IN CONTRAST to traditional wireless cellular networks

Transcription

1 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH Jont Opportuntc Power Schedulng and End-to-End Rate Control for Wrele Ad Hoc Network Jang-Won Lee, Member, IEEE, Rav R. Mazumdar, Fellow, IEEE, and Ne B. Shroff, Fellow, IEEE Abtract It known that opportuntc chedulng that account for channel varaton due to moblty and fadng can gve ubtantal mprovement over nonopportuntc cheme. However, mot work on th ubject ha focued on ngle-hop cellular type of archtecture. The tuaton qute dfferent n ad hoc network due to the nherent multhop nature of tranmon. In th paper, we preent a jont opportuntc power chedulng and end-to-end rate control cheme for wrele ad hoc network. We model the tme-varyng wrele channel a a tochatc proce and formulate a tochatc optmzaton problem, whch am at maxmzng ytem effcency by controllng the power allocaton of each lnk and the data rate of each uer n the ytem. The jont power chedulng and rate control algorthm obtaned by ung tochatc dualty and mplemented va tochatc ubgradent technque. We llutrate the effcacy of our approach va numercal example. Index Term Ad hoc network, cro-layer optmzaton, flow control, power control, chedulng, wrele network. I. INTRODUCTION IN CONTRAST to tradtonal wrele cellular network n whch communcaton acheved between the fxed) acce pont and moble) node by ung ngle-hop tranmon, n wrele ad hoc network, communcaton typcally acheved between moble) node by ung multhop tranmon. Hence, ad hoc network have a ubtantally dfferent tructure from cellular ytem and network control cheme for one ytem are not typcally applcable for the other. In fact, n many way, the tructure of ad hoc network mlar to that of wrelne network uch a the Internet. For example, both ue multhop tranmon for communcaton and both Manucrpt receved November 2, 2004; reved Aprl 2, 2005, January 24, 2006, and February 15, Th work wa upported n part by KT Incorporated through the Yone Jont Reearch Center and by the Natonal Scence Foundaton under Grant ANI , ANI , and ANI The revew of th paper wa coordnated by Prof. D. O. Wu. J.-W. Lee wth the Center for Informaton Technology, Department of Electrcal and Electronc Engneerng, Yone Unverty, Seoul , Korea e-mal: jangwon@yone.ac.kr). R. R. Mazumdar wth Department of Electrcal and Computer Engneerng, Unverty of Waterloo, Waterloo, ON N2L 3G1, Canada e-mal: mazum@ece.uwaterloo.ca). N. B. Shroff wth the School of Electrcal and Computer Engneerng, Purdue Unverty, Wet Lafayette, IN USA e-mal: hroff@ecn.purdue. edu). Color veron of one or more of the fgure n th paper are avalable onlne at Dgtal Object Identfer /TVT requre decentralzed mplementaton. However, there alo ext fundamental dfference. For one, n wrelne network, node are tatc, wherea n ad hoc network, node could be moble, reultng n tme-varyng channel condton. In wrelne network, lnk have a fxed and ndependent capacty, wherea n wrele network, the capacty of each lnk may be tme varyng and dependent on other lnk. Hence, network control cheme for wrelne network alo cannot be drectly appled to wrele ad hoc network. Therefore, there a real need and alo gnfcant nteret n tudyng varou problem related to wrele ad hoc network, uch a routng, power control, and rate control [1] [9]. An mportant leon that we have learned from the tudy of cellular) wrele network that by approprately explotng the tme-varyng wrele channel, we can more effcently utlze rado reource, e.g., ncreae ytem performance and decreae the conumpton of reource. Hence, there ha been a great deal of effort on the development of opportuntc chedulng and power control cheme takng nto account thee varaton n the channel condton [10] [17]. The bac dea of opportuntc chedulng to allocate reource to lnk when they experence good channel condton whle not allocatng reource f the farne/performance requrement allow t) to lnk when they experence poor channel condton, thu effcently utlzng rado reource. In uch work, the author explot the varaton of the channel condton of each wrele lnk and develop optmal chedulng algorthm that maxmze the ytem performance whle atfyng a farne or performance contrant of each uer. They how that opportuntc chedulng cheme provde hgher ytem performance than nonopportuntc cheme that do not explot the varaton of the wrele channel. Opportuntc oluton,.e., oluton that explot the channel varablty, have prmarly been developed for cellular network [10] [17] whle there ha been vrtually no work on the ubject for wrele ad hoc multhop) network. In [1] [4], [6] [9], reource allocaton cheme n ad hoc network have been developed by aumng that the channel condton of each lnk contant. In [5], a jont power allocaton and routng cheme ha been developed for tme-varyng ad hoc network. Th cheme tablze the network f the nput data rate are wthn the capacty regon. However, t may not provde the optmal oluton n term of network performance e.g., maxmzng network throughput and mnmzng power conumpton), whch very mportant n wrele network /$ IEEE

