Does Full-Duplex Double the Capacity of Wireless Networks?

Does Full-Duplex Double the Capacity of Wieless Netwoks? Xiufeng Xie and Xinyu Zhang Univesity of Wisconsin-Madison Email: {xiufeng, xyzhang}@ece.wisc.edu Abstact Full-duplex has emeged as a new communication paadigm and is anticipated to double wieless capacity. Existing studies of full-duplex mainly focused on its PHY laye design, which enables bidiectional tansmission between a single pai of nodes. In this pape, we establish an analytical famewok to quantify the netwok-level capacity gain of full-duplex ove halfduplex. Ou analysis eveals that inte-link intefeence and spatial euse substantially educes full-duplex gain, endeing it well below in common cases. Moe emakably, the asymptotic gain appoaches when intefeence ange appoaches tansmission ange. Though a compaison between optimal half- and fullduplex MAC algoithms, we find that full-duplex s gain is futhe educed when it is applied to CSMA based wieless netwoks. Ou analysis povides impotant guidelines fo designing full-duplex netwoks. In paticula, netwok-level mechanisms such as spatial euse and asynchonous contention must be caefully addessed in full-duplex based potocols, in ode to tanslate full-duplex s PHY laye capacity gain into netwok thoughput impovement. I. INTRODUCTION Wieless netwoks have commonly been built on half-duplex adios. A wieless node cannot tansmit and eceive simultaneously, because the intefeence geneated by outgoing signals can easily ovewhelm the incoming signals that ae much weake, so called self-intefeence effect. Yet ecent advances in communications technologies showed the feasibility of canceling such self-intefeence, thus ealizing full-duplex adios that can tansmit and eceive packets simultaneously. Substantial eseach effot has focused on designing full-duplex adios by combining RF and baseband intefeence cancellation [], [], enabling bi-diectional tansmission between a single pai of nodes. Full-duplex technology is thus anticipated to double wieless capacity without adding exta adios [], [3], [4]. In pactice, howeve, wieless netwoks ae moe sophisticated than a single-link. Real-wold wieless netwoks, such as wieless LANs and mesh netwoks, ae distibuted in natue, involving multiple contention domains ove lage aeas, thus entangling both self-intefeence and inte-link intefeence. Hence, one may aise an impotant question: Does full-duplex double the capacity of such distibuted wieless netwoks? This pape establishes a compehensive analytical famewok to quantify the benefits of full-duplex wieless netwoks. Contay to widely held beliefs, the analysis leads to a igoous but negative answe: full-duplex adios cannot double netwok capacity, even if assuming pefect self-intefeence cancellation. The key intuition behind this esult lies in the spatial euse and asynchonous contention effects in wieless netwoks. Fist and foemost, thee exists a fundamental tadeoff between spatial euse and the full-duplex gain. Full-duplex allows a eceive to send packets concuently, but this exta sende expands the intefeence egion, which would have been eused even by a half-duplex adio. Conside the two-cell WLAN in Fig. (a). With half-duplex adios, link TX RX and TX RX do not conflict and can be activated concuently. Wheeas full-duplex enables RX to tansmit simultaneously TX TX Half-duplex netwok RX Full-duplex netwok RX RX RX TX TX (a) (b) Fig.. Spatial euse and asynchonous contention offsets full-duplex gain. Dotted cicles denote intefeence ange. Potocol model [5] is assumed. with TX, it suppesses both TX (whose eceive RX is intefeed by RX) and RX (who intefees with eceive RX). Hence, supisingly, full-duplex esults in the same netwok capacity as half-duplex in such cases. Second, to avoid collisions, distibuted wieless netwoks commonly adopt CSMA algoithms that ae inheently asynchonous and cannot guaantee a pai of tansceives can access the channel simultaneously to enable full-duplex tansmission. As exemplified in Fig. (b), two vetices of a link may belong to diffeent contention domains involving independent contendes. When TX gains channel access and stats tansmission, RX may be still in a long backoff stage (as it contends with moe intefees) and needs to wait fo its tun to tansmit. So, how will these two factos manifest and affect the capacity of lage-scale full-duplex wieless netwoks? Assuming the potocol model [5] fo channel access, we fist analyze the spatial euse in -D andom netwoks and deive the exact capacity gain of full-duplex ove half-duplex netwoks, as a function of, a paamete that govens spatial euse and eflects the diffeence between intefeence ange and tansmission ange. Then, assuming an oacle, synchonized schedule, we extend the analysis to -D netwoks to obtain an uppebound fo the full-duplex capacity gain. Ou analysis establishes novel models (e.g., full-duplex exclusive egion) that fundamentally diffe fom those in capacity modeling of half-duplex netwoks. The analysis poves that unde typical settings, e.g., =, the capacity gain is only.33 in -D netwoks and bounded by.58 in -D netwoks. The asymptotic gain appoaches.8 (i.e., full-duplex impoves capacity by only 8%) as 0. Futhemoe, we elax the assumption of oacle schedule with a distibuted, asynchonous MAC algoithm that allows a pai of full-duplex tansceives to contend fo channel access and tansmit packets independently, as if no mutual intefeence exists. Paametes of the algoithm ae contolled by a utility-maximization famewok, which can achieve optimal thoughput with popotional fainess guaantee. We compae the capacity of this full-duplex MAC with the coesponding utility-optimal half-duplex MAC [6]. Simulation esults show that asynchonous contention futhe educes the chance of fullduplex tansmission in CSMA netwoks. Ou analysis and simulation implies that it is non-tivial to tanslate full-duplex PHY-laye capacity gain to netwoklevel thoughput gain. Taditional CSMA-style MAC is no RX TX

