Cell Breathing Techniques for Load Balancing in Wireless LANs

Transcription

1 1 Cell rething Tehniques for Lod lning in Wireless LANs Yigl ejerno nd Seung-Je Hn ell Lortories, Luent Tehnologies Astrt: Mximizing the network throughput while providing firness is one of the key hllenges in wireless LANs WLANs). This gol is typilly hieved when the lod of the ess points APs) is lned. However, reent studies on opertionl WLANs hve shown tht AP lod is often sustntilly uneven. To llevite suh imlne of lod, severl lod lning shemes hve een proposed. These shemes, essentilly, require proprietry lient softwre or speilly-designed WLAN rds t the user omputers for ontrolling the user-ap ssoition. In this pper we present new tehnique tht hieves lod lning y reduing the ell size of ongested APs, whih is oneptully similr to the so-lled ell rething methods in ellulr networks. The proposed sheme does not require ny modifition t the user side neither the stndrd, ut it only requires the ility of dynmilly hnging the trnsmission power of the AP eon messges. Unlike existing ellrething methods, whih utilize lol optimiztion heuristis, we develop lgorithms tht gurntee to find the optiml eon power settings, whih minimize the lod of the most ongested APs, in polynomil time. We then onsider the prolem of network-wide min-mx lod lning. We prove tht this prolem is -hrd nd nnot e esily pproximted. In spite of this, we identify vrint of the prolem, termed minmx priority lod lning, nd present polynomil-time lgorithms to find optiml solutions. Extensive simultions show tht the performne of our ell-rething methods is omprle with or superior to the existing ssoition-sed methods. Keywords: Wireless Lol Are Networks WLAN), IEEE 2.11, Cell rething, Power Control, Lod lning, Firness, Comintoril Optimiztion. I. INTRODUCTION Reent studies [1], [2] on opertionl IEEE 2.11 wireless LANs WLANs) hve shown tht the trffi lod is often unevenly distriuted mong the ess points APs). In WLANs, y defult, user sns ll ville hnnels to detet its nery APs nd ssoites itself with n AP tht hs the strongest reeived signl strength inditor RSSI), while eing olivious to the lod of APs. As users re, typilly, not evenly distriuted, some APs tend to suffer from hevy lod while their djent APs my rry only light lod. Suh lod imlne mong APs is undesirle s it hmpers the network from fully utilizing the network pity nd providing fir servies to the users. In this pper, we present novel lod-lning sheme tht redues the lod of ongested APs y deresing their ell size nd foring the users ner the oundries of ongested ells to move to the neighoring less-ongested ells. We hieve suh ell dimensioning y ontrolling the trnsmission power of the AP eon messges, whih is oneptully similr to the so-lled ell rething method in ellulr networks [], [4]. In ontrst to previous studies on ell-rething tht mostly rely on lol optimiztion heuristis, we present n optiml ell dimensioning lgorithm tht finds deterministi min-mx lod lning solutions. Informlly, WLAN is lled min-mx lod lned, if it is impossile to redue the lod of ny AP without deresing the lod of other APs with equl or higher lod. Our pproh is prtiulrly ttrtive in tht it does not require neither user ssistne or stndrd modifition, unlike most existing proposls for WLAN lod lning. A. Lod lning vi User-AP Assoition Control Currently the IEEE 2.11 stndrd [5] does not provide ny stndrd method to resolve the lod imlne. To overome this defiieny, vrious lod lning shemes hve een proposed y oth the demi nd the industry. Most of these methods ommonly tkes the pproh of diretly ontrolling the user-ap ssoition y deploying proprietry lient softwre or speilly-designed WLAN rds t the user omputers. For instne, some vendors hve lredy inorported ertin lodlning fetures in their devie drivers, AP firmwres, nd WLAN rds [], [7]. In these proprietry solutions, the APs rodst their lod levels to users vi modified eon messges, nd eh user hooses the lest-loded AP. Severl studies []-[14] hve proposed vriety of ssoition metris insted of using the RSSI s the sole ssoition riterion. These metris typilly tke into ount suh ftors s the numer of users urrently ssoited with n AP, the men RSSI of users urrently ssoited with n AP, nd the ndwidth tht new user n get if it is ssoited with n AP, e.g., [], [9]. lhndrn et l. [1] proposed to ssoite user with the AP tht n provide miniml ndwidth required y the user. If there exist mny of suh APs, the one with the strongest RSSI is seleted. In [11], Velyos et l. introdued distriuted lod lning rhiteture where the lod of n AP is defined s the ggregted downlink nd uplink trffi through the AP. In [12], Kumr et l. proposed n ssoition seletion lgorithm whih is sed on the onept of proportionl firness to lne etween throughput nd firness. Most of these work heuristilly determine only the ssoition of newly rrived users. [1], [14] re exeptions. Tsi nd Lien [1] proposed to ressoite users when the totl lod exeeds ertin threshold or the ndwidth lloted to users drops elow ertin threshold. In [14], n on-line sheme tht periodilly optimizes the user-ap ssoition is proposed. This work lso proved strong orreltion etween firness nd lod lning, i.e., the fir servie is otined when the AP lod is lned. Although the user-ap ssoition ontrol pproh n hieve lod lning in WLANs, the requirement of deploying proprietry lient softwre/hrdwre on ll or most) users rises n ute question out its prtility. Tody, WLAN users frequently move etween different WLANs, suh s hotels, ir-

2 2 ports, shopping enters nd university mpuses. 1 Different networks re mnged y different orgniztions nd likely dopt different lod lning mehnisms. It is unrelisti to require the users to hve the pproprite lient modules for eh visiting network. This motivtes the need for new lod-lning sheme tht does not require ny proprietry lient module nor ny modifition of the stndrd.. Cell rething for Lod lning In CDMA ellulr networks, the overge nd pity of ell re inversely relted with eh other [15]. The inrese of the numer of tive users in ell uses the inrese of the totl interferene sensed t the se sttion. Therefore, in ongested ells, users need to trnsmit with higher power to mintin ertin signl-to-interferene rtio t the reeiving se sttion. As the users in ongested ell inrese their trnsmission power, they lso inrese their interferene to the neighoring ells sine ll ells use the sme frequeny nd in CDMA networks. As result, the overll network pity my derese []. Furthermore, sine the mximl trnsmission power of the users is ounded, the users who re fr from the se sttion my experiene poor servies. This so-lled ner-fr prolem my result in imlned ell hndoff oundries for reverse nd forwrd links, s the ltter is determined y the strength of the pilot signl of the se sttions, independent of the interferene [4]. In other words, the ell hndoff oundry of the reverse link is tighter thn tht of the forwrd link. To overome these prolems the ell rething pproh ws proposed y Togo et l.[] nd Jlli [4], independently. This pproh shrinks the ell size of ongested ells nd lnes the forwrd nd reverse link hndoff oundries y reduing the pilot signl trnsmission power of the orresponding se sttions. Some studies hve explored the enefit of omining the ellrething methods with other interferene mitigtion methods. For instne, in [1], Yng nd Ephremides presented solution for the ner-fr prolem, whih is sed on the omintion of ell-rething nd ndwidth spe prtitioning. Du et l.[17] proposed distriuted lod lning tehnique tht utilizes ole osilltion lgorithm. In [1], Sng et l. proposed method tht oordintes the pket level sheduling with ell-rething tehniques. Generlly speking, the existing ell-rething tehniques utilize proilisti lol optimiztion methods. Therefore, they do not provide ny gurntee on the qulity of the solutions. Sine the ells of WLANs re muh smller thn those of ellulr networks, the proilisti lol optimiztion methods my not work well in WLANs, s we will demonstrte lter in the pper with simple exmple. Moreover, these tehniques nnot e esily pplied to IEEE 2.11 WLANs, e.g., they require the hnge of sheduling lgorithms or the knowledge on the user lotion. This motivtes the design for new ell-rething method for WLANs whih finds deterministi glol optiml solutions. C. Min-Mx Lod lning Algorithms In priniple, our ojetive n e viewed s min-mx vrint of the unrelted prllel mhine sheduling prolem [19]. The ltter seeks for jo-mhine ssoition tht minimizes the mximl proessing time of ny mhine, for given set of In mny ples even free 2.11 ess is offered. Reently, some ities e.g., Phildelphi nd Sn-Frniso, delred their intention to uild free ity-wide 2.11 networks. jos, mhines, nd the required running time of eh jo on eh mhine 2. Sine there re extensive literture on prllel mhine sheduling prolems nd mx-min solutions, we disuss here only the most relevnt ones to our study. Most of the work on min-mx or mx-min) solutions ddress the prolem of finding fir ndwidth llotion to set of pre-determined routes in wired network [2], [21]. Seleting routes for the mx-min fir ndwidth llotion is muh hrder prolem nd hs een studied in [22], [2]. Megiddo [22] ddressed the single-soure frtionl flow prolem nd presented polynomil time lgorithm tht finds n optiml mx-min fir solution. Extending this work, Kleinerg et l. [2] onsidered the se tht onnetion is routed long single pth. In prtiulr, their pproh n e pplied to lod onserving instnes of the unrelted prllel mhine sheduling prolem, where eh jo imposes the sme lod on the suset of mhines on whih it n e run. They rgued tht oordinte-wise onstnt-ftor pproximtion nnot e found for this prolem nd presented prefix-sum 2-pproximtion lgorithm, in whih for every integer