Near-Optimal Relay Station Placement for Power Minimization in WiMAX Networks

Transcription

1 Nea-Optimal elay Station lacement fo owe Minimization in WiMAX Netwok Dejun Yang, Xi Fang an Guoliang Xue Abtact In the IEEE 80.16j tana, the elay tation ha been intouce to inceae the coveage an the thoughput of WiMAX netwok. The placement of the elay tation play a citical ole in the ytem pefomance an theefoe aw temenou attention fom the eeach community. In thi pape, we tuy the elay tation placement poblem in the WiMAX netwok, with the coopeative communication a the elaying tategy. In paticula, given a bae tation an a et of ubcibe tation, we etemine the location of a elay tation, an the et of ubcibe tation uing the elay tation. The objective i to minimize the maximum tanmiion powe among all the ubcibe tation while atifying the ata ate equiement of the ubcibe tation. We evelop a nea-optimal algoithm to olve thi poblem an pove that the maximum tanmiion powe compute by ou algoithm i at mot opt + ε, whee opt i the maximum tanmiion powe in the optimal olution an ε > 0 i an abitay contant. The expeiment how that we can amatically euce the maximum tanmiion powe by eploying the elay tation accoing to ou algoithm. I. INTODUCTION WiMAX, hot fo Wolwie Inteopeability fo Micowave Acce, i an emeging technology to povie high pee boaban intenet acce. ae on IEEE tana [], a ingle WiMAX towe can blanket a aiu of 30 mile with wiele acce, outitancing WiFi by mile. In the baic WiMAX netwok, thee ae two type of entitie: bae tation (S) an ubcibe tation (). The S contol the channel uage an allocate eouce to. To meet the gowing capacity eman an futhe exten the netwok coveage into the emote ual aea, elay tation (S) wee intouce in IEEE 80.16j [3]. A typical WiMAX netwok with all thee entitie i hown in Fig. 1. Since the S ae involve in the tanmiion between the S an the, elaying tategie ae of inteet. Coopeative Communication (CC) i one of uch avance elaying tategie. Exploiting the natue of boacat an the elaying capacity of othe noe to achieve the patial iveity, CC ha been hown to have the potential to inceae the channel capacity between two wiele evice. In thi pape, we incopoate CC into the WiMAX netwok. Thi incopoation ha been alo coniee in [1, 4 9, 11, 1]. While mot peviou tuie elate to S placement only conie the icete S placement (except [4, 1]), whee the S can be place only at a et of caniate location, we Yang, Fang an Xue ae all with the CIDSE Dept. at Aizona State Univeity, Tempe, AZ {ejun.yang, xi.fang, xue}@au.eu. Thi eeach wa uppote in pat by NSF gant an AO gant W911NF The infomation epote hee oe not eflect the poition o the policy of the feeal govenment. S Fig. 1: An IEEE 80.16j WiMAX netwok conie the continuou cae, whee the S can be place anywhee. In thi pape, we tuy the S placement poblem in a WiMAX netwok with CC a the elaying tategy. Specifically, given a S an a et of, we nee to etemine the location of an S an the ubet of uing the S fo coopeative elaying, uch that all the ata ate equiement of the ae atifie an the maximum tanmiion powe among the i minimize. Thi poblem i vey challenging ue to two fact: 1) the each pace of the S location i continuou an the numbe of the poible location i infinite; an ) the et of exploiting the S an the location of the S have to be jointly coniee. The main contibution of thi pape ae following: S We tuy the S placement in WiMAX netwok, whee the S can be place anywhee in the egion. The moel i moe geneal compae with the peviou tuie on the S placement poblem. We eign a nea-optimal algoithm to etemine the location of the S an the et of exploiting the S fo CC. We how that the olution etune by ou algoithm i optimal if the ganulaity of the powe ajutment i given. In aition, ou algoithm can be eaily extene to the geneal cae, whee fobien aea ae coniee. The et of the pape i oganize a follow. We biefly uvey the elate wok on S placement poblem in Section II. In Section III, we peent the ytem moel an efine the poblem tuie in thi pape. In Section IV, we eign an appoximation algoithm to olve the poblem an pove it nea-optimality. We evaluate the pefomance of the popoe algoithm though extenive expeiment in Section V. In Section VI, we conclue thi pape.

