University of Alberta. Library Release Form. Title of Thesis: Joint Bandwidth and Power Allocation in Wireless Communication Networks

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1 Unversty of Alberta Lbrary Release Form Name of Author: Xaowen Gong Ttle of Thess: Jont Bandwdth and Power Allocaton n Wreless Communcaton Networks Degree: Master of Scence Year ths Degree Granted: 2010 Permsson s hereby granted to the Unversty of Alberta Lbrary to reproduce sngle copes of ths thess and to lend or sell such copes for prvate, scholarly or scentfc research purposes only. The author reserves all other publcaton and other rghts n assocaton wth the copyrght n the thess, and except as heren before provded, nether the thess nor any substantal porton thereof may be prnted or otherwse reproduced n any materal form whatever wthout the author s pror wrtten permsson. Xaowen Gong 2nd Floor ECERF Edmonton, Alberta Canada T6G2V4 Date:

2 Unversty of Alberta Jont Bandwdth and Power Allocaton n Wreless Communcaton Networks by Xaowen Gong A thess submtted to the Faculty of Graduate Studes and Research n partal fulfllment of the requrements for the degree of Master of Scence. Department of Electrcal and Computer Engneerng Edmonton, Alberta Summer 2010

3 Unversty of Alberta Faculty of Graduate Studes and Research The undersgned certfy that they have read, and recommend to the Faculty of Graduate Studes and Research for acceptance, a thess enttled Jont Bandwdth and Power Allocaton n Wreless Communcaton Networks submtted by Xaowen Gong n partal fulfllment of the requrements for the degree of Master of Scence. Chntha Tellambura and Sergy A. Vorobyov (Co-supervsors) Ehab S.Elmallah (External) Ynd Jng Date:

4 Abstract Wreless communcaton networks have recently attracted sgnfcant research attenton. As a crtcal ssue for mprovng network performance, effcent and ntellgent resource allocaton strateges have been ntensvely studed. Ths thess conssts of two studes on resource allocaton strategy, specfcally, jont bandwdth and power allocaton strategy, for wreless communcaton networks where both bandwdth and power are constraned resources. In the frst study, jont bandwdth and power allocaton strategy s proposed for wreless mult-user networks wthout relayng and wth decode-and-forward relayng based on three system-wde objectves. It s shown that the formulated resource allocaton problems are convex and, thus, the optmal solutons can be obtaned effcently usng convex optmzaton technques. Admsson control based on the jont bandwdth and power allocaton strategy s further consdered. A greedy search algorthm s developed for solvng the admsson control problem effcently, and the optmalty condtons of the greedy search algorthm are derved and shown to be mld. In the second study, jont bandwdth and power allocaton strategy s presented for maxmzng the sum ergodc capacty of secondary users under fadng channels n cogntve rado networks. Optmal bandwdth allocaton s derved n closed-form for any gven power allocaton. The structure of optmal power allocaton under each combnaton of four types of power constrants s derved. Usng these structures, effcent algorthms are developed for fndng the optmal power allocatons. In summary, ths thess has proposed, analyzed and solved jont bandwdth and power allocaton problems n wreless communcaton networks.

5 Acknowledgements Iwould lke to express apprecaton to my co-supervsors, Professor Chntha Tellambura and Professor Sergy A.Vorobyov for ther nsprng suggestons and persstent support throughout my research at the Unversty of Alberta. I feel fortunate to have studed under Chntha and Sergy s gudance at ths stage of my research career, durng whch I have learned many aspects of conductng research from ther knowledge and enthusasm n the feld of wreless communcatons. I am grateful to Professor Ha Jang for hs rewardng advces n our dscussons regardng my research. I also wsh to thank the commttee members, Professor Ehab S.Elmallah and Professor Ynd Jng, for ther valuable tme and efforts on revewng my thess. I want to show regard to all my colleagues and frends at the Unversty of Alberta, who have helped me n one way or another durng my graduate study there. I have been fortunate to know so many great people, wthout whom I would not have enjoyed my lfe n Edmonton and my research would not have been successful. The tme I have had wth them s an unforgettable experence n my lfe, and I hope we keep n touch n the future no matter where we are. My deepest grattude goes to my parents for ther endless love and support throughout my lfe. I owe them more than anyone else and they are the most mportant part of my lfe.

6 Contents 1 Introducton Motvaton Contrbuton Mathematcal Background Convex functons Convex optmzaton problems Lagrange dualty and Karush-Kuhn-Tucker condtons Solvng convex optmzaton problems Thess Outlne Jont Bandwdth and Power Allocaton wth Admsson Control n Wreless Mult-User Networks Wth and Wthout Relayng Introducton System Model Jont Bandwdth and Power Allocaton Sum capacty maxmzaton Worst user capacty maxmzaton Total network power mnmzaton Admsson Control Based on Jont Bandwdth and Power Allocaton Greedy search algorthm Complexty of the greedy search algorthm Optmalty condtons of the greedy search algorthm Smulaton Results

7 2.5.1 Jont bandwdth and power allocaton Greedy search algorthm Concluson Optmal Bandwdth and Power Allocaton for Sum Ergodc Capacty under Fadng Channels n Cogntve Rado Networks Introducton System Model Optmal Bandwdth Allocaton Optmal Power Allocaton Peak transmt power wth peak nterference power constrants Average transmt power wth average nterference power constrants Peak transmt power wth average nterference power constrants Average transmt power wth peak nterference power constrants Combnatons of more than two power constrants Smulaton Results Concluson Concluson and Future Work Concluson Future Work References 73

8 Lst of Fgures 2.1 Wreless mult-user network wthout relayng Wreless mult-user network wth relayng The range of c j /c to satsfy the condton C Sum capacty vs P R, W Worst user capacty vs P R, W Total network power vs c, W Admsson probablty vs capacty threshold Greedy search vs exhaustve search: wthout relayng Greedy search vs exhaustve search: wth relayng Cogntve rado network Optmal power allocaton under PTP + PIP constrants Optmal power allocaton under ATP + AIP constrants Optmal power allocaton under PTP + AIP constrants Optmal power allocaton under ATP + PIP constrants Optmal power allocaton under PTP + PIP + ATP + AIP constrants Sum ergodc capacty vs P pk Sum ergodc capacty vs P av Sum ergodc capacty vs Q pk Sum ergodc capacty vs Q av Sum ergodc capacty vs W