the sum of the first oordintes of the lulted mhine lod vetor sorted in inresing order is t most twie the sum of the first oordintes of the optiml min-mx frtionl ssignment. Another importnt study is the user-ap ssoition ontrol sheme presented in [14]. This study n e mpped to the generl unrelted prllel mhine sheduling, where eh jo my hve different running time on eh mhine. It presents minmx lod lning lgorithm tht ensures oordinte-wise - pproximtion rtio s ompred to the optiml min-mx frtionl solution. Notie tht ll of these methods require omplete ontrol on the jo-mhine ssoition. This ssumption is fesile for the user-ap ssoition ontrol shemes. However, suh freedom of ssoition ontrol is unville in the ellrething pproh, whih only impliitly ontrols the user-ap ssoition y djusting the ells oundries. This rises the need for new min-mx lod lning lgorithms for the ellrething pproh. D. Our Contriutions In this pper we present new lod lning sheme for IEEE 2.11 WLANs. The proposed sheme djusts the size of ells y hnging the trnsmission power of the AP eon messges without hnging the trnsmission power of the dt trffi hnnel. While there exist similr pprohs in ellulr networks, to the est of our knowledge, we re the first who pplies the ell rething onept to IEEE 2.11 WLANs. More importntly, unlike the existing ell rething studies [],[4],[1], [17],[1], we tkle the hllenge of finding the deterministi glol optimum insted of relying on lol optimiztion heuristis. Our lgorithms re not tied to prtiulr lod definition, ut support rod rnge of lod definitions. We tret the lod of n AP s the ggregtion of the lod ontriutions of its ssoited users. The lod ontriutions my e s simple s the numer of users ssoited with n AP or n e more sophistited to tke ount of ftors like trnsmission it rtes nd trffi demnds. Our sheme does not require ny speil ssistne from users nor ny hnge in the stndrd. It only requires the ility of dynmilly hnging the trnsmission power of When given jo nnot e served y speifi mhine, infinite running time is ssumed on tht mhine.

3 K the AP eon messges. Tody, ommeril AP produts lredy support multiple trnsmission power levels, so we elieve this requirement n e reltively esily hieved vi AP softwre updte. Our lgorithms will e run on network opertion enter whih ollets the lod nd ssoition informtion from the APs vi suh methods s SNMP. Depending on the extent of the ville informtion, we onsider two knowledge models. The first model ssumes omplete knowledge, in whih the user-ap ssoition nd the orresponding AP lod re known priori for ll possile eon power ssignments. Sine suh informtion is not redily ville in urrent WLANs, we lso onsider the seond model, the limited knowledge model, in whih only informtion on the user-ap ssoition nd AP lod for the urrent eon power ssignment is ville. The lgorithms for the omplete knowledge model serve s uilding loks for the lgorithms for the more prtil limited knowledge model. We present our lgorithms in two steps. At first, we ddress the prolem of minimizing the lod of the most ongested APs, whose lod is lled the ongestion lod. We present two polynomil time lgorithms tht find optiml solutions, one for the omplete knowledge model, nother for the limited knowledge model. These results re intriguing, euse similr lod lning prolems, e.g., [14], re known to e strong -hrd. It is prtiulrly interesting tht polynomil-time optiml lgorithm exists for the limited knowledge model. Our lgorithms re rooted from simple oservtion tht s long s the urrent power setting domintes the optiml setting i.e., eh AP hs the sme or higher power level thn its power level in the optiml solution), n optiml solution n e otined y ertin sequene of power redution opertions. The lgorithms strt with the mximl power level t ll APs nd, itertively, redue the power of seleted set of APs. For the omplete knowledge se, we use the onept of ottlenek set. After reduing the power level of ll APs in the ottlenek set, the lod of eh AP is gurnteed to sty the sme or stritly lower thn the initil ongested lod efore the power redution. This property ensures monotoni onvergene to the optiml solution. For the limited knowledge se, we tke different pproh, termed optiml stte reording, in whih the power levels of the ongested APs re grdully redued until the power nnot e redued ny further, while the est solution found so fr is reorded. Seondly, we ddress the prolem of finding the min-mx lod lned solutions. We prove tht this is strong - hrd prolem nd there exists no good pproximtion lgorithm. More speifilly, we prove tht there exists no lgorithm tht gurntees ny oordinte-wise pproximtion rtio, nd the pproximtion rtio of ny prefix-sum pproximtion lgorithm is t lest, where is the numer of APs in the network. In spite of this, we identified vrint of this min-mx prolem, termed min-mx priority lod lning, whose optiml solution n e lulted in polynomil-time for oth knowledge models. Here, the AP lod is defined s n ordered pir of the ggregted lod ontriutions of its ssoited users nd unique AP priority. y deploying the optiml stte reording method, we were le to onstrut n lgorithm tht, itertively, ompute eh oordinte of the min-mx priority lod lned solutions. We, lter, show tht our lgorithms n e effiiently emedded into dptive on-line shemes. Through extensive simultions, we show tht the performne of our ell-rething methods is overll omprle with or superior to the existing ssoition-ontrol methods, irrespetive of network lod ptterns. In prtiulr, we ould hieve suh performne even with smll numer of power levels. Our min-mx priority lod lning lgorithms yield ner optiml results even for the non-priority min-mx prolem y rndomly hoosing AP priorities. Although we primrily fous on IEEE 2.11 WLANs, our shemes should e pplile to other wireless networks. Due to the spe limittion, we omit some proofs. II. THE NETWORK MODEL We onsider n IEEE 2.11 WLAN tht omprises set of ess points APs), denoted y. denotes the numer of APs. All APs re tthed to fixed infrstruture, whih onnets them to wired networks, e.g., the Internet. Eh AP hs ertin trnsmission rnge nd it n serve only those users tht reside in tht rnge. Eh AP is onfigured " to use one of trnsmission power levels, denoted y $#&% ')' +*-,, where the miniml nd mximl levels re denoted y /.1) ;: < >= nd, respetively. Eh power level is identified y its power index nd its /A@C trnsmission power is? times stronger thn its predeessor, where? is defined s? EDF.9;:HG 5.1I2,.1)2 i.e.,? J. euse? it follows tht K? for every 4#M% N'I' +*. This ssumption is onsistent with the trnsmission power level onfigurtions supported y ommeril AP produts [], [7]. We denote the trnsmission power of eh AP O #P y nd its orresponding power index y Q. For the ske of simpliity, we ssume tht the AP deployment ensures high degree of overlps etween the rnge of djent APs. Consequently, every user is overed y t lest one AP R.1I2 even when ll APs re trnsmitting t the miniml power level. We define the network overge re to e the union of the trnsmission rnges of ll APs in. We use S to denote the set of ll users in the network overge re nd use ST to denote their numer. We ssume tht users hve qusi-stti moility pttern. In other words, users re free to move from ple to ple, ut they tend to sty in the sme lotions for long period. This ssumption is ked up y reent nlysis of moile user ehvior [1], [2]. At ny given time, eh user is ssoited with single AP. Eh AP periodilly trnsmits eon messges for dvertising its presene. When user enters WLAN, the user initites snning opertion, in whih it sns ll hnnels i.e., listening for the eon messges) for identifying ll APs in its reh. Then, sed on the RSSI s of the eon messges, the user ssoites itself with the AP tht hs the strongest RSSI. Whenever the hnnel qulity deteriortes elow ertin threshold, e.g., due to the user movement, the user initites new snning opertion nd it my ssoite itself with different AP. The RSSI tht user UV#WS senses for n AP OX#Y is denoted y Z\[^]. It depends on the trnsmission power of AP O,, nd _ the signl > ttenution, whih is denoted y [^],. We onsider only the signl ttenution i.e., Z [^] [^] tht results from long-term hnnel ondition hnges, suh s pth-loss nd slow fding. We ssume tht during the short period of time for exeuting our lgorithms the signl ttenution etween eh user-ap pir does not hnge. As the user-ap ssoition depends on the RSSI s, we n divide the network overge re into disjoint ells. A ell of n AP O defines