(a) Space eusable by othe TX Reusable by othe RX Uneusable by othe TX/RX (b) Fig.. Full-duplex tansmission modes: (a) bidiectional tansmission and (b) womhole elaying. longe the optimal solution fo full-duplex adios due to the spatial euse and asynchonous contention poblems. Netwok designes need to einvent the medium access potocols to fully exploit the potential of full-duplex communications. The emainde of this pape is stuctued as follows. Sec. II intoduces the backgound, model and motivation behind ou analysis. Sec. III analyzes the capacity gain of full-duplex ove half-duplex netwoks. Then, Sec. IV quantifies the asynchonous contention effect in full-duplex CSMA netwoks. Sec. V discusses elated wok and Sec. VI concludes the pape. A. Netwok model II. BACKGROUND AND MOTIVATION In this section, we pesent the essential models and assumptions undelying ou analytical famewok. Full-duplex communications. A full-duplex adio can tansmit and eceive diffeent packets at the same time. Its intefeencecancellation hadwae and baseband signal pocessos can effectively isolate the intefeence fom tansmitted signals to eceived ones. In pactice, pefect isolation is infeasible due to RF leakage. Even with sophisticated hadwae, a full-duplex link can achieve aound.84 capacity gain compaed with a half-duplex link []. In this pape, howeve, we will assume pefect full-duplex adios to isolate the PHY-laye atifacts, and focus on the netwok-level capacity gain instead. Fom a netwok pespective, full duplex links can opeate in two modes []. Bidiectional tansmission mode (Fig. (a)) allows a pai of nodes to tansmit packets to each othe simultaneously. It is applicable to WLANs with uplink/downlink taffic, o multi-hop netwoks with bidiectional flows. Womhole elaying mode (Fig. (b)) enables a eceiving node to fowad packets to anothe node simultaneously, thus acceleating data tanspotation in multi-hop wieless netwoks. Altenatively, full-duplex adios can send busy-tones while eceiving, thus peventing hidden teminals []. Such schemes undeutilize the capacity as the busy-tone contains no data, and thus they ae beyond the scope of this pape. Topology and intefeence model. We conside ad-hoc netwoks whee nodes ae andom unifomly distibuted in a egion of fixed aea, which can be a unit line segment in one dimension, o a unit squae in two dimensions. We extend the widely adopted potocol model [5] to epesent the adjacent link intefeence in full-duplex netwoks. Denote X i as the location of node i. Any tansmission fom node i to j can be successful iff X i X j, and k j, X k X j > ( + ). Hee and ( + ) ae the tansmission ange and intefeence ange, espectively. In multi-hop netwoks, to ensue end-toend connectivity between any two nodes, is a function of n, the numbe of nodes in the netwok, and must satisfy log n = Θ( n ) [5]. (a) (b) Fig. 3. Spatial euse in (a) half-duplex and (b) full-duplex netwoks. Dotted lines denote intefeence ange. B. Factos affecting the full-duplex gain Wheeas full-duplex has the potential to double the capacity of a single pai of nodes, we identify the following netwoklevel factos which degade the capacity gain when multiple contending links ae entangled. Spatial euse. Full-duplex allows a eceive to become an active tansmitte simultaneously, but this ceates moe intefeence and expands the effective spatial occupation of a pai of nodes, compaed with a half-duplex link. As shown in Fig. 3(a), fo a half-duplex link, the eceive is passive and anothe eceive of a neaby link can be placed close to it without mutual intefeence. Similaly, tansmittes of two links can be placed close by. Hence, within the union of the tansmitte and eceive s intefeence ange, thee exists a sizable faction of space that can be eused by nodes of othe links. In contast, if full-duplex adios ae employed and evey node is a tansmitte and eceive, then no space is eusable by othe nodes (Fig. 3(b)). In othe wods, fullduplex cannot double the numbe of concuent tansmissions in a netwok. Theefoe, despite the doubling of single-link capacity, full-duplex sacifices spatial euse and consequently, may not double the netwok-wide capacity. Asynchonous contention. Wieless LANs and multi-hop netwoks commonly adopt the asynchonous CSMA based contention algoithms. Two vetices of a link ae sepaated in space, and may be contending with diffeent set of neighbos. Hence, they may have diffeent channel status (idle/busy) at any time instance. Tansitions between channel status ae also inheently asynchonous due to unpedictable dynamics in each neighbohood. Even when both ae idle, the two nodes cannot tansmit immediately they must espect the CSMA backoff mechanism and wait fo thei backoff countes to expie in ode to avoid collision. And again, the backoff countes ae asynchonous a node with moe contendes tends to have a lage backoff counte, and needs to wait fo a longe idle peiod befoe tansmission. Howeve, to maximize the full-duplex gain, a pai of nodes must be able to synchonize thei tansmissions, i.e., ensuing both channels ae idle and backoff countes fie at the same time. Yet such synchonization equies both nodes to be awae of each othe s channel status in each time slot, which is infeasible in pactice. Theefoe, when applied to eal-wold CSMA netwoks, full-duplex adios need to inevitably espect the asynchonous contention, which educes the chance of fullduplex tansmissions and the capacity gain. Othe factos. Diffeent tansmission modes may affect fullduplex gain diffeently, as exemplified in Fig.. Assume = 0, i.e., intefeence ange equals tansmission ange. When womhole elaying is applied fo a single linea flow, the nomalized end-to-end thoughput can be (fo evey 4 hops, thee can be tansmittes), in contast to 3 fo half-duplex