9 Chapter 1 Introducton Recently, wreless communcaton networks have attracted a lot of research efforts. The man nterest concerns developng effcent and ntellgent resource allocaton strateges. Ths thess focuses on the resource allocaton n wreless communcaton networks. 1.1 Motvaton Wreless communcaton networks serve as essental means to carry out data communcatons among multple users. One of the crtcal ssues n wreless communcaton networks s the effcent allocaton of avalable rado resources n order to mprove network performance. Future wreless networks, such as cellular and ad hoc networks, are expected to provde users wth relable data transmssons at hgh rates. Thus, t s a challengng task to acheve a system-wde goal for the network, whle users ndvdual Qualty of Servce (QoS) requrements also need to be satsfed. Intellgent resource allocaton schemes should capture the tradeoff between user-centrc constrants and a partcular network-centrc objectve. Moreover, there exsts a conflct between the ncreasng demand for wreless servces and the avalablty of rado resources, ncludng both bandwdth and power resources. The overly crowded spectrum allocaton charts gven by Federal Communcatons Commsson (FCC) ndcates that the rado spectrum avalable for emergng wreless applcatons s scarce. Transmsson power s also a constraned resource at wreless devces due to ther fnte battery energy and hardware constrants. Therefore, there s a strong motvaton for 1

10 the research on effectve resource management and dstrbuton that can make the best use of lmted rado resources. Furthermore, effcent resource utlzaton s desrable for explotng the dynamcs and dversty nature of wreless mult-user networks. Snce conventonal fxed resource allocaton schemes are desgned regardless of the tme-varyng characterstc of wreless channel condtons, they certanly can not acheve hgh effcency. On the contrary, a dynamc resource allocaton scheme can take full advantage of the channel dversty among users by dstrbutng resources adaptvely accordng to ther avalable channel state nformaton (CSI) and, thus, enhance the performance substantally. Numerous works have been conducted on the resource allocaton of wreless communcaton networks. Power allocaton strateges have been a research focus for both energy effcency and nterference management. Power control technques for nterference-lmted networks, such as cellular networks, have been studed ntensvely n the lterature (see, for example, [1]- [6]), and amed at achevng optmal network performance whle guaranteeng a target sgnal-to-nterference-plus-nose rato (SINR) for each user. On the other hand, jont bandwdth and power allocaton strategy has receved much less attenton [10]- [12]. In fact, the jont allocaton of bandwdth and power s especally crtcal n practcal wreless networks, where both the avalable transmsson power of ndvdual nodes and the total avalable bandwdth for all users are lmted. Due to the lmted resources n wreless networks, there are stuatons where not all users can be satsfed wth ther QoS requrements and, therefore, admsson control should be carred out to determne whch users can be admtted nto the network. Power allocaton wth admsson control have been nvestgated for nterference-lmted networks [1], [3], [13], [14], where a removal approach has been proposed for removng users untl the remanng users n the network are feasble. 1.2 Contrbuton Motvated by the need to mprove the effcency of the conventonal dsjont allocaton strateges for bandwdth and power resources, ths thess ams at studyng the fundamental performance lmts of jont bandwdth and power allocaton strategy for wreless communcaton networks where both bandwdth and power are constraned resources. In partcular, the jont bandwdth and power allocaton strategy s studed for two setups. 2

11 In the frst setup, jont bandwdth and power allocaton strategy s proposed for wreless mult-user networks wthout relayng and wth decode-and-forward relayng based on () sum capacty maxmzaton; () worst user capacty maxmzaton; and () total network power mnmzaton. The formulated resource allocaton problems are shown to be convex and, thus, can be solved effcently usng convex optmzaton technques. Due to lmted resources, the network may not be able to support all users wth ther QoS requrements. Therefore, admsson control based on the jont bandwdth and power allocaton strategy s further consdered, whch ams at maxmzng the number of users that can be admtted nto the network. Snce fndng the optmal soluton to the admsson control problem has hgh complexty, a suboptmal greedy search algorthm s developed for solvng t effcently. The optmal condtons of the greedy search algorthm are derved and shown to be mld. In the second setup, jont bandwdth and power allocaton strategy s presented for maxmzng the sum ergodc capacty of secondary users (SUs) under fadng channels n cogntve rado networks. Optmal bandwdth allocaton s derved n closed-form for any gven power allocaton. The structures of the optmal power allocaton under all possble combnatons of four types of power constrants are derved, whch ndcate the possble numbers of users that transmt at nonzero power but below ther correspondng peak power, and show that other users do not transmt or transmt at ther correspondng peak power. Effcent algorthms are developed based on these structures for fndng the optmal power allocatons. The solutons and algorthms obtaned n both works acheve sgnfcant performance mprovements compared to conventonal methods, whch s verfed by numercal results provded n smulatons. 1.3 Mathematcal Background In ths secton, we brefly ntroduce convex optmzaton prelmnares, whch serve as the man mathematcal tool for studyng the resource allocaton problems n ths thess. The modelng, desgn, and optmzaton of wreless communcaton networks rely on optmzaton theory, whch has found a wde range of applcatons n wreless communcatons and networkng. Convexty and non-convexty s the great watershed n optmzaton theory. It s recognzed that non-convex optmzaton problems are computatonally dffcult 3