4 n o r z ˆ ` 4 the region in whih AP O hs the strongest RSSI. Note tht the ell of n AP O is susumed in its trnsmission rnge nd it depends not only on the trnsmission power of O, ut lso tht of the other APs in O s viinity. The trnsmission it-rte for user-ap pir is determined y the Signl-to-Noise Rtio SNR), whih is the strength of the reeived signl over the umulted strength of the other interfering trnsmissions nd the kground noises. Users who re ssoited with the sme AP my trnsmit with different it rtes. Eh user ontriutes ertin mount of lod on its serving AP, nd the lod on n AP is the ggregtion of the lod ontriutions of its ssoited users. Our lgorithms do not require ny expliit lod definition nd just ssume tht the lod of n AP O, denoted y `, is the sum of the lod ontriutions of its ssoited users. We use ] [ to denote the lod ontriution of user U on n AP O. We ssume tht this ontriution is onstnt, so tht the lod of eh AP O$# is ` [edafng ] [, where S denotes the set of users ssoited with O. The APs tht experiene the mximl lod re lled the ongested APs nd their lod, termed ongestion lod, is denoted y h. Other APs with lower lod re lled non-ongested APs. Our flexile lod model n ommodte mny ommonlyused lod definitions, inluding the numer of users ssoited with n AP s well s more dvned lod definitions tht my tke ount of the effetive trnsmission it-rte or the verge trffi demnd. It n lso del with the multiplitive user lod ji [edaf g ontriutions, i.e., sides. Tle I summrizes the key nottions. Symol Semntis The set of ll ess points APs). The ottlenek set of APs. The set of ongested APs. The set of fixed APs. ] [, y pplying kilem to oth p q ) s;tvu w Attenution of AP x s signl deteted y user y. { The mximl trnsmission power index. w u t } Lod ontriution of user y to AP x. w Trnsmission power of AP x. ~ w Trnsmission power index of AP x ~ w _ ƒ ƒ z,. ˆ tvu w Signl strength of ˆT +ŠŒ AP x reeived y user y. A network stte, xv ~ wž-. A reorded network stte. The set of ll users. w The set of users ssoited with AP x. ;w The lod of AP x. The network ongested lod. The ongestion lod of the reorded stte. The AP lod vetor, TALE I NOTATIONS. Š v >. III. THE CELL REATHING APPROACH In this setion we present the si onept tht underlies our pproh. We lso ddress some prtil spets nd the lgorithmi hllenges tht our pproh enounters. In this study, we ssume the presene of Network Opertion Center NOC). APs report to the NOC out their ssoited users, their lod nd dditionl relevnt informtion. The NOC exeutes our lgorithm nd onfigures the APs ordingly. š In Setion III we explin why we onsider onstnt lod ontriutions of the users lthough we llow hnges of the AP trnsmission power. AP trnsmission rnge AP ell size ) All APs trnsmit with the sme power level. AP trnsmission rnge AP ell size ) AP trnsmit with lower power level thn APs nd. Fig. 1. lning the AP lod y djusting their trnsmission power. A. The Conept of Cell rething Our sheme redues the lod of ongested APs y reduing the size of the orresponding ells. This fores users ner the ongested ells oundries to shift their ssoition to djent less-ongested) APs. Suh ell dimensioning n e otined, for instne, y reduing the trnsmission power of the ongested APs, s we illustrte in Exmple 1. Exmple Consider WLAN with three APs, O, nd œ tht trnsmit with mximl power.9;: nd let s ssume tht they re ssoited with, nd users, respetively, s depited in Figure 1-). In this exmple, we define the lod of n AP to e the numer of its ssoited users. Clerly, hs muh higher lod thn the other two APs. Now, y reduing the trnsmission power of, the ell size of is lso redued nd four of the users ssoited with suffer from low signl qulity. These users initite snning opertions tht use them to shift to djent APs. As result, the numer of users ssoited with the three APs re now ž, Ÿ nd ž, respetively, s illustrted in Figure 1- ), nd the AP lod eomes more evenly distriuted. Reduing the trnsmission power of n AP ffets the hnnel qulity of ll of its ssoited users, nd this effet is not limited to those users tht we intend to shift. The users who remin ssoited with the onsidered AP lso experiene lower hnnel qulity nd my hve to ommunite t lower it rte thn efore. This my result in longer trnsmission time of user trffi, whih effetively inreses the user lod ontriutions on the AP, if the AP lod is determined y onsidering not only the numer of users ut lso the effetive user throughput. Thus, we my end up with inresing the lod of more APs rther thn reduing the lod of the ongested APs. We overome this prolem y the segregtion etween the trnsmission power of the dt trffi nd tht of the AP eon messges. On one hnd, the trnsmission it-rte etween user nd its ssoited AP is determined y the qulity of the dt trffi hnnel. Trnsmitting the dt trffi with mximl power 4 mximizes the AP-user SNR nd the it-rte. On the other hnd, eh user determines its ssoition y performing snning opertion, in whih it evlutes the qulity of the eon messges of the APs in its viinity. y reduing the eon messges power level of ongested APs, we, prtilly, shrink the size of their ells nd, onsequently, disourge new user ssoition. This onept of ontrolling the ells dimensions y dpting power levels of the eon messges is termed ell rething. The segregtion etween the power levels of the dt trffi nd the eon messges is the only modifition tht Power ontrol of the AP dt trffi n e done, seprtely, for reduing inter-ap interferenes. Suh power llotion is eyond the sope of this pper.

5 5 1 2 ) AP nd AP hve the sme power level. 1 2 ) AP hs lower power level thn AP, p =p -1). Fig. 2. Exmple of n exeution of the greedy lgorithm. we require from APs. We elieve this n e reltively esily hieved y softwre updte.. Triggering User-AP Assoition Chnges In the long run, the method desried ove lnes the AP lod y disourging new user ssoition with ongested APs. However, sine the ell rething method does not expliitly ontrol the user-ap ssoition, it my not provide immedite relief to the ongested APs. It is euse, one the ssoition deision is mde, user stys onneted with the sme AP s long s it experienes stisftory hnnel qulity, regrdless of the reeived eon messge strength. For immedite lod redution, we need to enourge the users in the ongested ells to invoke the snning opertions. One method is reduing the dt trffi power levels of the ongested APs for short period, whih will trigger the snning mehnism for the users ner the oundries of the ongested ells. Another method is sending dis-ssoition messges to some or ll users who re ssoited with the ongested APs. The ltter method provides the flexiility to selet prtiulr users to hnge their ssoition without ffeting others. In the reminder of this pper, we ssume tht proper method is used to trigger ssoition shifts fter the ell rnge of n AP is ltered nd limit our disussion just to the power level of the eon messges. Tht is, when we sy trnsmission power, we men only the trnsmission power of eon messges. C. Algorithmi Chllenges One my onsider greedy lgorithm tht, redues the power level of the ongested APs until ny of the ongested APs rehes to the miniml power level. Note tht sine this lgorithm ttempts to shift users from ongested APs to their neighors, the set of ongested APs nd their lod my hnge during the exeution of the lgorithm. As we demonstrte in Exmple 2, even in very simple se, the greedy lgorithm my fil to find the optiml solution. Moreover, it n e shown tht in some ses the finl ongestion lod is even higher thn the initil ongestion lod. Exmple Consider WLAN with two APs, denoted s O nd, nd two users U nd U. User U n only e tthed to O nd it yields lod of. User U n e tthed to oth APs nd hooses n AP with higher power level, while it hooses O in se of tie. It produes lod of on its ssoited AP. We ssume tht, initilly, oth APs trnsmit with the mximl power level, i.e., Q Q, nd therefore oth users re ssoited with O whose lod eomes ž, s shown in Figure 2-). To lne the lod, the greedy lgorithm redues the power level of O nd s result U hnges the ssoition to. Now the lod on the two APs re nd, respetively, s depited in Figure 2- ). At the next moment, the lgorithm will redue the power of, whih is now the ongested AP, nd UL will e shifted k to O. The lgorithm ontinues to redue the power levels of the APs, until they oth trnsmit with the miniml power level. In the finl setting, the lod on O will e ž nd hs no lod, whih is oviously not the optiml solution. Exmple 2, demonstrtes the need for more sophistited lgorithms. IV. MINIMIZING THE CONGESTION LOAD This setion presents two lgorithms for minimizing the AP ongestion lod, one for the omplete knowledge CK) model, nother for the limited knowledge LK) model. A. The Prolem Sttement Definition 1 A network stte) : A network stte is defined s the eon messge power indies Q of ll the APs O$#X, i.e., HO Q Œ O$# «ª Q # % N'I' *,. For simpliity we denote stte y HO Q,. A network stte determines the rnges of ll AP ells. When we ssume tht the users re lwys ssoited with the AP whose eon signl hs the strongest RSSI, it lso determines the user-ap ssoition. Therefore, it lso determines the set of ongested APs,, nd their ongestion lod, h. Notie tht user my hnge its ssoition only when the network stte hnges, whih is termed stte trnsition. Let us now define our ojetives. Definition 2 AP Congestion Lod Minimiztion) : The AP ongestion lod minimiztion prolem seeks for network stte tht minimizes the AP ongestion lod. We ddress this prolem in two types of networks depending on the informtion ville. Definition The Complete Knowledge CK) Model) : A network hs omplete knowledge when the ville informtion omprises the signl ttenution, H[^], nd the lod ontriution, ] [, for every user-ap pir, user U # S nd n AP O #. A omplete knowledge model is fesile when ll users ollet the RSSI informtion from ll of the nery APs nd send the informtion to the NOC. Suh feture is suggested, for instne, in the IEEE 2.11-k proposl [24]. Unfortuntely, this feture is urrently not ville in most existing WLANs. We use this model minly s uilding lok of the limited knowledge solution. Definition 4 The Limited Knowledge LK) Model) : A network hs limited knowledge when the ville informtion omprises only the set of users tht re urrently ssoited with eh AP nd the lod ontriutions, ] [, of eh user U on its ssoited AP O. In the omplete knowledge model, the NOC n priori determine the user-ap ssoition in ll possile sttes without tully hnging the network stte. This llows the NOC to perform n off-line lultion of desired stte nd to diretly onfigure the APs with the orresponding power levels of tht stte. Suh lultion is not possile in the limited knowledge model. Nevertheless, we show in the following tht for oth models the optiml network sttes n e found.. Preliminry Oservtions We present some fundmentl oservtions tht relevnt to our lgorithms. In prtiulr, we study the reltionship etween the AP power redution nd the network stte trnsition.