2 802 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 In th paper, we tudy a jont opportuntc power chedulng and end-to-end rate control problem n wrele ad hoc network. By opportuntc power chedulng, we mean chedulng whch uer) to pck n a tme lot a well a determne the power level at whch the elected uer) hould tranmt. In partcular, we formulate an optmzaton problem that am at maxmzng network performance, whch defned a the weghted um of the utlte of all uer and power conumpton of all node n the ytem, wth contrant on the data rate of each uer and the power conumpton at each node. By olvng th problem, we develop a jont opportuntc power chedulng and end-to-end rate control algorthm n whch each uer adjut t data rate baed on feedback from the ytem, and the ytem allocate tranmon power to each lnk conderng the demand on rate allocaton and the channel condton of the lnk. To olve the problem, we wll ue mlar methodologe to thoe that have been ued n [17], uch a the dual approach and the ue of a tochatc ubgradent algorthm. However, the target ytem.e., multhop network) and the objectve.e., jont opportuntc power chedulng and end-to-end rate control) of th paper are dfferent from thoe.e., ngle-hop network and opportuntc power chedulng) of [17]. Hence, the algorthm that ha been developed n [17] cannot be appled to olve our problem, and thu, n th paper, we wll develop a new algorthm. Our algorthm can be decompoed nto the ytem algorthm that performed by the ytem to determne tranmon power level of each lnk n each tme lot and the uer algorthm that performed by each uer wth t local nformaton to determne t tranmon data rate n each tme lot. Moreover, f each lnk orthogonal to each other, the ytem algorthm can be further decompoed nto the node algorthm that performed by each node wth t local nformaton. Hence, n th cae, our algorthm can be ealy mplemented n a dtrbuted way. However, for a general nterference model, we need a central controller to collect the nformaton of each lnk and perform the ytem algorthm, whch mght not be applcable n ome ad hoc network. Thu, n ome cae, our algorthm may not be utable for beng mplemented n practce. However, even for uch cae, nce our algorthm can provde an upper bound on the achevable performance of the ytem, t can be ued a a benchmark for the performance of other dtrbuted algorthm. The ret of th paper organzed a follow: In Secton II, we decrbe the ytem model condered n th paper and formulate the optmzaton problem. In Secton III, we preent the jont opportuntc power chedulng and rate control algorthm, and n Secton IV, we preent the algorthm for a pecal cae where all lnk n the ytem are orthogonal. Numercal reult are provded n Secton V. Fnally, we conclude n Secton VI. II. SYSTEM MODEL AND PROBLEM We conder a wrele ad hoc network that cont of a et of node N and a et of lnk L. We denote a lnk from the tranmtter at node to the recever at node j a, j) and a et of lnk that emanate from node a L out. Note that a lnk an Fg. 1. Vrtual tranmtter and vrtual recever. abtract repreentaton of communcaton between two node. There are a et of uer M n the network, and each uer m ha t fxed routng path for communcaton. We denote a et of lnk that uer m ung for t communcaton a V m and a et of uer that are ung lnk, j) a M,j. Each uer m ha a utlty functon U m x m ), where x m t data rate. We aume that U m a contnuou and trctly concave functon of x m. Each node ha a maxmum tranmon power level P T. In wrele network, the channel condton of each lnk tme varyng and can be modeled a a tochatc proce. We allow the channel condton of each lnk to vary acro tme lot and model t a a tatonary tochatc proce. In a tme lot, the ytem aumed to be n one of everal poble tate, n whch each tate repreent one of everal poble level of channel condton for all lnk. Each tate take a value from a fnte 1 et S. We denote the probablty that the ytem n tate a π. In ad hoc network, dependng on the multple-acce cheme of the ytem, a node can tranmt data to or receve data from multple node multaneouly e.g., code-dvon multple-acce cheme), or t can tranmt data to or receve data from only one node at a tme e.g., tme-dvon multpleacce cheme). In addton, dependng on the duplex cheme of the ytem, a node can receve data whle t tranmt data.e., full duplex), or t cannot receve data whle t tranmt data.e., half duplex). To model thee varou ytem n a unfed framework, we defne a vrtual tranmtter and a vrtual recever correpondng to each lnk, a hown n Fg. 1. In th fgure, t, j) denote a vrtual tranmtter of lnk, j), and r, j) denote a vrtual recever of lnk, j). Note that although a node can have multple vrtual tranmtter or multple recever, t may only have one phycal tranmtter or one phycal recever dependng on the ytem;.e., vrtual tranmtter t3, 4) and t3, 5) and vrtual recever r1, 3) and r2, 3) may correpond to one phycal tranmtter and one phycal recever, repectvely. We defne the path gan from a vrtual tranmtter t, j) to a vrtual recever rm, n) when the ytem tate a G t,j),rm,n). By approprately ettng G t,j),rm,n), we can model varou ytem. When a node cannot tranmt and receve data multaneouly, we et G t,j),rm,n) =,f = n,. When a node can tranmt data to or receve data from only one node at a tme, we et G t,j),rm,n) =, f = m and j n, orf m and j = n,. 1 Note that th aumpton not retrctve, nce n a real ytem, the channel condton of a lnk mapped nto a level et wth a fnte number of level by ung quantzaton.