12 to solve and, thus, have receved lmted attenton [5]- [7]. Convex optmzaton, however, s much more appealng [15]- [18], snce many problems can be dentfed or formulated as convex optmzaton problems and ther optmal solutons can be computed relably and effcently through establshed technques such as nteror pont methods [8], even f they nvolve nonlnear objectves and constrants. Apart from computatonal effcency, convex optmzaton also offers theoretcal advantages by gvng nsghtful nterpretatons for optmal solutons such as, for example, Lagrange dualty. The avalablty of software packages for solvng convex optmzaton problems, such as [9], further enhances the popularty of convex optmzaton Convex functons Consder a functon f : R n R defned on a convex set D. We say f s a convex functon f for any two x, y D, the followng nequalty holds for any α [0, 1] f(αx + (1 α)y) αf(x) + (1 α)f(y). (1.1) A geometrc nterpretaton of (1.1) s that the plot of f along the lnear nterval from x to y s below the lne segment connectng (x, f(x)) and (y, f(y)). We say f s concave f f s convex. Suppose f s frst-order dfferentable. The frst-order convexty condton states that f s convex f and only f the followng nequalty holds for any two x, y D f(y) f(x) + f(x) T (y x). (1.2) Suppose f s second-order dfferentable,.e., the second-order dervatve exsts. The second-order convexty condton states that f s convex f and only f the second-order dervatve s postve semdefnte,.e., 2 f(x) 0. (1.3) Some basc convex functons nclude lnear functons, exponental and logarthmc functons, and power functons. There are some operatons that preserve the convexty of convex functons, ncludng addton, nonnegatve scalng, and pontwse maxmum. 4

13 1.3.2 Convex optmzaton problems Mathematcally, an optmzaton problem can be wrtten n the followng standard form mn x D f 0 (x) (1.4a) subject to (s.t.) f (x) 0, = 1,, m (1.4b) h (x) = 0, = 1,, l (1.4c) where f 0 denotes the objectve functon, f denotes the -th nequalty constrant functon, h denotes the -th equalty constrant functon, and D denotes the doman of the optmzaton problem. A pont x D s feasble f t satsfes the constrants (1.4b) and (1.4c). The problem (1.4a)-(1.4c) s feasble f there exsts at least one feasble pont, and s nfeasble f there does not exst any feasble pont. A feasble pont s optmal, denoted by x, f f 0 (x) f 0 (x ) holds for any feasble pont x. The optmal value of the problem (1.4a)-(1.4c), denoted by v, s defned as the value of the objectve functon at the optmal pont,.e., v = f 0 (x ). The problem (1.4a)-(1.4c) s a convex optmzaton problem f the objectve functon and nequalty constrant functons are convex, and the equalty constrant functons are lnear Lagrange dualty and Karush-Kuhn-Tucker condtons The Langrangan L : R n R m R l R assocated wth the problem (1.4a)-(1.4c) s defned as m l L(x, λ, µ) = f 0 (x) + λ f (x) + µ h (x) (1.5) =1 =1 where λ [λ 1 λ m ], µ [µ 1 µ l ], λ s the Lagrange multpler assocated wth the th nequalty constrant, and µ s the Lagrange multpler assocated wth the th equalty constrant. The Lagrange dual functon s defned as ( m g(λ, µ) = nf L(x, λ, µ) = nf f 0 (x) + λ f (x) + x D x D =1 ) l µ h (x). (1.6) It can be seen that the optmal value v of the problem (1.4a)-(1.4c) s lower bounded by the dual functon,.e., g(λ, µ) v, for any λ 0 and any µ. The tghtest lower bound =1 5

14 for v can be obtaned by solvng the dual problem defned as follows max λ,µ g(λ, µ) (1.7a) s.t. λ 0. (1.7b) Note that the dual problem (1.7a)-(1.7b) s always convex regardless of the convexty of the orgnal problem (1.4a)-(1.4c). The orgnal problem (1.4a)-(1.4c) s also called the prmal problem n ths context. Let d denote the optmal value of the dual problem (1.7a)-(1.7b). Then v d s defned as the optmal dualty gap between the prmal problem (1.4a)-(1.4c) and the dual problem (1.7a)-(1.7b). If the prmal problem (1.4a)-(1.4c) s convex, the optmal dualty gap s zero,.e., v d = 0, and we say that strong dualty holds. Usng the property of strong dualty, f the prmal problem (1.4a)-(1.4c) s convex, t can be solved equvalently by solvng the dual problem (1.7a)-(1.7b). Suppose the objectve functon f 0 and the nequalty functons f, = 1,, m are dfferentable. The optmal soluton of the prmal problem (1.4a)-(1.4c) and the dual problem (1.7a)-(1.7b), denoted by x and (λ, µ ), respectvely, satsfy the followng Karush-Kuhn- Tucker (KKT) condtons f (x ) 0, = 1,, m (1.8a) h (x ) = 0, = 1,, l (1.8b) λ 0, = 1,, m (1.8c) λ f (x ) = 0, = 1,, m (1.8d) m l f 0 (x ) + λ f (x ) + µ h (x ) = 0. (1.8e) =1 =1 In general, the KKT condtons are only necessary condtons for the optmal solutons x and (λ, µ ). However, f the prmal problem (1.4a)-(1.4c) s convex, the KKT condtons are both necessary and suffcent condtons for the optmal solutons and, therefore, solvng for the KKT condtons s equvalent to solvng the prmal problem (1.4a)-(1.4c). 6

15 1.3.4 Solvng convex optmzaton problems Although ntensve study has been done on analyzng the propertes of varous classes of optmzaton problems and developng algorthms for solvng them, optmzaton problems are generally computatonally dffcult to solve, even f the objectve and constrant functons are smooth. The effcency of computng the optmal solutons of general optmzaton problems depends on dfferent factors, ncludng the partcular forms and structures of the objectve and constrant functons, and the numbers of the varables and constrants. However, there exst few classes of optmzaton problems that can be relably and effcently solved by effectve algorthms, even f the problems nvolve a large number of varables and constrants. Convex optmzaton problems can serve as an example of such problems. Analytcal solutons of convex optmzaton problems can be obtaned, f possble, usng Lagrange dualty or the KKT condtons. However, general analytcal formulas for the optmal solutons are not avalable and, therefore, effectve methods lke nteror-pont methods should be used. Interor-pont methods can solve a convex optmzaton problem n an almost constant number of teratons regardless of the structure of the problem. In practce, a convex optmzaton problem wth hundreds or even thousands of varables and constrants can be solved effcently on a desktop computer n a few tens of seconds. Therefore, once we can recognze and formulate a research problem as a convex optmzaton problem, we can clam that we have found a method to solve ths research problem. 1.4 Thess Outlne Ths thess studes resource allocaton, specfcally, bandwdth and power allocaton n wreless communcaton networks. The outlne of each chapter s gven below. Chapter 1 provdes the motvaton, contrbuton, and outlne of the thess, and ntroduces basc convex optmzaton theory. Chapter 2 presents jont bandwdth and power allocaton strategy for wreless multuser networks wthout relayng and wth decode-and-forward relayng by takng nto account three network performance measures,.e., the sum capacty, the worst capacty, and the total network power consumpton. The admsson control problem based on the jont bandwdth and power allocaton strategy s further consdered. A greedy search algorthm s developed 7