6 Q Definition 5 A Set ± Power Redution) : A set ± power redution ±"² ) uses stte trnsition, in whih ll APs in ± redue their power indies y one level nd other APs mintin their urrent power levels. Lemm For set ± power redution, the only possile ssoition hnges re for the users who re ssoited with APs in ± to shift to the APs in the set ³ ±. Tht is, there re no ssoition hnges within the set ± neither the set «³P ±. From Lemm 1 follows Corollry 1. Corollry A set power redution of ll APs does not hnge the user-ap ssoition. )2µ) 2e ¹ We define tht stte I2µº domintes stte if for every AP O #+ stisfies tht Q» 2e ¹. y definition stte domintes itself. The stte HO¼ Q ½ $;,, in whih the power indies of ll APs is, domintes ll other sttes. This stte is lled s the mximl power stte. We define the network stte I2µº ¾¹À s n dominted optiml stte, if it is dominted y nd lso minimizes the ongestion )2eº lod of the APs mong ll network sttes dominted y. An AP O #+ is termed nhor if its power index )2eº QC¾ÁÀ in )2eº the optiml stte 1¾¹À is the sme s its power index)2µ) Q in. Finlly, I2µ) we denote the lod on n AP O in stte nd ¾¹À y ` nd Ǹ¾¹À O, orrespondingly. I2µ) Consider n initil stte nd let 1¾ÁÀ e its dominted optiml stte. From Lemm 1, it is sfe to perform set power redution opertion s long t the onsidered set does not ontin ny nhor AP. This oservtion is formulted I2µ) s follow: Lemm Consider n initil network stte nd let ¾¹À e one of its dominted I2µº optiml stte. Let  e the set of the nhor2e ¹ nodes in. Then, for every suset ±>à ijXÂ, the stte, otined y performing set ± power redution opertion, domintes the stte ¾ÁÀ. From Lemm 2 follows Corollry )2eº 2. Corollry As long s is not optiml, there exits sequene of set power redution opertions tht onverges to n optiml stte y reduing the power levels of non-nhor APs. The set of non-nhor APs n e otined y tking the set of ongested APs. Sine ¾¹À is not priori known, we nnot tell when n optiml solution is otined. Thus, ontinuing the set power redution opertions of the ongested APs my inrese the lod on some other APs to h or even higher, whih my prevent onvergene to n optiml stte, s demonstrted in Exmple 2. To overome this prolem, we use the following two pprohes. The first pproh, termed ottlenek set power redution, gurntees tht the ongested lod never inreses. We use this pproh in the omplete knowledge se to perform sequene of set power redution opertions tht monotonilly onverges to n optiml stte. The seond pproh, termed optiml stte reording, keeps reords of the est stte found so fr. We use this method in the limited knowledge se nd we prove tht this method lso finds the optiml solution. C. The ottlenek Set We define ottlenek set Å Ã s the miniml set of APs tht ontins ll ongested APs with lod h ) s well s ll APs whose lod my elevte to h or ove due to set power redution opertion. Formlly, the set Å is defined in reursive mnner s follows: Definition I2µº The ottlenek set) : Consider n initil stte, lod ` for eh AP O«#Æ nd ongestion lod h. We define with lod h GivenÉ@L stte s set of ongested APs in AO ht,, i.e., APs nd set, the stte is otined y set power redution opertion from the initil stte. The set omprises of the nd ll the other APs whose lod hve hnged from nd is equl to h or higher t. É@LÊ AOC OT#+ ª`» h, )2µ) The ottlenek set Å of the stte is defined s Å for the first index Ë suh tht É@L or ontins ny AP with miniml trnsmission power. Exmple : Consider the WLAN used in Exmple 2. When two APs hve the sme power level, s depited in Figure 2-), reduing the power level of the AP O dereses its lod from ž to, while it inreses the lod of the AP only to. Thus, the ottlenek set t tht moment ontins only O. When O hs one power level lower thn, s illustrted in Figure 2-), the power redution of redues its lod from to nd inreses O s lod to ž. In this se, the ottlenek set ontins oth APs. In the se of omplete knowledge, the ottlenek set of given network stte n e esily lulted. First, we lulte the RSSI etween eh AP O nd eh user U, denoted y Z [e]. This informtion enles us to determine the initil user- AP ssoition, the lod ` )2eº of eh AP O, nd the mximl lod. For given for every h É@L. From these we n lso lulte the set, we n lulte redued reeived signl Z [^] ± AP O&#Ì É@L nd every user U ssoited with É@L AP O, ssum- I2µ) is one level ing tht the power index of every AP OÍ#4 lower thn the one in the initil stte,. From these lultions, we identify ll users who will hnge their ssoition nd É@Lwe evlute the lod of the APs whih re not inluded in. Then we onstrut the set tht ontins the nd ll APs not ) whose new lod is h or higher. We exeute this proess itertively until ny termintion ondition is met. Upon termintion, the ottlenek set Å is otined. A forml desription of the ÂOœ ÎAÏ¹Ï ÐLÐAœŒ Ñ/ÐÏ routine is given in Figure. Rell tht this routine does not tully hnge the network stte I2µ) nd just lultes the ottlenek set of the initil stte y simulting the network stte hnges. From Defintion results Lemm. Lemm : Consider stte with ottlenek set Å nd its ongestion lod h. The set Å is the miniml set of APs tht ontins ll ongested APs. The lod of every AP O #W stys the sme or stritly less thn h fter the set Å redution opertion. Moreover, ny other set ± tht hve the ove property must inlude Å. LetŒ "Òµ, e sequene of sttes generted y sequentil ottlenek set power redution opertions nd let h Ò e the ongestion lod of stte "Ò. Lemm 4:Ah Ò, is monotoni non-inresing sequene. Furthermore, Lemm enles us to present in Theorem 1 strong property of ottlenek sets tht is essentil for our optimlity proofs. I2µº Theorem Consider ny su-optiml stte tht domintes n optiml 2^ ¹ stte 9¾ÁÀ nd let Å e its ottlenek set. Then, the I2µ) stte otined y set Å power redution opertion on lso domintes 1¾¹À.

