmnt on th amount of data offloadd to WF, so as to rduc cllular ntwork congston, and

2 2 thy ar wllng to ncras th capacty of WF to mt ths rqurmnt. W thn study th amount of capacty ndd to provd offload guarants for onln schdulng polcs. W propos a smpl onln schdulng polcy and prov that, n ordr to offload at last β as much data as th optmal offln polcy, our polcy nds to ncras th capacty by approxmatd. Th valu 2(β ) ofβ s chosn by th srvc provdr basd on ts rqurd offloadng guarants. On th othr hand, vn whn APs only us a fxd transmsson rat, th othr commonlyusd round-robn, max-wght, and proportonal far polcs nd to ncras th capacty by at last β to provd th sam guarant. In othr words, our polcy only nds half as much capacty to provd th sam prformanc guarant. W furthr prov that no polcy can guarant offloadng β data wth lss than 2(β ) capacty, and thrfor our polcy achvs th optmal trad-off btwn capacty and prformanc. Thortcal analyss only shows that th worst-cas prformanc of our polcs s bttr than that of th thr commonly-usd polcs. W furthr conduct smulatons to valuat th prformanc of schdulng polcs for a randomly gnratd systm. Smulaton rsults show that our polcs stll outprform th othr thr on avrag. Th fact that our polcy s sgnfcantly bttr than ths wdly usd polcs n WF schdulng, n trms of both thortcal bounds and smulaton rsults, furthr hghlghts that dlayd WF offloadng s fundamntally dffrnt problm from tradtonal WF schdulng. Th rst of th papr s organzd as follows: Scton II rvws som xstng work on WF offloadng. Scton III ntroducs our systm modl and problm formulaton. Scton IV ntroducs som basc lnar programmng proprts that ar vtal to ths work. Scton V studs th amount of offloadd data by onln schdulng polcs whn th APs hav unt capacty. Scton VI furthr studs th cas whn wrlss srvc provdrs can ncras th capacty of WF to provd prformanc guarants. Scton VII studs th prformancs of thr commonlyusd polcs. Scton VIII compars th compttv rato of our proposd polcs wth th thr commonly-usd polcs. Scton IX provds som practcal mplcaton of our polcy. Scton X provds th smulaton rsults. Fnally, Scton XI concluds ths papr. II. ELATED WOK Many xprmntal studs hav shown that mobl offloadng s promsng. Gass and Dot [9] compar WF and 3G ntwork through xprmnts and show that WF s abl to download mor data than 3G ntwork vn f though connctng tm s shortr. Balasubramanan t al. [0] study th avalablty of 3G and WF ntwork from movng cars n thr cts, and fnd that WF suffrs gratly from lmtd connctvtly. Thy thn propos a systm calld Wfflr to sgnfcantly mprov th amount of offloadd traffc. L t al. [3] study th WF offload prformanc through an xprmnt wth 00 Phon usrs n Soul, and obsrv that WF can upload about 65% of th traffc. Mota t al. [] study th WF hotspots avalablty durng bus routs n Pars, and show that currnt WF n Pars can offload up to 30% of mobl traffc. Addtonal work such as [2], [3] study th AP sd. Dmatto t al. [2] study how many APs ar rqurd to covr a mtropoltan ara offloadng. Trstan t al. [3] propos to upgrad th ntwork capacty n a slctd numbr of locatons, calld Drop Zon. Thy dsgn nfrastructur placmnt algorthm whch trs to rduc th AP numbr. It shows that by upgradng lss than 000 nfrastructurs across US wll upload 50% of data. Howvr, thr s stll no rsarch on upgradng th APs to guarant th offload data rato. Survys n [4], [5] provd som rsults on how much tm usrs ar wllng to wat for dffrnt applcatons. L t al. [6] analyz how much conomy bnft can b gnrat by dlayd offloadng and uss ral tracs for numrcal analyss. Mhmt and Spyropoulos [6] us a quung modl for dlayd mobl offloadng and analyz th man dlay as a functon of numbr of usrs and AP avalablty. Ca and tc. [7] propos an mchansm to ncourag usrs to partcpat n dlayd WF offloadng by rward. Thus th dlayd mobl offloadng problm s actually worth consdrng. An mportant challng for mobl offloadng s th unknown moblty pattrns of mobl usrs. Thr ar svral studs that focus on drvng modls for moblty pattrns [7] [9]. Chung and Huang [8] study th WF offloadng problm by formulatng th problm as a fnthorzon Markov dcson procss by usng th prdcton n [9]. L t al. [20] study usng a small st of mobl usrs to offload data, and propos a polcy basd on submodular optmzaton. Whtbck t al. [2] consdr usng offloadng to rduc th burdn n broadcastng mssags. Hou t al. [22] propos a transport layr protocol to ntgrat 3G and WF ntworks for vhcular ntwork accss. Barbr t al. [23] propos a systm dsgn for mobl offloadng wth pco bas statons. Bnns t al. [24] and Sngh t al. [25] consdr th problm of ntwork slf-organzng for offloadng traffc. Blgr Ytm and Martonos [26] propos offln schdulng polcs for WF offloadng. Ths studs assum that usr moblty follows som wll-dfnd random procss. In ral lf, usr moblty may b non-rgodc, and ths studs cannot b appld. In contrast, our work ams to maxmz th total offloadd data wthout any assumptons on usr moblty. III. SYSTEM MODEL W consdr a systm whr mobl usrs mov wthn th ara of a cllular ntwork. In ordr to rduc th congston of th cllular ntwork, th cllular oprator dploys a numbr of WF hotspots wthn th rgon. W us I to dnot th st of mobl usrs and M to dnot th st of WF APs. Mobl usrs may ntr th systm at dffrnt tms and at dffrnt locatons. Upon ntrng

