Hanif D. Sheralit. and foremost, we show that performance gap between. time in solving the optimization problem. II.

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1 CROSS-LAYER OPTMZATON FOR UWB-BASED AD HOC NETWORKS Y Sh* * Y. Thomas Hou* Hanf D. Sheralt Sastry Kompella' The Bradley Department of Electrcal and Computer Engneerng, Vrgna Tech, Blacsburg, VA The Grado Department of ndustral and Systems Engneerng, Vrgna Tech, Blacsburg, VA + nformaton Technology Dvson, Naval Research Laboratory, Washngton, DC t Technque (RLT) [13]. Durng each teraton of the branch-and-bound procedure, t s mportant (from computaton perspectve) to select a partton varable. We propose a partton varable selecton polcy not only based on the relaxaton error, but also based on ther relatve sgnfcance n our problem. t turns out that such polcy offers a soluton much faster than the polcy that s solely based on relaxaton error. n our numercal results, we further explore the propertes of ths cross-layer optmzaton problem. Frst and foremost, we show that performance gap between a cross-layer formulaton and a decoupled-desgn s huge, thereby underlnng the mportance of cross-layer optmzaton. n addton, we observe that the number ndex Terms-Nonlnear programmng, optmzaton, of tme slots does not need to be large to have nearcross-layer desgn, ultra-wdeband (UWB), ad hoc net- optmal performance. Ths result s mportant n practce wors, data rate utlty, schedulng, power control, routng. as fewer number of slots wll lead to less computaton tme n solvng the optmzaton problem. Abstract n ths paper, we consder a UWB-based ad hoc networ and study how to maxmze data rate utlty for a group of communcaton sessons. We formulate the data rate utlty problem nto a nonlnear programmng (NLP) problem through a cross-layer approach by tang nto consderaton of schedulng and power control at ln layer and routng at networ layer. The man contrbuton of ths paper s the development of a soluton procedure based on branch-and-bound framewor. We employ a powerful optmzaton technque called reformulaton lnearzaton technque (RLT) to obtan a lnear relaxaton, whch s a ey component requred n the branch-andbound framewor. Numercal results demonstrate the mportance of cross-layer approach and offer some mportant nsghts. NTRODUCTON n ths paper, we consder a group of sourcedestnaton communcaton pars, whch we call sessons, n UWB-based ad hoc networs. The objectve s to maxmze the total utlty, where a sesson's utlty s defned as the log functon of ts data rate []. Clearly, ths optmzaton problem nvolves ssues from dfferent layers,.e., ln layer schedulng, power control, and networ layer routng. The ln layer schedulng component deals wth how to use tme slots for transmsson and recepton. The power control component consders how much transmsson power a node should use n a partcular tme slot. Fnally, the routng problem at the networ level consders what set of paths the data from a source node should tae to ts destnaton node. We am to nvestgate the problem through formal nonlnear optmzaton technque. The outcome of ths effort wll fll n a crtcal theoretcal gap on ths problem. t wll also contrbute some mportant understandng, some of whch were overlooed by pror research efforts. The man contrbuton of ths paper s the development of a soluton procedure to the nonlnear programmng problem based on the branch-and-bound framewor [] and a powerful technque called Reformulaton-Lnearzaton. NETWORK MODEL AND OPTMZATON SPACE We consder an ad hoc networ consstng of N nodes and L un-drectonal source-destnaton communcaton sessons over a two-dmensonal area. We now tae a closer loo at each components of ths cross-layer optmzaton problem. Schedulng. At the ln level, the schedulng problem deals wth how to coordnate transmsson among the nodes n each "tme slot." An