2 II. ELATED WOK Thee have been many tuie on the S placement in WiMAX netwok [1, 4 6, 8, 9, 1]. In [1], Chang et al. popoe a elay eployment mechanim that etemine the eploye location of the S o that the ata ate equiement of the can be atifie while the netwok thoughput can be ignificantly impove. It i note that the popoe mechanim cannot povie any pefomance guaantee. In [8], Lu et al. jointly coniee the S an the S eployment poblem with the objective to maximize the ytem capacity. They fomulate thi poblem via an intege linea pogamming (IL) moel an popoe a two-tage heuitic algoithm. Late on, they tuie the S placement poblem une the eployment buget containt [9]. They fomulate thi poblem into an IL, pove it N-hane, an popoe a geey heuitic to povie a ub-optimal olution to it. In thei ytem moel, they aume that the numbe of caniate S location i finite, which enable them to fomulate the poblem into IL. Thei moel i funamentally iffeent fom the moel in thi pape, whee we conie a continuou pace. None of the above wok coniee CC fo elaying tategy. In [4], Lin et al. tackle the tak of S placement an elay time allocation to maximize the ytem capacity in WiMAX netwok, by incopoating coopeative elaying technologie. They fomulate the poblem into an optimization poblem. Howeve, the unning time fo olving thi optimization poblem i not guaantee to be polynomial. Along thi line, they popoe a ual-elay achitectue, whee the communication i elaye via two available S coopeatively [5]. The goal of thei wok wa to eploy a minimum numbe of S at a et of peefine caniate location uch that the ata ate equiement of the can be atifie. They fomulate the poblem an olve it though a twophae heuitic algoithm. In [6], they tuie the poblem of joint S placement an banwith allocation, whee the S can only be eploye at cetain caniate location. They fomulate the poblem a a mixe intege nonlinea pogamming poblem an popoe Genetic Algoithm bae heuitic. Allowing the S to be eploye anywhee, Yang et al. [1] coniee the S placement poblem to atify the ata ate equiement of all uing a minimum numbe of S. They pove the N-hane of the poblem an popoe efficient appoximation algoithm. III. SYSTEM MODEL AND OLEM DEFINITION In thi pape, we conie a tatic netwok coniting of thee type of entitie: the ae Station (S), the elay Station (S) an the Subcibe Station (). In the netwok, one S an a et S = { 1,,, n } of n ae given an fixe. All the an S ae locate on a D Eucliean plane. Each i S ha a ata ate equiement c i 0. Let c = (c 1, c,..., c n ) enote the ata ate equiement vecto. Let i enote the tanmiion powe of i an enote the tanmiion powe of the S. We aume iffeent can acce the S at thei aigne channel with Othogonal 1 (a) (a) lot 1 lot lot 3 lot 4 fame 1 fame (b) Fig. : A thee-noe example fo CC 1 lot 1 lot lot 3 lot 4 fame 1 fame Fig. 3: Multi- Coopeative Communication Fequency-Diviion Multiple Acce (OFDMA) technique. The channel popagation moel can be moele a follow. Let N 0 enote the abient noie. When a noe u tanmit a ignal to noe v at powe u, the ignal-to-noie atio (SN) at noe v, enote a SN uv, i SN uv =, (1) N 0 (u, v) whee (u, v) i the Eucliean itance between noe u an v an i the path lo exponent which i between an 4 in geneal, epening on the chaacteitic of the communication meium. Fo tanmiion moel, we fit peent a imple moel without Coopeative Communication (CC). Diect Tanmiion When the ouce noe iectly tanmit to the etination, the achievable ata ate fom to i u C DT (, ) = W log (1 + SN ). () whee W i the banwith