3 nodes. Hence, womhole elaying leads to a capacity gain of.5. On the othe hand, bidiectional tansmission mode can be applied to bi-diectional flows, inceasing the capacity to 3, potentially doubling the capacity of two linea flows. The case is diffeent when vaies and changes the spatial euse accodingly. Fo instance, if =, womhole elaying becomes infeasible as the fist tansmitte intefees the second tansmitte s eceive, although bidiectional tansmission still helps in impoving end-to-end thoughput of multiple flows. Theefoe, both the tansmission mode and intefeence ange paamete have significant impacts on the full-duplex gain. In what follows, we igoously analyze the capacity gain of full-duplex ove half-duplex netwoks by incopoating the above factos. Sec. III bounds the capacity gain assuming synchonized full-duplex tansmissions. Sec. IV futhe elaxes the assumption with an asynchonous channel contention model. III. FULL-DUPLEX GAIN: ASYMPTOTIC ANALYSIS In this section, we chaacteize the asymptotic capacity gain of full-duplex ove half-duplex ad hoc netwoks, as the numbe of nodes n. Each node is a souce node and its destination is andomly chosen. Netwok capacity λ is conventionally defined as the maximum data ate that can be suppoted between evey souce destination pai [5]. Let D denote the aveage distance between a souce node and its destination, d the aveage distance of a hop, then the aveage numbe of hops in each end-to-end flow is D/d, and each flow equies a MAC-laye thoughput of λd/d. Since thee ae n flows in this netwok, total demand fo MAC-laye thoughput is nλd/d. Let N(d) denote the maximum numbe of simultaneous tansmission links. It is a function of hop distance d, which affects the spatial euse between links. Suppose the data ate of any tansmission link is W, then the maximum suppotable MAC-laye thoughput is W N(d). The demand fo MAC-laye thoughput cannot exceed the suppotable amount, thus: A. -D Chain Netwok λ = W nd max (dn(d)) () 0<d In this section, we fist deive tight capacity bounds fo onedimensional full-duplex and half-duplex netwoks in Lemma and Lemma, espectively. Then we chaacteize the fullduplex capacity gain in Theoem. Simila to the model in [7], we assume all nodes ae unifomly and andomly distibuted within a -D netwok of unit length. Then we patition the unit length egion into a set log (n) n of bins. Length of each bin is l(n) = K, K is chosen to ensue that tansmission ange l(n), so that a node in one bin can always each any node in an adjacent bin. As poved in [8], as n, thee is at least one node in each bin with high pobability. Lemma The pe-flow capacity of a -D full-duplex andom netwok is λ F = W nd + Poof: We teat the two full-duplex modes sepaately. () (i) Bidiectional tansmission mode. Conside a full-duplex bidiectional tansmission pai A B with link distance d. As shown in Fig. 4, to avoid intefeence, the distance between A and the ight-most node of any tansmission pai left to A B must be lage than ( + ). d ( ) A d B ( ) ( ) d Fig. 4. Exclusive egion fo -D bidiectional tansmission mode. Theefoe, each tansmission pai must at least occupy a line segment of length ( + ) + d. By dividing the netwok size (unit length) by this minimum length, maximum numbe of simultaneous tansmission pais is uppe bounded by (+ )+d. In bidiectional mode, each tansmission pai can suppot data ate of W. Combining Eq. (), we have the uppebound of pe-flow capacity in bidiectional mode: λ B W nd max (3) 0<d ( + ) + d Fom Eq. (3), we obseve that when d =, the netwok has the maximum thoughput, which means the uppebound has the same fom as Eq. () in Lemma. Fo the capacity lowebound, we constuctively schedule flows following the schedule patten in Fig.4. Note that it is a andom netwok, and distance between any two nodes will toleate a small distubance of twice the bin size. Thus each tansmission will tavel a distance in the inteval of [ l, ], and the distance between two adjacent tansmission pais should be in the inteval of [( + ) l, ( + )]. Simila to [7], in ode to make the distubance negligible, we can choose a pope bin size l(n) which ensues tansmission ange l(n), let K = /l(n), K should be a lage constant, then the lowebound becomes: λ B W ( K ) nd ( + ) + ( K ) W nd + Since this constuctive lowebound equals the uppebound, we have the pe-flow capacity in Lemma. (ii) Womhole elaying mode. A womhole elaying tiple A R B consists of 3 nodes: a half-duplex tansmitte A, a fullduplex elay node R, and a half-duplex eceive B. Conside two adjacent womhole elaying tiples A R B and A R B as shown in Fig. 5. The link distance of A R and R B equals d. To maximize spatial euse (i.e., packing the maximum numbe of links inside the unit length netwok), eceives B and B (o equivalently tansmittes A and A ) must be placed adjacent to each othe. In addition, the distance between elay R and eceive B must be lage than the intefeence ange ( + ). d ( ) A R d A B B R Fig. 5. Exclusive egion fo -D womhole elaying mode. Consequently, thee can be at most one tiple within any line segment of length ( + ) + d. Since each tiple contains two tansmission links, evey two links must occupy a line segment of minimum length ( + ) + d, which is the same as in the bidiectional tansmission mode. Then with a simila deivation