3 3 th systm, a mobl usr spcfs th amount of data, dnotd by C, that t nds to obtan, and a dadln T. Th mobl usr movs around th systm and trs to obtan data from WF APs whnvr possbl. At tm T, th mobl usr downloads all th rmanng data from th cllular ntwork drctly. W assum that tm s slottd and numbrd as t =,2,... Th locaton of a mobl usr may chang from tm to tm and t dtrmns th connctvty and channl capacty btwn APs and tslf. Snc APs do not hav usrs locaton nformaton n advanc, channl condtons ar unprdctabl. Each AP maks schdulng dcsons basd on th past transmsson hstory and currnt channl condtons. W us K mt to dnot th channl capacty btwn AP m and mobl usr at tm t. If cannot b connctd to m at tm t, w hav K mt = 0. Thr hav bn som advancmnts n mult-homng, whr a mobl usr can b connctd to multpl APs smultanously. Our modl can asly accommodat mult-homng by allowng a usr to hav K mt > 0 for multpl APs. W assum ach usr can b connctd to at most H APs at any gvn tm. Also, snc cannot download any data pror to ts ntranc, and t wll us th cllular ntwork to download data aftr ts dadln, w st K mt = 0 for all t pror to s ntranc or aftr ts dadln. W normalz th systm so that 0 K mt H for all,m,t. Thrfor, at ach tm t, ach clnt can at most obtan on unt of data. Onc clnt s connctd to AP m at th bgnnng of tm slot t, th tm slot s fully occupd by clnt, rgardlss whthr or not AP m s transmttng data to clnt. AP m mploys som schdulng polcy to dtrmn th porton of tm t spnds transmttng to mobl usr durng tm slot t, dnotd by X mt. Th amount of data that mobl usr obtans from AP m at tm t s thn K mt X mt. Our goal s to dsgn a schdulng polcy that maxmzs th total amount of data that ar dlvrd through WF, whch, n turn, mnmzs th amount of data through th congstd cllular ntwork. Snc ach mobl usr nds to obtan C data, w formulat th followng lnar programmng problm: Offload: Max mtx mt K mt () s.t. mt X mt K mt C, I, (2) X mt, m M,t, (3) X mt 0, I,m M,t. (4) W us Γ opt to dnot th optmal valu of mt X mtk mt n th abov problm. Whl ths problm can b solvd by standard lnar programmng tchnqus, dong so rqurs th knowldg of th ntranc tms and locatons of all mobl usrs at tm 0, whch s mpractcal. Instad, w am to drv onln polcs that choos th valus of X mt solly basd on systm hstory up to tm t. W us η to dnot an onln polcy. W lt Γ η () b th valu of mt X mtk mt undr polcy η, gvn K mt and C. W assum that th cllular oprator may b abl to ncras th capacty of WF hotspots by, for xampl, upgradng APs or obtanng mor spctrum. Whn th capacty of WF hotspots s ncrasd by, th channl capacty btwn and m at tm t bcoms K mt. Equvalntly, w can also dscrb th systm as on wth channl capacty K mt, but th AP can spnd an amount of tm transmttng to clnts n ach slot, that s, X mt. Thrfor, w consdr th followng lnar programmng problm whn th capacty s ncrasd by : Offload(): Max mt X mt K mt (5) s.t. mt X mt K mt C, I, (6) X mt, m M,t, (7) X mt 0, I,m M,t. (8) Lt Γ η () b th valu of mt X mtk mt for th Offload() problm undr polcy η. W valuat th prformanc of η by ts compttv rato, whch s dfnd slghtly dffrntly from most xstng ltratur. Dfnton : A polcy η s sad to b (,β)-compttv f Γ opt /Γ η () β, as mn I C, for all systms. W not that whn =, th corrspondng β bcoms th compttv rato commonly dfnd n xstng ltratur. Our dfnton s rchr n that t charactrzs th amount of capacty ndd to provd prformanc guarants. Snc th vry rason of usng WF offloadng s that th cllular ntwork s congstd, th oprator may hav a hard rqurmnt on th amount of data bng offloadd through WF, and t s wllng to purchas bttr qupmnts and mor spctrum to achv ths rqurmnt. In ths cas, t nds to know how much capacty s ndd. Suppos th optmal offln polcy can offload all data through WF, and th oprator rqurs a porton /β of th data to b offloadd, our dfnton thn rvals that th capacty nds to b ncrasd by so that th mployd polcy s (, β)-compttv. W summarz th notatons w usd n th papr n Tabl I. IV. PELIMINAY Ths scton ntroducs som basc thorms that wll b usd n ths papr. A standard form of lnar programmng problm (LP)