of the channel. Fo each i S, let i T enote the minimum tanmiion powe equie ( to ) atify C DT ( i ) c i, i.e., i T = N 0 ( i, ) c i W 1. Without lo of geneality, we aume that 1 T T... n T. (3) Fo CC tategy, we conie ecoe-an-fowa (DF) moe in thi pape. DF moe can be bet ecibe with a well-known thee-noe example in Fig., whee i the ouce noe, i the elay noe an i the etination noe. Decoe-an-Fowa (DF) Fo ecoe-an-fowa moe, the elay noe ecoe an etimate the ignal tanmitte by the ouce noe in the fit time lot an then tanmit the etimate ata to the etination in the econ time lot. The achievable ata ate fom to i C DF (,, ) = W min{ log (1 + SN ), (b) log (1 + SN + SN )}.(4) Now we tun to conie the cae whee the S i hae by multiple. A imple example i hown in Fig. 3. We enote S a the et of exploiting the S, calle haing et. Since the object of eploying an S intea of a S i to lowe the eployment cot, it i eaonable to aume that S i not

3 a poweful a a S, in the ene that it cannot communicate with all the imultaneouly. In thi cae, we aume that the S equally povie evice to all the in S. Thi can be achieve fo example by uing a eevation-bae TDMA cheuling. The S eve each in a oun-obin fahion. Each fame i eicate to a ingle fo CC. Each get eve evey S fame. Theefoe, the aveage achievable capacity fo each i S i C DF ( i,,) S. Heeafte, we omit an in the capacity expeion. Thu we have C( i ) = { CDF ( i) S, if i S, C DT ( i ), othewie. Given the location = (x, y ) of the S an the haing et S, let i (, S ) enote the minimum tanmiion powe of i equie to atify C( i ) c i accoing to (5). We tuy the elay Station lacement fo owe Minimization (SM) poblem, which i efine a follow. Definition 1 (SM(, S, c)): Given a S, a et S of, an a ata ate equiement vecto c, the S placement fo powe minimization poblem, enote by SM(, S, c), i to etemine the location = (x, y ) of an S an the haing et S, uch that max i S i (, S ) i minimize. Let opt enote the value of max i S i (, S ) in the optimal olution to SM(, S, c). IV. A NEA-OTIMAL AOXIMATION ALGOITHM The SM poblem conit of two main component: the haing et etemination an the S placement. In thi ection, we fit tuy the S placement poblem fo a given haing et, an then icu how to chooe the haing et. Finally, we peent ou algoithm fo the SM poblem. A. S lacement fo A Given Shaing Set Given a haing et S S, let (S ) enote the optimal value of max i S i (, S ) by placing the S, i.e., (S ) = min =(x,y ) max i S i (, S ). The S placement poblem fo a given haing et i efine a: Definition (S(, S, c )): Given a S, a haing et S of, an the coeponing ata ate equiement vecto c, the S placement fo the haing et poblem, enote by S(, S, c ), i to fin a location = (x, y ) of an S uch that max i S i (, S ) i minimize. Aume that the tanmiion powe i of each i S i given. lugging (4) into (5) an conieing the ate equiement C DF ( i ) c i, we have ( ) W S log 1 + N 0( i,) c i, ( ) W S log (6) 1 + i N 0( i,) + N 0(,) c i, fo each i S. Afte pefoming ome imple algebaic manipulation, we have ( i, ) γ i ( i ) = i N 0(4 c i S /W 1), (, ) γ i ( i ) = (7) i ), N 0 (4 c i S /W N 0 ( i,) 1 fo each i S. In othe wo, given i, the S mut be place at a location uch that (7) i atifie fo each i S. (5) Geometically, let D i (γ i ( i )) enote the ik centee at i with aiu γ i ( i ) an D (γ i ( i )) enote the ik centee at with aiu γ i ( i ). The S then mut be place in the inteection of D i (γ i ( i )) an D (γ i ( i )) fo all i S. ae on the above obevation, we can fin the value of (S ) uing biection