7 Å w w SetĈÓ" +ŠŒ Routine Cl ottlenek xv ~ wó Ž ÁÔ Ó ) Ô"Õ Ö //used for the termintion ondition. while Ô7Ø7Ù Ô7Ø Õ Ž ÛÚ nd x Ô7ØÁ ~ wýü Ø ÞŽ eß à ˆ ) do //Get the simulted stte Ø y performing ˆ set Ô7Ø Õ //power redution opertion from stte غáغâ. for every AP x Ô7Ø Õ ~ Ø let w ~ wäã Ó à for every AP x n ã Ô7Ø Õ ~ Ø let w ~ Ó // Evlute the new user ssoition nd ˆ Ø // ompute the lod on eh AP in stte. w Ø The lod of AP x ˆ Ø in stte. ÔØ ÔØ Õ Nå Š x^æ x ń ç w è Ø ç w Ü Ø Ø Õ end oé while Ô7Ø return o end Fig.. A forml desription of the ottlenek set lultion routine. )2eº Proof: Sine is su-optiml its ongestion lod is stritly higher thn the ongestion lod of 9¾¹À. From Lemm follows tht the set Å is the smllest set of APs tht ontins the ongested APs nd its power redution opertion does not inrese 2e the lod of ny other AP to h or higher. Consequently, is either ¾¹À or it domintes 1¾ÁÀ. D. The Complete Knowledge Algorithm We now present the omplete knowledge lgorithm. The lgorithm strts with the mximl power stte in whih ll APs trnsmit with the mximl power. Clerly, this stte domintes ll other stte nd in prtiulr the optiml sttes. The lgorithm, itertively, lultes the ottlenek set Å, s desried in Setion IV-C. sed on the lulted set Å, the lgorithm determines whether it needs to pply nother set power redution opertion or n optiml stte is found. To this end, it utilizes two termintion onditions. The first ondition heks if. Rell tht from Corollry 1 follows tht reduing the trnsmission power of ll APs does not hnge the user-ap ssoition. Consequently, it nnot redue the mximl AP lod. In relity, this ondition is stisfied when the lod of ll APs re lned nd ny power redution opertion will use some APs to e ongested. The seond ondition heks if the ottlenek set Å ontins n AP tht trnsmits with the miniml trnsmission power. In suh se, the power level of ll APs in Å nnot e eqully deresed nd the lgorithm hlts. Suh se, typilly, ours when the AP lod is not lned nd the lgorithm ttempts to redue the mximl lod y repetedly reduing the power of the ongested APs. A forml desription of the lgorithm is given in Figure 4 nd typil exeution of the lgorithm is illustrted in Exmple 4 Exmple 4: Consider WLAN with three APs, denoted s O, nd œ, nd four users, U U e êu ëµ êu A,. Let, i.e.three power level. All possile user-ap ssoition re depited in Figure 5-), in whih solid lines indite the defult ssoition when ll APs hve the sme power level nd dotted lines indite other possile ssoition. The numer on eh line indites the lod ontriution y user to the lod of the ssoited AP. For exmple, U n only e ssoited with O nd it yields lod of Ÿ. U ë n e ssoited with ll APs, nd its lod ontriution is. It hooses n AP with the highest power level, ut in se of tie it prefers œ nd if œ hs lower power level thn O nd it prefers. An sterisk t the numers indi- n Alg CK Min Congestion Lod Alg ) for every AP x n ~µw z ì"í let q î { x s î/ï>ð ñ ì ì/í while q î { x s î/ï7ð ñ ì Ž do ò óõô // Find ongested APs nd their lod. wö ;w p Š x^æ x n ç w ō + x {)ø Ô/ùõúûú {ºü í üõø ý ñ ü ú ÉŠŒ xµ ~ waž p Ž if Iō n ŽHþ exist x o s.t. ~µw ÞŽŽ then ì/í q î { x s Èÿ >ì else for every AP x o let ~ w ~ w ã à end if end while return ŠŒ xµ ~ wž æ x n end Fig. 4. A forml desription of the omplete knowledge ongestion lod minimiztion lgorithm : * * - Two level gp requred to hnge the user's ssoition. : ) The onsidered WLAN 4 15* 1 u AP Lod: ) After first itertion, ={} 2 1 u AP Lod: 7 12 : AP Lod: 9 1 : 4 u ) The initil stte 2 1 u d) After seond itertion, ={,} Fig. 5. Exmple of n exeution of the omplete knowledge lgorithm. tes tht gp of two power levels is required for shifting the orresponding user, e.g., UC hnges its ssoition from O to only if Q nd Q. At the initil stte, ll APs hve the sme power index,, nd the initil user-ap ssoition re given in Figure 5-), where the lod of eh AP is, nd, respetively. In the first itertion, the ottlenek set Å œa, nd the power index of œ is redued to. Consequently, UCë hnges its ssoition to. The new user-ap ssoition nd AP lod re shown in Figure 5- ). Note tht is still the ongested AP. However, more power redution of œ will hnge the ssoition of U to nd the lod of will eome. Thus, in the seond itertion the ottlenek set ontins oth œ nd, s illustrted in Figure 5-d). This is the lst itertion, sine now Q. Clerly, the finl stte, presented in Figure 5-d), minimizes the ongested lod, ut it does not lne the lod of the non-ongested APs. Theorem The omplete knowledge lgorithm lwys finds n optiml stte tht minimizes the system ongestion lod. Proof: The lgorithm strts with the mximl power stte tht domintes the optiml solution. From Theorem 1 follows tht s long s the lgorithm hsn t rehed the optiml stte the resulting stte, otined y ottlenek set power redution opertion, lso domintes the optiml solution. From Lemm 4 7

8 Q results tht during the exeution of the lgorithm the ongestion lod of the WLAN never inreses. So now we just hve to show tht the lgorithm stop with n optiml stte. Sine the numer of possile set power redution opertion is limited, The lgorithm must stop fter t most K¹ set power redution opertions. Now suppose tht the finl stte is not optiml. Sine the sequene of mximl lod h is non-inresing, we onlude Ò tht the lgorithm must hve stopped efore finding n optiml stte. The lgorithm stopped euse Å ontins n AP O with or Å. Rell tht in the first se, set Å power redution opertion nnot e done nd in the seond se, from Corollry 1 results tht suh opertion does not redue the ongestion lod. Thus, there is set ± tht does not ontin the ottlenek set Å nd its set power redution opertion redues the ongestion lod. However, from Lemm results tht suh set ± does not exist. It n e shown tht the omputtionl omplexity of the lgorithm is KŒ ë KŒ ST. Thus, from theoretil perspetive, the lgorithm hs pseudo-polynomil running time. In prtie, is smll vlue like, nd therefore, for ny prtil men, the lgorithm hs polynomil running time. E. The Limited Knowledge Algorithm We now present our limited-knowledge lgorithm tht finds n optiml stte in the limited knowledge se. Unlike the omplete knowledge se, we nnot lulte the ottlenek set in dvne. We overome this ostle y using Corollry 2. Aording to it, s long s network stte is su-optiml nd it domintes n optiml solution, sequene of set power redution opertions of ongested APs onverges to the optiml stte. This property rises the prolem of determining termintion ondition when n optiml solution is found. Without the termintion ondition, s demonstrted in Exmple 2, we my end up with su-optiml solution. To this end, we use n optiml stte reording pproh tht keeps reord of the network stte with the lowest ongestion lod found so fr. We define two vriles for reording. The first is tht keeps the reorded stte nd the seond is h tht keeps the ongested lod vlue of stte, termed the reorded ongestion lod. The lgorithm works s follows. It strts with the mximl power stte nd it initilizes the reorded stte,, nd the reorded ongestion lod, h, ordingly. Then, the lgorithm, itertively, lultes the set of ongested APs nd, s long s the set does not ontin ny AP with miniml power level, it performs set power redution opertion. After eh itertion the lgorithm evlutes the ongestion lod of the new stte nd if tht lod is lower thn the reorded ongestion lod, then the lgorithm updtes its vriles, nd h, orrespondingly. At the end, the lgorithm sets the AP power levels ording to the reorded network stte. A forml desription of the limitedknowledge lgorithm is given in Figure nd typil exeution is illustrted in Exmple 5 Exmple 5: Consider the WLAN used in Exmple 4. At the initil stte, ll APs hve the sme power index,, nd the initil user-ap ssoition nd the AP lod re shown in Figure 7-). Sine, œ is the ongested AP, the lgorithm redues its power index twie in two suessive itertions, s shown in Figures 7- ) nd 7-). After the first itertion the lod on œ is redued from A to nd the lgorithm keeps reord of this stte. After the seond itertion, eomes the ongested AP with n Alg LK Min Congestion Lod Alg ) for every AP x n ~µw z let ˆT +ŠŒ xµ ~vw Ž Èò óõô wö ;w ì"í q î ì/í { x s î/ï>ð ñ ì while q î { x s î/ï7ð ñ ì Ž ò óõô do wö ;w p Š x^æ x n ç ;w if exist x p ~ w