mportant constrant that must be met s that a node cannot send and receve data wthn the same tme slot. Gven the number of tme slots K, denote t the normalzed length for tme slot,.e., the length of tme slot over the total length of all dfferent tme slots. We have. K E t 1 1 Power Control. Power control problem deals wth how much power a node should employ to transmt data n a partcular tme slot. Denote p^j as the power that node expends n tme slot for sendng data to node j. Although a node cannot send and receve wthn the same tme slot, a node can transmt to multple nodes

2 wthn the same tme slot. The total power that a node can expend at tme slot must satsfy the followng power lmt [], Pj < Pmax, jct where T1 s the set of one-hop neghbors of node. Ths requrement comes from the power densty lmtaton of UWB,.e., YnomZj p$w < Qmax, where Pmax QmaxW- Qmax s the maxmum allowed transmsson gnom power spectral densty, gnom s the gan at some fxed nomnal dstance [], and W 7.5 GHz s the spectrum for the UWB networ. A wdely-used model for power gan s gj d7a, (1) where dj s the dstance between nodes and j and oc s the path loss ndex. Denote 1 as the set of nodes that can mae nterference at node and rt as the ambent Gaussan nose densty. Then the achevable rate from node to node j wthn tme slot s c tw lg2 ± + mp+'dq (2) Routng. The routng problem at the networ level consders, for a sesson 1, 1 < 1 < L, how to relay a rate of r(l) from source node s(l) to the destnaton node d(l). To tae advantage of the mult-path avalablty wthn an ad hoc networ (.e., networ dversty), we allow a source node to splt ts data nto sub-flows and tae dfferent paths to the destnaton node. Denote fj (1) the data rate that s attrbuted to the l-th sesson on ln (,j). f node s the source node s(l), then fj () (l) (3) jct f node s an ntermedate relay node,.e., 7 s(l) and 7 d(l), then we have the followng flow balance: j7&s(l) m7&d(l) E fj (1) 1:fTn(l) O () jc7; mct; f node s the destnaton node d(l), then 1 fmtn)r(l) (5) mct t can be easly verfed that f (3) and () are satsfed, (5) must be satsfed. As a result, there s no need to lst (5) n the formulaton once we have (3) and (). Snce the sum of data rates on ln (, J) cannot be greater than the ln capacty, we have s(l)#j,d(1)# K f -(l)<5c$j. E 1<l<L l. MAXMZNG DATA RATE UTLTY PROBLEM n ths secton, we formulate the maxmzng data rate utlty problem as an optmzaton problem. We use El, nr(l) as a utlty metrc n the networ optmzaton problem, although our proposed soluton procedure s general and can be appled to other utlty functons. The motvaton for ths choce s that such log-based utlty functon can mae a good compromse between farness and effcency [1]. To ensure that a node cannot send and receve wthn the same tme slot, we ntroduce the noton of selfnterference parameter gjj [7] wth the property of gjj > max{gj : E 7j} and ncorporate ths nto the bt rate calculaton n (2) c C.. tw,mc 3j,qCTjgmjPmq LqGCTj9Pj~q Thus, when any p >,.e., node j jq s transmttng to a node q, we have c&- even f ptj >. n other words, when node j s transmttng to any node q, the ln capacty on ln (, J) s effectvely shut down to. To wrte (6) n a more compact form, we re-defne 1 to nclude node. Thus, (6) s now n the same form as n (2). Denote y mc,qt9mpmnq, We have tw 1og2 1 + tw ow1g2 l gp %/ mc3,qct. YmPm/ 1 + r,w + y ~~~ The maxmum utlty problem (MUP) can now be formulated as follows. Maxmum Utlty Problem (MUP): Max s.t. L Zlnr(l) 11 K E t l (1<<N1<<K) S Pj<Pmax jet yf 5 9Ympmq (1<<N1<<K) mcl,qetm c tw lg2 (1+ W+ jpj) K - l (1 < < Nj ET,1 < < K) (7) s(l)#j,d(l)# 5 fw>j(l)>o (1<<N jet) 1<<L (6)