16 to solve the admsson control problem effcently. The complexty and optmalty condtons of the greedy search algorthm are nvestgated. Chapter 3 proposes jont bandwdth and power allocaton strategy for the sum ergodc capacty maxmzaton of SUs under fadng channels n cogntve rado networks. Optmal bandwdth allocaton s derved frst n terms of any gven power allocaton. Then optmal power allocaton s obtaned subject to each combnaton of four types of power constrants. Chapter 4 summarzes the results of the thess and proposes future work drectons. 8

17 Chapter 2 Jont Bandwdth and Power Allocaton wth Admsson Control n Wreless Mult-User Networks Wth and Wthout Relayng Equal allocaton of bandwdth and/or power may not be effcent for wreless mult-user networks wth lmted bandwdth and power resources. Jont bandwdth and power allocaton strateges for wreless mult-user networks wth and wthout relayng are proposed n ths chapter for () the maxmzaton of the sum capacty of all users; () the maxmzaton of the worst user capacty; and () the mnmzaton of the total power consumpton of all users subject to rate requrements. It s shown that the proposed allocaton problems are convex and, therefore, can be solved effcently. Moreover, the admsson control based jont bandwdth and power allocaton s consdered. A suboptmal greedy search algorthm s developed to solve the admsson control problem effcently. The condtons under whch the greedy search s optmal are derved and shown to be mld. The performance mprovements offered by the proposed jont bandwdth and power allocaton are demonstrated by smulatons. The advantages of the suboptmal greedy search algorthm for admsson control are also shown. The rest of ths chapter s organzed as follows. Secton 2.1 gves the overvew of the 9

18 related lterature and summarzes the contrbutons. System models of mult-user networks wthout relayng and wth decode-and-forward relayng are gven n Secton 2.2. In Secton 2.3, jont bandwdth and power allocaton problems for the three aforementoned objectves are formulated and solved for both types of networks wth and wthout relayng. Admsson control problem based on jont bandwdth and power allocaton s formulated n Secton 2.4, where the greedy search algorthm s also developed and nvestgated for both types of systems wth and wthout relayng. Numercal results are reported n Secton 2.5, followed by concludng remarks n Secton Introducton It has been shown that the effcency of wreless communcatons can be mproved by usng relays [19]- [20]. In a relay-asssted communcaton system, the data transmtted from a source s forwarded va relayng to the correspondng destnaton. Snce relay-asssted communcaton has sgnfcant advantages such as extended coverage and enhanced communcaton qualty, relay networks are consdered promsng canddates for future wreless networks. One crtcal ssue n relay networks s the effcent allocaton of avalable rado resources to enhance the performance of relayng. Therefore, numerous works have been done on the resource allocaton for relay networks (see, for example, [21]- [33]). Note that [22]- [30] as well as most of the exstng works consder a sngle user,.e., a sngle source-destnaton par, whle only a few works have studed resource allocaton for mult-user relay networks. Power allocaton amng at optmzng the sum capacty of multple users for four dfferent relay transmsson strateges has been studed n [31], whle an AF based strategy n whch multple sources share multple relays usng power control has been developed n [32], [33]. In practcal wreless networks where both the avalable transmsson power of ndvdual nodes and the total avalable bandwdth of the network are lmted, jont bandwdth and power allocaton should be consdered [10] [12]. It s worth notng that most of the works mentoned above on the resource allocaton for relay networks have assumed equal and fxed bandwdth allocaton for the one-hop lnks from a source to a destnaton. In fact, t s neffcent to allocate the bandwdth equally when the total avalable bandwdth s lmted. Therefore, jont bandwdth and power allocaton s mportant for both networks 10

19 wth and wthout relayng. Varous performance metrcs for resource allocaton n mult-user networks have been consdered. System throughput maxmzaton and the worst user throughput maxmzaton are studed usng convex optmzaton n [15]. Sum capacty maxmzaton s taken as an objectve for power allocaton n [31], whle max-mn SNR, power mnmzaton, and throughput maxmzaton are used as power allocaton crtera n [32]. In some applcatons, certan mnmum transmsson rates must be guaranteed for the users n order to satsfy ther qualty-of-servce (QoS) requrements. For nstance, n realtme voce and vdeo applcatons, a mnmum rate should be guaranteed for each user to satsfy the delay constrants of the servces. However, when the rate requrements can not be supported for all users, admsson control s adopted to decde whch users to be admtted nto the network. The admsson control n wreless networks typcally ams at maxmzng the number of admtted users and has been recently consdered n several works. A snglestage reformulaton approach for a two-stage jont resource allocaton and admsson control problem s proposed n [34], [35], whle another approach s based on user removals [1], [3], [13], [14], [36]. To the best of our knowledge, admsson control based on jont bandwdth and power allocaton has never been consdered. In ths chapter 1, the problem of jont bandwdth and power allocaton for wreless multuser networks wth and wthout relayng s consdered, whch s especally effcent for the networks wth both lmted bandwdth and lmted power. The jont bandwdth and power allocaton are proposed to () maxmze the sum capacty of all users; () maxmze the capacty of the worst user; () mnmze the total power consumpton of all users. The correspondng jont bandwdth and power allocaton problems can be formulated as optmzaton problems that are shown to be convex. Therefore, these problems can be solved effcently by usng convex optmzaton technques. The jont bandwdth and power allocaton together wth admsson control s further consdered, and a greedy search algorthm s developed n order to reduce the computatonal complexty of solvng the admsson control problem. The optmalty condtons of the greedy search are derved and shown to be mld. 1 Ths work has been presented n [37], [38] and [39]. 11