4 as above, we can obtain the same capacity uppebound and lowebound, which concludes the poof fo Lemma. We note that fo each womhole elaying tiple A R B, the distance between A and B must be lage than ( + ) to avoid intefeence, i.e., d > ( + ). As capacity is maximized when d =, this implies womhole elaying can achieve the same capacity as bidiectional tansmission only if 0 <, which is also the pactical ange fo [9]. Lemma The pe-flow capacity of a -D half-duplex andom netwok is λ H = W nd (4) + Poof: Fo half-duplex adios, the distance between RX and TX fom diffeent tansmission pais must be lage than ( + ). Theefoe, the minimum length of line segment occupied by each tansmission pai is L H = ( + ), as shown in Fig. 6. Fig. 6. T d R R T ( ) Exclusive egion fo a -D half-duplex netwok. Since each half-duplex tansmission pai only contains one tansmission link, the maximum numbe of tansmission links that can be accommodated in the unit length netwok is: (+ ). Combining with Eq. (), we have the capacity uppebound of the half-duplex scheme: λ H W nd max d (5) 0<d ( + ) When d =, λ H achieves its maximum, which leads to the uppebound fo Eq. (4) in Lemma. With a simila method as in the full-duplex case, we can constuct the half-duplex lowebound: λ H W K nd ( + ) W nd (6) + which, combined with the uppebound, leads to Lemma. Based on Lemma and Lemma, we can obtain the fullduplex capacity gain of a -D andom netwok. Theoem The full-duplex gain of a -D andom netwok is G = λ F = + λ H + = + (7) + Fig. 7. Full-Duplex Capacity Gain.5.4.3.. 0 0.4 0.8..6 Full-duplex gain in -D andom netwoks. Implication: Fig. 7 plots the full-duplex gain unde vaying. When = 0, the asymptotic gain is, even though it can be lage than in cetain finite cases (Sec. II-B). The gain inceases with and appoaches as, when the entie netwok falls in one contention domain. Howeve, in the common case of 0, the gain is no lage than.33, fa fom doubling netwok capacity. B. Full-Duplex Gain in -D Netwoks In this section, we bound the full-duplex capacity gain by compaing capacity uppebound of full-duplex netwoks and lowebound of half-duplex netwoks. ) Exclusive egion fo full-duplex nodes: We extend the classic appoach [5] of deiving capacity uppebound to the fullduplex case. Netwok capacity is popotional to the numbe of simultaneous tansmission links that can be accommodated. Thus an uppebound can be obtained by dividing total netwok aea by a link s exclusive egion, the aea no othe active tansmitte o eceive can euse. We assume all links opeate in the bidiectional tansmission mode. A node is both TX and RX at the same time, hence the distance between any two nodes (except those belonging to the same bidiectional link) must be lage than intefeence ange ( + ). Based on this obsevation, we define the exclusive egion in Fig. 8, which is the union of two cicles with adius R = (+ ). Fig. 8. d ( ) R Exclusive egion fo a full-duplex bidiectional tansmission pai. Unde this definition, if two exclusive egions do not ovelap, the distance between thei nodes will be lage than (+ ) = R, i.e., they do not intefee with each othe. Convesely, if any two exclusive egions ovelap, the minimum distance between two nodes fom each egion will be smalle than ( + ), esulting in intefeence. Theefoe, the sufficient and necessay condition fo intefeence-fee tansmission in bidiectional mode is that no two exclusive egions ovelap. ) Capacity uppebound of -D full-duplex andom netwoks: Conside a -D andom netwok whee n full-duplex nodes ae distibuted unifomly within a unit squae. Its capacity uppebound can be chaacteized as follows: Theoem The pe-flow thoughput capacity λ F of -D fullduplex andom netwok is uppe bounded by W λ F < nd( + ) 4 π accos( + ) + + (+ ) fo lage n. Poof: Fist, we can deive the aea S F of the exclusive egion defined above, with R = (+ ) : S F = πr (R accos( d R ) d R ( d ) ) By dividing the netwok aea by S F, we can deive the uppebound of the maximum numbe of simultaneous tansmission pais. In the bidiectional tansmission mode, one tansmission pai contains two tansmission links, thus the maximum numbe of simultaneous tansmission links is N F = /S F. Then fom Eq. (), we can deive the uppebound of pe-flow capacity of -D full-duplex andom netwoks unde a given d: W ndr max 0<d ( d R π accos( d R ) + d R ) (8) ( d R )