4 4 I M m t K mt X mt H C T Y mt, Z η Γ opt Γ η() Γ η() β s: and ts dual s TABLE I NOTATIONS Mobl usr st Sngl usr WF AP st Sngl AP Tm slot Normalzd channl capacty btwn AP m and usr at tm t Porton of tm AP m spnds on usr at tm slot t Th maxmum numbr of APs that a usr can connct to Data amount that usr nds Tm that usr swtch from WF to cllular ntwork Dual varabls Polcy Capacty ncrasng rat Optmal valu of () n Offload Valu of () n Offload undr polcy η Valu of (5) n Offload() undr polcy η Valu of max(γ opt/γ η()) (P) : Max s.t. (D) : Mn s.t. n c x, = n a j x b j, n, = x 0, m b j y j, j= m a j y j c, j m, j= y j 0. W hav th followng two fundamntal thorms: Thorm (Wak Dualty [27]): Lt {x } n and {y j } m satsfy th constrants of th prmal (P) and th dual (D) LPs, rspctvly, thn: n c x = m b j y j. j= Thorm 2 (Complmntary Slacknss [27]): Lt {x } n and {y } m satsfy th constrants of th prmal (P) and dual (D) LPs, rspctvly. Furthr, {x } and {y } hav th followng proprts: If x > 0, thn c m j= a jy β c for som β > ; If y > 0, thn n = a jx = b j ; Thn: m n b j y j β c x. j= = V. COMPETITIVE ATIO WITH UNIT CAPACITY In ths scton, w dscuss th spcal cas wth =. W frst show that whn K mt s thr 0 or, any workconsrvng polcy s (, 2)-compttv. W thn study th cas whn K mt can b any ral numbr n [0,]. W propos a smpl onln schdulng polcy and prov that t s (, )-compttv. A. Prformanc of Work-Consrvng Polcy undr On-Off Channls W frst consdr th cas whr K mt s thr 0 or, whch s usually rfrrd as On-Off channls, and w say that clnt s connctd to AP m at tm t f K mt =. W study th prformanc of work-consrvng schdulng polcy, undr whch ach AP m slcts to srv on connctd clnt that has yt to rcv all th data, as long as thr s on, and only dls whn all connctd clnts hav alrady rcvd all thr data. Thorm 3: Any work-consrvng polcy s (, 2)- compttv wth ON-Off channls. Proof: Th offload problm s shown as () to (4), and ts dual s (D) : Mn Y mt + C Z, mt (9) s.t. Y mt +K mt Z K mt,,m,t, (0) Y mt 0, m,t, () Z 0,, (2) whr Y mt s th dual varabl for ach constrant n (2), and Z s th dual varabl for ach constrant n (3). W st X mt = f clnt s srvd by AP m at tm slot t, and X mt = 0 othrws. W st Y mt = f AP m schduls a clnt at tm t, and Y mt = 0 f m dls at t. W st Z = f clnt hav rcvd all ts data bfor ts dadln, and Z = 0 othrws. W wll us Thorm 2 to stablsh th thorm. Frst, w show that X mt, Y mt, and Z satsfy th constrants (2) (3), and (0). (2) and (3) ar satsfd bcaus ach AP schduls at most on clnt at any tm, and t nvr schduls clnts that hav alrady rcvd all thr data. Gvn,m,t, f K mt = 0, thn (0) s satsfd snc Y mt and Z ar non-ngatv. (0) also holds f K mt = and Y mt =. Fnally, f K mt = and Y mt = 0,.., AP m dos not schdul any clnt at tm t, thn all clnts connctd to AP m at tm t must hav alrady rcvd all thr data. Hnc, Z = and (0) stll holds. Nxt, w vrfy th complmntary slacknss condtons. If Z > 0, thn clnt obtans all ts data, and mt X mtk mt = C. If Y mt > 0, thn AP m schduls som clnt at tm t, and X mt =. In addton, f X mt > 0, thn K mt =. Thus, 2K mt = 2 Y mt + K mt Z K mt. By Thorm 2, w know mt Y mt+ C Z 2 mt X mtk mt. Furthr, Γ opt mt Y mt + C Z, by Thorm, and hnc any workconsrvng polcy s (, 2)-compttv.

5 5 B. Onln Algorthm for Gnral Channls W now dscuss th gnral cas n whch K mt can b any ral numbr btwn 0 and. W propos an onln schdulng algorthm and prov that t s (, )- compttv. In our algorthm, APs kp track of and updat a varabl Z for ach clnt. Z s ntally st to 0. If ach tm t, ach AP m chooss to srv th clnt that maxmzs K mt ( Z ), and dlvrs K mt data to th clnt. Each tm clnt obtans K mt data from an AP m, Z wll b updatd as Z ( + K mt ) + K mt. At C (d )C tm t, f thr ar multpl APs that srv at th sam tm t, whch s possbl undr mult-homng, Z wll b m:m srvs updatd as Z (+ K mt m:m srvs )+ K mt C (d )C. Hr d s a valu only usd n calculaton and t s st to b (+/C mn ) Cmn. W show th valu chosn for d s rasonabl n proof of Lmma. AP m broadcasts th updatd Z to all APs. Algorthm formally dscrbs th algorthm. In Algorthm, w also ntroduc two othr varabls, X mt and Y mt. Ths two varabls ar only usd to stablsh th compttv rato, and ar not ndd n actual mplmntatons. Algorthm : Intally, X mt = 0, Y mt = 0, Z = 0. 2: C mn mn C,d (+/C mn ) Cmn. 3: for ach tm slot t do 4: for ach AP m do 5: m argmax{ m:m srvs K mt( Z )}. 6: f K m mt( Z m ) > 0 thn 7: Y mt K m mt( Z m ). 