independent links: a new approach to increase spatial reuse in wireless networks

Computer Networks 57 (2013) 1880–1893

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Computer Networks

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Independent links: A new approach to increase spatial reuse inwireless networks

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E-mail address: [email protected] (J.F. de Rezende).

Alexandre A. Pires, José F. de Rezende ⇑Programa de Engenharia Elétrica, COPPE, Universidade Federal do Rio de Janeiro, Caixa Postal 68504, 21941-972 Rio de Janeiro, RJ, Brazil

a r t i c l e i n f o

Article history:Received 27 March 2012Received in revised form 9 February 2013Accepted 24 March 2013Available online 1 April 2013

Keywords:Wireless mesh networksSpatial reusePower controlCarrier sensing

a b s t r a c t

This paper explores a new method of improving the performance of CSMA-based wirelessnetworks. The main proposal is to achieve a higher degree of spatial reuse by increasing thenumber of independent links. A pair of links is independent if the mutual interference gen-erated by simultaneous transmissions on these links can be tolerated by all devicesinvolved. In this paper, we derive the conditions for independence of links, and show thatmaximizing their number in a network is a complex problem. These insights were used todevelop an efficient and decentralized heuristic, whose performance is evaluated by sim-ulation. This mechanism outperforms recent literature proposals on a large number of ran-dom topologies.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

CSMA-based wireless networks, such as IEEE 802.11 [1],have been widely used to provide access to the Internet. Asa result of the growing popularity of this technology, sev-eral works have focused on increasing the capacity of adhoc and mesh networks based on this standard. One ofthe main lines of research on this topic is the developmentof mechanisms to increase spatial reuse in order to achievea high capacity even in scenarios of intense co-channelinterference.

In a wireless network, one of the main problems to besolved is to precisely recognize the appropriate degree ofspatial reuse to be achieved. Basically, the problem consistsof maximizing the number of simultaneous transmissionswithout causing frame collisions due to excessive co-chan-nel interference. In CSMA-based networks, access to theshared channel is regulated through carrier sensing, whichis used to identify when the channel should be consideredbusy by comparing the received power level to a certainthreshold. Thus, adjusting the carrier sensing threshold

CSth has been the focus of several studies with the purposeof achieving this proper degree of spatial reuse.

Another key factor for the efficient reuse of a wirelesschannel is the transmission power Pt used by each node.For a transmission to be successful, it is necessary thatthe SINR (Signal to Interference plus Noise Ratio) on a gi-ven receiver be at least equal to a certain threshold b. Thus,both the power used by the transmitter and the accumu-lated power generated by interfering nodes may affectthe reception of the frame, demonstrating the importanceof adjusting the transmission power.

Although CSth and Pt represent the main parameterscapable of regulating the degree of spatial reuse, thereare significant differences to be taken into account in thedevelopment of adjustment mechanisms for each one ofthem. Unlike CSth, there is a natural optimal value of Pt,at least from the perspective of energy efficiency. Given alink, this value corresponds to the minimum power re-quired for correct frames decoding at the intended recei-ver, considering only the presence of a certain level ofnoise. In fact, it is common sense that the use of the lowestpossible power levels would lead to a lower average inter-ference in all receivers, helping to maintain the requiredSINR in each one of them.

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A.A. Pires, J.F. de Rezende / Computer Networks 57 (2013) 1880–1893 1881

However, recent research indicates that it would bepossible to achieve higher spatial reuse using more thanthe minimum required transmission power. The work in[2] suggests the application of a factor k P 1, which wouldbe multiplied by the minimum required power level, todetermine the transmission power to be employed. Thisfactor is a function of the distance between transmitter(TX) and receiver (RX) for each link. By optimizing theparameters used for the calculation of k, the study suggeststhe possibility of gains of about 15% over an existingmechanism.

In this paper, we propose a different technique to adjustthe transmission powers and then the threshold CSth of thenodes. Although the scheme leads to power levels abovethe minimum required, as in [2], the method makes adjust-ments in Pt to increase the number of independent links. Itis shown that the power to be used in each node should notsimply be adjusted depending on the distance between TXand RX, but should also take into account other factors,such as the SINR required at each node and the possibilityof collisions.

We conclude that the problem of maximizing the num-ber of pairs of independent links is computationally com-plex, opting for developing a heuristic. The results arethen compared with a static configuration of Pt and CSth,as used in the 802.11 standard, and the analytical methodproposed by [2], resulting in significant throughput gainsover both.

The remainder of this paper is organized as follows. InSection 2, we review the related literature on spatial reuse.Section 3 defines the conditions that guarantee the inde-pendence between two links. The problem of maximizingindependent links in a given network is exposed in Sec-tion 4. Our proposed heuristic is defined in Section 5. InSection 6, we describe the simulation environment andthe obtained results. Finally, Section 7 summarizes thework and presents the conclusions.

2. Related works

In recent years, several studies have been devoted toincreasing the capacity of CSMA-based wireless networks.Most of them propose this increase via spatial reuse, usu-ally focusing on the adjustment of either Pt, CSth or both.