metho within the ange [ S min, S max], whee S min i the maximum value among the oot of the following equation γ i ( i ) + γ i ( i ) = ( i, ), fo all i S, (8) an S max = max i S i T. Intuitively, on one han, (S ) houl not be le than the tanmiion powe of any i S if i excluively employ the S. On the othe han, (S ) houl not be moe than the maximum of the minimum iect tanmiion powe of all i S. Othewie, it i againt the oiginal intention of eploying the S. Fo each value [ S min, S max], we et i to an check if the ik D i (γ i ( i )) an D (γ i ( i )) fo all i S have an inteection. Note that auming that the have the ame tanmiion powe oe not change the maximum tanmiion powe. The algoithm fo the S poblem i peente in Algoithm Algoithm 1: S(, S, c ) A the inteection of ik D i (γ i ( S max)) an D (γ i ( S max)) fo all i S ; if A i empty then etun (, ) an ; L S min, U S max; while L + ε < U o L+U ; A the inteection of ik D i (γ i ( )) an D (γ i ( )) fo all i S ; if A i nonempty then U apx (S ) ; (x, y ) any point in A; ele L ; en etun (x, y ) an apx (S ) Lemma 1: Fo any given ε > 0, Algoithm 1 compute a location (x, y ) an apx (S ) that i at mot (S ) + ε, in O(n 3 (log S max + log 1 ε )) time. oof: In Line 3, we compute min, which can be finihe in O(n) time. The while-loop (Line 4 to Line 11) execute at mot log S max S min ε = O(log S max + log 1 ε ) time. Fo each execution, Line 6 ominate the unning time, which can be finihe in O(n 3 ) time uing the algoithm in [10]. Theefoe the unning time of Algoithm 1 i O(n 3 (log S max + log 1 ε )). We next pove that apx (S ) (S ) + ε. Since L i the lowe boun of (S ), (S ) L. Theefoe apx (S ) = U L + ε (S ) + ε, conieing the topping conition of the while-loop.. Shaing Set Detemination In oe to etemine the haing et, we fit intouce thee lemma howing a few popetie of the pob-

4 lem. Given a haing et S S, let (S ) = max{ (S ), max i S S i T }. We aume that thee exit at leat one i S, uch that ({ i }) < i T. Lemma : Thee exit an optimal haing et S opt, which i of the fom { 1,,..., k 1, k }, fo ome 1 k n. oof: Aume that an optimal haing et S i not of the fom { 1,,..., k 1, k }. Thee mut exit at leat one j S, uch that j T k T. Let j be the mallet value atifying thi conition. We contuct a new haing et S opt S by emoving the { j+1, j+,..., k }. It i clea that we have (S opt ) < (S ). Theefoe, we have max{ (S opt ), j T } max{ (S ), j T }. We pove that S opt i alo an optimal haing et, which i of the fom { 1,,..., j, j 1 }. Let S k enote the et of of the fom { 1,,..., k }. Coeponingly, let c k enote the ata ate equiement vecto of S k. Lemma implie that it uffice to conie S i fo 1 i n to fin an optimal haing et. To imply the notation, we ue i to enote (S i ). It i clea that we have 1 < <... < n. (9) Now (S i ) = max{ i, i+1 T }. Seaching equentially fo the optimal value of k take O(n) time. iection metho can impove the time complexity to O(log n). Lemma 3 an Lemma 4 guie the each iection uing biection. Lemma 3: Fo any 1 < k n, if k > k T, then (S k 1 ) < (S k ). oof: We pove thi lemma by howing that max{ k 1, k T } < max{ k, k+1 T }. y aumption, we know that k T < k. y (9), we have k 1 < k. Theefoe we have (S k 1 ) = max{ k 1, k T } < k max{ k, k+1 T } = (S k). Lemma 4: Fo any 1 k < n, if k < k T an k+1 k+1 T, then (S k+1) (S k ). oof: y (3), we have k+ T k+1 T. Thu we have (S k+1 ) = max{ k+1, k+ T } k+1 T max{ k, k+1 T } = (S k ). C. Algoithm Detail Having tuie the S poblem an the haing et etemination poblem, we ae eay to peent