8: X m mt. 9: nd f 0: nd for : for ach clnt do 2: Z Z ( + m:m srvs K pt. (d )C 3: f p,s t X psk ps > C thn 4: X mt C p,s<t X psk ps 5: nd f 6: nd for 7: nd for p:p srvs K pt m:m srvs K pt C ) +, m srvrs In Algorthm, ach of Y mt and X mt s only updatd at tm slot t, whl Z may b updatd n many dffrnt tm slots. W not that th valu of Z s non-dcrasng n ach updat. At tm t, t s possbl that alrady obtans most of ts data and only nds lss than m:m srvs usr K mt data to complt ts download. In ths cas, APs us only a fracton of a tm slot to dlvr all rmanng data that nds. Stp 4 addrsss ths cas, and th total amount of offloadd data s mt X mtk mt. Lmma : Lt Z (t) b th valu of Z at th nd of tm slot t. Thn, ( d )(d Z (t) m,s t X mt K mt C ). (3) Proof: W prov (3) by nducton on t. Whn t = 0, Z (t) = 0 = ( d )(d0 ), and (3) holds. Suppos (3) holds for all tm bfor s. Consdr tm t = s+. If s not schduld at s+, X mt = 0 for all m at t = s+ and Z (s+) = Z (s). Hnc (3) holds. On th othr hand, f s schduld by AP p at tm s+, Z (s+) =Z (s) (+ (d ) (d + p:p srvs K p(s+) p:p srvs )+ K p(s+) C (d )C m,t s X mt K mt C p:p srvs K p(s+) (d )C = (d ) [d X mt K mt m,t s C (+ )(+ p:p srvs K p(s+) C ) p:p srvs K p(s+) ) ] C It s asy to vrfy that ln(+x)/x s dcrasng whn x [0,]. Thus ( + y) ( + x) (y/x) for x y. Lt p:p srvs y = K p(s+) and x =. W thn hav C C mn Z ((s+) [(d X mt K mt m,t s C )(+ C mn ) (d ) p:p srvs K p(s+)c mn C ] call that th valu of d s (+/C mn ) Cmn. Thus Z (s+) (d ) (d X mt K mt m,t s+ C ), and (3) holds. By nducton, (3) holds for all t. Thorm 4: Algorthm s (, )-compttv. Proof: Th offload problm and and ts dual ar statd as () to (4), and (9) to (2), rspctvly. W prov Algorthm s (, )-compttv by th followng stps: Frst, w show that th dual solutons {Y mt } and {Z } satsfy constrants (0) to (2). Snc m argmax{k mt ( Z )}, w hav: K m mt( Z m ) K mt ( Z ),,m,t. Furthr, by stp 7 n Algorthm, w hav: Y mt +K mt Z K mt K m mt( Z m )+K mt Z K mt K mt ( Z )+K mt Z K mt = 0. Thus (0) s satsfd. It s asy to chck that Y mt and Z ar non-ngatv, and () and (2) hold.

6 6 Scond, w show that X mt satsfy constrants (2) to (4). Stp 4 nsurs that (2) holds. By Lmma, Z (t) < only whn dos not rcv all ts data at tm t. Hnc, X mt s updatd only f th total rcvd data of clnt s lss than ts C, whch s p,s<t X psk ps < C, whch maks (3) and (4) hold. Thrd, w show that vry tm stps 7, 8, and 2 ar nvokd, th rato btwn th chang of th dual objctv functon (9) and chang of th prmal objctv d functon () s d. W not that w gnor th chang of () by stp 4 now, whch wll b takn nto account latr. Whn AP m schduls at tm t, X mt s ncrasd from 0 to, and () s ncrasd by m:m srvs K mt. Manwhl, (9) s ncrasd by m:m srvs + K mt ( Z (t ) )+C (Z (t ) m:m srvs K mt C m:m srvs K mt ) (d )C =(+ d ) m:m srvs K mjt. Thus th rato btwn chang of (9) and () s + d = d d. LtΓ opt b th optmal valu of (), Γ dual,η b th valu of (9) undr Algorthm, and Γ prm,η b th valu of () undr Algorthm wthout stp 4. W hav stablshd that Γ opt Γ dual,η = d d Γ prm,η, whr Γ opt Γ dual,η bcaus of Thorm. Fnally, w addrss th nflunc of stp 4. Stp 4 s only nvokd whn m obtans all ts data,.., p,s X m ps K m ps = C m. By Lmma, stp 4 s nvokd at most for on tm slot for ach clnt. Furthr, whn stp 4 s nvokd, () dcrass by no mor than m:m srvs K mt. Lt Γ prm,η b th valu of () undr Algorthm wth stp 4. W now hav Γ prm,η Γ prm,η ( d C mn ) Γ opt d ( C mn ). Snc d, as C mn, Algorthm s (, )-compttv. VI. COMPETITIVE ATIO FO VAIABLE CAPACITY In th prvous scton, w obtan a (, )- compttv onln algorthm. Thus, Algorthm guarants to offload 63% as much data as an optmal offln algorthm dos. Howvr, ths also ndcats that, whn th optmal offln algorthm offloads all data, our algorthm may mss almost 37% of th data. Thn, how much capacty s ndd to guarant offloadng, say, 95% of th data? W wll focus on ths problm n ths scton. It s frst of ntrsts to study whthr t s fasbl to ncras th capacty by tms so as to guarant an onn algorthm can always offload as much data as an optmal offln algorthm wth unt capacty dos. Or, wth our trmnology, to study whthr thr xsts a (, )-compttv polcy. Th followng xampl shows that (, )-compttv polcy dos not xst, for any. Exampl : Fx. Consdr a systm consstng of N = + clnts and on AP wth On-Off channls. Each clnt has a fl sz of C. On of th clnts, say, clnt, ntrs th AP covrag ara at tm and lavs at tm C, whl all othr clnts ntr th AP covrag ara at tm and stay forvr. W hav T = for all clnts. Th optmal offln polcy schduls clnt n tms t C, and thn schduls othr clnts aftr t = C. Hnc, th optmal offln polcy offloads all data. On th othr hand, snc onln polcs do