In [3], the authors performed a joint analysis of theinfluence of CSth and Pt on the performance of an 802.11network. This study is extended in [4], which considersthe possibility of using multiple transmission rates. Theyhave used a honey-grid model of interference, where ahexagonal arrangement of interfering transmitters is posi-tioned around each node. The six interfering transmittersare at the same distance from the node of interest and fromeach other, as guaranteed by the hexagonal shape. If thedistance is slightly greater than the carrier sensing range,none of them could sense the others transmissions, leadingto the analytical calculation of a worst-case interference.The same interference model is used in [5], which studiesthe adjustment of Pt, CSth and transmission rate. However,the study argues that the adjustment of Pt would be moreeffective than CSth’s, using examples of specific network

topologies. Thus, they propose an algorithm for simulta-neous adjustment of Pt and transmission rates.

In [6], an analytical model is proposed to investigate theeffect on the performance of adjusting both Pt and CSth. Theinterference model assumes randomly located nodes, fol-lowing a bidimensional Poisson distribution with constantdensity. The accumulated interference in a certain node iscalculated analytically by means of an integral that com-putes the effects of interfering nodes positioned in ringsof increasing radii from the node of interest.

This latter work is extended in [7] by means of an algo-rithm capable of adjusting the transmission power and thecarrier sensing threshold. Basically, nodes should exchangetables containing information about the neighborhood,allowing the construction of a minimum spanning tree ineach one of the nodes. The tree is then used to determinethe values of Pt and CSth. This work, along with [6], has be-come the main reference in the area since then. As far asthe adjustment of transmission power is concerned, bothsuggest the use of the lowest power levels capable ofestablishing communication in order to increase spatialreuse.

In [8], the authors argue about the inherent difficultyof determining optimal CSth since the number of interfer-ing nodes at each moment is random. Thus, they proposea trial-and-error method through the monitoring of statis-tics on successful transmissions on the neighborhood. Agame theory model is used in [9] to show that the localadjustment of CSth may lead to an undesirable Nash equi-librium, causing great unfairness. An analytical model toset appropriate CSth values is proposed in [10], which ad-dresses the interference issue in a similar manner to thatused in [6,7].

In [11], the authors devote themselves to determining asafe carrier sensing range through an analytical model.With this range, identified by a factor that multiplies themaximum operating distance of network links, they couldvirtually eliminate the hidden node problem. However, theissue of spatial reuse is not directly addressed.

Contrary to the idea of using the minimum requiredpower on each link, the work in [2] proposes the use ofhigher power levels to increase the spatial reuse. Usingan analytical approach, they propose a method to adjustPt depending on the TX–RX distance. Thus, the power ofeach link should be calculated by kðdsrÞ � Ptmin

, where theparameters of the k(d) function are determined throughoptimization. Once optimal values for Pt are determined,the method of adjustment of CSth proposed in [6] with min-or modifications is employed. With this setting globallydetermined and using a derived expression for the aggre-gate capacity of the network, they use MATLAB to obtainthe results, which indicate a better performance than thatobtained in [6].

In [12], the authors propose a different method of car-rier sensing which takes advantage of the fact that theminimum required SINR depends on the arrival order ofthe frames. In practice, the scheme leads to an indirectadjustment of CSth, but the issue of power adjustment isnot addressed. An evaluation of the performance increaseachieved with an optimal adjustment of CSth appears in[13]. However, a method of adjustment is not shown, and

f

b

r1

2r2s

s1

ce

a

d

Fig. 2. Generic topology.

1882 A.A. Pires, J.F. de Rezende / Computer Networks 57 (2013) 1880–1893

the network performance is evaluated for different thresh-old values on regular topologies.

The authors of [14] introduce the concept of ceased-areas, defined as the region around a TX–RX pair whereall other nodes are required to keep silent. Using this in-sight, they developed a method of power control that triesto minimize the average size of these areas, aiming toachieve a higher spatial reuse degree.

3. Independent links

The performance gain due to spatial reuse may be ex-plained by the possibility of communications occurringsimultaneously over the network. Thus, the problem ofdetermining the optimum degree of reuse must go throughthe study of the conditions that make links independent ofeach other.

Consider the two links displayed in Fig. 1. Let us assumethat a = 3.0, b = 10, and the used transmission power is theminimum necessary for communication Ptmin in each link.In this scenario, it is easy to verify that a collision occursif a DATA frame is sent on link 1 during the transmissionof an ACK on link 2. This results in a SINRACK of 8 at s2,which causes the loss of the ACK frame. It thus appears thatsuch links do not meet the requirements of mutualindependence.

Now, let the nodes s1 and s2 use a power Ps, and r1 and r2

use Pr. For this particular topology, if Pr = 1.25 � Ps the SINRfor the above situation becomes 10. Indeed, the use of thispower configuration guarantees the required SINR at allnodes, and so simultaneous transmissions of any frames.

In order to formally establish the conditions of indepen-dence, we propose considering the generic topology exhib-ited in Fig. 2. The distances between each of the nodes arerepresented by a, b, c, d, e and f.