5 Then we need to find the optimal value of d which leads to the maximum pe-flow thoughput λ F. Let x = d/r, C = W nd, and λ F = f(x): x f(x) = C π accos(x) + x (9) x The fist ode deivative of f(x) is: x+x ( x) x df(x) dx = C π accos(x) + (π accos(x) + x (0) x ) Since x = d/r, 0 < x <, we have df(x) dx > 0. Theefoe, f(x) is a monotonic inceasing function of x when 0 < x <. Because x = d/r and d, the netwok achieves maximum pe-flow thoughput when d =. With this obsevation and Eq. (8), we can obtain the netwok capacity in Theoem. Based on Theoem, combined with the connectivity equiement that = Θ( log(n)/n), we can obtain a capacity scaling law fo andom full-duplex netwoks: Coollay Fo a andom full-duplex netwok, λ F (n) = O(/ n log(n)) as n This implies that full-duplex may impove netwok capacity by at most a multiplication facto. Its asymptotic capacity emains the same as that of a half-duplex netwok [5]. 3) Full-duplex gain in -D egula netwoks: To make a fai compaison, ideally full-duplex should be evaluated against half-duplex capacity in the same topology. Howeve, the exact capacity of a -D andom netwok is always difficult to obtain. Theefoe we fist focus on a egula lattice netwok: nodes ae located at junction points, and the distance between adjacent junction points equals tansmission ange. With simila method fo the -D andom netwok, we can deive the fullduplex capacity uppebound fo -D lattice netwok: Lemma 3 The pe-flow thoughput capacity λ F of -D fullduplex egula lattice netwok is uppe bounded by λ F < W nd (+ ) (π accos (+ ) )+ + fo lage n. Poof: In a lattice netwok, the exclusive egion should contain a set of squae cells with side length. Given the necessay and sufficient condition fo intefeence-fee tansmission (Sec. III-B), we can find the aea of the exclusive egion SF R in such a netwok: SF R = S F. Since the link distance d can only be, we can deive the full-duplex uppebound fo lattice netwok similaly to the andom netwok case and obtain Lemma 3. In addition, we can obtain a lowebound of half-duplex peflow capacity in lattice netwoks by constucting an achievable schedule. Lemma 4 The pe-flow thoughput capacity λ H of -D halfduplex egula lattice netwok is lowe bounded by λ H > W nd max(, + )( + ) () fo lage n. Poof: A -D egula netwok is equivalent to multiple -D egula netwoks placed in paallel. To constuct a feasible schedule, we divide time into slots: even slots ae used to tansmit data hoizontally, and odd ones fo vetical tansmission. Thus all nodes in the netwok can be ensued the same chance to send data. Fig. 9 illustates a snapshot of the constucted schedule. d ( ) l ( ) Fig. 9. Half-duplex capacity lowebound. To avoid intefeence, the distance between paallel flows cannot be smalle than +. Meanwhile, in the lattice netwok, the distance between two nodes cannot be smalle. Theefoe the minimum distance between paallel flows should be max (, + ), and thee can be only one tansmission in each ectangle block of side lengths ( + ) and max (, + ), thus the space occupied by each tansmission link is: max(, S H = + )( + ) () By dividing the unit squae aea by aea of the space occupied by each tansmission link, we can obtain the suppotable numbe of simultaneous tansmission links: ( N H max(, + )( + ) ) (3) Fom Eqs. (3) and () we obtain the pe-flow capacity lowebound λ H of -D lattice netwok in Lemma 4. Finally, by compaing full-duplex capacity uppebound (Lemma 3) and half-duplex lowebound (Lemma 4), we can bound the full-duplex capacity gain fo -D lattice netwoks. Theoem 3 Fo a -D egula lattice netwok, full-duplex capacity gain of is uppe bounded by G L = λ F < max(, + )( + ) λ H (+ ) (π accos (+ ) )+ + This esult implies that the uppebound of full-duplex gain G L is detemined only by, and unelated to numbe of nodes n o tansmission ange. Fig. 0 plots the elationship between G L and. Fig. 0. Full-Duplex Capacity Gain.6. 0.8 0.4 0 0 0.4 0.8..6 Uppebound of full-duplex gain in -D lattice netwoks. Implication: In geneal, G L inceases with, because fo lage, the faction of space eusable by neighboing links

6 deceases in half-duplex netwoks (as illustated in Fig ), and thus full-duplex advantage is moe pominent. Howeve, G L is always below in the pactical ange of 0, which matches ou intuition in Sec. II. Note that the fluctuation of G L is caused by the ceiling functions in Theoem 3. In cetain cases, G L can fall below when the spatial euse effect ovewhelms the benefit of full-duplex tansmission. = 0 = ( + ) ( + ) Reusable space Reusable space Fig.. Fo half-duplex, with lage, a smalle faction of space can be eused by neighboing links. 4) Full-duplex gain in -D andom netwoks: In this section, we fist constuct a capacity lowebound fo -D half-duplex andom netwoks, and then compae it with the uppebound in Theoem in ode to bound the full-duplex capacity gain. Lemma 5 Fo lage n, the thoughput capacity λ H of -D half-duplex andom netwok is lowe bounded by W λ H nd( + ) (4) Poof: We can patition the netwok into egula cells (Fig. (a)), each being a squae with side length l(n) = log(n) K n, K >. As poved in [8], fo lage n with high pobability thee is at least one node in each cell. Then we can constuct a schedule by expanding the -D schedule in Lemma, as illustated in Fig. (b). We schedule nodes along a ow (o column) of cells. Multiple -D schedules ae placed paallelly, and the distance between them must be lage than ( + ). Time is slotted to schedule tansmissions along ows and columns, simila to the poof fo Lemma 4. ln ( ) Fig.. (a) Cell patition Paallel -D Schedule ( ) (b) Lowebound constuction Constucting a capacity lowebound fo -D andom netwoks. As nodes ae andom unifomly distibuted, the distance between paallel -D schedules has a small distubance of twice the side length of a cell, and falls in the ange of [( + ), ( + ) + l(n)]. In ode to cancel the small distubance caused by topology andomness, we can choose a lage tansmission ange = K l(n), whee K is a lage constant. Then the distubance is negligible, and the numbe of simultaneous paallel -D schedules is lowe bounded by: ( + )(n) + (n) K ( + )(n) (5) Combining Eq. (5) with the capacity of -D half-duplex netwok in Lemma, we can deive the capacity lowebound λ H in Lemma 4. Fom Lemma 5 we can also obseve that this constuctive lowebound follows the scaling law in [5]. By synthesizing Theoem and Lemma 5, we can easily pove: Theoem 4 The full-duplex capacity gain of a -D andom netwok is uppe bounded by G R = λ F 4 < λ H π accos( + ) + (6) + (+ ) fo lage n. Implication: Simila to egula netwoks, the uppebound of full-duplex gain in andom netwok depends on. Fig. 3 plots G R as vaies. Unde a typical setting of =, G R is only.58, and appoaches.8 as 0, i.e., tansmission ange appoaches intefeence ange. Note that in pactical wieless netwoks, can be vey close to 0, especially fo links with low bit-ate (and thus longe tansmission ange) [9]. Fig. 3. Full-Duplex Capacity Gain.8.6.4. 