20 2.2 System Model Wthout Relayng Consder a wreless network, whch conssts of M source nodes S, M = {1, 2,, M}, and K destnaton nodes D, K = {1, 2,, K}, as shown n Fg The network serves N users U, N = {1, 2,, N}, where each user represents a one-hop lnk from a source to a destnaton. The set of users whch are served by S s denoted by N S. N S = {j 1,j,j +1} S... 1 S... S M U 1 U 2 U j 1 U j U j+1 U N 1 U N D D K Fg Wreless mult-user network wthout relayng. A spectrum of total bandwdth W s avalable for the transmsson from the sources. Ths spectrum can be dvded nto dstnct and nonoverlappng channels of unequal bandwdths, so that the sources share the avalable spectrum through frequency dvson and, therefore, do not nterfere wth each other. Let P S and W S denote the allocated transmt power and channel bandwdth of the source to serve U. Then the receved SNR at the destnaton of U s γ D = P ShSD W SN 0 (2.1) where h SD denotes the channel gan of the source destnaton lnk of U and W S N 0 stands for the power of addtve whte Gaussan nose (AWGN) over the bandwdth W S. channel gan h SD The results from such effects as path loss, shadowng, and fadng. Due to the fact that the power spectral densty (PSD) of AWGN s constant over all frequences wth the constant value denoted by N 0, the nose power n the channel s lnearly ncreasng wth the channel bandwdth. It can be seen from (2.1) that a channel wth larger bandwdth ntroduces hgher nose power and, thus, reduces the SNR. 12

21 Channel capacty gves an upper bound on the achevable rate of a lnk. Gven γ D, the source destnaton lnk capacty of U s C SD It can be seen that W S spectral effcency and, thus, C SD = W S log(1 + γ D ) = W S log (1 + P S hsd W S N 0 ). (2.2) characterzes channel bandwdth, and log(1 + γ R ) characterzes characterzes data rate over the source destnaton lnk n bts per second. Moreover, for fxed W S, CSD s a concave ncreasng functon of P S. It can be also shown that C SD γ D s a concave ncreasng functon of W S s a lnear decreasng functon of W S. Indeed, t can be proved that CSD functon of P S Wth Relayng and W S jontly [11], [12]. for fxed P S, although s a concave Consder L relay nodes R, L = {1, 2,, L} added to the network descrbed n the prevous subsecton and used to forward the data from the sources to the destnatons, as shown n Fg Then each user represents a two-hop lnk from a source to a destnaton va relayng. To reduce the mplementaton complexty at the destnatons, sngle relay assgnment s adopted so that each user has one desgnated relay. Then the set of users served by R s denoted by N R. The relays work n a half-duplex manner due to the practcal lmtaton that they can not transmt and receve at the same tme. A twophase decode-and-forward (DF) protocol s assumed,.e., the relays receve and decode the transmtted data from the sources n the frst phase, and re-encode and forward the data to the destnatons n the second phase. The sources and relays share the total avalable spectrum n the frst and second phases, respectvely. It s assumed that the drect lnks between the sources and the destnatons are blocked and, thus, are not avalable. Note that although the two-hop relay model s consdered n the paper, the results are applcable for mult-hop relay models as well. Let P R and W R denote the allocated transmt power and channel bandwdth of the relay to serve U. The two-hop source destnaton lnk capacty of U s gven by { C SD = mn{c SR, C RD } = W S log (1 + P S ) hsr, W R log (1 + P R )} hrd where C SR and C RD U, respectvely, and h SR W S N 0 W R N 0 (2.3) are the one-hop source relay and relay destnaton lnk capactes of and h RD denote the correspondng channel gans. 13

22 N S = {j 1,j,j +1} S... 1 S... S M U j 1 1st phase U 1 U 2 U j U j+1un 1 U N R 1... R... R L N RL = {2,j +1,N} 2nd phase D D K Fg Wreless mult-user network wth relayng. C RD It can be seen from (2.3) that f equal bandwdth s allocated to W S can be unequal due to the power lmts on P S lnk capacty C SD s constraned by the mnmum of C SR and W R, CSR and and P R. Then the source destnaton and C RD. Note that snce all users share the total bandwdth of the spectrum, equal bandwdth allocaton for all one-hop lnks can be neffcent. Therefore, the jont allocaton of bandwdth and power s necessary. 2.3 Jont Bandwdth and Power Allocaton Dfferent objectves can be consdered whle jontly allocatng bandwdth and power n wreless mult-user networks. The wdely used objectves for network optmzaton are () the sum capacty maxmzaton; () the worst user capacty maxmzaton; and () the total network power mnmzaton. In ths secton, the problems of jont bandwdth and power allocaton are formulated for the aforementoned objectves for both consdered systems wth and wthout relayng. It s shown that all these problems are convex and, therefore, can be effcently solved usng standard convex optmzaton methods Sum capacty maxmzaton In the applcatons wthout delay constrants, a hgh data rate from any user n the network s preferable. Thus, t s desrable to allocate the resources to maxmze the overall network performance, e.g., the sum capacty of all users. 14

23 Wthout Relayng In ths case, the jont bandwdth and power allocaton problem amng at maxmzng the sum capacty of all users can be mathematcally formulated as max {P S,W S} N s.t. C SD (2.4a) P S P Sj, j M (2.4b) N Sj N W S W. (2.4c) The nonnegatvty constrants on the optmzaton varables {P S, W S } are natural and, thus, omtted throughout the paper for brevty. In the problem (2.4a) (2.4c), the constrant (2.4b) stands that the total power at S j s lmted by P Sj, whle the constrant (2.4c) ndcates that the total bandwdth of the channels allocated to the sources s also lmted by W. Note that snce C SD s a jontly concave functon of P S and W S, the objectve functon (2.4a) s convex. The constrants (2.4b) and (2.4c) are lnear and, thus, convex. Therefore, the problem (2.4a) (2.4c) tself s convex. Usng the convexty, the closed-form optmal soluton of the problem (2.4a) (2.4c) can be found as t s shown below. It s worth notng that the optmal soluton demonstrates that for a set of users served by one source, the sum capacty maxmzaton based allocaton strategy allocates all the power of each source only to one user, that s, the user wth the hghest channel gan. Therefore, t results n hghly unbalanced resource allocaton among the users. The followng proposton descrbes the result formally. Proposton 2.1: The optmal soluton of the problem (2.4a) (2.4c), denoted by {P S W S N }, s P S = P S, W S / I, where P S, = W h SD P S / j I hsd j Pj S, I, and P S = W S = 0, s the total power of the source servng U,.e., P S = PSk for N Sk, and I = { = arg max j NSk h SD j, k M}. Proof: We frst gve the followng lemma. 15

