Optimum Association of Mobile Wireless Devices with a WLAN 3G Access Network

Transcription

1 Optmum Assocaton of Moble Wreless Devces wth a WLAN 3G Access Network K. Premkumar Appled Research Group Satyam Computer Servces Lmted, Bangalore Emal: kprem@ece.sc.ernet.n Anurag Kumar Dept. of Electrcal Communcaton Engneerng Indan Insttute of Scence, Bangalore Emal: anurag@ece.sc.ernet.n Abstract In ths paper, we consder the problem of assocaton of wreless statons STAs wth an access network served by a wreless local area network WLAN and a 3G cellular network. There s a set of WLAN Access Ponts APs and a set of 3G Base Statons BSs and a number of STAs each of whch needs to be assocated wth one of the APs or one of the BSs. We concentrate on downlnk bulk elastc transfers. Each assocaton provdes each STA wth a certan transfer rate. We evaluate an assocaton on the bass of the sum utlty of the transfer rates and seek the utlty maxmzng assocaton. We also obtan the optmal tme schedulng of servce from a 3G BS to the assocated STAs. We propose a fast teratve heurstc algorthm to compute an assocaton. Numercal results show that our algorthm converges n a few steps yeldng an assocaton that s wthn 1% n objectve value of the optmal obtaned through exhaustve search; n most cases the algorthm yelds an optmal soluton. I. INTRODUCTION In thrd generaton 3G wreless cellular networks, moble multmeda servces requrng data rates of the order of 100s of Kbps are envsaged. The sophstcated, wde coverage nfrastructure of wreless cellular networks s expensve and t s dffcult to meet the ever ncreasng traffc demands. Wreless Local Area Networks WLANs, on the other hand, are easy to nstall and can coexst wth a cellular network. WLANs are beng wdely deployed n publc areas, such as campuses, hotels and offces. Thus, a seamless ntegraton of WLANs and 3G cellular networks s mmnent. In ths paper, we focus on the problem of optmal assocaton of a number of wreless statons STAs wth an access network comprsng WLAN Access Ponts APs and 3G Base Statons BSs. In many practcal wreless systems ncludng the IS-856 HDR system, an STA s assocated wth the BS from whch t can receve the hghest sgnal to nterference plus nose rato SINR. In WLANs, each STA detects ts nearby APs and assocates tself wth the AP from whch t has the strongest receved sgnal strength, whle gnorng load consderatons. In ths paper, we formulate an assocaton problem n whch STAs assocate wth BSs or APs so as to maxmze a network objectve functon. Recently, there has been consderable work done n the area of optmal assocaton and resource allocaton [1],[10],[14],[3]. Hanly [10] and Yates and Huang [14] studed the base staton assgnment problems for a CDMA cellular system. They provded a combned power control and cell ste selecton algorthm that mnmzes the transmt power whle meetng the uplnk target SINR. Lee, Mazumdar and Shroff [11] studed the downlnk rate and power allocaton and base staton assgnment problem, agan for a CDMA cellular system. They formulated the power allocaton and base staton assgnment problem based on a prcng approach. Kumar and Kumar [1] formulate the optmal assocaton problem for WLANs as a sum utlty maxmzaton problem and obtan solutons for some smple stuatons. Bejerano et. al. [3] also consder the same problem. They propose the dea of fractonal assocaton, formulate a lnear programmng LP problem and relate the result of the LP problem to the nteger soluton usng graph theoretc deas. In ths paper, we study the assocaton and schedulng problem when there