s.t. ì"í ) then q î { x s Èÿ >ì // Set power indies of ll APs to z nd evlute AP lod. else end if end if end while ˆ return. end p à // Derese power indies of ll APs in y // nd evlute AP lod. for every AP x p ~ w let ~ w ã à ò óõô wö ;w if Ž then ˆ ŠŒ xv ~ waž Fig.. A forml desription of the limited-knowledge ongestion lod minimiztion lgorithm AP Lod: 7 12 : : 4 4 ) After seond itertion, D={} u ) initil stte, D={} AP Lod: u AP Lod: AP Lod: 9 1 : ) After first itertion, D={} : 4 4 d) After rd itertion, D={} 2 1 u 2 1 u Fig. 7. Exmple of n exeution of the limited knowledge lgorithm. ongested lod. In the third itertion the lgorithm redues its power index nd U ë hnges its ssoition ordingly. As result, œ eomes the ongested AP gin. Sine it trnsmits with miniml power, the itertion loop ends. At the end, the lgorithm onfigures the APs with the reorded stte, shown in Figure 7-). Theorem : The limited-knowledge lgorithm lwys finds n optiml stte tht minimizes the network ongestion lod. Proof: The lgorithm strts with the mximl power stte tht domintes ny optiml solution. Sine, the lgorithm keeps reord of the stte with the miniml ongested lod, we just hve to show tht it rehes n optiml stte efore it stops. Now suppose in ontrst tht the lgorithm hve not found n optiml stte during its exeution. Rell tht Corollry 2 lims tht s long s the lgorithm hven t rehed n optiml stte, the sequene of sttes otined y itertively reduing the power of the ongested APs lso domintes the optiml stte. Conse-

9 h h Ò 9 quently, the lgorithm stops with su-optiml stte tht domintes the optiml stte. However, the lgorithm hlts when ny ongested AP trnsmits with miniml power. Thus the lod on this AP nnot e redued y further power redution opertions. This implies tht either the finl stte is optiml or it does not dominte n optiml stte, whih ontrdits our ssumption tht the lgorithm stopped efore finding n optiml stte. V. FINDING A MIN-MAX LOAD ALANCED STATE The lgorithms presented in Setion IV minimize the network ongestion lod ut they do not neessrily lne the lod of the non-ongested APs, s demonstrted in Exmples 4 nd 5. In this setion, we onsider the min-mx lod lning solutions tht not only minimize the network ongestion lod ut lso lne the lod of the non-ongested APs. This ojetive is formlly defined in Setion V-A. Unfortuntely, this prolem is -hrd nd it is hrd to find even n pproximted solution. In spite of this, we introdue vrint of the min-mx prolem, termed min-mx priority-lod lning prolem, whose optiml solution n e found in polynomil time. We present our lgorithm for this prolem in Setion V-. Our solution is given for the limited-knowledge model, oviously, it n e used for omplete-knowledge model s well. A. The Prolem Sttement A ommonly used pproh to evlute the qulity of lodlning method is whether it genertes min-mx lod lned solution [14], [2]. Informlly, we sy tht network stte is min-mx lod lned if there is no wy to redue the lod of ny AP without inresing the lod of nother AP with sme or higher lod. We define the lod vetor, ` ŒK KŒKÞ `),, of stte to e the -tuple onsisting of the lod of eh AP sorted in deresing order. Definition 7 Min-Mx Lod lned Network Stte) : A fesile network stte is lled min-mx lod lned if its orresponding lod vetor ` ŒKŒK KŒ `º, hs the sme or lower lexiogrphil vlue thn ny other lod vetor ŒK KŒKÞ ` ±, of ny other fesile stte ±. In other words, if h h is the lod vetor of min-mx lod lned stte, there exists n index suh tht ` Ò ` ± nd for every index Ë, it follows tht `. ` ± h ± where We now show tht the prolem of finding min-mx lod lned stte is -hrd. Furthermore, we prove tht even simpler prolem, i.e., the prolem of identifying the miniml set of ongested APs for known miniml ongestion lod, is y itself -hrd. Theorem 4: Consider WLAN nd let h e known lower ound on its ongestion lod tht n e otined y the ell rething pproh. Then, identifying network stte tht minimizes the numer of ongested APs is -hrd, even for instnes with only two power levels. Proof: For the ske of simpliity, we ssume tht the lod generted y user on its ssoited AP my e zero. The prolem stys NP-hrd lso when the lod of user is stritly positive. We prove this theorem y reduing ny instne of the miniml dominting set MDS) [2] prolem to the onsidered stte seletion prolem. Rell tht the MDS prolem of given grph Î is known -hrd prolem tht seeks for the smllest suset ² suh tht every node in 4³ hs t lest one neighor in. Consider grph _ nd let à ` ± h ± e the set of nodes tht ontins node nd ll its neighors in the grph. We onstrut WLAN with users, denoted y S, nd È APs, denoted y, tht n trnsmit in one of two power level, i.e.,. For eh node P#, we define n AP O # nd user U #ÈS. User U n e ssoited with ny AP O " suh tht ± # #. If the power index of AP O is or ll APs O, ± #, hve power index, then user U is ssoited with AP O nd produes lod of. Otherwise, user U is ssoited with one of the other AP O, ± # $ ³ %, with power level nd it does not yield ny lod on this AP. Note tht in our onstrution the lod of n AP my e either or zero. We lim tht the grph &" '_ hs dominting set of size if nd only if there is network stte suh tht APs hve lod nd ll other APs hve lod. If the grph hs dominting set à of size, then we ssign power index to every AP O, #) nd to the other APs. Sine is dominting set, result tht ll users will e ssoited with APs O, #*. Thus, the numer of APs with lod is. Now suppose tht there is power level seletion tht yields APs with lod. Clerly, eh user U is ssoited with one AP O, ± # #. Thus we just hve to show tht the lod of this AP must e. If U is ssoited with AP O then, y definition, the lod of O is. Otherwise, U is ssoited with n AP O, ± # # ³W +¼,. Thus, the power index of this AP must e nd onsequently lso user U is ssoited with AP O. Therefore, the lod of AP O must e. From the ove follows tht for t lest one of the orresponding APs hs lod. In eh set other words, the APs with lod define dominting set of size for the grph nd this ompletes our proof. Corollry : The prolem of finding min-mx lod lned stte is -hrd. The ove proofs re sed on the redution from the miniml dominting set MDS) prolem. Rell the MDS prolem is not just hrd to lulte, it is lso hrd to pproximte nd there is no lgorithm tht n find n pproximtion rtio smller then œ kilem¼õè, for some œ &, unless Æ [25]. y using this redution nd the hrdness property of the MDS prolem, it n e shown tht our min-mx lod lning prolem is lso hrd to pproximte. Theorem 5: There exists no polynomil lgorithm tht n ensure? -oordinte-wise pproximtion solutions to our minmx lod lning prolem for ny?, unless ÍV. Theorem : There exists no polynomil lgorithm tht n ensure? -prefix-sum pproximtion solutions to our min-mx lod lning prolem for? œ KHk)l^m, for some œw, unless Í. In spite of the -hrdness, we now turn to present vrint of the min-mx lod lning prolem tht its optiml solution n e lulted in polynomil time. In this prolem, we ssume tht eh AP OW# hs unique priority, lso termed weight,, # %º^')'û *, tht indites the AP s importne. In this study we do not ddress the prolem of lloting priority to the APs. In the following we give new AP lod definition, termed priority lod, for this prolem. Definition A priority lod of n AP) : Consider n AP O # with priority, nd let [^daf g ] [ e the ggregted lod of ll of its ssoited users. The priority lod of AP O, denoted y `, is defined s the ordered pir `,. For simpliity we refer to n AP priority lod y the AP lod. We sy tht AP O hs higher lod thn AP if ` \ hs -,