3 13 fj(l)) (l) S fj(l)- E fm(l) jctf mctf (1 < < L, s(l)) jct m#d(l) j# s(l) r(1), fj(1) sc O or 1, tc: Pcj (l1<l<l,1l<<n, 7 (1)) d (l)) >O (l <l<l, l<<n)z1d(1)) j E T, j7s(1)) / > O ( < < N, j E T) < c < K) To remove the non-polynomal term n (7), we use the lnear rate-snr property that s unque to UWB. That s, we have a lnear approxmaton for log functon,.e., ln(1 + x) x when < x << 1. We have c.,whch s equvalent to n 2 W+ gj.jp Wc +Y c-- P g'jp'c' n 2 P Fnally, wthout loss of generalty, we let t have the followng property: t <_ t2 <_. <K t. Ths orderng wll help speed up the computaton. Wth the above re-formulatons, we now have a revsed MUP (or R-MUP) formulaton, whch s a nonlnear programmng (NLP) problem and s NP-hard n general [3]. n the next secton, we develop a soluton procedure based on branch-and-bound [] and the novel Reformulaton-Lnearzaton Technque (RLT) [13] to solve ths NLP problem. T1 gj V. A SOLUTON PROCEDURE TO R-MUP We fnd that branch-and-bound approach s most effectve n solvng our problem. Under the so-called branch-and-bound approach, we am to provde a (1 -) optmal soluton, where E s a small postve constant reflectng our desred accuracy n the fnal soluton. ntally, branch-and-bound analyzes all varables n nonlnear terms (denote these varables as partton varables) and determnes the value ntervals for these varables. By usng some relaxaton technque, branch-and-bound obtans a lnear programmng (LP) relaxaton for the orgnal NLP problem; ts soluton provdes an upper bound (UB) to the objectve functon. Wth the relaxaton soluton as a startng pont, branch-and-bound uses a local search algorthm to fnd a feasble soluton to the orgnal NLP problem, whch provdes a lower bound (LB) for the objectve functon. f the obtaned lower and upper bounds are close to each other,.e., LB > (1 -)UB, we are done. f the relaxaton errors for non-lnear terms are not small, then the lower bound LB could be far away from the upper bound UB. To close ths gap, we must have a tghter LP relaxaton,.e., wth smaller relaxaton errors. Ths could be acheved by further narrowng down the value ntervals of partton varables. Specfcally, branch-and-bound selects a partton varable and dvdes ts value nterval nto two ntervals by ts value n the relaxaton soluton. Then the orgnal problem (denoted as problem 1) s dvded nto two new problems (denoted as problem 2 and problem 3). Agan, branch-and-bound performs relaxaton and local search on these two new problems. Now we have LB2 and UB2 for problem 2 and LB3 and UB3 for problem 3. Snce the relaxatons n problems 2 and 3 are both tghter than that n problem 1, we have UB2,UB3 < UB1 and LB2,LB3 > LB1. The upper bound of the orgnal problem s updated from UB UB1 to UB max{ UB2, UB3} and the lower bound of the orgnal problem s updated from LB LB1 to LB max{lb2, LB3}. As a result, we now have smaller gap between UB and LB. f LB > (1 -)UB, we are done. Otherwse, we choose a problem wth the maxmum upper bound and perform partton for ths problem. Note that durng the teraton process for branch-andbound, f we fnd a problem z wth (1 -)UBz < LB, then we can conclude that ths problem cannot provde much mprovement on LB. That s, further branch on ths problem wll not yeld much mprovement and we can thus remove ths problem from further consderaton. Eventually, once we fnd LB > (1 -)UB or the problem lst s empty, branch-and-bound procedure termnates. t has been proved that under very general condtons, a branch-and-bound soluton procedure always converges [13]. n the rest of ths secton, we develop mportant components n the branch-and-bound soluton procedure. ntal Value ntervals for Partton Varables. For R-MUP, t, pj, cj, r(l), and Y are the varables that are n nonlnear terms whose value ntervals are canddates to be parttoned. t s easy to obtan the followng < PK, < PmaX, < C- < bounds: < t < K+1' Pmax W W1gj 192 ',q and <Yy<Pmax E gm,. We now develop an upper bound for r(l). Snce r(l) should be no more than the maxmum transmsson rate from source node s(l) and the maxmum recevng rate to destnaton node d(l), we analyze these two end rates ndvdually. At source node s(l), ts transmsson upper bound Cs(l) can be calculated by havng node s(l) transmt to ts nearest neghbor wth pea power on all tme slots. Assumng the nearest neghbor of s(l) s j, we have Cs(l) 9Wlog2 (+ 9),jjmax). At destnaton node d(l), t turns out that an upper bound for recevng rate Cd(l) s acheved when each node m E fd(l) transmts to d(l) wth pea power n all tme s