not know whch clnt s connctd to th AP only n tms t C, and thy can at most offload C < NC data n tms t C, thy cannot guarant to offload all data. Whn th capacty s ncrasd by, th corrspondng offload problm s dscrbd n (5) (8). In ths scton, w frst propos two onln polcs and study thr compttv ratos. W thn drv a thortcal lowr-bound for th compttv rato of all onln polcs. Fnally, w study th compttv rato of th round robn polcy. A. Prmal-Dual Schdulng Polcy Th prmal-dual (PD) schdulng polcy s vry smlar to Algorthm. It s dscrbd as Algorthm 2. Th only dffrncs ar that w choos d = ( + /C mn ) Cmn/, and w assgn X mt = f s schduld by m at tm t. Algorthm 2 Prmal-Dual Algorthm : Intally, X mt = 0, Y mt = 0, Z = 0. 2: C mn mn C,d (+/C mn ) Cmn/. 3: for ach tm slot t do 4: for ach AP m do 5: m argmax{k mt ( Z )}. 6: f K m mt( Z m ) > 0 thn 7: Y mt K m mt( Z m ). 8: X m mt. 9: nd f 0: nd for : for ach clnt do 2: Z Z ( + m:m srvs K mt. (d )C 3: f p,s t X psk ps > C thn m:m srvs K mt C ) + 4: X mt C p,s<t X psk ps. m : m srvs 5: nd f 6: nd for 7: nd for p:p srvs K mt W now study th compttv rato of PD. Lmma 2: Lt Z (t) b th valu of Z at tm slot t. W hav ( d )(d Z (t) m,s t X mt K mt C ). (4)

7 7 Proof: W prov (4) by nducton on t. Th proof s smlar to that of Lmma. / Thorm 5: PD s (, )-compttv. It s [ / ] approxmatly (,+ 2 )-compttv. / Proof: W prov PD s (, [ / ] )-compttv by th followng stps: Frst, {Y mt } and {Z } satsfy constrants (0) (2). Th proof s th sam as th proof for Thorm 4. Scond, w show that X mt satsfy constrants (6) (8). Stp 4 nsurs (6). Furthr, by Lmma 2, Z (t) < only f m,s t X mtk mt < C. Thrfor, a clnt s only schduld whn t s yt to rcv all ts data, whch nsurs (7) and (8). Thrd, w show that whnvr stps 7, 8, and 2 ar nvokd, th rato btwn th chang of (5) and th d (d ) chang of (9) s. W gnor th chang of (5) du to stp 4 now. Suppos clnt s schduld by AP m at tm t. W hav m:m srvs X mt =, and (5) s ncrasd by m:m srvs K mt. On th othr hand, (9) s ncrasd by m:m srvs + K mt ( Z (t ) )+C (Z (t ) m:m srvs K mt C m:m srvs K mt ) (d )C =(+ d ) m:m srvs K mt. Thus th rato btwn th chang of objctv functons (5) and (9) s (+ d )/ = d (d ). Fnally, w consdr th nflunc of stp 4. Stp 4 s only nvokd whn m obtans all ts data,.., p,s X m ps K m ps = C m. By Lmma 2, stp 4 s nvokd at most onc for ach clnt. Furthr, whn stp 4 s nvokd, (5) dcrass by no mor than m:m srvs K m m mt, snc w normalzd our systm such that K mt H. Thrfor, throughout th systm lftm, th rato of dcras causd by stp 4 s no mor than C mn. As C mn, d /. By Thorm and th abov / argumnts, w stablsh that PD s (, [ / ] )- compttv. W can approxmat follows: [ ] =+ + ( + / [ / ] 2! ! ) = + + 2! ! = by Taylor srs as + 2, orgnal approxmaton Fg.. Illustraton of th approxmaton. whn. Thus, th compttv rato s approxmatly(,+ / 2 ). Fg. plots and + [ / 2. As can b sn n th ] fgur, + 2 s a vry accurat approxmaton vn for small. B. Last Progrss Frst Schdulng Polcy Whn mplmntng PD, APs nd to kp track of an artfcal varabl Z and updat th valu of Z wth ach othr. Furthr, PD nds to know th valu of C mn to st d. Blow, w dscrb an approxmaton of PD that s smplr and dos not rqur any nformaton xchang. Usng an argumnt smlar to th proof of Lmma 2, w approxmat Z (t) by ( d )(d X mt K mt m,s t C /, and x by +x, for all 0 < x <. Wth ths approxmatons, w hav K mt ( Z ) K mt [ ( d )(d K mt d (/ ). W furthr approxmat d by (d ) K mt( m,s<t X ms K ms C X ms K ms m,s<t C ) )] m,s<t X msk ms C ). Snc PD maks AP m schdul th clnt wth th largst K mt ( Z ), w can approxmat PD by makng ach AP m schdul th clnt wth th largst m,s<t K mt ( XmsKms ). Furthr, w not that ( m,s<t XmsKms C C ) s th porton of data that s yt to obtan. Thrfor, ths polcy smply schduls th clnt wth th largst product of channl capacty and porton of undlvrd data, both valus ar radly avalabl at APs. Th polcy s summarzd n Algorthm 3 and s calld Last Progrss Frst (LPF). It can b asly mplmntd n a fully dstrbutd fashon. Fnally, w not that whn all clnts nd to obtan th sam amount of data,.. C C, thn LPF bcoms th sam as th wll-known