s a heterogenous access nfrastructure comprsng 3G BSs and WLAN APs. Several STAs seek to connect to a 3G cellular network or to a WLAN wth the objectve of performng elastc downloads. We formulate the assocaton problem as a cooperatve sum utlty maxmzaton problem. The utlty obtaned by each STA s the of the TCP bulk download rate obtaned by the STA n an assocaton. Our work dffers from [10] and [14] snce these papers are concerned wth fxed rate servces each wth a target E b /N 0 ; ths yelds the problem of settng transmtter power so as to acheve target carrer-to-nterference rato CIR. In [11] the problem of jont power and rate control whch maxmzes the downlnk expected throughput s studed. Our present work s an extenson of the work reported n [1] where the problem of optmal assocaton wth WLAN APs was consdered, whle here we consder optmal assocaton n a heterogenous network of WLAN and 3G cellular network. We formulate the problem and propose an teratve greedy search algorthm for obtanng a good assocaton n a few teratons. We also gve a suffcent condton to check f the assocaton s optmal. An extensve numercal study of our algorthm s provded. The rest of the paper s organzed as follows. In Secton II, we descrbe the WLAN and 3G system wth respect to the assocaton, physcal rates, and the throughput. We defne the system utlty and state the optmal assocaton problem based on the system utlty n Secton III. We analyze the optmum assocaton problem n Secton IV. We present the algorthm to determne a good assocaton n Secton V. A suffcent condton on the optmalty of the soluton gven by /06/$20.00 c 2006 IEEE Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 2006 proceedngs. Authorzed lcensed use lmted to: IEEE Xplore. Downloaded on February 21, 2009 at 03:32 from IEEE Xplore. Restrctons apply.

2 the algorthm s also dscussed. We provde numercal results n Secton VI. Secton VII concludes the paper. II. ASSOCIATION, PHYSICAL RATES, AND THROUGHPUT There are l WLAN APs ndexed by the set L = {1, 2,...,l} and b BSs ndexed by the set B = {l +1,l+2,...,l+ b}. Let the set N be L B.Letn = l + b. Therearem STAs ndexed by the set M = {1, 2,...,m}. Leta N be the AP or BS to whch STA s assocated. Let A =a 1,a 2,...,a m denote an assocaton vector,.e., A A= N m = {a 1,a 2,...,a m : a N, M}.LetS j be the set of STAs assocated wth AP/BS j, j N and let = S j, be the number of STAs assocated wth AP/BS j, j N. We focus on downlnk elastc bulk data transfers to the STAs e.g., TCP controlled fle transfers. We assume the channel gans to be statc over the tme scale of the optmzaton. Gven an assocaton A = a 1,a 2,...,a m,letr a be the raw physcal data rate wth whch STA s assocated wth a, a N a Lor a B. Snce we are consderng a CDMA cellular network, f a B, then n general r a wll depend on the nterference and hence the assocatons of the other STAs to BSs. A. Physcal rates: WLAN AP to a STA Let STA be assocated wth AP j at the raw physcal data rate r j C. For example, IEEE b supports the set of physcal data rates C = {1, 2, 5.5, 11} Mbps. These rates are acheved dependng on the dstance between the AP and the STA. For smplcty, we assume that the WLAN coverage s such that an STA that can be assocated wth a WLAN AP does so at the maxmum physcal data rate. We also assume that the cochannel APs are placed suffcently far apart such that there s no ntercell cochannel nterference [12]. B. Physcal rates: 3G BS to a STA Let G j be the channel gan whch ncludes dstance loss, shadowng and fadng between STA and BS j. LetP j be the power