Let the path loss exponent be equal to a, the minimumrequired SINR b, and the power level of each node be givenby Ps1 ; Ps2 ; Pr1 and Pr2 . Considering the simultaneoustransmission of DATA frames in the two links, and assum-ing that noise is much lower than the powers involved, thetwo conditions expressed in Eq. (1) need to be met so thatboth frames are correctly received.

Ps1Ps2

da

� �aP b

Ps2Ps1

cb

� �a P b

8<: ð1Þ

Using Rs1 ;s2 ¼ Ps1=Ps2 , the conditions can be summarizedas indicated in Eq. (2), which can be defined as a unidirec-tional independence condition, ensuring simultaneoustransmissions of frames.

dd

s1 s2 r2r1

2d

Fig. 1. Particular topology.

bad

� �a6 Rs1 ;s2 6

1b

cb

� �að2Þ

Similarly, in order to guarantee the possibility of simul-taneous transmissions of any frames from all nodes in-volved, we can define the four conditions that establishindependence between two links, hereafter called inde-pendence conditions, according to the following equation:

b ad

� �a6 Rs1 ;s2 6

1b

cb

� �ab a

f

� �a6 Rs1 ;r2 6

1b

eb

� �ab a

e

� �a6 Rr1 ;s2 6

1b

fb

� �a

b ac

� �a6 Rr1 ;r2 6

1b

db

� �a

8>>>>>>><>>>>>>>:

ð3Þ

In order to discuss the implications of the above inde-pendence conditions, we shall consider a network consist-ing of terminals geographically distributed in an area inwhich each transceiver may choose a transmission powerfrom a predefined discrete set of power levels betweenthe limits Pmin and Pmax.

Given a particular link v, all other links in the networkcan be classified according to their dependency relation-ship with it. Each link will be dependent or independentof v, depending on whether or not it meets the previouslyexposed independence conditions.

In addition, as the transmission power used by the ter-minals has a limited variation, it may be possible to iden-tify links among those independent of v, whoseindependence is distance-based. We can say that v0 is dis-tance-based independent of v, if this independence condi-tion always holds regardless of the power levels used bythe terminals that compose v and v0.

Similarly, one can eventually identify links whosedependence on v is also distance-based. These links will al-ways be dependent on v, whatever the used power levels,as long as they are within certain limits.

This means that, from the point of view of a given link v,each one of the other links in the network can be classifiedinto four distinct sets, according to their condition ofdependency on the link under consideration. The first set,called Id, corresponds to the links that will always be inde-pendent of v (distance-based independence). The secondset, Dd, is composed of links dependent on v, whose depen-dence is also distance-based.

Finally, the sets Ip and DP are composed respectively byindependent and dependent links of v, but whose condi-tions are valid only for the power levels currently usedby the terminals involved. In other words, at least one


combination of power levels exists to be used by the termi-nals involved that would reverse the condition of the con-sidered link in respect to v (making a link in Ip dependenton v, or a link in DP independent of v).

The following subsections formally define the condi-tions used to classify links according to the above sets.

3.1. Distance-based independence

When using a graphical representation, the indepen-dence between two links can be seen as a multiple-solu-tion operating region in relation to the transmissionpowers used by the terminals involved. Taking the firstcondition of independence, we can rewrite it as expressedin the following equation:

Ps1 P b ad

� �aPs2

Ps1 61b

cb

� �aPs2

(ð4Þ

Using a Cartesian plane, one can identify thismultiple-solution operating region, that is, the values thatPs1 can assume as a function of Ps2 that keep the conditionvalid, as shown in Fig. 3. In the figure, tan hsup

� �¼

1b

cb

� �a; tanðhinf Þ ¼ b a

d

� �a, and any valid solution lies in the re-gion between the two lines. Considering also that thepower levels are always chosen in a discrete set betweenPmin and Pmax, it is possible to clearly identify the two oper-ating points ðPs1 and Ps2 Þ that satisfy the condition. InFig. 3, four discrete power levels were used.

In distance-based independent links, the region be-tween the two lines shown in Fig. 3 completely enclosethe set of all possible operating points (a grid of N2

p points,where Np is the number of discrete power levels used). Thiscondition can be expressed by the following equation:

b ad

� �a6

PminPmax! d P bPmax

Pmin

� �1aa

1b

cb

� �a P PmaxPmin! c P bPmax

Pmin

� �1ab

8>><>>: ð5Þ

The other conditions of independence can be dealt within the same way, that is, by establishing operating regionsof solution for three new Cartesian planes: Pr1 � Ps2 ,Ps1 � Pr2 and Pr1 � Pr2 . By performing a similar proce-dure and simultaneously ensuring compliance with all

θθsup

inf

Ps1

Ps2

Fig. 3. Defining a solution region.

conditions, we can summarize the distance-based inde-pendence condition in a single expression given in thefollowing equation:

minðc;d; e; f ÞP bPmax

Pmin

� �1a

maxða; bÞ ð6Þ

3.2. Distance-based dependence

Similarly, we can use the same approach to determinethe conditions that define the dependency relation of linksbased on the distance between them, that is, regardless ofpower levels adopted in each terminal involved.