8 8 Max-Wght schdulng polcy. Howvr, as w wll show n Scton VII-B, Max-Wght polcy can b much wors than LPF whn dffrnt clnts hav dffrnt C. Algorthm 3 Last Progrss Frst : for ach tm slot t and ach AP m do 2: m argmax{k Amount of undlvrd data mt C }. 3: AP m transmts to clnt m at tm t. 4: nd for C. A Lowr Bound on Compttv ato In th prvous scton, w show that th compttv rato of our onln schdulng polcy s approxmatly (,+ 2 ). In ths scton, w ar ntrstd n th bst compttv rato that onln polcs can achv. Thorm 6: For any gvn ǫ > 0, thr xsts a postv numbr 0, such that, whn > 0, no polcy has bttr compttv rato than (,+ 2 ǫ ). Proof: Consdr a systm consstng of N clnts wth sam fl szc and on AP wth On-Off channls. Assum thr ar k clnts that wll mov around th ara and N k clnts stay n th sam plac. For clnt, f k, K t = n tm t C, n othr words, clnt s connctd to AP n tm [,C]; clnt 2 s connctd to AP n tm [,2C]; clnt 3 s connctd to AP n tm [,3C]; tc. If > k, clnt s connctd to th AP forvr. T = for all clnts. If th conncton of all clnts ar known n advanc, th optmal offln polcy wth unt capacty s to schdul th clnts n th ordr of,2,3,...,n. Th optmal offln polcy s thn abl to transmt NC amount of data. On th othr hand, onln polcs cannot know K mt n advanc. Also, all clnts hav th sam fl sz. Thrfor, whn th systm capacty s ncrasd by tms, th bst that th AP can do s to vnly dstrbut ts capacty among all connctd clnts. Th AP dlvrs a total C amount of data n th frst C tm slots, and clnt rcvs C N amount of data. In th frst 2C tm slots, clnt 2 rcvs C N + C N amount of data... In th frst k C tm slots, clnt k rcvs C N + C C N N k+ amount of data. Th othr clnts rcv all thr data. Thus, th AP can at bst dlvr Γ η () = { C C N k+ N }+{C N }+...+{C }+(N k)c amount of data. N + C N + C N Lt N = (k +). Snc N < N <... < compttv rato s: β = NC Γ η () > = > N N k+ (+k)k 2 +(N k) (k +) (k+) k+ (+k)k (k+) (k+) (+k)k 2 +((k +) k) (k +) 2 +((k +) k) k 2k = + 2(k +)( ) 2k( )+k k 2k > + 2(k +) 2 2k+k > + k 2k k 2k/ = + 2(k +)2 2(k +) = + 2 ( 2(k +) + k (k +) ) N k+, th For suffcntly larg k and, w hav 2(k+) < ǫ/2, k (k+) < ǫ/2, and thrfor β > + 2 ǫ. Ths complts th proof. Snc PD s approxmatly (,+ 2 )-compttv, Thorm 6 dmonstrats that PD ndd achvs th optmal trad-off btwn capacty and th amount of offloadd data. VII. COMPTITIVE ATIOS OF OTHE POLICIES In Scton V-A, w hav shown that th compttv rato of any work-consrvng polcy s at last (,2) wth On-Off channls. In comparson, th compttv rato of Algorthm s (, ) (,.58). It appars that th compttv rato of any work-consrvng polcy s clos to that of Algorthm. W now study th compttv rato of work-consrvng polcs whn th capacty s ncrasd by tms. In partcular, w stablsh a lowr-bound on compttv rato for th followng thr polcs. A. ound obn Schdulng Polcy Wth round robn polcy (), ach AP vnly dstrbuts ts capacty among all connctd clnts. Thorm 7: ound robn schdulng polcy cannot hav bttr compttv rato than (,+ ). Proof: Gvn, w construct a systm wth on AP and N clnts as follows: C = NC, and C = C, for all. K t = for t C, and K t = 0 for t > C. For, K t = for all t. In othr words, clnt s connctd to AP n tm [,C ], whl all othr clnts ar connctd to th AP forvr. T = for all clnts. Th systm s shown s Fg. 2. Th optmal offln polcy s frst to srv clnt from tm to NC. Thn clnt 2 to N wll b srvd. Thus th optmal offln polcy s abl to dlvr all C + N =2 C amount of data wth unt capacty, whl round robn

9 9 β = MC +C +NC MC +C = + N M + + as N. C=NC, T=[, NC] C2=C, T2=[, nf] C3=C, T3=[, nf] Fg. 2. Systm Constructon for ound obn.... CN=C, TN=[, nf] transmts C =2 C amount of data wth capacty. Th compttv rato of round robn s thn at last: as N. N + N β = C + N =2 C = C N + N =2 C N +(N ) +(N ) +, B. Max-Wght Polcy = N +(N ) +(N ) Wth max-wght (MW) polcy, of all connctd clnts, ach AP slcts th on wth maxmum valu of th product of K mt and th clnt s rmanng data. Thorm 8: Th max-wght polcy cannot hav bttr compttv rato than (,+ ). Proof: Gvn, w construct a systm wth on AP and N + clnts as follows: C = MC+C, and C = C, for all. K t = for all t, K t = for t NC. In othr words, clnt s connctd to AP forvr, whl all othr clnts ar connctd to AP n tm [,NC]. Hr w choos N = M/, and T = for all clnts. Th optmal offln polcy s frst to offload data rqustd by clnt 2 to N+ n tm [,NC]. Thn th polcy wll offload data rqustd by clnt. In ths cas, th polcy s abl to dlvr all C + N =2 C = MC+C+NC amount of data wth unt capacty. Wth capacty, MW polcy frst transmts MC data to clnt untl t has rmanng data rqust of C, whch s th sam as th othr clnts. It taks NC amount of tm to transmt MC amount of data. Snc clnt,, has a dadln of NC, thy wll no longr gt data through WF. Thn WF wll transmt th rmanng C data to clnt. Thus th total amount of data transmttd s MC + C. Th compttv rato of MW polcy cannot b bttr than: C. Proportonal Far Schdulng Polcy Wth proportonal far (PF) polcy, of all connctd clnts, ach AP slcts th on wth th maxmum valu K of mt Throughput of clnt. Thorm 9: Th proportonal far polcy cannot hav bttr compttv rato than (,+ ). Proof: Gvn, w construct