A New Competitive Ratio for Network Applications with Hard Performance Guarantees

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1 A Nw Compttv Rato for Ntwork Applcatons wth Hard Prformanc Guarants Han Dng Dpartmnt of ECE Txas A&M Unvrsty Collg Staton, TX 77840, USA Emal: I-Hong Hou Dpartmnt of ECE Txas A&M Unvrsty Collg Staton, TX 77840, USA Emal: Abstract Ntwork applcatons n hghly mobl systms nd to mploy onln algorthms that do not rly on prcs prdctons about futur vnts In ordr to mt hard prformanc guarants n th prsnc of unknown futur vnts, common practc s to add rdundancy to th systms In ths papr, w dfn a nw compttv rato that rflcts th amount of rdundancy ndd to nsur som gvn prformanc guarants W study two spcal applcatons, namly, onln ob allocatons n cloud computng and onln schdulng n dlayd mobl offloadng, and propos onln algorthms for ach of thm W prov that our algorthms ar optmal By comparng our algorthms wth commonly usd othr polcs, w also show that our polcs nd much lss rdundancy than thos commonly usd polcs to provd th sam dgr of prformanc guarants I INTRODUCTION Many xstng studs us compttv rato to valuat th prformanc of onln algorthm that nds to mak dcsons wthout any knowldg of futur vnts Tradtonally, compttv rato s dfnd as th worst-cas rato btwn th prformanc of an onln algorthm, and that of th optmal soluton For xampl, Moharr, and Sanghav and Shakkotta [] hav studd th compttv ratos of onln polcs for ob allocatons n cloud computng, and proposd an onln algorthm that guarants to allocat as many obs as th optmal polcy Mao, Koksal, and Shroff [2] hav studd th packt schdulng problm n mult-hop ntworks wth nd-to-nd dadln guarants Thy proposd an onln schdulng polcy that has a compttv rato of O(P logp), manng that th polcy s guarantd to dlvr /O(P logp) as many packts as th optmal polcy, whr P s th lngth of th longst path n th ntwork Whl xstng studs on compttv ratos rval mportant nsghts about onln algorthms, thy ar nsuffcnt to captur srvc rqurmnts of many mrgng ntwork applcatons For xampl, by only guarantng to allocat 63% as many obs as th optmal polcy, th algorthm n [] may nd to drop up to 37% of obs In a ntwork whos longst path has lngth 0, th polcy n [2] may drop mor than 90% of packts Most mrgng ntwork applcatons cannot survv such hgh drop ratos In ordr to mt th strngnt srvc rqurmnt of ntwork applcatons, currnt practc s to add mor rdundancy to th systm, such as ncrasng th numbr of srvrs n data cntrs, and ncrasng ntwork bandwdth Motvatd by ths obsrvaton, w dfn a nw compttv rato that capturs th mnmum amount of rdundancy ndd to provd a crtan dgr of srvc guarants Our rcnt studs [3], [4] hav valuatd compttv ratos of onln algorthms for two spcfc applcatons: onln ob allocaton n cloud computng, and onln packt schdulng for dlayd mobl offloadng W dmonstrat that both applcatons can b formulatd as lnar programmng problms that bar som smlarts wth th wll-studd onln packng problm [5] For both applcatons, addng rdundancy nto th systm s quvalnt to rlaxng som constrants n thr corrspondng lnar programmng problms W thn propos onln algorthms and dmonstrat that thy ar optmal, or clos to optmal, n vw of our nw dfnton of compttv rato Ths papr provds a brf ovrvw of th studd applcatons and th tchncal approach of valuatng th nw compttv rato II AN EXAMPLE OF ONLINE PACKING PROBLEM WITH STANDARD COMPETITIVE RATIO