4 slots [1] and we have Cd(l) mte d (l) w log, (1 ± gm,d(l)pmax TjW + Pmax Z1eTd(l) g,d(l) ) Based on the rate analyss at source node s(l) and destnaton node d(l) for the l-th sesson, we have < r (l) <_ Mnt C,(1), Cd(l) } Lnear Relaxaton. Durng each teraton of the branch-and-bound procedure, we need a lnear relaxaton to obtan an upper bound of the objectve functon. For the polynomal term, we propose to employ Reformulaton-Lnearzaton Technque (RLT) [13]. For the non-polynomal term (.e., log term), we propose to employ three tangental supports, whch s a convex envelope lnear relaxaton. We frst show how RLT can obtan a lnear relaxaton for a polynomal term. Specfcally, Yjcj, pjcj and tp n () are polynomal terms. RET enables to use new varables to replace those polynomal terms and add lnear constrants for these new varables, thus relaxng nonlnear constrant nto lnear constrants. As an example, we ntroduce a new varable u for Yc..e., Yc. Assume <tj* (Yj)L < Yj < (Y)u J jj,.e. and (Cj)L < CKj < (cj)u, we have [y1 (Y)L [Cj- (C)L] >_ O, [Yj Y )L * (Cj) ]>O [(Y)U -y ] [Cj-((Cj)L] >, and [(Y)uu-Y] [(Cj)U -cj] >. From the above relatonshps, we obtan the followng lnear constrants (also called RLT constrants [13]) for u'. (C)Ly (Cj)L. yj (y)l c + (Yj)U cj + (y)l c + (Cj)U. y (y)u C + (c )U. y < (Y)L * :j u > (Yj)U (C )L (C)L u >(y)(c) K (YL)- (Cj)U j W < (Y)U* (C )U Through ths relaxaton, we can replace Yc.3 wth ujj n () and addng RLT constrants for uj nto the RMUP formulaton. Followng the same toen, we can have lnear relaxaton for all polynomal terms. Now we show how to obtan a lnear relaxaton for non-polynomal term. We can denote hl n r(l) for ln r(l). Note that the functon y nx, over sutable bounds of x, can be bounded by four segments (or a convex envelope), where segments,, and are tangental supports and segment V s the chord (see Fg. 1). n partcular, three tangent segments are at (XL, n XL), x xl xu(lnxulnxl) (/n/3), and (xu, nxu), where s the horzontal locaton for the pont ntersects extended tangent segments and ; segment V s the segment that jons ponts (XL, n XL) and (xu, n xu). The convex a XU -XL xl D Fg. 1. A convex envelope for y x xu n x. regon defned by the four segments can be descrbed by the followng four lnear constrants. XL Y-X < XL(nXL- 1) / y _ K< (nf- 1) xu *y - < xu(lnxu -1) nx L-XL * nxu As a result, the non-polynomal (log) term can also be relaxed nto lnear objectve and constrants. Local Search Algorthm. For a problem z, the local search algorthm determnes a feasble soluton tz based on the relaxaton soluton z. Denote t, p, and f as the vectors for varables t,pj. and ft(l), respectvely. n our local search, we set t t. Note that n R-MUP, we ntroduced the noton of selfnterference parameter to remove the bnary varables n MUP. Then n p, t s possble that P > and p>o for certan node wthn a tme slot. Therefore, t s necessary to fnd a new p from p by changng some P to such that no node can transmt and receve or pm wthn the same tme slot. We follow the followng two gudelnes when we set such transmsson power to. Frst, we try to mantan the same connectvty n /z as that n fz wherever possble. Second, we try to splt the total tme slots used at a node nto two groups of equal length wherever possble, one group for transmsson and the other group for recevng. More detals can be found n [1]. After we obtan t and p for oz, we can compute c from (7). Then an optmal routng soluton f (under t and p) can be obtaned by solvng a concave optmzaton problem through standard approach [1]. Addtonal Detals. Note that branch-and-bound chooses a partton varable n the nonlnear term wth the maxmum relaxaton error, where the relaxaton error for a nonlnear term s the dfference between the value of ths term and the value of ts correspondng new varable (XU-XL)Y+(lnXL-nxu)X>Xu