10 higher lexiogrphilly vlue thn ` É -, õ, i.e., one of the following two ondition stisfied: 1) V, or 2) ½ nd,,9. Thus, our new ojetive is finding network stte tht provides min-mx priority lod lned solution. Sine there re no two APs with the sme priority, results tht there re no two AP with the sme priority) lod. This ensures the following useful property. Property With the priority lod definition, t ny network stte, the set of ongested APs lwys ontins single AP.. The Min-Mx Algorithm We now present our min-mx lgorithm for the limited knowledge model. The lgorithm itertively finds min-mx priority-lod lned stte tht yields the optiml lod vetor, h. At ny itertion, #%º^')'û È *, we ll routine, similr to the lgorithm presented in Setion IV-E, to lulte network stte tht minimizes the priority-lod of the -th oordinte of the lod vetor. The routine needs to stisfy two requirements: Requirement 1): The initil stte of eh itertion,, must dominte the optiml stte. Requirement 2): The lulted network stte t the -th itertion should not ffet inrese) the lod of the APs tht their lod hve lredy een determined y the previous itertions. To meet Requirement 1), the lgorithm strts with the mximl power stte in the first itertion nd we need to ensure tht eh itertion ends with dominting stte of the optiml solution. Moreover, to meet Requirement 2), we define set of fixed APs,., whose lod hve lredy een determined y previous itertions. Initilly, the set. is empty nd t eh itertion new AP is dded to it, until. ontins ll the APs. We define the ongesting lod, h, s the mximl lod on ny non-fixed AP. From Property 1 follows tht t ny given time there is only single non-fixed AP, termed the ongested AP, /, tht rries the ongesting lod. At eh itertion the lgorithm invokes the ä 1 ËÁCË Ë2HÐ Âv 4 /^ËÁLOHÏ Ð routine for minimize the -th oordinte of the lod vetor nd let s ssume tht Requirement 1) is stisfied t the eginning of the itertion. The routine uses three reording vriles. The reorded ongestion lod, h, keeps the ongested lod vlue of the optiml stte found so fr. The reorded stte vrile,, reords the first disovered stte with ongestion lod h. While, the reorded ongestion AP, /, identifies the ongested AP with this lod. At the eginning, the routine initilizes the reording vriles, h nd / with the itertion initil stte, its ongested lod nd the ongested AP, ordingly. Then, the routine, itertively, lultes the ongested AP / nd it stops if AP / is lredy trnsmitting with miniml power level. If not, the routine performs power redution opertion of / nd evlutes the ongestion lod h s well s the lod of the fixed APs in the new stte. It stops if one of the fixed APs suffers from elevted lod. This ensures Requirement 2) to preserve the lod of the fixed APs. Otherwise, if stte with lower ongested lod is disovered the routine keeps reord of this stte y updting the reording vriles. At the end, the routine sets the AP power levels ording to the reorded network stte nd returns the reorded stte nd the orresponding ongested AP /. The lter is dded to the set of fixed APs nd the lgorithm invokes the routine gin for minimizing the lod of the Aä³XÏ-5 n Alg Min Mx Priority Lod lning Alg ˆ +ŠŒ ) xv ~vw z Ž æú x n r Ö r while Ù n Ž Iˆ do Áq Ž LK Minimize m Coordinteˆ r ) r r å Š q end while ˆ return end Routine LK Minimize m Coordinteˆ غáغâ r ) ˆ Ȉ Ø)á Ø)â ò óõô wö Õ7 ;w q q x9 ƒ ú ƒ x n ã r ç ;w ì"í q î { x s î/ï7ð ñ ì while ì/í q î { x s î/ï7ð ñ ì Ž do if ~;: ) then ì/í q î { x s ÿ < >ì else ~ : ~ : ã à // Power redution & lod evlution. Èò óõô w ö Õ7 ;w q x= ƒ ú ƒ x n ã r ç w if exist x r s.t. ;w ;w ì/í ) then q î { x s Èÿ < >ì else if Ž then ˆT ŠŒ xv ~vw Ž- q q end if end if end while ˆ Return, q ). end // Chek if fixed AP lod ws inresed. // Chek if etter network stte ws found. Fig.. A forml desription of the Min-Mx priority-lod lning lgorithm. AP Lod: : 4 ) After first itertion, F={} 2 1 u AP Lod: : 4 ) After seond itertion, F={,} 2 1 u Fig. 9. Exmple of n exeution of the min-mx priority lod lning lgorithm. oordinte. A typil exeution is illustrted in Exmple nd forml desription of the lgorithm is given in Figure. Exmple : Consider the WLAN used in Exmple 4. In this se, the first invotion of the ä 1 ËÁCË Ë2HÐ Âv 4;/^Ë-LOHÏ Ð routine returns the network stte depited in Figure 9-). As demonstrted in Exmple 5, this stte domintes ny other stte tht minimizes the first oordinte of the lod vetor. The seond invotion of the 1 1 ËÁCË Ë>2HÐ Âv 4;/eËÁLOHÏ Ð routine returns the stte shown in Figure 9-), whih is the only minmx lod lned stte of this network. We now prove tht the proposed lgorithm finds the optiml lod vetor. Lemm 5: Consider ny itertion tht stisfies Requirement 1). Then, t the end of the itertion, the reorded stte minimizes the -th oordinte of the lod vetor, without hnging the lod of the fixed APs. It lso stisfies Requirement

11 ` ` ` ` h ` 11 1) t the eginning of the -th itertion. Proof: Eh itertion strts y preserving the initil stte nd it stop when the lod of fixed AP hs hnges. Thus t ny given time mintins the lod of the fixed APs. Sine -oordinte. Thus we only hve to show tht is the smllest deteted ongestion lod, lso minimizes the preserves Requirement 1) t the eginning of the -th itertion. From our ssumption, results tht the itertion strts with stte tht domintes the optiml solution. is the first deteted stte tht minimizes the ongestion lod nd stisfies Requirement 2). Thus, it domintes ny stte tht minimizes the first oordintes nd, onsequently, it stisfies Requirement 1). Theorem 7: The lgorithm lwys finds min-mx prioritylod lned solution. Proof: This is diret result from indutive pplition of Lemm 5 for ll of the oordintes of the optiml lod vetor. As finl point, it n e shown tht the omplexity of the lgorithm is K^ KST. VI. THE ON-LINE STRATEGY Exeution of the optimiztion lgorithms desried in the previous setions eh time user rrives or deprts my use frequent ssoition hnges nd potentil disruption of on-going user sessions. To void this, we propose n on-line strtegy tht strikes lne etween the frequeny of the ssoition hnges nd the optimlity of the network stte in terms of lod lning. Our on-line strtegy is omintion of the glol optimiztion, whih re desried in the previous setions, nd lotion optimiztion. The lol optimiztion dels with dynmi user rrivls nd deprtures, while the glol optimiztion is invoked periodilly or whenever the lol optimiztion fils to mintin lod-lned stte. The on-line strtegy uses three onfigurtion prmeters whih re miniml lod threshold, ell dpttion threshold?, nd time nd? determine the invotion ondition of the lol optimiztion to prevent the lol optimiztion from unneessrily invoked for negligile gins, where frequent invotions my use servie interruption to tive ontrols the frequeny of the periodi invotion of the glol optimiztion. The lol optimiztion lgorithm is fundmentlly different from the glol optimiztion lgorithm in tht it not only dereses the AP power level ut lso inreses it. The detils re given elow. For eh AP OP#Y, we define set of ll of its neighoring APs nd let A e the verge lod of the APs in. When the lod of n AP O redues for whtever reson e.g., user movements or lol optimiztion opertions y other APs), the lgorithm heks whether the new lod ` stisfies the ell enlrgement ondition, whih is if ny AP é# hs `^ & nd lso ` ³? RKCA. If this ondition is met nd the power index, Q, of the AP O is not mximl, i.e., Q, the lgorithm inreses the AP O s power level y one. On the flip side, when the lod of the AP O inreses, the lgorithm heks the ell redution ondition, whih is if ` & nd lso. If the ondition is met nd the power in- Mê\D?é9KEA dex of the AP O is not miniml, the lgorithm redues the AP O s power level y one. If ny of the two onditions is stisfied ut the lol lgorithm nnot djust the AP power level, the glol optimiztion lgorithm is invoked. In ddition, the glol optimiztion periodilly is lled in time units. VII. SIMULATION RESULTS In this setion we ompre the performne of our sheme with two existing methods, whih re the Strongest-Signl-First SSF) method nd the ssoition ontrol method proposed in [14]. The SSF method is the defult user-ap ssoition method in the IEEE 2.11 stndrd nd is the sme s our sheme with single power level. The method in [14] determines the user-ap ssoition to hieve the mx-min fir ndwidth llotion insted of mking ssoition deisions purely sed on the signl strength. Sine the mx-min firness prolem is NP-hrd, it first lultes the frtionl optiml solution whih we ll FRAC) under the ssumption tht user n simultneously ssoite with multiple APs, nd then otins the integrl solution whih we ll INT) vi rounding to stisfy the single ssoition onstrint. This prtiulr method is hosen s performne enhmrk euse the hrteristis of its solutions re known. The FRAC solution provides the strit performne upper ound i.e., lowest possile ongested lod) nd the INT method gurntees 2-pproximtion solution 5. Furthermore, s