ANALYSIS W frst gv an ovrvw of th basc prmal-dual approach for studyng th compttv rato W focus on th ad-aucton problm, whch s a spcal cas of th onln packng problm [5] Thr ar a st of n buyrs, and ach buyr has a budgt of B() Sllr has a collcton of m products whch arrvs on by on Whn a product arrvs, ach buyr wll provd ts bd b(, ) for ths product Th sllr chooss a buyr and slls th product to t wth prc b(,) W us y(,) to b th ndcator functon that product s sold to buyr Th goal s to maxmz th total rvnu of th sllr Thr ar two constrants of th problm Frst, buyrs cannot spnd mor than thr budgts Scond,

2 ach product cannot b allocatd to mor than on buyr Thus w hav th followng lnar programmng: Packng Max b(, )y(, ) () st, Th dual of Packng s: Covrng b(,)y(,) B(), n, (2) y(,), m, (3) y(,) 0,, (4) Mn B()x()+ z() (5) st b(,)x()+z() b(,),,, (6) x(),z() 0,, (7) Whn a nw product arrvs, a nw st of constants b(,) appar for th Packng problm, and th sllr nds to dtrmn a nw st of varabls y(,) At th man tm, a nw st of constrants n th form of (6) appar for th Covrng problm, and th sllr nds to dtrmn a nw st of varabls z() and updats xstng ons x() It s wll-known that any fasbl soluton of Covrng has an obctv functon, B()x()+ z(), that s no smallr than that of th optmal soluton of Packng Th standard prmal-dual approach ams to dtrmn and updat varabls x(),y(,) and z() so as to mantan a dsrabl rato btwn ts solutons to Covrng and Packng A dtald dscrpton of th approach s shown n Algorthm Algorthm Algorthm for Onln Ad-aucton Problm : Intally, x = 0, y = 0 2: for ach nw product do 3: argmax b(,)( x()) 4: f x( ) < thn 5: z() b(,)( x( )) 6: x( ) x( )(+ b(,) B( ) )+ b(,) (c ) B( ) 7: y(,) 8: Product s assgnd to buyr, charg th buyr th mnmum btwn b(,) and ts rmanng budgt Thorm ( [5]): Algorthm s ( /c)( Q max )- compttv, whr c = ( + Q max ) ( Qmax ), Q max = max, { b(,) B() } Whn Q max 0, th compttv rato gos to ( /) III ONLINE JOB ALLOCATION WITH HARD ALLOCATION RATIO REQUIREMENT In ths scton, w dscuss th onln ob allocaton problm for cloud computng For xampl, n YouTub, or Ntflx, ach vdo s rplcatd only n a small numbr of srvrs, and ach srvr can only srv a lmtd numbr of strams smultanously Whn a usr maks a rqust to watch a vdo, a srvr wll b slctd for ths rqust If no srvr can procss ths rqust, th rqust s droppd, whch can sgnfcantly mpact usr satsfacton Th systm prformanc dpnds on th numbr of rqusts that can b allocatd For cloud computng, mrly achvng a compttv rato of ( /) s not accptabl, as vn a small porton of droppd rqusts can rsult n sgnfcant rvnu loss Thrfor, w nd a dffrnt concpt of compttv rato to valuat onln polcs A Systm Modl W formulat th problm of maxmzng th numbr of allocatd rqusts as a lnar programmng problm W us J to dnot th st of srvrs, and I = {,2,} to dnot th arrval squnc of obs A srvr has a total amount of C unts capacty Each ob can b srvd by a subst of srvrs If K =, thn ob can b srvd by srvr ; othrws, cannot b srvd by Th corrspondng srvr subst s rvald whn ob arrvs Th systm thr allocats t to a srvr or drop t If ob s allocatd to srvr, thn ob taks on unt of capacty n W us X to dnot th assgnmnt of th obs If X =, thn ob s assgnd to srvr If X = 0, ob s not assgnd to srvr Smlarly to scton II, w can us lnar programmng to formulat ths problm Allocaton: Max K X (8) st K X C, J, (9) K X, I, (0) X 0, I, J () In ordr to allocat a hgh rato of all rqusts wthout knowng any nformaton about futur rqusts, currnt practc s to ncras th capacty of srvrs by addng rdundancy Suppos that th systm capacty s ncrasd by R tms Thn th problm of maxmzng th numbr of srvd obs can b formulatd as follows: Allocaton(R): Max K X (2) st K X RC, J, (3) K X, I, (4) X 0, I, J (5)