transmtted from BS j to STA and let P j be the total power transmtted from BS j to STAs S j.let P T be the maxmum transmt power of any BS. If N 0 s the one sded nose power spectral densty and W s the system bandwdth, then the receved downlnk SINR of STA when assocated wth BS j s gven by Γ j = G j P j k =j G kp k + N 0 W. 1 assumng that the users of the same cell use orthogonal spreadng codes, and gnorng ntracell nterference caused by multpath. We follow the model gven n [13] for computng the transmsson rate as a functon of SINR and system bandwdth. If the physcal rate r j s used from BS j to STA then the energy-per-bt to nose power spectral densty rato of STA when assocated wth BS j s gven by W r j Γ j 2 In order for STA to be able to decode the BS s sgnal wth an acceptable probablty of error ɛ, t s necessary that the energy-per-bt to nose power spectral densty rato exceeds γ, where, for a gven dgtal modulaton scheme, γ = γɛ s a threshold whch s determned by the probablty of error ɛ. Then from Eqn. 2, the achevable physcal rate s gven by r j = W γ Γ j = αγ j 3 where α = W/γ. It should be noted from Eqn. 3 that, wth γ fxed, the achevable physcal rate r j between STA and BS j s lnearly proportonal to the SINR Γ j. C. TCP Bulk Throughput We consder persstent, TCP controlled local area elastc data transfers to all m users. Let θ j be the TCP throughput to STA when assocated wth AP or BS j from server n the local area. Assumng that when an STA assocates wth an AP t does so at a fxed rate e.g., 11 Mbps and the packet error rate s neglgble, t can be shown that the TCP bulk transfer download throughput s of the form θ j = Θ 0 / f STAs are assocated wth the AP wth whch STA s assocated, where Θ 0 s a number that depends on the IEEE MAC parameters and the TCP packet length see [8] & [9]. For example, wth a physcal data rate of 11 Mbps, TCP packet sze of 1500 B, RTS/CTS for data packets and basc access for TCP acks, Θ 0 = 4.3 Mbps. In a pedestran or statc scenaro, when an STA s assocated wth a BS j we assume that the target E b /N 0, γ s acheved and hence the packet error rate s suffcently low e.g., Then θ j = r j. In the case of a 3G cellular network, t s shown n [2] that the optmal sum physcal rate can be acheved f each BS transmts to only one data user at a tme regardless of the topoy of the network,the presence of voce users n the cell, and user locatons and wth maxmum power when there are only data users and no voce users,.e., the multple users assocated wth BS j are served n a tme dvson multple access TDMA mode. It s to be noted that HDR uses TDMA. We thus take P j = P T, M, j B and P j = P T, j Band a slotted system for 3G cellular network where the users are served one at a tme [2]. Let be the fracton of tme STA s servced by BS j. Snce, we consder local area bulk TCP throughput for low packet error rates, θ j = r j s the average TCP throughput that STA gets on assocaton wth BS j. Note that under ths model r j does not depend on the assocaton. We defne an m n matrx R, wherer j, M, j B, represents the physcal rate that STA gets from BS j and r j =Θ 0, M, j Ls the bulk TCP throughput that STA gets when t s the only STA assocated wth an AP. III. THE DOWNLINK OPTIMAL ASSOCIATION PROBLEM When user s assocated wth AP or BS j and obtans throughput θ j we evaluate the utlty obtaned by user as 2003 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 2006 proceedngs. Authorzed lcensed use lmted to: IEEE Xplore. Downloaded on February 21, 2009 at 03:32 from IEEE Xplore. Restrctons apply.