Analyzing the Cartesian plane Ps1 � Ps2 , it is possible toidentify three situations in which the set of valid solutionswould be empty, that is, where independence conditionsare impossible to be met. The first one refers to a situationin which the region above the line of slope b a

d

� �a has nointersection with the set of potential operating points.Mathematically, this hypothesis can be expressed by thefollowing equation:

bad

� �a>

Pmax

Pmin! d <

bPmin

Pmax

� �1a

a ð7Þ

The second situation concerns the case of the area be-low the line of slope 1

bcb

� �a which has no intersection withthe set of solutions defined by discrete power levels used,which is represented by the following equation:

1b

cb

� �a<

Pmin

Pmax! c <

bPmin

Pmax

� �1a

b ð8Þ

Finally, there is still the situation in which the regionabove the line of slope b a

d

� �a has no intersection with theregion below the line 1

bcb

� �a, which is represented by thefollowing equation:

bad

� �a>

1b

cb

� �a! cd < b

2aab ð9Þ

It is worth noting that this last condition is the only oneactually independent of the power levels used, and thatcould really be called ‘‘distance-based dependence’’. Theother two cases refer to a combination of distance andpower limits. However, this distinction does not seem tobe relevant, since there will always be limits to the powerlevels used.

By establishing similar conditions to the Cartesianplanes Pr1 � Ps2 , Ps1 � Pr2 and Pr1 � Pr2 , the conditions ofdistance-based dependence can be summarized as shownin Eq. (10). If one of them holds, the links will be dis-tance-based dependent.

minðc;d; e; f Þ < bPminPmax

� �1a

maxða; bÞ; or

minðcd; ef Þ < b2aab

8<: ð10Þ

3.3. Estimation of the amount of independent links

Considering a CSMA-based wireless network distrib-uted in a flat area, the conditions of independence


previously defined can be used to obtain an estimation ofthe number of independent links in the absence of a powercontrol mechanism.

Let y = min (c,d,e, f) be the distance between two links,hence, Eq. (11) is valid.

ay

� �a¼max a

c

� �a; a

d

� �a; a

e

� �a; a

f

� �ah iyb

� �a ¼min cb

� �a; d

b

� �a; e

b

� �a; f

b

� �ah i ð11Þ

Thus, if Ps1 ¼ Pr1 ¼ Ps2 ¼ Pr2 ¼ Pfixed, the conditions in Eq.(3) can be expressed in a single inequality that shows themore restrictive comparison, as shown in the followingequation:

bay

� �a

6 1 61b

yb

� �að12Þ

Through the introduction of a variable x = max(a,b), anew simplification can be performed, resulting in the fol-lowing equation:

y P b1=ax ð13Þ

The right side of the condition of Eq. (13) represents adistance limit from each of the two terminals that establisha link. A close link can only be considered independent ifboth link terminals are located beyond this limit. If y,which corresponds to the distance between a neighbor linkand the considered link, is below this limit, at least one ofthe conditions of independence will not be valid, therebymaking the links dependent on each other.

Thus, the union of the area defined by two circles withcenters in the communicating terminals, whose radii areequal to y = b1/a x, is a region of interest, within whichthere can be no independent links. This region is shownin Fig. 4, and its area A can be easily calculated by theuse of simple geometry, resulting in the expression of thefollowing equation:

A¼ 2b2=a p�arccos1

2b1=a

� �� þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4b2=a�1

q2

8<:

9=; �x2¼/ðb;aÞ �x2 ð14Þ

The definition of this region of dependence is impor-tant, since it allows us to establish a function able to esti-mate the amount of independent links in a given network.

y

y

s r

x

Fig. 4. Dependence region of a link ðy ¼ b1axÞ.

The variable x, which is needed to calculate the area ofthe dependence region A, needs to be mentioned. Asx = max(a,b), where a is the distance between the TX andRX of the considered link and b of the nearby link, it followsthat x can assume a different value for each one of theneighbor links. However, as we solely want to have an esti-mate of the number of independent links in the network,this analysis will take the expected values of x and A, whichcan be easily determined as described below.

Consider a network with m links distributed over a flatarea of size S. Thus, the density of links in the network isgiven by r m

S . If the mean area of the dependence regionis given by �A, one can approximate the average amountof independent links �I by the following equation:

I ¼ rðS� AÞ ð15Þ

If there are m links in the network, the mean total num-ber of independent links in a network is given by the fol-lowing equation:

IT ¼mI2

ð16Þ

The division by 2 is needed to eliminate duplicity, dueto reciprocal counting of links independent of each other.Since m = rS, the function capable of estimating the num-ber of independent links results in the following equation:

I ¼ r2SðS� AÞ2

ð17Þ

Now, it is necessary to determine the mean area A. As Ais a function of the random variable x;A can be calculatedby means of the integral expressed by the followingequation:

A ¼Z xmax

0AðxÞ � f ðxÞdx ¼ /ðb;aÞ

Z xmax

0x2 � f ðxÞdx ð18Þ

where, f(x) is the probability density function of the vari-able x = max (a,b) and xmax corresponds to its maximumvalue (which is the maximum range of a link). In order todetermine f(x), we initially find the related probability dis-tribution function F(x), using the following equation:

FðxÞ ¼ P½maxða; bÞ 6 x� ¼ P½a 6 x� � P½b 6 x� ¼ x2

x2max

ð19Þ

By considering that the range of network links has auniform distribution up to the limit xmax; P½a 6 x� ¼P½b 6 x� ¼ x

xmax.