24 Lemma 2.1: The optmal soluton of the problem max {p,w } s.t. N N p p w w N ( w log 1 + h ) p w (2.5a) (2.5b) (2.5c) whch s denoted by {p N }, s p k = p, w k = w, and p = w = 0, k, where k = arg max N h. Proof of Lemma 2.1: Consder f N = {1, 2}. Then the problem (2.5a) (2.5c) s equvalent to max p 1 p, w 1 w ( g(w, p) = w log 1 + h ) ( 1p 1 + (w w 1 ) log 1 + h ) 2(p p 1 ). (2.6) w 1 w w 1 Assume wthout loss of generalty that h 1 > h 2. Consder f the constrants 0 p 1 p and 0 w 1 w are nactve at optmalty. Snce the problem (2.6) s convex, usng the Karush-Kuhn-Tucker (KKT) condtons, we have ( log 1 + h 1p ) 1 w1 h 1p 1 w1 + h 1p 1 ( h1 p 1 = y w 1 ) y ( log 1 + h 2(p p 1 ) ) h 2 (p p 1 w w1 + ) w w1 + h 2(p p 1 ( ) h2 (p p 1 ) ) w w1 = 0 (2.7a) h 1 w1 h 2 (w w1 w1 + h 1p ) 1 w w1 + h 2(p p 1 ) = 0. (2.7b) where y(x) log(1 + x) x/(1 + x). Snce y(x) s monotoncally ncreasng, t can be seen from (2.7a) that h 1 p 1 w 1 = h 2(p p 1 ) w w1. (2.8) Combnng (2.7b) and (2.8), we obtan h 1 = h 2, whch contradcts the condton h 1 > h 2. Therefore, at least one of the constrants 0 p 1 p and 0 w 1 w s actve at optmalty. Then t can be shown that p 1 = p and w 1 = w. Note that ths s also the optmal soluton f h 1 = h 2 s assumed. Furthermore, ths concluson can be drectly extended to the case of N > 2 by nducton. Ths completes the proof. 16

25 Now we are ready to show Proposton 2.1. It can be seen from Lemma 2.1 that P S P S, I, and P S = 0, / I. Then the problem (2.4a) (2.4c) s equvalent to max {W S} I s.t. I W S log ( h SD 1 + P S W SN 0 ) = (2.9a) W S W. (2.9b) Snce the problem (2.9a) (2.9b) s convex, usng the KKT condtons, we have ( ) ( ) log 1 + P S h SD P S h SD N0 N0 + P S λ P S h SR = y λ = 0, I (2.10a) N0 W S W S h SD W S W I W S = 0 (2.10b) where λ denotes the optmal Lagrange multpler, and y(x) log(1 + x) x/(1 + x). Snce y(x) s monotoncally ncreasng, t follows from (2.10a) that P S h SR W S N0 = P S j W S j h SR j Solvng the system of equatons (2.10b) and (2.11), we obtan W S I. Ths completes the proof. Wth Relayng N0,, j I 1, j. (2.11) = W h SD P S / j I h SD j Pj S, The sum capacty maxmzaton based jont bandwdth and power allocaton problem for the network wth DF relayng s gven by max C {P S,W S,P R,W R} SD (2.12a) N s.t. P S P Sj, j M (2.12b) N Sj P R P Rj, j L (2.12c) N Rj N N W S W (2.12d) W R W. (2.12e) 17

26 Introducng new varables {T N }, the problem (2.12a) (2.12e) can be equvalently rewrtten as mn {P S,W S,P R,W R,T } N T (2.13a) s.t. T C SR 0, N (2.13b) T C RD 0, N (2.13c) the constrants (2.12b) (2.12e). Note that the constrants (2.13b) and (2.13c) are convex snce C SR and C RD (2.13d) are jontly concave functons of P S, W S and P R, W R, respectvely. The constrants (2.13d) are lnear and, thus, convex. Therefore, the problem (2.13a) (2.13d) tself s convex. It can be seen that the closed-form optmal soluton of the problem (2.13a) (2.13d) can not be obtaned due to the couplng of the constrants (2.13b) and (2.13c). However, the convexty of the problem (2.13a) (2.13d) allows to use standard numercal convex optmzaton algorthms for solvng the problem effcently [8]. Intutvely, the sum capacty maxmzaton based allocaton for the network wth DF relayng should not result n as unbalanced resource allocaton as that for the network wthout relayng. It s because the channel gans n both transmsson phases for the networks wth relayng affect the achevable capacty of each user. Below we gve the condtons under whch the sum capacty maxmzaton based resource allocaton strategy for the network wth relayng does not allocate any resources to some users. In partcular, f two users are served by the same source and the same relay, and one user has lower channel gans than the other user n both transmsson phases, then no resource s allocated to the former user. The result can be formally stated n terms of the followng proposton. Proposton 2.2: If h SR h SR j then Pj S = W S j = P R j = W R j = 0. Proof: It can be seen that and h RD h RD j where {, j} N Sk and {, j} N Rl, C SD + C SD j = mn{c SR, C RD } + mn{c SR, C RD } mn{c SR j j + Cj SR, C RD + Cj RD } (2.14) When P S j = W S j = P R j = W R j = 0, t follows from Lemma 2.1 that the maxmum value of the rght hand sde of (2.14) s acheved and equals to C SD and, on the other hand, the left 18