3 W valuat th prformanc of onln polcs by comparng th numbr of allocatd obs undr onln polcy wth R tms th srvrs capacty aganst that undr offln polcy wth unt capacty Spcfcally, gvn K and {C }, lt Γ opt b th optmal valu of K X n Allocaton, and Γ η (R) b th valu of K X n Allocaton(R) undr polcy η W dfn th compttv rato as follows: Dfnton : An onln polcy η s sad to b (R,θ)- compttv f thr xsts a functon Θ(C mn ) such that for all {K } and {C } wth C C mn, Γ opt /Γ η (R) Θ(C mn ), and Θ(C mn ) θ as C mn B Schdulng Polcy In ths scton, w wll propos an onln polcy for th onln allocaton problm usng prmal dual mthod as ntroducd n Scton II Th dual problm of Allocaton can b wrttn as: Dual(Allocaton): Mn C α + β, (6) st α +β, I,K = (7) α 0,, (8) β 0,, (9) Intally, th systm sts α, β, and X to b 0 Whn a ob arrvs, th polcy chcks th valus of α for all srvr wth K =, and slcts srvr whch has th mnmum valu of α If α <, ob s allocatd to srvr, and α wll b ncrasng y ba crtan rato as α (+ ) +, whr d = (+/C ) RC, C (d )C for all Th valu of d s chosn to achv th optmal compttv rato If α, ob s dscardd Th polcy s shown as Algorthm 2 [3] Algorthm 2 PD Algorthm for Onln Job Allocaton : Intally, α = 0, β = 0, X = 0 2: d (+/C ) RC, 3: for ach arrvng ob do 4: argmn K=α 5: f α < thn 6: β α 7: α α (+ )+ C (d )C 8: X 9: Job s assgnd to srvr 0: ls : Dscard ob C Compttv Rato W hav th followng thorms on th compttv ratos of onln polcs Th dtald proof can b found n [3] Thorm 2: Algorthm 2 s (R, R R )-compttv Proof: W prov Thorm 2 by thr stps: Frst, w show that all constrants n Dual(Allocaton) ar satsfd Intally, α and β ar 0 By stp 7 n Algorthm 2, α s non-dcrasng, and thus (8) holds From stp 5 and stp 6 w know thatβ 0 constant (9) holds If α, constrant (7) holds If α <, β = α α Thus α +β, constrrant (7) holds Scond, w show that all constrants n Allocaton(R) ar satsfd By nducton, w can prov that α [n] = ( d )(dn/rc ), whr α [n] s th valu of α aftr n obs hav bn allocatd to srvr Thn aftr RC obs hav bn allocatd to srvr, α = Snc stp 5 n Algorthm 2 shows that obs can b allocatd to srvrs whn α <, w know that constrant (3) s satsfd Nxt, w drv th rato btwn X and C α + β Both formulas ar ntally 0 W now consdr th amounts of chang of ths two formulas whn a ob arrvs W us P(R) to dnot th chang of X, and D to dnot th chang of C α + β If ob s dscardd, thn {X }, {α } and {β } rman unchangd, and thrfor P(R) = D = 0 On th othr hand, consdr th cas whn ob s assgnd to srvr W hav X = and P(R) = W also hav D P(R) = D =C ( α + )+ α C (d )C =+ d = d d = (+/C ) RC (+/C ) RC Whn w mposs a lowr bound on C by rqurng C C mn, for all, and lt C mn, w hav D P(R) R R, whnvr a ob s allocatd to som srvr Thrfor, w hav, C α + β X = R R Fnally, by wak dualty, Algorthm 2 s (R, R R )- compttv Thorm 3: Any onln polcy cannot b bttr than (R, R R )-compttv From Thorm 3 and 2, w know that our proposd polcy achvs th optmal bound W also valuat th nw compttv rato for two commonly usd allocaton polcs, namly, on th shortst quu (JSQ) and on th most rsdu quu(jmq) JSQ allocats ach arrvng ob to th srvr wth th fwst obs JMQ allocats ach arrvng ob to th srvr wth most rmanng spac, whch s th srvr capacty mnus th numbr of allocatd obs n ths srvr