0 0.4 0.8..6 Uppebound of full-duplex gain in -D andom netwoks. IV. FULL-DUPLEX GAIN UNDER ASYNCHRONOUS CONTENTION In this section, we intoduce a full-duplex MAC that confoms to the asynchonous contention mechanism of pactical CSMA netwoks. We fist descibe the potocol opeations, and then build a distibuted optimization famewok that adapts the opeations to achieve optimal netwok thoughput. We will compae the capacity of this full-duplex MAC with an optimal half-duplex MAC. A. Full-duplex MAC: model and potocol The poposed full-duplex MAC etains pimitive opeations (e.g., caie sensing and backoff) of the widely-adopted 80. MAC, but with two featues specific to full-duplex: (i) While in eceiving mode, a eceive can continue sensing its channel status [0]. It maks a busy channel if the channel is occupied simultaneously by nodes othe than its own tansmitte. (ii) While in eceiving mode, a eceive can tansmit back to the sende if it senses an idle channel and finishes backoff. Hee we only conside the bidiectional tansmission mode. Fig. 4 illustates a typical channel contention pocedue fo the full-duplex MAC. As the tansmitte and eceive s MAC opeations cannot be synchonized, they need to contend fo channel access independently. Full-duplex oppotunity occus only when thei tansmissions ovelap. The contention pocedue is simila to 80. CSMA, except that each pai of tansmitte/eceive ae awae that they do not intefee with each othe. Specifically, befoe tansmission, a node needs to sense the channel fo a DIFS duation []. Afte sensing an

7 RX RX TX TX Fig. 4. sensing backoff tansmission backoff tansmission tansmission full-duplex oppotunity time Flow of opeations in full-duplex MAC. idle channel, it stats backoff and waits fo an additional idle duation of B time slots, with B andomly chosen fom [0, CW]. It feezes the backoff if the channel becomes busy again, and esumes othewise. Upon completing the backoff, it begins tansmission immediately. CW, the backoff window size, is eset to CW min upon each successful tansmission, and doubled upon failue until eaching a maximum value of CW max []. To ensue a fai compaison between half-duplex and fullduplex netwok capacity, we assume pefect caie sensing fo both, i.e., a tansmitte is awae of all othe tansmittes that can intefee with its eceive. This complies with the potocol model, which assumes no hidden teminal and exposed teminal poblems. In pactical half-duplex netwoks, hidden teminals can be significantly educed using the RTS/CTS message exchange befoe data tansmission [9]. Exposed teminal poblem can be solved appoximately by building a conflict gaph offline [], which specifies the intefeence elation between links. Simila mechanisms can be applied to full-duplex netwoks. To ealize RTS/CTS fo full-duplex bidiectional tansmission, an active tansmitte should be able to decode RTS equests fom the eceive. If the tansmitte is not disupted by stong intefees to the RX TX link, then it can tempoaily suspend its data tansmission, and feeds back a CTS packet to the RX instead. RX then stats tansmission and TX esumes its tansmission. To combat the exposed teminal, an offline conflict gaph can be used, similaly to half-duplex netwoks []. Even if two tansmittes can sense each othe, they can still send packets concuently if thei mutual intefeence is much weake compaed with the signal stength at the eceive of each. B. Utility-optimal full-duplex MAC: optimization fomulation and distibuted solution We conside the poblem of optimizing MAC-laye thoughput capacity fo a netwok containing a given set Γ of links that un the above asynchonous contention potocol. Evey link is a single-hop connection and has a countepat of evese diection, thus Γ is an even numbe and the numbe of nodes equals Γ. Denote S as the set of independent sets (each is a subset of non-intefeing links within Γ). In full-duplex mode, a link and its evese countepat can belong to the same independent set. At any time instance, the set of tansmitting links coespond to one independent set. A MAC scheduling algoithm can be chaacteized by π s, the faction of time each independent set s S is scheduled. Each link e may appea in multiple independent sets, and its thoughput ρ e equals the sum time of all these sets. Ou objective is to map the above asynchonous contention potocol to such a scheduling algoithm, and optimize its paametes to maximize netwok thoughput subject to a fainess constaint. This is equivalent to a utility optimization poblem: max U(ρ e ) (7) e Γ s.t. ρ e π s, e Γ (8) e s,s S π s = (9) s S When U(ρ e ) = log(ρ e ), the objective is poven to achieve optimal thoughput with popotional fainess guaantee [6] among e Γ. Simila to half-duplex netwoks, finding the optimal schedule is an intactable poblem. Howeve, the poblem can be appoximately solved by eplacing