ou algoithm fo the the SM poblem, which i hown in Algoithm. The time complexity an the pefomance of Algoithm can be guaantee by the following theoem. Theoem 1: Fo any given ε > 0, Algoithm compute a haing et S an a location (x, y ), which guaantee the maximum tanmiion powe at mot opt + ε, in O(n 3 log n(log 1 T + log 1 ε )) time. oof: To pove the unning time, it uffice to note that the while-loop ominate the unning time an will execute at mot log n time, ince it i bae on biection metho. In aition, Algoithm 1 i calle uing each iteation. ae on Lemma 1, the unning time of Algoithm i O(n 3 log n(log 1 T + log 1 ε )). We aume that apx = max{ apx t, t+1} T i the maximum tanmiion powe compute by Algoithm an opt = Algoithm : SM(, S, c) 1 L 1, U n; while L < U o 3 k L+U ; 4 (x, y, k ) S(, S k, c k ); 5 if k > k T then 6 U k 1; 7 ele 8 (x, y, k+1 ) S(, S k+1, c k+1 ); 9 if k+1 k+1 T then L k + 1; 10 ele beak; 11 en 1 en 13 S ; 14 if L = U then 15 (x, y, L ) S(, S L, c L ); 16 if L L T then 17 S S L ; i max{n 0 ( i, ) ( 4 ci S /W 1 ) 18, ) N 0 ( i, ) (4 ci S /W N 0(,) 1 }, i S ; 19 en 0 en 1 i i T, i S S ; etun (x, y ), S, an i fo all i S max{ l, l+1 T } fo ome 1 l n. Obviouly, we have t l. We pove the pefomance fo two cae. Cae 1: t = l. If apx t t+1, T we have apx = apx t l + ε max{ l, l+1 T } + ε = opt + ε, whee the fit inequality follow fom Lemma 1. If apx t < t+1, T we know that apx = max{ apx t, t+1} T = t+1 T = l+1 T max{ l, l+1 T } = opt. Theefoe we have apx = t+1 T = l+1 T = opt. Cae : t < l: It i clea that t + 1 l. We have t+1 T < apx t+1 t+1 + ε l + ε, (10) whee the fit inequality follow fom Lemma 3, the econ inequality follow fom Lemma 1 an the thi inequality follow fom (9). In aition, we have apx t t + ε < l + ε, (11) whee the fit inequality follow fom Lemma 1 an the econ inequality follow fom (9). Combining (10) an (11), we get apx = max{ apx t, t+1} T l + ε max{ l, l+1} T + ε = opt + ε. Thi complete the poof. emak 1: Accoing to the IEEE 80.16j tana [3], the powe level ajutment unit i 0.5. Theefoe Algoithm can compute an optimal olution to the SM poblem by etting ε equal to the coeponing powe value. emak : Algoithm can be eaily extene to the geneal cae, whee fobien aea have to be coniee, fo example, lake, ive an oa.

5 Maximum owe (W) SM anom Numbe of (a) SM v. anom owe Impovement Numbe of (b) Impact of n owe Impovement n = 5 n = 10 Fig. 4: Expeiment eult (c) Impact of owe Impovement () Impact of A. Expeiment Setup V. EXEIMENTAL ESULTS We conie a et of anomly itibute in a quae of ize The S i locate at (0, 0). We et W = 10 MHz an N 0 = 10 9 W. The ata ate equiement i unifomly itibute ove [1, 10] Mbp. The tanmiion powe of the S i 0 W [3] an =.5 fo mot expeiment. Fo each etting, we an 1000 intance an aveage the eult. Fo all the expeiment, the pefomance metic inclue the maximum tanmiion powe an the powe impovement, efine a T apx max, whee T max T max = max i S i T. To the bet of ou knowlege, ou wok i the only one on the SM. Hence, we peent a anom algoithm, enote a anom, fo the ake of compaion. In anom, the location of the S i fit anomly electe within the quae. The haing et i then choen anomly until no can be ae without intoucing any tanmiion powe highe than it minimum iect tanmiion powe.. eult Analyi Compaing with anom: Fig. 4(a) how the compaion between SM an anom. A expecte, we ee that SM amatically outpefom anom, with the powe impovement anging fom 84.