27 hand sde of (2.14) also equals to C SD. Therefore, the maxmum value of C SD + Cj SD acheved when P S j = W S j = P R j = W R j = 0. Ths completes the proof. s Worst user capacty maxmzaton Farness among users s also an mportant ssue for resource allocaton. If the farness ssue s consdered, the achevable rate of the worst user s commonly used as the network performance measure. In ths case, the jont bandwdth and power allocaton problem for the network wthout relayng can be mathematcally formulated as max mn {P S,W S} N CSD s.t. the constrants (2.4b) (2.4c). (2.15a) (2.15b) Smlar, for the networks wth relayng, the jont bandwdth and power allocaton problem can be formulated as max mn {P S,W S,P R,W R} N CSD s.t. the constrants (2.12b) (2.12e). (2.16a) (2.16b) Introducng a varable T, the problem (2.16a) (2.16b) can be equvalently wrtten as mn T (2.17a) {P S,W S,P R,W R,T } s.t. T C SR 0, N (2.17b) T C RD 0, N (2.17c) the constrants (2.12b) (2.12e). (2.17d) Smlar to the sum capacty maxmzaton based allocaton problems, t can be shown that the problems (2.15a) (2.15b) and (2.17a) (2.17d) are convex. Therefore, the optmal solutons can be effcently obtaned usng standard convex optmzaton methods. The next proposton ndcates that the worst user capacty maxmzaton based allocaton leads to absolute farness among users, just the opposte to the sum capacty maxmzaton based allocaton. The proof s ntutve from the fact that the total bandwdth s shared by all users, and s omtted for brevty. 19

28 Proposton 2.3: In the problems (2.15a) (2.15b) and (2.16a) (2.16b), the capactes of all users are equal at optmalty. Proof: Consder the problem (2.15a) (2.15b). Assume that the capacty of one user s larger than the mnmum capacty among the capactes of other users at optmalty. Then we can always take an arbtrary small amount of bandwdth allocated to ths user and reallocate t to the user(s) wth the mnmum capacty such that the mnmum capacty of all users s ncreased. Ths contradcts the optmalty assumpton. Thus, the capactes of all users are equal at optmalty n the problem (2.15a) (2.15b). Smlarly, t can be shown that all users acheve the same capacty at optmalty n the problem (2.16a) (2.16b). Ths completes the proof Total network power mnmzaton Another wdely consdered desgn objectve s the mnmzaton of the total power consumpton of all users. Ths mnmzaton s performed under the constrant that the rate requrements of all users are satsfed. The correspondng jont bandwdth and power allocaton problem for the network wthout relayng can be wrtten as mn {P S,W S} N P S (2.18a) s.t. c C SD 0, N (2.18b) the constrants (2.4b) (2.4c) (2.18c) where c s the mnmum acceptable capacty for U, whle the same problem for the network wth relayng s mn (P S {P S,W S,P R,W R} N + P R ) (2.19a) s.t. c C SR 0, N (2.19b) c C RD 0, N (2.19c) the constrants (2.12b) (2.12e) (2.19d) where the constrants (2.19b) and (2.19c) ndcate that the one-hop lnk capactes of U should be no less than the gven capacty threshold. Smlar to the sum capacty maxmzaton and worst user capacty maxmzaton based allocaton problems, the problems 20

29 (2.18a) (2.18c) and (2.19a) (2.19d) are convex and, thus, can be solved effcently as mentoned before. 2.4 Admsson Control Based on Jont Bandwdth and Power Allocaton In the mult-user networks under consderaton, admsson control s requred f a certan mnmum capacty must be guaranteed for each user. Thus, we next consder admsson control problem for both systems wth and wthout relayng. Wthout Relayng The objectve of admsson control s to maxmze the number of users whose capacty requrements can be satsfed subject to the bandwdth and power constrants of the network. The admsson control problem based on jont bandwdth and power allocaton n the network wthout relayng can be mathematcally expressed as max I (2.20a) {P S,W S},I N s.t. c C SD 0, I (2.20b) the constrants (2.4b) (2.4c) (2.20c) where I stands for the cardnalty of I. Note that the problem (2.20a) (2.20c) can be solved usng exhaustve search among all possble subsets of users. However, the computatonal complexty of the exhaustve search can be very hgh snce the number of possble subsets of users s exponentally ncreasng wth the number of users, whch s not acceptable for practcal mplementaton. Therefore, we develop a suboptmal greedy search algorthm that sgnfcantly reduces the complexty of solvng the admsson control problem (2.20a) (2.20c). 21

30 2.4.1 Greedy search algorthm Gven that all power constrants and capacty requrements are satsfed, the mnmum total bandwdth requred to support a set of users I can be defned as G(I), where G(I) mn {P S,W S} I W S (2.21a) s.t. c C SD 0, I (2.21b) the constrant (2.4b). (2.21c) The followng proposton provdes a necessary and suffcent condton for the admssblty of a set of users. Proposton 2.4: A set of users I s admssble f and only f G(I) W. Proof: It s equvalent to show that there exsts a feasble pont {P S, W S I} of the problem (2.20a) (2.20c) f and only f G(I) W. If {P S, W S I} s a feasble pont of the problem (2.20a) (2.20c), then snce t s also a feasble pont of the problem (2.21a) (2.21c), we have G(I) I W S W. If we have G(I) W, then the optmal soluton of the problem (2.21a) (2.21c) for I, denoted by {P S, W S I}, s a feasble pont of the problem (2.20a) (2.20c) snce I W S = G(I) W. Ths completes the proof. Proposton 2.4 s nstrumental n establshng our greedy search algorthm, whch removes users one by one untl the remanng users are admssble. The worst user,.e., the user whose removal reduces the total bandwdth requrement to the maxmum extent, s removed at each greedy search teraton. In other words, the removal of the worst user results n the mnmum total bandwdth requrement of the remanng users. 2 removal crteron can be stated as n(t) arg max n N (t 1) (G(N (t 1)) G(N (t 1) \ {n})) = arg mn n N (t 1) Thus, the G(N (t 1) \ {n}) (2.22) where n(t) denotes the user removed at the t-th greedy search teraton, N (t) N (t 1) \ {n(t)} denotes the set of remanng users after t greedy search teratons, and the symbol \ stands for the set dfference operator. Note that, ntutvely, N (t) can be nterpreted as the best set of N t users that requres the mnmum total bandwdth among all possble sets of N t users from N, and 2 Note that the approach based on user removals appears n dfferent contexts also n [1], [3], [13], [14], [36]. 22