the objective function with: max V U(ρ e ) π s log(π s ) (0) e Γ s S whee V is a positive constant. Because s S π s log(π s ) s S S log( S ) = log( S ), it can be easily seen that the appoximation deviates fom the optimum by at most log( S ) V, which is negligible fo lage V. In what follows, we deive a distibuted asynchonous MAC by solving the appoximated optimization poblem, using a subgadient method in a simila manne to the utility-optimal half-duplex MAC in [6]. The above appoximated optimization (0) can be easily poven to be convex, and its Lagangian is: L(ρ, π, q, β) = V U(ρ e ) π s log(π s ) e Γ s S + q e π s ) () π s q e ρ e β( e s,s S s S with dual vaiables q e and β. By solving the KKT condition following a simila pocedue to [6], we can obtain the optimal values of β and π s as: ( ) β = log q e ) () s S exp( e s πs Π e exp(q e ) = s S Π (3) e:e sexp(q e ) In addition, a subgadient of the dual vaiable q e is: q e = [(V/q e ) ] Q π s (4) s:s S,e s whee Q denotes the pojection to [q min, q max ], the ange of q e. The dual vaiable q e can be intepeted as a vitual queue fo each link. The subgadient adaptation (4) conveges to the optimal q e fo the utility optimization poblem (0) [6]. Moe impotantly, it is fully distibuted fo each link e. Suppose time is divided into fames, each stating when a packet is geneated o successfully tansmitted. In each fame, a link e only needs to obseve the numbe of successful tansmissions (which accounts fo the tem s:s S,e s π s), and adapt q e following (4). It has been poven that, by unning a CSMA potocol with packet geneation ate p e fo each link e and exponentiallydistibuted packet duation with mean µ e, the stationay distibution of π s is [3]: Π e p e µ e π s = s S Π (5) e:e sp e µ e

8 Algoithm Utility-maximizing full-duplex CSMA.. Fo each time fame, geneate a packet with pobability p e and exponentially distibuted duation (in tems of the numbe of slots) with mean µ e.. Continuously un the full-duplex CSMA/CA potocol in Sec. IV-A. 3. if tansmission completes o new packet aives then 4. Update q e accoding to: 5. q e q e + α(v/q e K e ) Q 6. Set p e and µ e accoding to p e µ e = exp(q e ) endif 7. goto. Thus, to achieve optimal utility using CSMA, it is sufficient to adapt q e and set p e and µ e in each time fame such that exp(q e ) = p e µ e (c.f., Eq. (3)). With this obsevation, we obtain the adaptive CSMA Algoithm fo each full-duplex link. Hee α is a small step size, and K e the numbe of seved packets in the time fame. The algoithm essentially contols the ate of each link in ode to achieve the optimal utility when combined with the undelying full-duplex CSMA contention algoithm that avoids collision. Note that Q can be defined accoding to pactical values of µ e and λ e in 80. netwoks. C. Full-duplex gain: expeimental simulation We compae the achievable capacity of the above full-duplex CSMA algoithm with a utility-optimal half-duplex algoithm [6] which is deived fom an optimization famewok simila to (7), but does not allow a eceive to sense o tansmit while eceiving packets. Both algoithms ae implemented in a C++ based discete event simulato with MAC/PHY paametes (e.g., time slot, CW ange) consistent with 80.g [6]. All links ae assumed to have a capacity of 6Mbps without ate adaptation. By default equals. We conside two types of topologies: (i) Multi-cell WLANs with vaying link distance and bipola Poisson distibution of nodes [4]. APs locations follow Poisson distibution with a given density, and each AP is paied with a client in a andom diection with a given distance d. (ii) Ad-hoc netwoks with andom unifom distibution of node locations in a fixed -D aea. Given node density (aveage numbe of neighbos within tansmission ange), the topology geneato keeps andom tials until obtaining a topology whee all nodes ae connected. Without loss of geneality, each node is paied with one neighbo fo bidiectional tansmission, and MAC-laye capacity (total thoughput of all links) is used as pefomance metic. Effects of asynchonous contention. We study the asynchonous contention effect (Fig. (b)) by simulating the above utility-optimal CSMA in an ad-hoc netwok with 00 nodes, density 6. Fig. 5(a) shows the distibution of the numbe of concuent tansmittes ove 0 5 time slots. Due to asynchonous contention, bidiectional links often cannot be scheduled concuently. Coupled with the effects of spatial euse, full-duplex esults in only.47 aveage gain ove half-duplex mode. One may ague that the contention ovehead can offset the impovement of concuency owing to full-duplex. Fig. 5(b) Faction of slots 0.8 0.6 0.4 0. half-duplex full-duplex Faction of slots 0.8 half-duplex 0 0 4 6 8 0 6 8 0 4 6 8 Numbe of concuent TXs Numbe of concuent TXs (a) (b) Fig. 5. Distibution of the numbe of concuent tansmittes in each slot: (a) utility-optimal, distibuted, asynchonous CSMA; (b) oacle schedule that andomly picks tansmittes (no contention ovehead). Faction of links 0.8 0.6 0.4 0. 0.6 0.4 0. 