a systm wth on AP and N + clnts as follows: C = MC, and C = C, for all. K t = for t MC, K t = for all t. In othr words, clnt s connctd to AP n tm [,MC], whl all othr clnts ar connctd to AP forvr. Hr w choos N = M, and T = for all clnts.. Th optmal offln polcy s frst to srv clnt n tm to MC. Aftr tm NC, th othr clnts wll b srvd. Thus th optmal offln polcy s abl to dlvr all C + N =2 C = MC +NC amount of data wth unt capacty. Wth capacty, from tm [,MC], PF polcy dlvrs MC N+ data for ach clnt. Aftr tm MC, all clnts xcpt clnt ar connctd and PF polcy wll dlvr th rmanng data of thm. Thus PF polcy dlvrs a total amount of MC N+ + NC data. Th compttv rato of PF polcy cannot b bttr than: as N. β = MC +NC MC N+ +NC = + N +N N N+ +N VIII. PEFOMANCE COMPAISON For a larg, / ( / ) s vry clos to +, and t may sm that th compttv rato of, MW, and PF s clos to that of PD. Howvr, w should ntrprt our rsults on compttv ratos as follows: Suppos t s rqurd that onln polcs nd to offload at last β as much data as th optmal offln polcy wth unt capacty, for som gvn β >. Wth PD, th systm nds to ncras ts capacty by approxmatly 2(β ) tms. On th othr hand, vn wth th smpl On-Off channls, th, MW, and PF polcs stll nd at last β capacty to achv th rqurmnt. In othr words,pd nds about half as much capacty to provd th sam guarant as th othr thr polcs. Thus, our polcy s much mor prfrabl to provd strngnt prformanc guarants. Fg 3 llustrats th capacty rqurmnts for dffrnt β. In ordr to guarant to offload at last 95% as much

10 0 qurd capacty PD, MW, PF lowr bound /β Fg. 3. Capacty rqurmnts of dffrnt polcs. data as th optmal offln polcy, PD nds to ncras th capacty by 9.7 tms, whl th othr polcs nds to ncras th capacty by at last 9 tms. IX. PACTICAL IMPLICATION In ths scton, w wll dscuss th major practcal mplcatons of our work. Frst, our work can b usd for systm plannng. Whn a wrlss oprator s to dploy publc WF APs, t can us th macro-scal statstcs of systm hstory to stmat th amount of rsourc ndd for th offln polcy to offload a crtan amount of data. For xampl, suppos w hav collctd th moblty pattrns of all clnts n a spcfc ara and th clnts dmand of th past 30 days. W us K mt as clnts conncton status to ndcat th moblty pattrn. W us C to rprsnt th dmand. Suppos th wrlss oprator nds to offload a total of C 0 amount of data. Hr C 0 C. Wth th abov nformaton, th wrlss oprator can thn drv th mnmum rqurd rsourc 0 by smply solvng th followng lnar programmng problm. Mn 0 s.t. mtx mt K mt C 0, I, m M,t, X mt K mt C, I, mt X mt 0, m M,t, X mt 0, I,m M,t. Assumng th statstcs of usr moblty and dmand do not chang too much, th wrlss oprator s abl to stmat th avrag total rsourc rqurmnt wth th past nformaton. Howvr, ths macro-scal statstcs bcom much lss usful whn t coms to onln packt schdulng. For xampl, whn two clnts ntr a coff shop wth WF at th sam tm, t s vry dffcult to prdct whch of thm wll lav th coff shop frst. In ths cas, t s not guarantd to offload C 0 amount of data wth th gvn 0 rsourc. Our drvaton of prformanc bound s ndd basd on ths dffculty. On th othr hand, t s possbl to formulat th schdulng problm as a Markov dcson procss (MDP) problm usng past statstcs [8]. Howvr, ths formulaton typcally rqurs ach AP to solv a hgh-dmnsonal optmzaton problm for vry packt transmsson, and ncurs prvntvly hgh computaton complxty. Thrfor, n many practcal scnaros, smpl onln polcs that do not rly on past statstcs ar ndd. Now, lts say that th wrlss oprator stmats that th offln polcy nds an amount of 0 rsourc to offload th dsrabl amount C 0 of traffc. How much rsourc dos a smpl onln polcy nd to guarant offloadng 0.95C 0 traffc? Thorm 5 rvals that th answr s Th scond mportant mplcaton concrns th comparson aganst othr poplar polcs ncludng round robn, max-wght, and proportonal far polcs. Our study rvals that our proposd polcy only nds 50% as much rsourc as th othr thr to provd th sam dgr of prformanc guarants. In othr words, n vw of compttv rato, mplmntng our polcy s as ffctv as doublng capacty. It s wll known that 2X2 MIMO has th potntal of doublng transmsson rat. Thrfor, our polcy offrs th sam prformanc mprovmnt as mplmntng 2X2 MIMO. X. SIMULATION In ths scton, w valuat th prformanc of th two algorthms w proposd as wll as that of th othr thr polcs dscussd n Scton VIII. W construct a systm wth 9 APs (3 by 3 grd). Each AP s 000 mtrs away from ts narst nghbor, and has a transmsson rang of 400 mtrs. Thr ar 200 clnts whch ar dvdd nto two groups: Th frst group s 00 statonary clnts that ar unformly dstrbutd wthn th covrag ara of APs. Th scond group s 00 mobl usrs whos locatons ar chosn unformly at random at ach tm t. In ach group, th -th clnt has C = 00, and T = , f 95, and C = 0,000, T = 5000( 95) f 95 < 00. Fg. 4 shows a snapshot of th locatons of all clnts, whr th crcls rprsnt th covrag ara of APs. Th channl gan s dtrmnd by both pathloss and aylgh fadng. Th pathloss factor btwn an AP and a clnt s computd by mn{,/(dstanc/80) 2 }. Th aylgh fadng factor s computd as a 2 +b 2, whr both a and b ar Normal random varabls wth man 0 and varanc. Fnally, th channl gan s th product of ths two factors. W consdr both On-Off channls and gnral channls. Wth On-Off channls, w consdr th channl to