3 LEE et al.: POWER SCHEDULING AND RATE CONTROL FOR WIRELESS Ad Hoc NETWORKS 803 By ung th vrtual tranmtter and recever model, we can defne the gnal-to-nterference-plu-noe rato SINR) of each lnk, j),.e., γ,j, when the ytem n tate a γ,j P )= G t,j),r,j) P,j θ M m=1,m M n=1,n j G tm,n),r,j) P m,n + n,j 1) where P,j power allocaton for lnk, j) when the ytem n tate ; G t,j),rm,n) path gan from a vrtual tranmtter t, j) to a vrtual recever rm, n) when the ytem n tate ; n,j background noe of lnk, j) when the ytem n tate ; P =P,j ),j) L power allocaton vector for all lnk when the ytem n tate ; θ orthogonalty factor 0 θ 1). A n [2] and [3], we aume that the data rate that can be acheved a lnear functon of the SINR. The data rate r,j for lnk, j) when the ytem n tate thu defned a r,j P ) = Wγ,j P ) 2) where W the bandwdth of the ytem. We ue a weghted um of the utlte of all uer and the average power conumpton of all node a a meaure of ytem performance. The ytem performance thu defned a F x, P )= a m U m x m ) π b,j P,j 3) m M S,j:,j) L where a m and b,j are nonnegatve contant, x =x m ) m M, and P =P,j ),j:,j) L, S. Then, the optmzaton problem that we tudy n th paper formulated a P) max F x, P ) x, P.t. x mn m x m x max m, m M π r,j P ) x k,, j) L S k M,j P,j P a, N π S j:,j) L out P P, N, S where x mn m and x max m are mnmum and maxmum data rate for uer m, P a the maxmum average power conumpton of node, P =P,j ) j:,j) L out, P = {P,j ) j:,j) L out j:,j) L P out,j P T, 0 P,j P T, j :, j) Lout }, and P T the maxmum tranmon power lmt of node. Hence, by olvng th problem, we can obtan the jont power chedulng and rate allocaton that maxmze the ytem performance under contrant on the mnmum and maxmum data rate of each uer.e., the frt contrant), the capacty of each lnk.e., the econd contrant), the average power conumpton, and the maxmum tranmon power lmt of each node.e., the thrd and fourth contrant). In Secton III, we wll develop the algorthm that olve problem P). An mportant queton that not the focu of th paper how doe one enure feablty,.e., how do we enure that there ext a power chedulng and rate control polcy that atfe the contrant. Th can be acheved by adoptng an approprate call admon control trategy. However, the development of uch a trategy outde the cope of th paper. Intead, n th paper, we wll aume that the ytem ha an admon control polcy n place that enure feablty and focu on the power chedulng and rate control problem. III. ALGORITHM When olvng problem P), f we already knew the underlyng probablty dtrbuton for the ytem tate.e., π, S), the problem would be equvalent to a determntc optmzaton problem. However, n practce, we do not have uch aprorknowledge. Thu, we need to develop an algorthm that wll work even wthout uch aprorknowledge of the underlyng probablty dtrbuton for the ytem tate. To th end, we conder the dual of problem P) and develop the algorthm that olve the dual problem even wthout uch a pror knowledge. However, we mut note that problem P) may not be a convex optmzaton problem. Th mple that there mght be a dualty gap between problem P) and t dual, and thu, we cannot guarantee to obtan the optmal power chedulng and rate allocaton that olve problem P) by ung the dual approach. We wll come back to th ue later n th ecton; ntead, here, we frt focu on olvng the dual problem of problem P). We defne a Lagrangan functon aocated wth problem P) a L x, P, µ, λ) = a m U m x m ) π b,j P,j m M S,j:,j) L + µ,j π r,j P ) x k,j:,j) L S k M,j + λ P a π P,j N S j:,j) L out = a m U m x m ) π b,j P,j m M S N j:,j) L out + π µ,j r,j P ) S N j:,j) L out µ,j x m m M,j:,j) V m + λ P a π P,j N S j:,j) L out

4 804 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 = π a m U m x m ) µ,j x m S m M,j:,j) V m + b,j P,j + µ,j r,j P ) N j:,j) L out λ P,j ) + λ P a N where µ =µ,j ),j:,j) L, λ =λ ) N, P = P ) N, S, L out a et of lnk that emanate from node, M,j a et of uer that are ung lnk, j), and V m a et of lnk that uer m ung. Then, the dual problem defned a where D) Q µ, λ) = mn Q µ, λ) µ 0, λ 0 max L x, P, µ, λ) 4) x X, P P X = {x m ) m M x mn m x m x max m, m M}, and P = { P ) N, S P P, N, S}. We frt conder the problem n 4) for gven µ and λ. Snce L x, P, µ, λ) eparable n, x µ) =x m µ)) m M and P µ, λ) = P µ, λ)) S where P µ, λ) = P µ, λ)) N ) olve the problem n 4) f and only f x m µ) = arg max {a m U m x m ) x mn m x m x max m,j:,j) V m µ,j x m } m M 5) and 6), hown at the bottom of the page, obtaned