% to 91.6%. Impact of the numbe of : We vaie n fom 5 to 30 with the tep of 5. Fig. 4(b) how the eult of the expeiment. We can ee that the powe impovement eceae with the inceae of n. The eaon i that the numbe of in the haing et i limite an oe not inceae with the inceae of n. Howeve, with the inceae of n, the gap between T max an max{ t, T t+1} i naowe. Impact of the tanmiion powe of the S: To check the impact of, we vaie it fom 10 to 0 with tep of. We tete two cae, whee n = 5 an n = 10. Fig. 4(c) how the eult of the expeiment. We obeve that the powe impovement emain almot the ame fo iffeent value of. Thi obevation how that the bottleneck of the powe impovement i the, not the S. Impact of : We alo conucte expeiment on iffeent value of. The eult ae hown in Fig. 4(). We obeve that when the envionment i elatively clean (fee of noie), eploying an S oe not impove the maximum tanmiion powe. Only when the path lo i evee to cetain extent ( =.5) can the maximum tanmiion powe be amatically euce by eploying an S. In aition, if the envionment become noie, the powe impovement eceae. VI. CONCLUSIONS In thi pape, we have tuie the S placement poblem in the WiMAX netwok. We incopoate the coopeative communication into the elaying tategy. In paticula, given a S an a et of, we etemine the location of an S an the ubet of uing the S. The objective i to minimize the maximum tanmiion powe among all the while atifying the ata ate equiement of the. We have evelope a nea-optimal algoithm to olve thi poblem an have pove that the maximum tanmiion powe compute by ou algoithm i at mot opt + ε, whee opt i the maximum tanmiion powe in the optimal olution an ε > 0 i an abitay contant. The expeiment howe that we can amatically euce the maximum tanmiion powe by eploying the S accoing to ou algoithm. EFEENCES [1] C.-Y. Chang, C.-T. Chang, M.-H. Li, an C.-H. Chang, A novel elay placement mechanim fo capacity enhancement in IEEE 80.16j WiMAX netwok, in oc. IEEE Intenational Confeence on Communication (ICC 09), 009, pp [] IEEE Tak Goup, IEEE , Oct [Online]. Available: [3] IEEE elay Tak Goup, IEEE 80.16j-009, May 009. [Online]. Available: [4]. Lin,.-H. Ho, L.-L. Xie, an X. Shen, Optimal elay tation placement in IEEE 80.16j netwok, in oc. ACM Intenational Confeence on Wiele Communication an Mobile Computing (IWCMC 07), 007, pp [5], elay tation placement in IEEE 80.16j ual-elay MM netwok, in oc. IEEE Intenational Confeence on Communication (ICC 08), 008, pp [6]. Lin,.-H. Ho, L.-L. Xie, X. Shen, an J. Tapolcai, Optimal elay tation placement in boaban wiele acce netwok, IEEE Tan. Mobile Comput., vol. 9, no., pp , Feb [7]. Lin, M. Mehjoo,.-H. Ho, L.-L. Xie, an X. Shen, Capacity enhancement with elay tation placement in wiele coopeative netwok, in oc. IEEE Wiele Communication an Netwoking Confeence (WCNC 09), 009, pp [8] H.-C. Lu an W. Liao, Joint bae tation an elay tation placement fo IEEE 80.16j netwok, in oc. IEEE Global Telecommunication Confeence (GLOECOM 09), 009, pp [9] H.-C. Lu, W. Liao, an F. Lin, elay tation placement tategy in IEEE 80.16j WiMAX netwok, IEEE Tan. Commun., vol. 59, no. 1, pp , Jan [10] N. Megio, The weighte eucliean 1-cente poblem, Mathematic of Opeation eeach, vol. 8, no. 4, pp , Nov [11] D. Yang, X. Fang, an G. Xue, OA: Optimal elay aignment fo capacity maximization in coopeative netwok, in oc. IEEE Intenational Confeence on Communication (ICC 11), 011, to appea. [1] D. Yang, X. Fang, G. Xue, an J. Tang, elay tation placement fo coopeative communication in WiMAX netwok, in oc. IEEE Global Telecommunication Confeence (GLOECOM 10), 010, pp. 1 5.