11 500 AP Statonary Clnts Mobl Clnts MW PF PD LPF Offload ato Fg. 4. Locaton of APs and clnts. Offload ato MW PF PD LPF AP Capacty Fg. 5. Prformanc comparson for On-Off channls AP Capacty Fg. 6. Prformanc comparson for gnral channls. Furthr, w not that our polcs outprform th othr thr polcs n both scnaros. Th thortcal analyss n Scton VI only provs that th worst-cas prformanc of our polcs s bttr than that of th othr polcs. Ths smulaton rsults furthr suggst that our polcs ar stll mor prfrabl on avrag. Wth On-Off channls, w notc that th data offload wth our polcs at ar no lss than that offload at 2. For xampl, whn = 2, our polcs ar abl to offload mor data than whn = 4. Th dffrnc btwn our polcs and th polcy bcoms vn largr whn gnral channls ar consdrd. Ths s bcaus th round robn polcy dos not consdr channl capacty, and wll us a larg amount of tm srvng clnts wth poor channl qualts. For both On-Off channls and gnral channls, our polcs outprform Max- Wght polcy. Proportonal far polcy dos not hav a good prformanc bcaus t trs to vn out ach clnt s porton of rcvd data. Whn clnts hav dffrnt data rqust and moblty pattrn, th polcy dos not optmz th whol systm s offload data amount. b ON f th channl gan s largr than /25, n whch cas w st K mt =. Th thrshold of /25 s chosn as th pathloss factor whn th dstanc s 400. Wth gnral channls, w st K mt to b th channl gan. For ach smulaton run, w comput th porton of data that ach polcy s abl to offload through WF. All smulaton rsults ar th avrag of 5 smulaton runs. Th smulaton rsults for both channls ar shown n Fgur 5 and Fgur 6, rspctvly. Th standard dvatons of our algorthms ar on th ordr of 0 4, whch shows that th dvaton of our algorthm s vry small. W notc that PD and LPF hav almost dntcal prformanc. call that LPF s dsgnd to b an approxmaton to PD wth smallr ovrhad and asr mplmntaton. Ths smulaton rsults confrm that t s ndd an accurat approxmaton. XI. CONCLUSION In ths papr, w study th dlayd mobl offloadng problm wth unprdctabl usr movmnt pattrn. W am to download as much data through WF as possbl. W prsnt two onln algorthms for th problm and study thr prformanc by comparng how much data thy ar abl to offload to th optmal offln polcy. Frst, w propos PD polcy and prov that t s approxmatly (, 2 )-compttv and achvs th optmal trad-off btwn capacty and amount of offloadd data. Scond, w propos an altrnatv LPF polcy that s asr to mplmnt and has almost dntcal prformanc as PD. W also provd that th tght bound of onln polcs s (, 2 ). Our polcs ar compard wth thr commonlyusd polcs, ncludng ound obn, Max-Wght, and Proportonal Far polcy, and and w prov that our polcs only nd half as much capacty to provd

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Martonos, Adaptv usag of cllular and WF bandwdth: an optmal schdulng formulaton, n Procdngs of th svnth ACM ntrnatonal workshop on Challngd ntworks. ACM, 202, pp [27] N. Buchbndr and J. (Sff) Naor, Th dsgn of compttv onln algorthms va a prmal: Dual approach, Found. Trnds Thor. Comput. Sc., vol. 3, no. 2R3, pp , Fb [Onln]. Avalabl: Han Dng rcvd hr M.S. dgr n Elctrcal and Computr Engnrng from Oakland Unvrsty, MI, USA n 202 and hr B.S. n Informaton Engnrng n Bjng Insttut of Tchnology, Bjng, Chna n Sh s now a PhD studnt at Txas A&M Unvrsty, USA. Hr rsarch ntrsts ar n wrlss and wrd ntworks and optmzaton. I-Hong Hou (S0-M2) rcvd th B.S. n Elctrcal Engnrng from Natonal Tawan Unvrsty n 2004, and hs M.S. and Ph.D. n Computr Scnc from Unvrsty of Illnos, Urbana-Champagn n 2008 and 20, rspctvly. In 202, h jond th dpartmnt of Elctrcal and Computr Engnrng at th Txas A&M Unvrsty, whr h s currntly an assstant profssor. Hs rsarch ntrsts nclud wrlss ntworks, wrlss snsor ntworks, ral-tm systms, dstrbutd systms, and vhcular ad hoc ntworks. Dr. Hou rcvd th C.W. Gar Outstandng Graduat Studnt Award from th Unvrsty of Illnos at Urbana-Champagn, and th Slvr Prz n th Asan Pacfc Mathmatcs Olympad.