by ung 1) and 2). Note that for a gven ytem tate, the above problem are determntc, and we can olve them wthout knowledge of the underlyng probablty dtrbuton. We now olve the dual problem D). However, to mnmze Q µ, λ), we need explct knowledge of the underlyng probablty dtrbuton, whch nfeable to obtan n practce. To overcome th dffculty, we ue a tochatc ubgradent method [18], [19] that defned by the followng teratve procee: [ ] + µ n+1),j = µ n),j αn) v µ n),j,, j :, j) L 7) and [ ] + λ n+1) = λ n) α n) v n) λ, N 8) where [a] + = max{0,a}, and v µ n),j and v n) λ are ome random varable. Let the equence of oluton µ 0), µ 1),..., µ n) and λ 0), λ 1),..., λ n) be generated by 7) and 8), repectvely. Let v µ n),j and v n) λ be choen uch that { E v µ n),j µ 0),..., µ n), λ 0),..., λ n)} = µ,j Q µ n), λ n)) and { E v n) λ µ 0),..., µ n), λ 0),..., λ n)} = λ Q µ n), λ n)) repectvely, where E{v} the expected value of a random varable v, and µ,j Q µ n), λ n) ) and λ Q µ n), λ n) ) are ubgradent of Q µ, λ) at µ = µ n) and λ = λ n) wth repect to µ,j and λ, repectvely. Then, v µ n),j and v n) λ are called tochatc ubgradent of Q µ, λ) at µ = µ n) and λ n) = λ wth repect to µ,j and λ, repectvely. By ung thee teratve procedure wth a equence of tep ze that atfy α n) 0, α n) =, n=0 and n=0 α n)) 2 < the dual varable µ n) and λ n) can be hown to converge to the optmal oluton that olve the dual problem D) wth probablty one [18], [19]. To apply th method to olve problem D), we need to know the tochatc ubgradent of Q µ, λ),.e., v µ n),j and v n), and they are obtaned a n [16] and [17] by λ v µ n),j = r,j n) and P n) µ n), λ n))) v n) λ = P a j:,j) L out P n),j x n) k k M,j µ n)) 9) µ n), λ n)) 10) where n) a ytem tate at teraton n, and x n) k µn) ) and P n) µ n), λ n) )= P n) µ n), λ n) )) N are oluton to the problem n 5) and 6), repectvely, wth µ = µ n) and λ = λ n). Note that th tochatc ubgradent method doe P µ, λ) = arg max P P N = arg max P P N j:,j) L out j:,j) L out b,j P,j + µ,j r,j P ) λ P,j G t,j),r,j) Wµ P,j,j M M m=1,m n=1,n j G tm,n),r,j) P m,n+ n,j b,j + λ )P,j, S 6)

5 LEE et al.: POWER SCHEDULING AND RATE CONTROL FOR WIRELESS Ad Hoc NETWORKS 805 not requre aprorknowledge of the underlyng probablty dtrbuton for ytem tate. It only requre the nformaton of the ytem tate at the current teraton, whch can be obtaned by meaurng the channel condton of each lnk at the current teraton. Thu far, we have developed an algorthm that olve the dual problem D) even wthout aprorknowledge of ytem tate. However, a we have mentoned before, due to the dualty gap, we cannot guarantee that the dual approach wll reult n an optmal oluton for our orgnal optmzaton problem P). Fortunately, we are able to how n the followng propoton that a the randomne n the ytem ncreae, the dualty gap decreae. Th mple that a the randomne n the ytem ncreae.e., a we have a fner granularty of the quantzaton for the channel condton of each lnk), the dual oluton approache that of the orgnal problem P). Propoton 1: Let upp ) be the optmal value of problem P) and nfd) be the optmal value of the correpondng dual problem. If up π m M U mx m ),j:,j) L b,jp,j ) 0 and up π r,j P ) k M,j x k ) 0,, j) L a S, then upp ) nfd) 0 a S. Proof: Th can be proven n the ame way a n [17, Proof of Prop. 1]. From th reult, we deduce that when the randomne of the ytem large, the ame power chedulng algorthm that ha been developed n th paper can be ued to obtan a good approxmaton to the optmal oluton, although the prmal problem cannot be formulated a a convex optmzaton problem. In Secton V, we provde numercal reult to llutrate the effectvene of our oluton. To mplement the propoed algorthm, we can decompoe t a a ytem problem that olved by the ytem and uer problem each of whch olved by each uer. Frt, baed on parameter n tme lot n,.e., = n), µ = µ n), and λ = λ n), the ytem olve the problem n 6) and obtan the power allocaton for each lnk n the current tme lot. Note that the problem n 6) may not be a convex optmzaton problem nce the objectve functon mght not be a concave functon. However, nce the objectve functon a convex functon n each varable P,j, the optmal oluton of the problem can be obtaned at an extreme pont of P [3]. Hence, f a node tranmt data, t tranmt data to only one lnk at a tme at t maxmum power level P T. By ung th property, the optmal oluton to the problem n 6) can be found ealy, although t not a convex optmzaton problem. The data rate for each