3 U θ j where U. s an ncreasng concave functon. We defne the system utlty as the sum of the ndvdual utltes of STAs each of whch s assocated wth ether a BS or an AP. We are nterested n fndng the optmal assocaton A that maxmzes the system utlty. A = argmax U θ a 4 M Based on the dscusson n Secton II-C, Eqn. 4 can be wrtten as A =argmax U θ j+ U θ j S j j B S j =argmax U + r j U j B S j S j =1, j B 5 The varables, j B, model the TDM schedulng of STAs assocated wth BS j. In ths paper, we consder U = see [1]. Thus, we have the optmum assocaton problem A =argmax Remarks 1: + j B S j r j S j =1, j B 6 It should be noted that for computng the optmal assocaton, the rates r j, M, j N should be known at a central devce. Note also that the soluton of ths problem wll also yeld,j B, S j. IV. ANALYSIS OF THE OPTIMUM ASSOCIATION PROBLEM In ths secton, we study the optmzaton problem Eqn. 6 we posed n Secton III. We dvde the above problem nto two sub-problems. The frst sub-problem s fndng the optmum partton of m STAs nto two groups, one comprsng STAs assocated wth the 3G cellular network, and the other comprsng those STAs assocated wth the WLAN. The second subproblem s fndng the optmum assocaton n each of the parttons. Let the subset of STAs to be assocated wth the WLAN be S W and let S W = m W. Also, let the subset of STAs to be assocated wth the 3G cellular network be S C where S C = M S W. The frst sub-problem wll yeld S W and S C. The second sub-problem requres only m W for the WLAN. But, S C s requred as a whole for the assocaton wth the 3G cellular network. A. Sub-Problem: Optmum Assocaton among APs Gven m W, we need to fnd the optmal m 1,m 2,...,m l such that m 1 + m m l = m W. For the utlty functon, we have max m 1,m 2,...,m l = m W {0, 1, 2,...,m W }, j L 7 We have an nteger programmng problem. Let us replace by /m W. Notce that ths does not change the optmzaton problem. We relax the nteger constrant by settng x j = /m W and lettng x j [0, 1]. Ths yelds the followng relaxed problem whose soluton wll provde an upper bound to the maxmzaton problem defned n Eqn. 7. max x j x 1,x 2,...,x l x j =1 x j x j 0, j L 8 The above problem s a concave maxmzaton problem wth lnear constrants. Solvng ths problem gves x j =1/l. Hence, m j = m W /l, j Lsolves the orgnal problem, defned n Eqn. 7, whenever these are ntegers. In such a case, the optmum s to equally dvde the STAs over the APs. B. Sub-Problem: Optmum Assocaton among BSs Gven S C,defnem C = S C. We need to fnd the optmal S j, j B, such that j B S j = S C. For the utlty functon, we have r j max Φ,S j, j B S j j B S j =1, j B 0, S j, j B j B S j = S C S j S k =, j k, j B,k B 9 where Φ =[ ], S j, j B. The above maxmzaton problem can be resolved nto two maxmzaton problems one over Φ and the other over the parttons S j, j Bof S C. Gven a partton of S C nto S j, j B, the maxmzaton problem over Φ, max Φ S j j B S j =1, j B r j 0, S j, j B Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 2006 proceedngs. Authorzed lcensed use lmted to: IEEE Xplore. Downloaded on February 21, 2009 at 03:32 from IEEE Xplore. Restrctons apply.

4 s a concave maxmzaton problem n s wth lnear constrants. It s easly seen that solvng ths problem yelds =1/ S j, S j, j B. To solve the second maxmzaton problem, we can now wrte Eqn. 9 as max S j, j B rj j B S j j B S j = S C S j S k =, j k, j, k B 11 The soluton to the above problem yelds the optmal S j, j B. Note that ths s a nonlnear combnatoral optmzaton problem, snce S j s are dscrete sets. C. Optmal Assocaton among APs and BSs: A Specal Case Motvated by the above dscusson, n ths subsecton we consder a specal scenaro of a smplfed formulaton. We assume that all the STAs are clustered e.g., as n an audtorum. In ths scenaro, we assume that all STAs see the same channel to BS j. Thus, for j B, we denote r j, Mby the common value Θ j. We smplfy the formulaton as follows. Motvated by the result n Secton IV-A, we seek a number m 0 0 of STAs that wll be assocated wth each AP, and a number of STAs that wll be assocated wth BS j. Ths yelds the followng problem. U = max m 0,, j B lm 0 lm 0 + j B = m m 0 + Θj j B m 0,, j Bare non negatve ntegers. 