0 0.5.5.5 3 Full-duplex thoughput gain Fig. 6. Distibution of thoughput gain. full-duplex plots the esults of an oacle, ound-based schedule with no contention ovehead. In each slot, it andomly selects tansmittes who do not intefee those links aleady selected, until no moe tansmittes can be selected. This schedule bette leveages the full-duplex advantage and impoves concuency, but the capacity gain is still fa below, as tansmittes ae still selected asynchonously, and the effect of spatial euse pesists. Capacity gain. Fig. 6 plots the distibution of thoughput gain ove all 00 links. The utility-optimal, popotionally fai schedule allocates thoughput quite diffeently fo halfand full-duplex netwoks. Wheeas some links eceive.5 thoughput gain, othes thoughput may be educed when using full-duplex. The aveage thoughput gain of all links is.46. With espect to the Jain s fainess index [6], half- and fullduplex has a fainess index of 0.58 and 0.56, espectively, which ae compaable. Oveall, full-duplex povides cetain capacity gain without noticeable sacifice of fainess, but again, the gain is well below. Effects of. Ou pio analysis identifies, the excess of intefeence ange ove tansmission ange, as a key paamete govening the full-duplex gain. Fig. 7 plots the mean thoughput gain unde vaying (eo bas show the std. ove 00 andom topologies). Consistent with ou theoetical analysis, the gain inceases as inceases. But unde pactical settings, e.g.,, the gain is fa below, and is below the theoetical uppebound pedicted in Sec. III-B. This essentially veifies ou analytical models, and shows the asynchonous contention futhe offsets the full-duplex gain in pactical, lagescale wieless netwoks. Effects of taffic locality in WLANs. In multi-cell wieless LANs, a client can abitaily appoach the AP, thus confining the tansmissions to a local aea and educing spatial euse advantage of half-duplex netwoks. Fig. 8 evaluates this effect by vaying the AP-client distance in a 50-cell netwok. When the link distance (nomalized w..t. tansmission ange) is close to 0, half-duplex netwoks cannot leveage the spatial euse advantage. Full-duplex oppotunities abound and ise the capacity gain close to. Howeve, in the common cases with link

9.7.6.8.5.4.6.3.4... 0 0.4 0.8..6 0. 0.3 0.5 0.7 0.9 Nomalized link distance Fig. 7. Effect of in ad-hoc Fig. 8. Full-duplex gain in multicell WLANs. netwoks. distance above 0.3, the gain is still fa below and deceases as link distance inceases to. Theefoe, full-duplex gain can be pominent in multi-cell WLANs, but only if the AP-client distance is much smalle compaed with the tansmission ange. Mean thoughput gain Mean thoughput gain V. RELATED WORK Full-duplex technology has aleady been poposed in boadband cellula netwoks (e.g., WiMax) to facilitate the fequencydevision mode. Cellula base stations tansmitting and eceiving adios can opeate simultaneously in two adjacent fequency bands, and a special hadwae filte called duplexe is used to mitigate leakage intefeence. Howeve, owing to lowe cost and bette suppot fo asymmetic uplink/downlink taffic, half-duplex time-devision mode is deployed in most cellula netwoks [5]. Recently, Choi et al. [] ealized singlechannel full-duplex though delicate antenna placement and self-intefeence cancellation, which inspied substantial wok (e.g., [], [6], [7]). Reseach along this diection mainly tagets the design and implementation of full-duplex PHY laye. The implication fo highe-laye (e.g., multi-hop outing [8], [9]) is lagely undeexploed. Since the landmak pape of Gupta and Kuma [5], substantial eseach has focused on analyzing wieless netwok capacity unde vaious topologies and PHY laye technologies (e.g., MIMO and diectional antennas [0]). Existing analysis unanimously assumes half-duplex adios, and tagets capacity scaling laws unde infinite numbe of nodes. In this wok, we have developed simple models to analyze the unique featues of full-duplex netwoks, and deive the capacity gain ove halfduplex netwoks. The utility-optimal full-duplex MAC in this pape shaes simila spiit with existing wok on utility-optimal CSMA [6], []. Howeve, ou objective is not to implement a new MAC, but athe to pefom a fai compaison between the optimal thoughput of half- and full-duplex netwoks, thus distilling the key factos that futue MAC potocols should take into account in ode to exploe the full-duplex gain. VI. CONCLUSION While it is tempting to believe that full-duplex can double wieless capacity, this pape dispoves the peception though asymptotic analysis and netwok optimization. Indeed, fo a single link, full-duplex may have a capacity gain of ove half-duplex, but in lage-scale wieless netwoks, spatial euse and asynchonous contention effects significantly undemine the actual benefits of full-duplex. Futue netwok designes need to eenginee the MAC potocols taking into these two factos, in ode to tanslate the PHY laye full-duplex gain into netwok laye thoughput impovement. Fo tactability, ou analysis has made a numbe of assumptions. The analysis of full-duplex capacity gain in -D netwoks assumes an oacle schedule that geedily enables bidiectional full-duplex tansmissions. Wheeas the utility-optimal MAC allows a mix of full-duplex and half-duplex tansmissions, it adopts a andomized CSMA-style schedule. As futue wok, we will deive the full-duplex capacity when an oacle, adaptive schedule is used. In addition, consistent with the potocol model, we assumed pefect caie sensing when compaing half- and full-duplex thoughput unde the utility-optimal MAC. Since pactical MAC potocols still suffe fom hidden- and exposed teminal poblems, both half- and full-duplex netwoks may undeutilize the capacity, possibly to diffeent extent. It would be inteesting to test the full-duplex gain unde these pactical conditions. REFERENCES [] J. I. Choi, M. Jain, K. Sinivasan, P. Levis, and S. Katti, Achieving Single Channel, Full Duplex Wieless Communication, in Poc. of ACM MobiCom, 00. [] M. Duate, C. Dick, and A. 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