lnk obtaned by ung 2) baed on the power allocaton. After allocatng power and data rate, baed on the power conumpton of each node and the aggregate data rate of the uer that are ung each lnk n the current tme lot, the ytem update parameter λ and µ for the next tme lot by ung 7) 10). Next, the data rate of each uer determned by the uer telf baed on t local nformaton. In tme lot n, each uer m determne t data rate x n) m by olvng the problem n 5). To olve the problem, the uer requre the um of µ n),j,, j :, j) V m,.e., the um of µ n),j correpondng to t routng path. Th can be obtaned by feedback from the node that are on t routng path. IV. SPECIAL CASE: SYSTEM WITH ORTHOGONAL LINKS If all lnk are orthogonal, we can et θ =0n 1), and the SINR of lnk, j) n 1) can be rewrtten a γ,jp,j) = G t,j),r,j) P,j n. 11),j Hence, there no nterference between lnk. In th cae, we can how that the ytem problem can be further decompoed a everal node problem where each node olve t own problem that requre only t local nformaton. Furthermore, problem P) become a convex optmzaton problem. Hence, there no dualty gap between problem P) and problem D), and we can alway obtan the optmal power chedulng and rate allocaton by ung the propoed algorthm. Frt, the problem n 6) can be rewrtten a P µ, λ) { = arg max P P = arg max P P Nj:,j) L out { N j:,j) L out b,j P,j +µ,j r,j P,j ) λ P,j Wµ,j G t,j),r,j) n j b,j } } λ )P,j, S. 12) The above problem eparable n. Hence, P µ, λ) = P µ, λ)) N maxmze t f and only f P µ, λ) = arg max P P j:,j) L out Wµ,j G t,j),r,j) n j b,j λ )P,j N, S. 13) Each node olve t problem [.e., the problem n 13)] and obtan power and rate allocaton for lnk that are emanatng from t. The oluton of the problem n 13) can be obtaned a P T, P,j = f j =arg max k L out 0, otherwe. Moreover, 9) can be rewrtten a v µ n),j = r,j n) P,j µ n) n), λ n))) { G Wµ t,k),r,k),k n b,k λ >0,k } G Wµ t,k),r,k),k n b,k λ,k x n) k k M,j µ n)). 14) To olve problem n 10) and 14), we only need local nformaton correpondng to node. Hence, node can update

6 806 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 TABLE I DATA RATE AND UTILITY DATA RATE/UTILITY) FOR THE MODE 1SYSTEM Fg. 2. Network model. parameter λ n) and µ n),j, j Lout, by olvng problem n 10) and 14), repectvely. Hence, the algorthm can be mplemented n a dtrbuted way, n whch each node determne power and rate allocaton for lnk that emanate from t and update parameter for the next tme lot by ung t local nformaton, and each uer determne t tranmon rate baed on feedback from the node that are on t routng path. TABLE II AVERAGE POWER CONSUMPTION FOR THE MODE 1SYSTEM V. N UMERICAL RESULTS In th ecton, we provde numercal reult to llutrate varou feature of our jont opportuntc power chedulng and rate control cheme. We conder a wrele ad hoc network n Fg. 2 that cont of x node and fve uer C 1 C 5 ). Each node ha maxmum tranmon power lmt P T =1 and maxmum average power conumpton P a =0.5. Each uer m ha a utlty functon U m, whch defned a U m x m ) = logx m ). Here, uer m maxmum and mnmum data rate are gven by x max m =10and x mn m =1, repectvely. We model the path gan from the vrtual tranmtter t, j) of lnk, j) to the vrtual recever rm, n) of lnk m, n),.e., G t,j),rm,n),a G t,j),rm,n) = 10K t,j),rm,n) d α,n 15) where d,n the dtance from node to node n, α a dtance lo exponent, and Kt,j),rm,n) a normally dtrbuted random varable wth mean 0 and varance σ 2 n decbel), whch repreent hadowng [20]. We et σ =4dB and α =4. We aume that each lnk, j) ha a fxed background noe n,j =10 7,, and the ytem ha a unt bandwdth,.e., W =1. We conder three dfferent ytem, whch are called mode 1, mode 2, and mode 3. In mode 1, we aume that all lnk are orthogonal,.e., θ =0. Hence, n thee ytem, the node can tranmt and receve data multaneouly. Moreover, t can tranmt data to or receve data from multple node. In mode 2 and mode 3 ytem, the node cannot tranmt and receve data multaneouly,.e., we et G t,j),rm,n) =, f = n,. However, n the mode 2 ytem, a node can tranmt data to or receve data from multple node, wherea n the mode 3 ytem, a node can tranmt data to or receve data from only one node at a tme,.e., n the mode 3 ytem, we et G t,j),rm,n) = f = m and j n, orf mand j = n,. We frt conder the mode 1 ytem. We mulate our cheme by varyng weght factor a m and b,j n 3). By adjutng thee weght factor, we can control the tradeoff between network performance total ytem utlty) and power conumpton. We et a m = a, m M, and b,j =1 a,, j :, j) L. A larger value of a mple that we put more weght on ncreang network performance and le weght on decreang power conumpton. We provde the acheved data rate and utlty for each uer n Table I and the average power conumpton for each node n Table II. When a =0, the objectve of our problem to obtan jont power chedulng and rate allocaton that mnmze the average total power conumpton. Hence, each uer allocated t mnmum data rate to mnmze power conumpton of the network. A the value of a ncreae, n general, the data rate of each uer alo ncreae, provdng hgher total ytem utlty. Th ncreae n network performance come from a correpondng ncreae n power conumpton of the network. Hence, a the value of a ncreae, the average power conumpton of each node alo ncreae. However, th ncreae lmted by the maxmum average power conumpton contrant. When a =1, the objectve of our problem to obtan jont power chedulng and rate allocaton that maxmze the total ytem utlty. Hence, each node conume t power a much a poble wthn t maxmum average power contrant to maxmze ytem performance. To how the performance gan of our opportuntc cheme over a nonopportuntc cheme, we now mulate a nonopportuntc cheme, n whch the tme-varyng channel condton of each lnk not exploted for reource allocaton. In th cheme, each uer ha a unt data rate, and each node allocate power to t lnk baed on the average channel condton of each lnk uch that each lnk can upport the aggregate data rate of the uer on that lnk. To th end, n each tme lot, the power allocaton P,j non for each lnk, j) obtaned by ung 2) and 11) a,j = n,j m M,j x m WE { } = G t,j),r,j) P non n,j M,j WE { G t,j),r,j) }

7 LEE et al.: POWER SCHEDULING AND RATE CONTROL FOR WIRELESS Ad Hoc NETWORKS 807 TABLE III COMPARISON OF THE AVERAGE POWER CONSUMPTIONS FOR THE MODE 1SYSTEM TABLE IV COMPARISON OF THE ACHIEVED DATA RATES FOR THE MODE 1SYSTEM Fg. 4. Rato of the total data rate of the opportuntc cheme to that of the nonopportuntc cheme. TABLE V COMPARISON OF THE TOTAL SYSTEM UTILITIES OF THE MODE 2 AND MODE 3SYSTEMS Fg. 3. Rato of the total power conumpton of the opportuntc cheme to that of the nonopportuntc cheme. where M,j the number of the uer that are ung lnk, j). Hence, n th cheme, each uer allocated the ame data rate a n our prevou mulaton wth a =0. We provde the average power conumpton for each node and the total average power conumpton of the network n each cheme n Table III. A hown n Table III, our opportuntc cheme requre le power from each node than t nonopportuntc counterpart to acheve the ame performance. We now mulate our opportuntc cheme wth x mn m =1, x max m =10, and a m =1for each uer m, and b,j =0for each lnk, j). We further et the maxmum average power conumpton P a for each node a the average power conumpton of each node n the nonopportuntc cheme n Table III,.e., P0 a =0.511, P1 a =0.818, and o on. Snce we et a m =1and b,j =0, our algorthm maxmze the um of the utlte of all uer wthout conderng power conumpton. However, the power conumpton n each node controlled by the contrant on t maxmum average power conumpton, whch the ame value a the average power conumpton of each node n the nonopportuntc cheme. Hence, th mulaton llutrate the performance gan of our cheme over the nonopportuntc cheme when both cheme conume the ame amount of power. Table IV how the acheved data rate of each uer n each cheme, and our cheme provde a hgher data rate to each uer than the nonopportuntc cheme. In Fg. 3 and 4, we mulate the ame cenaro for varou value of σ, whch the tandard devaton of Kt,j),rm,n) n 15), and we compare the rato of the total ytem power conumpton.e., the um of the power conumpton of each node) and the total ytem data rate.e., the um of the data rate of each uer) of our opportuntc cheme to thoe of the nonopportuntc cheme, repectvely. They how that a the randomne of the channel condton of each lnk ncreae, the performance gan of our opportuntc cheme over the nonopportuntc cheme alo ncreae. Hence, a the varaton of the channel condton of each lnk get larger, t more mportant to approprately explot t for ytem performance. In Table V, we mulate mode 2 and mode 3 ytem and compare ther performance. We et a m =1, m, and b,j = 0,, j :, j) L, and provde the total ytem utlte for each ytem by varyng the orthogonalty factor θ. A expected, the mode 2 ytem provde hgher total ytem utlte than the mode 3 ytem. Although we allow multple tranmon n the mode 2 ytem, due