12 We relax the above nteger constrants by takng /m = x j. The soluton of the relaxed problem wll provde an upper bound to the maxmzaton problem defned n Eqn. 12. Thus, the above optmzaton problem can be wrtten as U = mn x 0,x j, j B lx 0 x 0 + j B x j x j lx 0 Θ 0 x j Θ j j B lx 0 + j B x j =1 x 0 0,x j 0, j B 13 The above objectve s a convex functon of x j s and the constrants are lnear. Solvng the above problem gves the optmal soluton, x 0 = x j = Θ 0 lθ 0 + k B Θ k Θ j 14 lθ 0 + k B Θ. 15 k Hence consder, for j B, m 0 = m j = m lθ 0 + k B Θ Θ 0 16 k m lθ 0 + k B Θ Θ j. 17 k If m 0 and m j, j B, are ntegers then the above soluton s optmal and the STAs are dstrbuted over the APs and the BSs n proporton to the Θ j values. Even though Θ 0 s qute large, 4.3 Mbps for 11 Mbps physcal data rate [9], we see that a postve number of STAs are assocated wth the BSs. We look at the followng examples to understand the above scenaro. Example 1: We take m =6, l =2, and b =2.Letus consder the scenaro where Θ 0 = Θ 1 = Θ 2 = 4 Mbps, Θ 3 = 2Mbps, and Θ 4 = 2 Mbps. We obtan the optmal assocaton vector, A by enumeratng all possble assocatons. The optmal utlty based assocaton, A s found to be [ ]. It s noted that under optmal assocaton A, AP1 and AP2 servce two STAs and, BS3 and BS4 servce one STA each. Ths result s n conformance wth that of Eqns. 16 and 17. It s to be noted that the resultng m 0 and m j gven by Eqns. 16 and 17 are ntegers n ths case. If the STAs are assocated wth APs/BSs based on the rates r j as n HDR then the resultng assocaton A 1 assgns three STAs wth AP1 and the remanng three wth AP2. Though the rates that ndvdual STAs get when assocatng wth APs seem to be more, t s benefcal to assocate some STAs wth BSs as ths ncreases both the ndvdual throughput of the STAs and the system utlty. In assocaton A 1, the throughput of an STA s 1.33 Mbps whereas n assocaton A, the throughput of an STA s 2.0 Mbps. The system utlty for assocaton A 1 s whereas the utlty correspondng to the optmal assocaton, A s Example 2: We take m =9, l =2, and b =2.Letus consder the scenaro where Θ 0 = Θ 1 = Θ 2 = 4 Mbps, Θ 3 = 2 Mbps, and Θ 4 = 1 Mbps. The optmal assocaton vector, A obtaned through enumeraton for ths case s [ ].UnderoptmalassocatonA,AP1 and AP2 servce three STAs and, BS3 servce two STAs and BS4 servce one STA. In ths case, the m 0 and m j gven by Eqns. 16 and 17 are not ntegers and the optmal number of STAs assocated among the APs and BSs are dstrbuted n the rato of 3:3:2:1 nstead of 4:4:2:1. Ths soluton s obtaned by exhaustve search. We consder the assocaton A 1 whch assocates based on the rates r j four STAs wth AP1 and the remanng fve wth AP2 and compare t wth that of utlty maxmzng optmal assocaton, A. Ths example also shows that t s benefcal to assocate wth BSs as ths ncreases ndvdual STA s throughput and the overall system utlty. In assocaton A 1, the throughput of an STA s ether 0.8 Mbps or 1.0 Mbps whereas n assocaton A, the throughput of an STA s 1.0 Mbps or 1.33 Mbps. The system utlty for assocaton A 1 s whereas the utlty correspondng to the optmal assocaton, A s Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 2006 proceedngs. Authorzed lcensed use lmted to: IEEE Xplore. Downloaded on February 21, 2009 at 03:32 from IEEE Xplore. Restrctons apply.