4 Rqurd capacty PD JSQ, JMQ /β Fg Capacty rqurmnts of ob allocaton problm Thorm 4: JSQ and JMQ cannot b bttr than (R,+ R )-compttv-pr-sampl-path Our rsults can b ntrprtd as follows: Suppos th offln optmal polcy can allocat all obs To guarant that at last θ of th obs ar allocatd by onln polcs, our polcy nds to ncras th capacty by R = lnθ tms so that R R /( θ ) On th othr hand, JSQ and JMQ both nd to ncras at last (θ ) tms, whch s an ordr hghr capacty to achv th sam allocaton rato rqurmnt than our polcy W llustrat th capacty rqurmnt for dffrnt allocaton ratos n Fg IV ONLINE SCHEDULING FOR DELAYED MOBILE OFFLOADING In ths scton, w study anothr problm calld dlayd mobl offloadng Dlayd mobl offloadng ams to offload mobl traffc from cllular ntworks to WF ntworks so as to rduc th congston of cllular ntworks A Systm Modl W consdr a cllular ntwork systm n whch WF hotspots ar dployd by th cllular oprator to rduc th ntwork congston Mobl usrs ar movng wthn th ara of th ntwork Thy may ntr th systm at any tm and any locatons W us M to dnot th st of WF APs and I to dnot th st of mobl usrs Usr nformaton s rvald only aftr t ntrs th systm, that s, a mobl usr spcfs th amount of data, dnotd by C, that t nds to obtan, and a dadln T Hr w assum that tm s slottd and numbrd as t =,2, Usrs mov wthn th systm and trs to download as much data from WF APs as possbl bfor th dadln Aftr dadln, usrs downloads th rmanng data from cllular ntwork As th usrs ar movng n th systm, th connctvty and channl capacty btwn APs and usrs may chang from tm to tm A usr s only abl to connct to on AP at any pont of tm To rprsnt th connctvty of a clnt and AP m at tm slot t, w us K mt to dnot th channl capacty At tm t, f thr s no conncton btwn and m, w st K mt = 0 Snc usr cannot connct to any srvr bfor ts ntranc tm and aftr dadln T, w st thos corrspondng K mt to b 0 W normalz th systm so that0 K mt for all,m,t Whn clnt s connctd to AP m at tm t, th AP dtrmns th porton of tm t spnds on transmttng to n tm slot t, dnotd by X mt Thus, th amount of data that mobl usr obtans from m durng tm t s K mt X mt Our goal s to maxmz th total data dlvrd by WF, whch s th total amount of K mt X mt of all clnts from all APs bfor thr rspctv dadlns Smlar to th prvous scton, w assum that th cllular oprator may b abl to ncras th capacty of WF hotspots Whn th capacty of WF APs ar ncrasd by R tms, w can also dscrb th systm as ncrasng th srvng tm by R tms wth th orgnal capacty Th problm of maxmzng total offloadd traffc can b wrttn as th followng lnar programmng problm: Offload(R): Max mt K mt X mt (20) st mt K mt X mt C, I, (2) X mt R, m M,t, (22) X mt 0, I,m M,t (23) W us Γ opt to dnot th offln optmal valu of mt X mtk mt whn R = Lt Γ η (R) b th valu of mt X mtk mt for th Offload(R) problm undr polcy η W dfn th compttv rato blow: Dfnton 2: A polcy η s sad to b (R,β)-compttv f Γ opt /Γ η (R) β, as mn I C, for all systms B Schdulng Polcy In ths scton, w frst propos an onln polcy usng PD mthod Th dual problm to Offload(R) whn R = s: Dual(Offload): Mn Y mt + C Z, mt (24) st Y mt +K mt Z K mt,,m,t, (25) Y mt 0, m,t, (26) Z 0,, (27) Intally, X mt, Y mt, and Z = 0 ar st to b 0 APs kp track of and updat Z for ach clnt At ach tm t, ach AP m chooss a clnt whch has maxmum valu of K mt ( Z ) Lt ths clnt b m If K mt ( Z ) > 0, thn w st X m mt to b R AP m updats Z m by sttng