5 TABLE DESCRPTON OF 5 UN-DRECTONAL SESSONS N A 15-NODE NETWORK. Sesson ndex s(l) m d(l) Sesson ndex s(l) m d(l) * 2 * 5 1 * H Total Number of Tme Slots K Fg. 3. The total utlty as a functon of total avalable tme slots K for fve sessons. A 1,2, 6 13 Sesson 1: Node 9>15 -t 1 3, 1 Sesson 2: Node >5 j Sesson 3: Node 1>2 Fg. 2. A 15-node ad hoc networ wth 5 sessons. n the relaxaton soluton. f ths nonlnear term has multple varables, e.g., Yc, then we need to choose a partton varable from Y and c. Specfcally, f ((Y )U - (Y)L) mn{yj- (Y)L, (Y))U y} > ((cj )u- (cj)l) mn{cj (cj)l, (cj) c } we partton on Y and obtan two new value ntervals [(Yj)L, Y] and [Ya,(Yj)u]. Otherwse, we partton on c. For our specfc problem, by explotng the physcal nterpretaton of certan varable and weghng ts sgnfcance, further mprovement can be made on partton varable selecton polcy. For example, t s clear that varable t drectly affects the fnal soluton. As a result, the algorthm wll run much more effcently f we gve t hgher prorty when we choose a partton varable. Ths s precsely what we have done n our algorthm mplementaton, where we gve the hghest prorty to t, the second hghest prorty to p, then c. and consder y~~~~~~~~~~~~~~~~ last when we choose a partton varable. Note that ths choce wll not hamper the convergence property of the algorthm [13], although t wll yeld dfferent computatonal tme. There are two types of problems that can be elmnated before solvng ther LP relaxatons. n the frst case, f a problem s found to be nfeasble, then there s no need to solve a full scale LP relaxaton. For example, after we partton on p<, f a node must send and receve wthn the same tme slot n a new problem,.e., (PKj)L > and 2 5A Fg.. Optmal routng for three sessons wth K 6 (Pn)L >, then ths new problem must be nfeasble. n the second case, f a problem cannot provde sgnfcant mprovement, then there s no need to solve a full scale LP ether. For example, after we partton on r (l), f (1-6) EzL ln(r(l))u < LB, then ths new problem cannot provde sgnfcant mprovement and can be elmnated from problem lst. V. SMULATON RESULTS n ths secton, we present some mportant numercal results to offer further nsghts on the optmzaton problem. These results are mportant as they are not obvous from our theoretcal development of the soluton procedure n the last secton. We frst descrbe the smulaton settngs. We consder a randomly generated networ of 15 nodes deployed over a x area. There are 5 sessons (see Table and Fg. 2). All dstances are based on normalzed length n (1). The path loss ndex s a 2 and the nomnal gan s chosen as gnom.2. The power densty lmt Qmax s assumed to be 1% of the whte nose rt []. Schedulng. We frst nvestgate how the total utlty s affected when the total avalable tme slots K change. 1

6 TABLE TOTAL UTLTY AND RATE UNDER DFFERENT TME SLOT LENGTH ALLOCATON POLCES WTH K 6 AND L 3. Sesson 1: Node 9> Allocaton of Normalzed Tme Slot Length Optmal: (.1,.1,.1,.15,.21,.22) Equal: (.,.,.17,.17,.17,.17) Random: (.,.,.13,.,.,.29) Random: (.1,.15,.,.1,.1,.19) ZL lnr(l) l n r (l) a 21, 9 X1 (a) Mnmum-energy routng. Zlr(l) Sesson 3: Node 1>2 TABLE TOTAL UTLTY AND RATE UNDER DFFERENT ROUTNG STRATEGES FOR K 6 AND L 3. Routng Strategy Optmal Routng Mnmum-Energy Routng Mnmum-Hop Routng Sesson 2: Node >5,15 r(l) ,6,13 Sesson 1: Node 9>15 Sesson 2: Node >5 '1 One would expect the total utlty