So, f(x) is given by the following equation:

f ðxÞ ¼ ddx

FðxÞ ¼ 2xx2

maxð20Þ

Through Eq. (18), the mean area of the dependence re-gion is given by the following equation:

A ¼ /ðb;aÞ � 2x2

max

Z xmax

0x3 dx ¼ /ðb;aÞ � x

2max

2ð21Þ

In Fig. 5, one can verify that the expression used to esti-mate the number of pairs of mutually independent links isquite effective. Simulations were performed with a certainnumber of terminals randomly positioned on a flat area

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 50 100 150 200

Num

ber o

f ind

epen

dent

link

s

Number of potencial links

Simulation resultsProposed function

Fig. 5. Estimation of the amount of independent links.


and the average number of independent links for each sce-nario is totalized. Each point corresponds to 50 differenttopologies. The comparison with the function obtained toestimate the amount of independent links shows a verysmall error.

3.3.1. The influence of the power controlIn the previous section, an analytical model that corre-

lates the spatial distribution of links in the network to theconditions of independence among them was used to esti-mate the number of independent links in the absence of apower control mechanism. In this section, this same modelis extended to show the influence of power control on thenumber of independent links.

Consider a network in which terminals can use p differ-ent discrete power levels P0, P1, . . . , Pp�1. Additionally,these terminals use a power control scheme, such thatthe power level used on each link is the lowest amongthose that enable communication.

In this situation, the condition expressed in Eq. (13)must be adjusted as a consequence of the difference inpower levels used by the two-links candidate to mutualindependence. Considering the links 1 and 2, whose dis-tances between TX and RX are respectively a and b, andthe power levels used are Pi and Pj, we can adapt Eqs.(12)–(22).

bay

� �a

6Pi

Pj6

1b

yb

� �að22Þ

Thus, we can determine two additional conditions that,in the case where a power control scheme is used, areequivalent to the single condition in Eq. (13). These condi-tions are shown in the following equation:

y P bPj

Pi

� �1aa

y P b PiPj

� �1ab

8>><>>: ð23Þ

It is worth noting that it is not possible in this case todetermine the most restrictive condition. If a > b (or b > a)and Pi = Pj, that is, the difference between the link

distances, does not lead to the necessity of using a higherpower level in link 1 than the one used by link 2, the con-dition can be unified as in Eq. (13), using the variablex = max (a,b) in order to express the most restrictivecondition.

However, if a > b and Pi > Pj, we cannot determine themost restrictive condition a priori. Just consider the casewhere a = b + e, where e is a very small distance but enoughto force link 1 to use a power level Pi > Pj. In this case, eventhough max (a,b) = a, the most restrictive condition may be

related to b, since b PiPj

� �1ab can be greater than b

Pj

Pi

� �1aa,

depending on Pi, Pj and e values.Let Aði;jÞ be the mean area of the dependence region set

around a link that uses the power level Pi and all nearbylinks use the power level Pj. It is noteworthy that thedependence region is based on one of the conditions setin Eq. (23). It is not possible to define the most restrictivecondition without examining each case, depending on thevalues of a, b, Pi and Pj.

Consider a network composed of m links, and a numberof mi links use the power level Pi. One can define ki ¼ mi

m ,which represents the proportion of links that use that par-ticular discrete power level. Thus, if a particular link E usesthe power level Pi, one can estimate the number of inde-pendent links of E according to the following equation:

IðiÞ ¼ r �Xp�1

j¼0

kj½S� Aði;jÞ� ¼ r �Xp�1

j¼0

kjzði;jÞ ð24Þ

In this equation, an auxiliary variable zði;jÞ ¼ ½S� Aði;jÞ�was introduced, which could be seen as an independenceregion, as opposed to the dependence region A. Basically,it becomes a weighted average of the areas z(i,j) using theratios k as weights.

Thus, we can estimate the mean total number of inde-pendent links from Eq. (25), obtained in a similar mannerto Eq. (17).

IT ¼rS2�Xp�1

i¼0

kiIðiÞ ð25Þ

Lemma 3.1. If a link E is using a power level Pp, lower thanall previously existing power levels, the new mean area of itsdependence region A0ðp;jÞ may be higher than the previousAðp�1;jÞ; j 2 ð0;1;2; . . . ; p� 1Þ.

Proof. The area A(p,j) can be similarly calculated as in Eq.(14). The only difference refers to the fact that the radiusof the used circles were b

1a maxða; bÞ, and in the case using

a power control these circles have radii defined by themost restrictive condition in Eq. (23).