5 t to b Z m (+ K m mt )+ K m mt Hr d s st to b C (d )C m ( + /C mn ) Cmn AP m broadcasts th updatd Z m to all APs Th dtald aglorthm s dscrbd as Algorthm 3 Algorthm 3 PD Algorthm for Onln Mobl Offloadng : Intally, X mt = 0, Y mt = 0, Z = 0 2: C mn mn C,d (+/C mn ) Cmn/R 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 ) m mt 8: Z m Z m (+ K m mt )+ K C m (d )C m 9: X m mt R 0: AP m transmts to clnt m at tm t : f p,s t X m ps K m ps > C m thn 2: X m mt C m p,s<t X m ps K m ps K m mt C Compttv Rato Our prvous work [4] has stablshd th nw compttv ratos of Algorthm 3 and othr commonly usd polcs /R Thorm 5: Algorthm 3 s (R, R[ /R ] )-compttv It s approxmatly (R,+ 2R )-compttv Thorm 6: For any gvn ǫ > 0, thr xsts a R 0, such that, whn R > R 0, no polcy has bttr compttv rato than (R,+ 2R ǫ R ) /R Snc R[ /R ] + 2R Thorms 6 and 5 show that our proposd polcy s clos to th thortcal bound of th nw compttv rato W valuat thr commonly usd polcs, ncludng round robn (RR) polcy, max-wght (MW) polcy, and proportonal far (PF) polcy as comparson Wth round robn polcy (RR), ach AP vnly dstrbuts ts capacty among all connctd clnts 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 Wth proportonal far (PF) polcy, of all connctd clnts, ach AP slcts th on wth K th maxmum valu of mt Throughput of clnt W hav th followng thorm Thorm 7: RR, MW, and PF schdulng polcs cannot hav bttr compttv rato than (R,+ R ) Suppos on nds to guarant to offload at last θ as much data as th optmal offln polcy, PD algorthm nds to ncras capacty by approxmatly θ 2 tms, whl RR, MW, and PF polcs nd at lastθ capacty to achv th sam rqurmnt In othr words, PD Rqurd capacty PD RR, MW, PF /β Fg 2 Capacty rqurmnts of mobl offloadng problm nds about half as much capacty to provd th sam guarant as th othr thr polcs Fg 2 llustrats th dffrncs btwn ths polcs V CONCLUSION In ths papr, w us a nw dfnton of compttv rato to valuat onln polcs Instad of smply consdrng th prformanc rato of an onln polcy and th offln optmal polcy, w consdr how ncrasng rsourc can affct th prformanc rato W us two spcal cass to xplan ths quston In ob allocaton problm, w show that th commonly usd JSQ and JMQ polcs nd an ordr hghr capacty to achv th sam allocaton rato rqurmnt than our polcy In mobl offloadng problm, w show that th commonly usd RR, MW, and PF polcs nd at last doubl th capacty of our polcy to offload th sam amount of data REFERENCES [] S Moharr, S Sanghav, and S Shakkotta, Onln load balancng undr graph constrants, n ACM SIGMETRICS Prformanc Evaluaton Rvw, vol 4, pp , ACM, 203 [2] Z Mao, C E Koksal, and N B Shroff, Optmal onln schdulng wth arbtrary hard dadlns n multhop communcaton ntworks, IEEE/ACM Transactons on Ntworkng, vol 24, pp 77 89, Fb 206 [3] H Dng and I-H Hou, Onln ob allocaton wth hard allocaton rato rqurmnt, n Computr Communcatons (INFOCOM), 206 IEEE Confrnc on, p to appar, Aprl 206 [4] H Dng and I-H Hou, Onln schdulng for dlayd mobl offloadng, n Computr Communcatons (INFOCOM), 205 IEEE Confrnc on, pp , Aprl 205 [5] N Buchbndr, K Jan, and J S Naor, Onln prmal-dual algorthms for maxmzng ad-auctons rvnu, n Procdngs of th 5th Annual Europan Confrnc on Algorthms, ESA 07, (Brln, Hdlbrg), pp , Sprngr-Vrlag, 2007