5 V. A GREEDY SEARCH ALGORITHM Here, we provde an algorthm to compute a near optmal assocaton of STAs wth the access network comprsng 3G BSs and WLAN APs. We provde the followng theorem whch gves a suffcent condton for an assocaton to be optmal. Theorem 1: For a gven rate matrx, R, f the assocaton vector A satsfes θ a θ j, M, j N, 18 where θ j = rj, M, j N, then the assocaton vector A s a utlty maxmzng optmal assocaton vector. Proof: See Appendx. Remarks 2: Ths result says that an assocaton s optmal f the throughput an STA gets from the devce wth whch t s assocated s at least as large as the throughput t would otherwse get f ts assocaton s exchanged wth the assocaton of a devce assocated wth any other AP/BS. Ths can be used as a way to check f an assocaton s optmal. Ths condton s smlar to the Nash equlbrum of a pure strategy game. We now consder a crteron for lookng for the best assocaton among a partcular set of assocatons. Let UA denote the system utlty for the assocaton vector A = [a 1,a 2,...,a m ]. Let us denote the assocaton vector [a 1,a 2,...,a 1,k,a +1,...,a m ] by A k. It s to be noted that A k dffers from A only n the th poston. We thus have a collecton of assocatons, C = {A k, M, k L B}. Let the change n system utlty when STA s assocaton s changed from a to k be gven by, k =UA k UA. Lemma 1: The best assocaton n C s A k f and only f,k = arg max M,k N, k = arg max M,k N [ m k +1m k +1 +m k m k m a 1 m a 1 ] rk +m a m a + 19 r a where m a s the number of STAs assocated wth each AP/BS a under assocaton A. Proof: See Appendx. We now dsplay a greedy algorthm that computes a good assocaton. Algorthm: 1 Let h =0. M, assocate STA wth the AP/BS a = arg max k N {r k }. Let the assocaton vector be A 0. 2 Fnd the number of STAs assocated wth each AP/BS j. 3 Check f Theorem 1 s satsfed. If Theorem 1 s satsfed, then the assocaton A h s optmal. Go to step 5. If Theorem 1 s not satsfed, then the assocaton A h may or may not be optmal. 4 Usng Lemma 1, compute,k.ifk = a go to step 5. Else, assocate STA wth k keepng the assocaton of the rest of the STAs same. Let the new assocaton be A h+1.leth = h +1.Gotostep2. 5 A h s the assocaton yelded by the algorthm. VI. NUMERICAL EVALUATION OF THE GREEDY SEARCH ALGORITHM In ths secton, we study the algorthm for varous scenaros. For varous values of m, l, and b, we generate a random rate matrx R. Thefrstl columns are fxed at Θ 0 and the remanng b columns of R are generated usng unformly dstrbuted channel gans G j s. We fnd the optmal assocaton whch maxmzes the utlty by enumeratng all possble assocatons. We also run our algorthm and the result of our algorthm s compared aganst the optmal assocaton got by enumeraton. Though the enumeraton technque guarantees optmal assocaton, ts computatonal complexty s of the order l + b m = n m whch s prohbtvely hgh. We consder 6 dfferent cases of m, l, b. In each case, we compute the assocaton for 50 dfferent rate matrces R 12 n the case of 9,2,5. For each nstance of the rate matrx R, we observe the utltes of all n m possble assocatons and thus note down the maxmum and the mnmum utltes obtaned for each rate matrx R. Thus from the 50 nstances analyzed, we tabulate the range of the maxmum and the mnmum utltes obtaned. We also observe the number of computatons requred by the enumeraton technque, the average tme taken by the enumeraton technque, the average number of teratons taken by our algorthm averaged over the number of nstances analyzed, the average tme taken by our algorthm and the average percentage of dfference between the utltes of the optmal assocaton computed by the enumeraton technque and that gven by our algorthm. The observatons are shown n Table I. We note from Table I that there s a large gap between the mnmum and the maxmum utlty of assocatons, n every case, thus emphaszng the mportance of seekng an optmal assocaton. For a gven n, the average number of teratons ncreases wth m. We observe that the number of teratons taken by the algorthm to converge s sgnfcantly lower than the number of searches to be evaluated n the enumeraton technque. The tme taken for the algorthm to converge s neglgble, whereas the tme taken for the enumeraton technque s sgnfcantly hgher. We observe from Table I that for large number of STAs m cases 1,3,4, and 6 our algorthm converge to the optmum assocaton vector and for small number of STAs cases 2 and 5 our algorthm s found to converge to a suboptmal assocaton vector. It s also noted that n Case 2 m =7, l = 2, b = 4 the average dfference n the system utlty between the optmal assocaton vector and that obtaned by our algorthm s 0.1% and that n Case 5 m =9,l =2,b = 5, t s 0.15%. Ths could be explaned as follows. When m s small, the gradent search method does a coarse exploraton of the objectve functon and hence the algorthm converges to a local maxma. On the other hand for large m, the gradent search method carres out a fner exploraton and hence the algorthm converges to a better soluton Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 2006 proceedngs. Authorzed lcensed use lmted to: IEEE Xplore. Downloaded on February 21, 2009 at 03:32 from IEEE Xplore. Restrctons apply.