shown in [14] the integer solution INT) onverges with the frtionl one FRAC) s the numer of users inreses. Thus, one n not expet the proposed ell-rething sheme to outperform the optiml ssoition ontrol method tht hs signifintly higher degrees of freedom thn the former. The ojetive of the omprison with the INT solution is to show tht our ellrething hieves omprle performne to the ssoition ontrol method, without requiring speil lient softwre for ssoition ontrol on eh moile. Surprisingly, however, the simultion results indite tht the ell-rething sheme outperforms the INT solution in vrious lod onditions. The simultion setting is s follows. A totl of 2 APs re loted on 5 y 4 grid, where the distne etween two djent APs is set to 1 meters nd eh AP is equipped with 1 Mps khul link. Assuming tht pproprite frequeny plnning ws mde to eliminte the interferene mong APs, we simulte the IEEE 2.11 wireless link s follows. To determine the it rte etween user nd n AP, SNR is omputed nd the it rte is» hosen ordingly. The 11 Mps it rte is used when SNR»» 9 d, 5.5 Mps when SNR» 5 d, 2 Mps when SNR d nd 1 Mps when SNR 1d. We set the mximl trnsmission power to µ dm nd the miniml power to dm, while the intermedite power levels re determined y eqully dividing the dm gp y the numer of power levels simulted. Unless speified otherwise, 1 power levels re used. To simulte the indoor environment, we hose the pth loss exponent [27] of ž' ž, so tht the pth loss is omputed y Ý/H Ÿ^ 1³P 1K žn' ž1k k)l^mf/, where/ is the distne etween user nd n AP. The kground noise level is set to ³HGež dm. Under this hnnel model, the rnge of ell with the mximl power level is 15 meters nd tht with the miniml power level is 75 meters. Therefore, even with the miniml power level, the network hs no overge hole. All users re ssumed ompletely k-logged nd qusi-stti. The priority of eh AP is rndomly hosen. We orrow the system performne metri from [14] nd define the lod of n AP O y, \JI [^d7k g L Due to integrlity gp, 2-pproximtion is gurnteed only when the AP lod is ove ertin threshold whih is 1 in our setting). 4 [^]

12 SSF Min-Congestion Min-Mx INT FRAC SSF Min-Congestion Min-Mx INT FRAC. 1.5 Y. Y AP Index AP Index Fig. 1. Lod omprison in networks with à ; rndom users. Fig. 12. Lod omprison in hotspot networks with à ; users. Y AP Index SSF Min-Congestion Min-Mx INT FRAC Fig. 11. Lod omprison in networks with M rndom users. where 4 [^] with the AP O nd N is the it rte with whih the user U ommunites is the set of users tht re ssoited with AP O. Due to spe limittion, we provide only few hrts with typil results of our simultions. Figure 1 shows the simultion results when 1 users re rndomly distriuted within the retngulr ox tht onnets the oundry APs. 1 users were hosen to simulte modertely-loded network in whih the rtio of APs to tive users is O. The Y xis represents ` i.e., the AP lod) nd the X xis represents the AP index. Note tht the APs re sorted y their ` vlues in deresing order. Eh ` vlue is otined y verging simultion runs. Tht is, the ` for the AP index P is omputed y verging the ` of the P -th highest-loded AP of eh run. Only the points orresponding to the integer P indies re meningful ontinuous lines re drwn only for the presenttion purpose). The thik solid line represents our Min-Mx method nd the thin solid line represents our Min- Congestion method. While oth methods lwys generte the sme mximl ` vlue, it is shown tht the Min-Mx method genertes lod vetor with lower lexiogrphil order thn the Min-Congestion method. The thik dotted line represents the INT solution nd the horizontl thin dotted line represents the FRAC solution. oth the Min-Mx nd INT solutions re lerly etter thn the SSF solution. oth yield very similr mximl ` vlues, whih re out žqo7r higher then FRAC. We lso simulted the se of 5 users nd the result is shown in Figure 11. Now the gp etween the INT nd the FRAC is igger thn in the se of 1 users, nd our Min-Mx method outperforms the INT method with signifint mrgin. It is euse the performne of the INT method depends on the numer of users. Generlly less users mke the INT solution frther from the FRAC solution due to the igger rounding error. Interestingly, the reltive performne of our Min-Mx method ginst the FRAC method nd the SSF method, i.e., the gp etween Min-Mx nd FRAC nd the gp etween Min-Mx nd SSF, is not drstilly ffeted y the numer of users. The sme trend ws oserved in other simultion settings. For exmple, in the se of 2 users not shown due to the spe limit), the gp etween FRAC nd the our sheme stys round žqo7r. The stedy reltive performne ginst the FRAC solution regrdless of the network lod onditions i.e., numer of users) is one of the key strengths of the ell-rething method. We then onsider the se of unlned user distriutions. Only 2% of the users re rndomly distriuted nd the rest re onentrted on two hotspots whih do not overlp with eh other. Eh hotspot is irle-shpe re with 75 meter rdius, nd one hotspot ontins twie more users thn the other hotspot. This setting uses the hevy lod ondition in the hotspots. Figure 12 show the results when the totl numer of users is 1. It shows tht the Min-Mx method performs even etter in the presene of hotspots i.e., hevy lod ondition) nd lerly ets the INT method. So fr, we hve ssumed 1 power levels. To exmine the impt of the numer of power levels, we tried 4 different numers of power levels. The simultion results in the 1 rndom user se is shown in Figure 1. The impt of the numer of power levels eomes mrginl eyond ertin numer of power levels, whih is etween 5 nd 1 in the urrent simultion setup. While the omplete knowledge lgorithms ompute the optiml solutions t one shot, the limited knowledge lgorithms grdully hnge the power setting until they find the optiml solutions. To pture the overhed of the limited knowledge lgorithms, we ounted the frequeny of power djustments nd onsequent user movements during the simultion. The numer of power djustments determines the time to tke efore the system onverges on the optiml stte nd the numer of user movements deides the hndoff overhed. The olleted sttistis re summrized in Tle II. For eh tle entry, two numers re given; the first numer is for the limited knowledge Min-

13 1 Y AP Index 1 levelssf) 2 levels 5 levels 1 levels 2 levels Fig. 1. Impt of the numer of power levels on the performne of the Min- Mx method Cse Power djustments User movements 1 rndom 12.9.) ) 2 rndom ) ) 1 hotspot ) 94.4.) 2 hotspot ) ) TALE II RUN-TIME STATISTICS OF LIMITED KNOWLEDGE ALGORITHMS. formt: MIN-MAX MIN-CONGESTION) Mx method nd the seond numer is for the limited knowledge Min-Congestion method. Generlly the Min-Congestion method onverges firly quikly, while the Min-Mx method tkes little longer. For instne, if the power djustment intervl is 1 seond, the lod-lning in 1 rndom user network tkes out seonds y the Min-Congestion method, nd less thn 2 minutes y the Min-Mx method. The inrese of the user numer seems not neessrily inresing the onvergene time. The presene of hotspots does not neessrily men longer onvergene time either nd in the se of Min-Congestion, it even tkes less time to onverge. VIII. CONCLUSION We presented novel ell rething sheme for optiml lod lning in IEEE 2.11 networks. We provided rigorous nlysis of the prolem nd presented two lgorithms tht find network-wide deterministi optiml solutions. The first lgorithm minimizes the lod of most ongested APs) in the network, nd the seond lgorithm produes n optiml min-mx priority) lod lned solution. These optiml solutions re otined only with the miniml informtion whih is redily ville without ny speil ssistne from the users or modifition of the stndrd. We only ssumes the ontrol on the trnsmission power of the AP eon messges, whih should e possile with simple softwre updte of APs. The simultions show tht even smll numer of power levels, e.g., etween O to, is enough to hieve ner optiml lod lning solutions, whih is nother indition of the prtility of our sheme. In prtie, our ell-rething sheme n e deployed in network mngement tool of WLANs nd tivted eh time the APs experiene unlned lod. REFERENCES [1] M. lzinsk nd P. Cstro. 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