s a non-decreasng functon of K. Ths s because the more tme slots avalable, the more opportunty (or larger desgn space) for each node to avod nterference wth other nodes, and thus, ths yelds hgher utlty for the networ. However, there s a prce to pay for havng a large K. The larger the K s, the larger the total length of all tme slots n a cycle, and thus the larger delay at each node. Further, the larger the K, the larger the sze of our optmzaton problem, whch means more computatonal tme wll ncur. Therefore, we wsh to choose a sutably K that can provde near-optmal performance. The total utlty under dfferent K s shown n Fg. 3. Snce mult-hop s not allowed when K 1 (a node cannot send and receve wthn the same tme slot), the mnmum requred K s 2. As expected, the total utlty s a non-decreasng functon of K. But what really s nterestng s that there s a "nee" pont n ths performance fgure,.e., when K s beyond 6, there s hardly much ncrease n the total utlty. Ths suggests that for practcal purpose, t s suffcent to choose K 6 tme slots for ths partcular networ. We now show how the allocaton of length for each tme slot affects the performance. To plot the complete routng topology (mult-path for each sesson) legbly on a fgure, We use the frst three sessons (1 1, 2, 3) n Table for ths nvestgaton. Fgure shows an optmal routng for these three sessons. Clearly, an optmal routng for each sesson s a mult-path routng. Gven ths routng topology, Table shows the results under dfferent allocaton for normalzed tme slot length. The frst polcy s the optmal length allocaton obtaned va our soluton to the NLP problem, whch apparently 3 Sesson 3: Node 1>2 2, 9 1 (b) Mnmum-hop routng. Fg. 5. Other routng strateges for L 3 n the 15-node networ. offers the largest total utlty. The second tme slot length allocaton polcy s equal allocaton. Fnally, we also lst the performance under two random tme slot length allocatons. Note that equal allocaton s not a good polcy and has about the same performance as a random allocaton polcy. Routng. For a gven optmal tme slot length (obtaned through our soluton procedure), we now study the mpact of routng on our cross-layer optmzaton problem. n addton to our cross-layer optmal routng, we also consder the followng two routng approaches, namely, mnmum-energy routng and mnmum-hop routng. Under the mnmum-energy routng, the energy cost s defned as d2 for ln (, j). Table shows the results n ths study. Clearly, the cross-layer optmal routng outperforms both mnmum-energy and mnmum-hop routng approaches. Both mnmum-energy routng and mnmum-hop routng are mnmum-cost routng (wth dfferent ln cost). Mnmum-cost routng only uses a sngle-path,.e., mult-path routng s not allowed, whch may not provde good soluton. Moreover, t s very

7 lely that multple sessons share the same path (see path 7 -X -X 5 n Fg. 5(a)). Thus, the rates for these sessons are bounded by the capacty of ths path. V. RELATED WORK ACKNOWLEDGEMENTS The wor of Y.T. Hou and Y. Sh has been supported n part by NSF under Grant CNS-3739 and ONR Grant N The wor of H.D. Sheral has been supported by NSF under Grant REFERENCES An overvew of UWB technology s gven [9]. Phys- [1] M.S. Bazaraa, H.D. Sheral, and C.M. Shetty, Nonlnear Programmng: Theory and Algorthms, second edton, John Wley cal layer ssues assocated wth UWB-based multple & Sons, nc., New Yor, NY, access communcatons can be found n [], [5], [15] and [2] F. Cuomo, C. Martello, A. Baocch, and F. Caprott, "Rado references theren. n ths secton, we focus our attenton resource sharng for ad hoc networng wth UWB," EEE J on related wor addressng networng related problems on Selected Areas n Commun., vol., no. 9, pp , Dec. 2. for UWB-based networs. Garey and D.S. Johnson, Computers and ntractablty: A n [6], Neg and Rajeswaran frst showed that, n [3] M.R. Gude to the