31 G(N (t)) s the correspondng mnmum total bandwdth requrement. Thus, the stoppng rule for the greedy search teratons should be fndng such t that G(N (t 1)) > W and G(N (t )) W. In other words, N t can be nterpreted as the maxmum number of admssble users Complexty of the greedy search algorthm It can be seen from Proposton 2.4 that usng the exhaustve search for fndng the maxmum number of admssble users s equvalent to checkng G(I) for all possble I N and, therefore, the number of tmes of solvng the problem (2.21a) (2.21c) s upper bounded by N =d ( N ), where d denotes the optmal value of the problem (2.20a) (2.20c). On the other hand, t can be seen from (2.22) that usng the greedy search, the number of tmes of solvng the problem (2.21a) (2.21c) s upper bounded by t 1 =0 N. Therefore, the complexty of the proposed greedy search s sgnfcantly reduced as compared to that of the exhaustve search, especally f N s large and d s small. Moreover, the complexty of the greedy search can be further reduced. Then the lemma gven below s n order. Lemma 2.2: The reducton of the total bandwdth requrement after removng a certan user s only coupled wth the users served by the same source as ths user, and s decoupled wth the users served by other sources. Mathematcally, t means that G(I) G(I \ {n}) = G(I N S ) G(I N S \ {n}) for n N S, I N. Proof: Ths lemma follows drectly from the decomposable structure of the problem (2.21a) (2.21c), that s, G(I) = M G(I N S ). Ths completes the proof. Let N S (t) N S N (t) denote the set of remanng users served by S after t greedy search teratons. Then the followng proposton s of nterest. Proposton 2.5: The user to be removed at the t-th greedy search teraton accordng to (2.22) can be found by frst fndng the worst user n each set of users served by each source,.e., n S (t 1) arg max (G(N S (t 1)) G(N S (t 1) \ {n})) n N S (t 1) and then determnng the worst user among all these worst users. Mathematcally, t means that n(t) = n S (t 1) where ( arg max G(NS (t 1)) G(N S (t 1) \ {n S M (t 1)}) ). 23

32 Proof: Ths proposton follows from applyng Lemma 2.2 drectly to the removal crteron n (2.22). Ths completes the proof. Proposton 2.5 can be drectly used to buld an algorthm for searchng for the user to be removed at each greedy search teraton. It s mportant that such algorthm has a reduced computatonal complexty compared to the drect use of (2.22). As a result, although the number of tmes that the problem (2.21a) (2.21c) has to be solved remans the same, the number of varables of the problem (2.21a) (2.21c) solved at each tme s reduced, and s upper bounded by 2 max M N S Optmalty condtons of the greedy search algorthm We also study the condtons under whch the proposed greedy search algorthm s optmal. Specfcally, the greedy search s optmal f the set of remanng users after each greedy search teraton s the best set of users,.e., where N N (t) = N N t, 1 t N (2.23) arg mn I = G(I) s the best set of users. Let us apply the greedy search to the set of users N S served by the source S. The ( worst user,.e., the user n S (t) arg max n NS (t 1) G( NS (t 1)) G( N S (t 1) \ {n}) ) s removed at the t-th greedy search teraton, where N S (t) N S (t 1) \ { n S (t)} denotes the set of remanng users n the set N S arg mn I NS, I =j G(I) denote the best set of j users n N S. after t greedy search teratons. Also let N S,j The followng theorem decouples the optmalty condton (2.23) nto two equvalent condtons C1 and C2 per each set of users N S and, therefore, allows us to focus on equvalent problems n whch users are subject to the same power constrants. Specfcally, the condton C1 of the theorem ndcates that the set of remanng users n N S after each greedy search teraton s the best set of users, whle the condton C2 of the theorem ndcates that the reducton of the total bandwdth requrement s decreasng wth the greedy search teratons. hold: Theorem 2.1: The condton (2.23) holds f and only f the followng two condtons C1: NS (t) = N S,N S t, 1 t N S, M; C2: G( N S (t 2)) G( N S (t 1)) > G( N S (t 1)) G( N S (t)), 2 t N S, M. 24

33 Proof: We frst show that C1 and C2 are suffcent condtons. Defne V (n) G(N (t 1)) G(N (t)) for n = n(t), 1 t N. It follows from C2 that V ( n S (1)) > V ( n S (2)) > > V ( n S (N S )), M. Then usng Proposton 2.2, we have n(t) = arg max n N (t 1) V (n), 1 t N. Therefore, we obtan V (n(1)) > V (n(2)) > > V (n(n)). (2.24) It can be seen from C1 that N \ N N t N S = arg mn I NS, I =t G(N S \ I) = { n S (j) 1 j t }, M, where t N \ N N t N S. Then we have N \ N N t = { n S (j) 1 j t, M} and G(N ) G(NN t ) = t M j=1 V ( n S (j)). Therefore, we obtan {t M} = arg max {k }; k M k =t M j=1 V ( n S (j)). Snce t follows from C2 that V ( n S (1)) > V ( n S (2)) > > V ( n S (N S )), M, we have N \ NN t = arg max I N, I =t n I V (n) = {n() 1 t} = N \ N (t), where the second equalty s from (2.24). Ths completes the proof for suffcency of C1 and C2. We next show that C1 and C2 are necessary condtons by gvng two nstructve counter examples. Consder f C1 does not hold. Assume wthout loss of generalty that M = {1}. Then t can be seen that C1 s equvalent to the condton (2.23) and, therefore, the condton (2.23) does not hold, ether. Consder f C2 does not hold. Assume wthout loss of generalty that M = {2}, N S2 = 1 and G( N S1 (1)) G( N S1 (2)) > G(N S2 ) G( N S2 (1)) > G(N S1 ) G( N S1 (1)). Then we have N N 2 = N \ { n S 1 (1), n S1 (2)}, whle t follows from Proposton 2.2 that N (2) = N \ { n S1 (1), n S2 (1)}. Therefore, NN 2 N (2). Ths completes the proof for necessty of C1 and C2. Let h h SD /N 0 denote the channel gan normalzed by the nose PSD. Recall that c s the mnmum acceptable capacty for U. Defne F (p) as the unque soluton for w n the equaton ( c = w log 1 + h ) p w (2.25) gven h and c for any p > 0, whch represents the mnmum bandwdth requred by a user for ts allocated transmt power. Then the problem (2.21a) (2.21c) for the set of users N S 25