If the link E starts to use a new power level Pp lowerthan the existing ones, and there is a power control schemebeing used, the power level previously used by E was Pp�1,since P0 > P1 > P2 > � � � > Pp�1 > Pp. Thus, considering a linkclose to E, which uses a power level Pj, the conditionsstated in Eq. (23) result in Eq. (26). These are the conditionsthat define the Aðp;jÞ, before the event of E adopting a powerlevel Pp < Pp�1.


y P bPj

Pp�1

� �1aa

y P bPp�1

Pj

� �1ab

8>><>>: ð26Þ

As a first scenario, consider that the second condition inEq. (26) is the most restrictive one. In this case, it is easy tosee that the adoption of a level Pp < Pp�1 by E results in areduction in the value of the radius of the circles,and hence provides a reduced area for the value ofAðp;jÞ < Aðp�1;jÞ.

In the case where the most restrictive condition is thefirst one, the exchange of Pp�1 by Pp seems to increase theresulting value, which would lead to an increase in thearea. However, we must determine whether the conditionremains the most restrictive one even when the transmis-sion power is reduced.

The maximum possible reduction in the transmissionpower of a node is limited by the minimum powerrequired to maintain communication at link E, whosedistance is a. This minimum power is given by RXthaa

k , whereRXth is the minimum reception threshold in the presence ofnoise and k is the transmission gain.

So, using the smallest possible value of Pp, whichcorresponds to the maximum increase in the value referredto in the first condition of Eq. (26).

y P bkPj

RXthaa

� �1a

a P bkPj

RXth

� �1a

As Pj P RXthba

k ; ba6

kPj

RXth, we can write:

y P bkPj

RXth

� �1a

P b1ab

As Pp < Pj, we have b1ab > ðb Pp�1

PjÞ

1ab. Therefore, it appears

that the first condition remains more restrictive than thesecond, showing the possibility of the new area A0ðp;jÞ beinglarger than Aðp�1;jÞ.

Lemma 3.2. If the terminals on a network can switch to anew power level Pp, lower than the p of previously existingpower levels, the new estimated number of independentlinks IðpÞ can be lower than Iðp� 1Þ.

Proof. If a new power level is available, those links whoseminimum required power is less than Pp should adopt it.The number of independent links of a link that uses powerPp�1 is given by Eq. (24), that is,

Iðp� 1Þ ¼ rXp�1

i¼0

kizðp�1;iÞ

Let k0p 6 1 be the proportion of links that are using the levelPp. With the introduction of a new level Pp < Pp�1, a numberof terminals that once operated at level Pp�1 will adopt thenew power.

According to Lemma 3.1, it is possible that the reduc-tion of the transmit power level may lead to an increase inthe mean area of the dependence region. Thus, as z(i,j) =[S � A(i,j)], it is possible that z(p,i) is lower than z(p�1,i),resulting in IðpÞ < Iðp� 1Þ.

Theorem 3.3. If the terminals can switch to a new powerlevel Pp, lower than the p previously existing power levels,the new estimated total number of independent links I0T maybe lower than IT .

Proof. The estimated total number of independent linksusing p power levels (P0 to Pp�1) is

IT ¼rS2

Xp�1

i¼0

ki � IðiÞ

As exposed at the proof of Lemma 3.2, a fraction of thelinks that were using level Pp�1 will use the new powerlevel Pp. In accordance with Lemma 3.2, it is possible tohave IðpÞ < Iðp� 1Þ. So, it is also possible that I0T < IT .

The importance of Theorem 3.3 is to prove that tradi-tional power control schemes do not necessarily result ina larger number of independent links. Indeed, it is possiblethat more independent links exist if terminals use higherpower levels, even though lower levels may be used.

4. Maximizing independent links

The main idea explored in this paper is to increase spa-tial reuse by maximizing the number of independent links.However, this does not represent the optimal solution tothe spatial reuse maximization problem. The reason forthis is the cumulative interference generated by multiplesimultaneous transmissions. Two links can be independentof each other, and each of them can be independent of athird. However, it is possible that when these three linksare simultaneously used, the SINR becomes insufficient inone or more of the receivers involved. Thus, the proposalis to maximize link independence two-by-two in order toobtain a better spatial reuse. This is a much simpler prob-lem, although its solution is not necessarily optimal.

Let us define D = (V,E) as a link dependence graph,whose set of vertices V corresponds to all physical commu-nication links on a network. Thus, in a vertex v, defined bynodes ni and nj, only 2V if ni establishes a single-hop com-munication with nj, that is, ni and nj are correspondents. Anedge exists between two vertices if they have a relation-ship of dependency between them, which can be indirectlydefined by the conditions of independence listed in Sec-tion 3. If one of the conditions of independence is notmet, it is said that vi is dependent from vj and vice versa.

As indicated earlier, compliance with independenceconditions is a function of node positioning, the transmis-sion powers of the nodes and the required SINR values.Thus, the independent links maximization problem, con-sidering some specific topology and required SINR at eachnode, could be solved by following these steps: (i) deter-mine the dependence graph D considering that each nodeuses a certain transmission power, (ii) given D, determinethe maximum number of links mutually independent, thatis, which are not directly connected by edges belonging toE, and (iii) perform steps (i and ii) with all possible config-urations of power levels, determining the solution thatmaximizes the number of independent link pairs.