to the property of the optmal tranmon a we have tuded n Secton III, a node n the mode 2 ytem alway tranmt data to only one node at a tme. Hence, mode 2 and mode 3 ytem have the ame tranmon trategy, and the only dfference between them that a node n the mode 2 ytem can receve data from multple node at a tme, wherea a node n the mode 3 ytem can receve data from only one node at a tme. However, although a node allowed to receve data from multple node at a tme n the mode 2 ytem, when the orthogonalty factor large, the optmal trategy to receve data from only one node at a tme, nce a the orthogonalty factor ncreae, nterference between two tranmon alo ncreae. Hence, a the orthogonalty factor ncreae, the performance dfference between thee two

8 808 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 ytem decreae, and when the orthogonalty factor large, they provde almot the ame performance. VI. CONCLUSION AND FUTURE WORK In wrele network, the channel condton of the communcaton lnk tme varyng, and explotng th property mportant n mprovng the ytem performance. In th paper, we have modeled the channel condton a a tochatc proce and formulated a tochatc optmzaton problem that am at maxmzng the performance of the ytem whle atfyng the contrant on the data rate of each uer and power conumpton of each node. Our problem formulaton can be appled to varou mode of wrele ad hoc network by mply modfyng parameter ettng. By ung a dual approach and the tochatc ubgradent algorthm, we have developed a jont opportuntc power chedulng and end-to-end rate control algorthm for wrele ad hoc network. Numercal reult how that by ung our cheme,.e., by opportuntcally explotng the tmevaryng channel condton of each lnk, we can mprove ytem performance whle reducng power conumpton at each node n the ytem. However, a hown n th paper, to obtan the optmal oluton that explot ntantaneou tme-varyng channel condton of each lnk n ad hoc network, n general, we need to have a centralzed algorthm or pobly a dtrbuted algorthm wth a global nformaton, whch mght be nfeable n mot of tuaton n ad hoc network. Hence, t wll be a good future reearch topc to develop a dtrbuted algorthm, n whch each node requre only local nformaton and perform t own algorthm locally. Th dtrbuted algorthm may provde a uboptmal oluton. In th cae, the algorthm developed n th paper can be ued to meaure t effcency. [13] S. Bort and P. Whtng, Dynamc rate control algorthm for HDR throughput optmzaton, n Proc. IEEE Infocom, 2001, vol. 2, pp [14] S. S. Kulkarn and C. Roenberg, Opportuntc chedulng for wrele ytem wth multple nterface and multple contrant, n Proc. ACM Int. Workhop Model., Anal., and Smul. Wrele and Moble Syt., 2003, pp [15] Y. Lu and E. Knghtly, Opportuntc far chedulng over multple wrele channel, n Proc. IEEE Infocom, 2003, vol. 2, pp [16] J.-W. Lee, R. R. Mazumdar, and N. B. Shroff, Opportuntc power chedulng for mult-erver wrele ytem wth mnmum performance contrant, n Proc. IEEE Infocom, 2004, vol. 2, pp [17], Opportuntc power chedulng for dynamc mult-erver wrele ytem, IEEE Tran. Wrele Commun., vol. 5, no. 6, pp , Jun [18] P. Kall and S. W. Wallace, Stochatc Programmng. Hoboken, NJ: Wley, [19] Y. Ermolev, Stochatc quagradent method and ther applcaton to ytem optmzaton, Stochatc, vol. 9, no. 1/2, pp. 1 36, Mar [20] G. Stuber, Prncple of Moble Communcaton. Norwell, MA: Kluwer, Jang-Won Lee S 02 M 03) receved the B.S. degree n electronc engneerng from Yone Unverty, Seoul, Korea, n 1994, the M.S. degree n electrcal engneerng from Korea Advanced Inttute of Scence and Technology, Daejeon, Korea, n 1996, and the Ph.D. degree n electrcal and computer engneerng from Purdue Unverty, Wet Lafayette, IN, n In , he wa wth Dacom R&D Center, Daejeon. In , he wa a Potdoctoral Reearch Aocate wth the Department of Electrcal Engneerng, Prnceton Unverty, Prnceton, NJ. Snce September 2005, he ha been an Atant Profeor wth the Center for Informaton Technology, Department of Electrcal and Electronc Engneerng, Yone Unverty. H reearch nteret nclude reource allocaton, QoS and prcng ue, optmzaton, and performance analy n communcaton network. REFERENCES [1] T. ElBatt and A. Ephremde, Jont chedulng and power control for wrele ad-hoc network, n Proc. IEEE Infocom, 2002, vol. 2, pp [2] R. L. Cruz and A. V. Santhanam, Optmal lnk chedulng and power control n CDMA mult-hop wrele network, n Proc. IEEE Globecom, 2002, vol. 1, pp [3], Optmal routng, lnk chedulng and power control n mult-hop