6 TABLE I COMPARISON OF OPTIMAL ASSOCIATION BY ENUMERATION AND THE SOLUTION OF OUR ALGORITHM. Case m, l, b # of nstances System Utlty Range Enumeraton Algorthm Average Analyzed No. of Tme Average Tme % error from Maxmum Mnmum evaluatons = n m ms # Iteratons ms optmal value. 1 10,1, ,2, ,1, ,1, ,2, ,1, It s to be noted f Theorem 1 s satsfed then the soluton s optmal. On the other hand, f Theorem 1 s not satsfed then the soluton of the algorthm may not be optmal. We found that n Case 4 of Table I, Theorem 1 s satsfed for 7 out of 50 rate matrces and n cases 1 and 6, Theorem 1 s satsfed for 4 out of 50 rate matrces. In cases 2, 3, and 5, Theorem 1 s not satsfed for any rate matrx. VII. CONCLUSION In ths paper, we studed the optmal assocaton of STAs n a heterogenous wreless access network comprsng WLAN APs and 3G BSs. We formulated the problem as one of maxmzng the total utlty provded to the users, where the utlty of a user s evaluated as the of the elastc download throughput that the user gets. We studed some characterstcs of the general soluton, and derved the soluton for a smple specal case. Then we provded a heurstc greedy search algorthm that yelds an optmal or near optmal assocaton for a large number of problems for whch we could obtan the exact soluton by exhaustve search. In our ongong work we are studyng other algorthmc technques, such as the genetc algorthm. Our future work wll nclude optmal assocaton under user moblty, the effect of network prcng, and decentralzed algorthms for optmal assocaton. APPENDIX PROOF OF THEOREM 1 Let us rewrte Eqn. 4 motvated by [3] for the utlty as follows. Let f j be the fracton of tme STA s assocated wth AP/BS j. Let the matrx F be [f j ]. Ths yelds the followng problem. U = max f j r j F M j N M f j =1, j N f j 0, M, j N. 20 If the matrx F that solves ths problem s such that f j =0, j a, M, then we have solved Eqn. 4,.e., A A whch s the optmal assocaton. The above problem s a concave maxmzaton problem wth lnear constrants. The Lagrangan of the above problem s LF,λ,ν = λ j M j N + M j N Dfferentatng w.r.t. f j L f j = r j gves f j r j j N M f j ν j f j 21 k N f λ k j + ν j, M, j N r k 22 Snce we are maxmzng a strctly concave functon over lnear constrants, F s the optmzer f there exsts λ,ν that satsfes the followng KKT condtons r j k N f k r k λ j + ν j = 0, M, j N 23 ν j 0, M, j N 24 ν j f j = 0, M, j N. 25 Now consder an assocaton A = a 1,a 2,...,a m and examne F such that f a > 0, andf j = 0, j a, M. For such a soluton to be a KKT pont we need { 0 f j = a ν j = 0 f j a, M, j N 26 Eqn. 23 gves f k r k = k N r j λ j ν j, M, j N 27 For the set of STAs, S p, assocated wth AP/BS p N, we have from Eqn. 27, f p r j r p =, S p, j N 28 λ j ν j When j = p, we need ν p =0, S p.thsgves λ p = 1 f p, S p, p N 2007 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 2006 proceedngs. Authorzed lcensed use lmted to: IEEE Xplore. Downloaded on February 21, 2009 at 03:32 from IEEE Xplore. Restrctons apply.