Analyzing this sequence of steps, we are able to verifythe computational unfeasibility of this solution. Step (i)implies checking the conditions of independence for allpairs of links that belong to V. In a network with m links,this means carrying out the tests for each mðm�1Þ

2 combina-tion. Note here that the number of possible links in the net-work also varies quadratically with the number of nodes.Step (ii) can be reduced to the known problem of deter-mining the maximum independent set, whose solution inpolynomial time is unknown. Finally, the third step re-quires us to explore all possible combinations of power.Considering Np discrete power levels, this results in Nn

p

combinations to be tested, where n is the number of nodes.

4.1. Preliminary heuristic

Faced with the complexity of the independent links max-imization problem, we have opted for the development of aheuristic to be executed by nodes in a distributed fashion.The goal is to increase the number of existing pairs of inde-pendent links by adjusting the transmission power of somenodes. In order to evaluate the effectiveness of the proposalin increasing the aggregate throughput of a network by aug-menting the number of independent links, we developed asimple preliminary heuristic. The main idea was to use it asa proof of concept, evaluating its results, and only then to pro-ceed to the elaboration of a more complete mechanism.

The proposed method requires a periodic exchange ofspatial reuse frames. Basically, each node must send toits neighbors: (a) the identification of its correspondentnodes; (b) an estimate of the distance for each of them;(c) the required SINR for reception (for itself and its corre-spondent nodes); and (d) the transmission powers used incommunicating with its correspondents.

The estimated distance (dest) referenced in (b) may bedetermined by the frame reception power when the powerlevel used by the corresponding transmitter is known. By

using a log-distance propagation model, dest ¼ GPtPr

� �1a. The

reception of a spatial reuse frame itself is an opportunityto determine this distance since it is always sent at fullpower. To mitigate problems related to channel instability,it is convenient to use a moving average for estimating thedistance from several consecutive observations. It is worthmentioning that this distance may not represent a reliableestimate of the actual distance between nodes, but it canbe considered as an equivalent distance for propagationpurposes. Actually, the only thing that really matters inestablishing the independence between links is the meanattenuation level between nodes, since it determines thecontribution to the SINR in each receiver.

The basis of the heuristic is that each one of the linksshould elect a certain dependent link, that is, a neighborlink in the dependence graph D, in an attempt to adjustpower levels that can generate independence. Obviously,the graph D here has only a local significance. Each link,through one of its nodes, determines neighbor dependentlinks. Once this link is determined, the optimization iscomplete, that is, a particular configuration of power levelsfor each one of the four nodes involved that generatesindependence is chosen. If more than one solution exists,

lower power levels are used and both links are marked toprevent a new change in power levels. If the attempt fails,links remain available to participate in new optimizationattempts.

Briefly, the steps to be periodically performed by a nodeof each link are as follows:

1. Determine its neighbor links in the graph D and theirrespective equivalent distances b, c, d, e and f.

2. Select the link to perform the optimization.3. Explore all possible combinations of power levels

between the four nodes, using the conditions explainedin Eq. (3), to verify independence.

4. If independence can be achieved, announce to theneighbor link, and both adopt the power levels chosen(i.e. minimum power levels that generate indepen-dence) and are marked, avoiding new optimization fora certain period of time. Moreover, the neighbor set ofD for each link involved is updated, excluding oneanother.

5. After performing the above steps or being notified of anoptimization performed by a neighbor link thatinvolves itself, the nodes of the link adjust their respec-tive values of CSth according to the new set of neighborsin D.

It is enough that one of the nodes of each link runs theheuristic, considering that the required information will betransmitted to its correspondent node. The first step startswith the reception of spatial reuse frames from each one ofthe neighbors. For each received frame, the dependencytest is carried out, thereby building the local view of graphD. If one of the distances (c,d,e, f) cannot be estimated,being the node out of reach, the worst case is assumedfrom the standpoint of interference generated by it. Thus,the maximum range is used as an estimated distance,determined by the maximum power level and the recep-tion threshold corresponding to the basic transmissionrate.

The selection of the neighbor link to be optimized isbased on higher probability to achieve independence. Ana-lyzing the independence conditions, it appears that thesmaller the value of b, the shorter the neighbor link, thatis, the more likely it is to become independent of the linkof reference. Likewise, the higher the values of c, d, e andf, the lower the mutual interference between the links, alsofavoring independence. Thus, the chosen method to selectthe link to optimize is to compare b

minðc;d;e;f Þ for each of theneighbors in the graph D. The attempt at optimization willbe performed with the one that presents the lowest ratio. Ifthere are m links in the network, the worst case is repre-sented by (m � 1) comparisons.

The third step is performed by testing all possible combi-nations of power to the terminals involved. Considering Np

discrete levels of power, it results in N4p combinations (256

configurations in our evaluation). The fourth step is trivial.Finally, in step 5, each node of the link adjusts its own

CSth. The idea here is that the node should have its accessto the medium denied only if one of the nodes of a neigh-bor link in D is transmitting. Thus, the worst case refers tothe farthest node from those belonging to the neighbor

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Fig. 6. Preliminary heuristic.


links in D. For this, the node uses the distances c, d, e and fto estimate the respective reception power of each nodewhose link is a neighbor in D. It is used as CSth the smallestvalue of these powers minus a safety margin.