wrele network, n Proc. IEEE Infocom, 2003, vol. 1, pp [4] Y. Qu and P. Marbach, Bandwdth allocaton n ad hoc network: A prce-baed approach, n Proc. IEEE Infocom,2003,vol.2,pp [5] M. J. Neely, E. Modano, and C. E. Rohr, Dynamc power allocaton and routng for tme varyng wrele network, n Proc. IEEE Infocom, 2003, vol. 1, pp [6] R. Bhata and M. Kodalam, On power effcent communcaton over mult-hop wrele network: Jont routng, chedulng and power control, n Proc. IEEE Infocom, 2004, vol. 2, pp [7] Y. Y and S. Shakkotta, Hop-by-hop congeton control over a wrele mult-hop network, n Proc. IEEE Infocom, 2004,vol.4,pp [8] X. Ln and N. B. Shroff, Jont rate control and chedulng n multhop wrele network, n Proc. IEEE CDC, 2004, vol. 2, pp [9], The mpact of mperfect chedulng on cro-layer rate control n multhop wrele network, n Proc. IEEE Infocom, 2005, vol. 3, pp [10] X. Lu, E. K. P. Chong, and N. B. Shroff, Opportuntc tranmon chedulng wth reource harng contrant n wrele network, IEEE J. Sel. Area Commun., vol. 19, no. 10, pp , Oct [11] X. Lu, Opportuntc chedulng n wrele communcaton network, Ph.D. dertaton, Purdue Unv., Wet Lafayette, IN, [12] X. Lu, E. K. P. Chong, and N. B. Shroff, A framework for opportuntc chedulng n wrele network, Comput. Netw., vol. 41, no. 4, pp , Mar Rav R. Mazumdar S 80 M 83 SM 94 F 05) wa born n Bangalore, Inda. He receved the B.Tech. degree n electrcal engneerng from the Indan Inttute of Technology, Bombay, Inda, n 1977, the M.Sc. DIC degree n control ytem from Imperal College, London, U.K., n 1978, and the Ph.D. degree n ytem cence from the Unverty of Calforna, Lo Angele UCLA), n He currently a Profeor of electrcal and computer engneerng and holder of a Unverty Reearch Char wth the Unverty of Waterloo, Waterloo, ON, Canada. From 1985 to 1988, he wa an Atant Profeor wth the Department of Electrcal Engneerng, Columba Unverty, New York, NY, where he wa alo a member of the Center for Telecommuncaton Reearch. From 1988 to 1996, he wa a Profeor wth the Natonal Inttute of Scentfc Reearch INRS)-Telecommuncaton, whch a graduate reearch nttute afflated to the Unverty of Quebec, Montreal, QC, Canada. He alo held an nvted appontment wth the Department of Electrcal Engneerng, McGll Unverty, Montreal. From 1996 to 1999, he wa a Profeor of mathematc wth the Unverty of Eex, Colcheter, U.K. From 1999 to 2004, he wa a Profeor of electrcal and computer engneerng wth Purdue Unverty, Wet Lafayette, IN. He ha held vtng poton and abbatcal leave wth UCLA; the Unverty of Twente, Enchede, The Netherland; the Indan Inttute of Scence, Bangalore; and the Ecole Natonale Supereure de Telecommuncaton, Par, France. H reearch nteret are n wrele and wrelne network; applcaton of game theory to networkng; appled probablty, queung theory, and tochatc analy wth applcaton to traffc engneerng, tattcal flterng theory, and mathematcal fnance. Dr. Mazumdar a Fellow of the Royal Stattcal Socety. He a member of the workng group WG6.3 and 7.1 of the Internatonal Federaton of Informaton Proceng and a member of the Socety for Indutral and Appled Mathematc and the Inttute of Mathematcal Stattc.

9 LEE et al.: POWER SCHEDULING AND RATE CONTROL FOR WIRELESS Ad Hoc NETWORKS 809 Ne B. Shroff S 91 M 93 SM 01 F 07) receved the Ph.D. degree from Columba Unverty, New York, NY, n He ha been wth Purdue Unverty, Wet Lafayette, IN, nce 1994, where he currently a Profeor of electrcal and computer engneerng. H reearch nteret pan the area of wrele and wrelne communcaton network. He epecally ntereted n fundamental problem n the degn, performance, control, ecurty, and prcng of thee network. Dr. Shroff an Edtor of the IEEE/ACM TRANSACTIONS ON NETWORKING and the Computer Network Journal and a pat Edtor of the IEEE COMMUNICATIONS LETTERS. He wa the Techncal Program Co-Char for IEEE INFOCOM 03, the Panel Co-Char for ACM Mobcom 02, the Program Co-Char for the Sympoum on Hgh-Speed Network, Globecom 2001, and the Conference Char for the 14th Annual IEEE Computer Communcaton Workhop. He wa the co-organzer of the Natonal Scence Foundaton NSF) Workhop on Fundamental Reearch n Networkng n Aprl He receved the IEEE INFOCOM 2006 Bet Paper Award, the IEEE IWQoS 2006 Bet Student Paper Award, the 2005 Bet Paper of the Year Award for the Journal of Communcaton and Networkng, the 2003 Bet Paper of the Year Award for Computer Network, and the NSF CAREER award n H INFOCOM 2005 paper wa alo elected a one of two runner-up paper for the Bet Paper Award.