Theory of NP-completeness, pp. 25-2, W. H. contrast to prevously publshed results, the throughput Freeman and Company, New Yor, NY, for UWB-based ad hoc networs ncreases wth node [] EEE 2.15 WPAN Hgh Rate Alternatve PHY Tas Group 3a, densty. Ths result demonstrates the sgnfcance of pub/tg3a. html. physcal layer propertes on networ layer metrcs such [5] EEE Journal on Selected Areas n Communcatons - Speas networ capacty. n [2], Cuomo et al. studed a cal ssue on Ultra-Wdeband Rado n Multaccess Wreless Communcatons, Guest Edtors: N. Blefar-Melazz, M.G. D multple access scheme for UWB. The mpact of routng, M. Gera, H. Luedger, M.Z. Wn, and P. WthngBenedetto, however, was not addressed. ton, vol., no. 9, Dec. 2. The most relevant research to ths wor are [7] [6] R. Neg and A. Rajeswaran, "Capacty of power constraned adhoc networs," n Proc. EEE NFOCOM, pp. 3-53, Hong and []. n [7], Neg and Rajeswaran studed how Kong, Chna, March 7-,. to proportonally maxmze rates n a sngle-hop UWB [7] R. Neg and A. Rajeswaran, "Schedulng and power adaptaton networ. n contrast, we consder a mult-hop networ for networs n the ultra wde band regme," n Proc. EEE envronment where routng s also part of the crossglobecom, pp , Dallas, TX, Nov. 29-Dec. 3,. layer optmzaton problem. As a result, our problem [] G.L. Nemhauser and L.A. Wolsey, nteger and Combnatoral Optmzaton, pp , John Wley & Sons, New Yor, NY, s consderably more dffcult. n [], Radunovc and Le Boudec explored the same problem as we studed [9] R.C. Qu, H. Lu, and X. Shen, "Ultra-wdeband for multple access communcatons," EEE Communcatons Magazne, n ths paper. The authors studed some smple cases vol. 3, ssue 2, pp. -7, Feb. 5. and attempted to generalze the results from these smple [1] B. Radunovc and J.-Y. Le Boudec, "Rate performance objeccases to a networ wth general topology. Many of these tves of mult-hop wreless networs," n Proc. EEE NFOresults are heurstc n nature. On the other hand, n ths COM, pp , Hong Kong, Chna, March 7-,. B. Radunovc and J.-Y. Le Boudec, "Optmal power control, [] paper, we have developed a formal soluton procedure and routng n UWB networs," EEE J Selected schedulng, to solve the complex cross-layer optmzaton problem Areas n Commun., vol. 22, no. 7, pp. 52-7, Sept.. nstead of resortng to heurstcs. [] A. Rajeswaran, G. Km, and R. Neg, "A schedulng framwor V. CONCLUSONS for UWB & cellular networs," n Proc. Frst nternatonal Conference on Broadband Networs (Broadnets), pp , San Jose, CA, Oct ,. [13] H.D. Sheral and W.P. Adams, A Reformulaton-Lnearzaton Technque for Solvng Dscrete and Contnuous Nonconvex Problems, Chapter, Kluwer Academc Publshers, Dordrecht/Boston/London, [1] Y. Sh, Y.T. Hou, and H.D. Sheral, "Nonlnear Optmzaton of Data Rate Utlty Problem for UWB-based Ad Hoc Networs," Techncal Report, the Bradley Department of Electrcal and Computer Engneerng, Vrgna Tech, Blacsburg, VA, Oct. 5. Avalable at n ths paper, we studed the data rate utlty problem n a UWB-based ad hoc networ. We formulated the problem nto a nonlnear programmng wth consderaton of ln layer and networ layer varables. We proposed a soluton procedure based on branchand-bound framewor. A powerful technque that we Research.html. employed n branch-and-bound framewor s the refor- [15] M. Wn and R. Scholtz, "Ultra-wde bandwdth tme-hoppng mulaton lnearzaton technque, whch s able to replace spread-spectrum mpulse rado for wreless multple-access communcatons," EEE Trans. on Commun., vol., ssue, nonlnear terms n the constrants wth lnear ones , Aprl. Numercal results demonstrate the sgnfcance of cross- [] pp. H. Zhang and J. Hou, "Capacty of wreless ad-hoc networs layer consderaton and offers nsghts on the mpact of under ultra wde band wth power constrant," n Proc. EEE each component n the optmzaton space. NFOCOM, pp , Mam, FL, March 13-17, 5.