7 Thus, λ p = f p 1 = S p, S p, p N 29 Then from Eqn. 23, t s clear that ν j = λ j r j = m a r a f a r a ra To ensure ν j 0, we need ra m a Thus we have a KKT pont f ra m a m a rj rj PROOF OF LEMMA 1 r j 30 M, j N. M, j N. The system utltes correspondng to the assocatons A and A k are gven by UA = rqp m p p N q S p rqp UA k = Hence, m p p N,p =a,p =k q S p rqa + q S a,q = + q S k {} m a 1 rqk m k +1 UA k UA = m k +1m k +1 +m k m k m a 1 m a 1 rk +m a m a + r a Thus, n a collecton of assocatons C, A k s one of the best assocatons f UA k UA UB UA, B C. Thus the maxmzer of Eqn. 19 gves the best assocaton n C. [8] R. Bruno, M. Cont, and E. Gregor, Throughput Analyss of TCP clents n W-F hot spot Networks, In WONS, January [9] George Kurakose, Anurag Kumar and Vnod Sharma, Analytcal Models for Capacty Estmaton of IEEE WLANs usng DCF for Internet Applcatons, Submtted. [10] S. V. Hanly, An algorthm for combned cell-ste selecton and power control to maxmze cellular spread spectrum capacty, IEEE Journal on Selected Areas n Communcatons, vol. 19, no. 7, pp , Sept [11] J-W Lee, R. R. Mazumdar, and N. B. Shroff, Jont Resource Allocaton and Base-Staton Assgnment for the Downlnk n CDMA Networks, IEEE/ACM Transactons on Networkng, 2006 to appear. [12] Manoj Panda, Anurag Kumar, and S. H. Srnvasan, Saturaton Throughput Analyss of a System of Interferng IEEE WLANs, Proc. of WOWMOM, [13] A. J. Vterb, Prncples of Spread Spectrum Communcatons, Addson Wesley, Readng, Massachusetts, [14] R. D. Yates and C.-Y. Huang, Integrated power control and base staton assgnment, IEEE Transactons on Vehcular Technoy, vol. 44, no. 3, pp , Aug REFERENCES [1] Anurag Kumar and Vnod Kumar, Optmal Assocaton of Statons and APs n an IEEE WLAN, Natonal Conf. on Communcatons, [2] A. Bedekar, S. C. Borst, K. Ramanan, P. A. Whtng, and E. M. Yeh, Downlnk Schedulng n CDMA Data Networks, Report PNA-R9910, CWI, Oct [3] Y. Bejerano, S.-J. Han, and L. Lee, Farness and Load Balancng n Wreless LANs Usng Assocaton Control, Proc. ACM Mobcom, pp , Sep. Oct [4] T. Bonald, S. C. Borst, N. Hegde, and A. Proutère Wreless Data Performance n Mult-cell Scenaros, Proc. of ACM SIGMET- RICS/PERFORMANCE, [5] T. Bonald, S. C. Borst, and A. Proutère Inter-cell Schedulng n Wreless Data Networks, Proc. European Wreless, [6] T. Bonald and A. Proutère, Wreless Downlnk Data Channels: User Performance and Cell Dmensonng, Proc. of ACM Mobcom, [7] M. Bottglengo, C. Casett, C.-F. Chassern, and M. Meo, Short-term Farness for TCP Flows n b WLANs, Proc. of IEEE Infocom, Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 2006 proceedngs. Authorzed lcensed use lmted to: IEEE Xplore. Downloaded on February 21, 2009 at 03:32 from IEEE Xplore. Restrctons apply.