4.2. Preliminary results

In order to evaluate the performance of the preliminaryheuristic, we performed simulations of ad hoc 802.11 net-works using an enhanced version of ns-2 [15]. Basically, anSINR-based interference model was implemented in ns-2version 2.31 to account for the cumulative interferencecaused by simultaneous transmissions.

Each simulation scenario consists on a flat square areaof side L = 1000 m, in which m TX–RX pairs are positionedaccording to a two-dimensional uniform random distribu-tion. Each link has a uniformly distributed distance to thelimit of transmission range. The number of flows, m, variesfrom 4 to 16. The traffic pattern is constant-bit rate (CBR)and the propagation model is the two-ray ground. Allnodes use omnidirectional antennas and the physicaltransmission rate is 18 Mbps. Receivers are exposed to awhite thermal noise of power �100.6 dBm. The frame sizeis 1024 bytes. Each simulation lasts 30 s.

Fig. 6 shows the results obtained for fixed Pt and CSth, theproposal in [2] and our preliminary heuristic. Each pointconsists of the average aggregate throughput from 100 runson 50 random scenarios, with a 95% confidence interval.Fig. 6b exhibits the results in the presence of multipath fad-ing, modeled according to a Rice distribution [16]. Whetheror not considering the presence of fading, one can verify thatthe heuristic presents a good performance.

Next, we evaluated the performance of the preliminaryheuristic in multi-hop scenarios by using the same config-uration parameters of the first evaluation. In these scenar-ios, the terminals were placed in random topologies. Thenumber of nodes was set to 50, that is, 25 traffic sourcesand 25 sinks. The number of hops for each flow can varyfreely, according to the specific topology of the scenario.The routes were previously calculated by a shortest-pathfirst algorithm and statically inserted in each node, thusavoiding the influence of the routing protocol.

In order to evaluate the performance on different net-work densities, we varied the maximum neighborhood de-gree (d) allowed in the topology generation. So, terminalsare positioned freely in the area, ensuring, however, thatnone of them has more than d neighbors in its communica-tion range.

Thus, given a communication range area of size pR2max,

we have inside it at least two nodes, and a maximum of(d + 1) nodes. Considering a linear distribution betweenthe minimum and maximum limits, the size of the totalarea used was chosen for each value d, according to Eq.(27). This size allows us to keep the density of the wholenetwork in a value close to the mean density of the nodes’neighborhoods.

A ¼ 2pR2max

dþ 3nnodes ð27Þ

where Rmax represents the communication range of a node,and nnodes is the total number of nodes, which is set to 50.

Thus, as d increases, the total area is reduced, resulting in adenser topology.

Another modification from the first evaluation schemeis the traffic pattern used. Instead of using CBR/UDPsources, we use FTP/TCP traffic sources. The goal is toachieve a better utilization of network capacity, using theadaptive nature of TCP. In a scenario of multiple streamsand multiple hops, in which flows share links with eachother, the tendency is that better performance is achievedwhen the sources regulate their flows in accordance withpacket losses.

Finally, as a result of the large variability in results dueto the use of multi-hop communication, it was necessary toincrease the number of simulated scenarios for each d to300. Thus, the achieved 95% confidence intervals are smallenough for comparison purposes.

Fig. 7 shows the achieved aggregate throughput as afunction of the maximum allowed neighborhood degree.Note that both proposals, the one described in [2] andthe preliminary heuristic, presented a considerable perfor-mance deterioration, resulting in a lower aggregatethroughput than the one obtained by the fixed Pt and CSth

scheme.The explanation for this lies in the significant increase in

the number of active links in relation to the first evaluation.


In the single-hop scenarios, each simulation had up to 16 ac-tive unidirectional links, due to the CBR/UDP traffic.

In the multi-hop scenarios with FTP/TCP traffic, we

could have up to 1225 bidirectional links nnodesðnnodes�1Þ2

� �. In

practice, the average number of active links in each simu-lation resulted in about 120 bidirectional links, which isstill well above the first evaluation.

The basic idea of the preliminary heuristic, and also ofthe method proposed in [2], consists of (a) adjusting thepower levels to make more links independent of each otherand (b) adjusting the CSth in order to take advantage of thenew relations of independence. Thus, the medium accessmechanism becomes more aggressive, wasting less timeon unnecessary back-offs, tending to increase the aggre-gate throughput. However, it imposes a higher risk of col-lisions, which leads to a reduction in the networkperformance.

The key question then becomes whether the mecha-nism generates a sufficient number of independent linksfrom the total number of active links, so that the balancebetween reducing back-off time and increasing collisionsbrings a benefit to the aggregate throughput. The poor per-formance achieved evidences that both mechanisms fail toachieve that balance in a multi-hop scenario, where thenumber of active links is very high.

So, we decided to develop a new heuristic, more com-plex than the preliminary one, but more efficient in pro-ducing independent links, allowing gains even inscenarios where the density of active links is high.

5. Independent links heuristic

This new heuristic is based on the idea of increasing thenumber of independent links by considering a mesh net-work topology in which each node has multiple neighborsand can communicate with any of them. This mechanism ishereafter referred as ILH, the acronym for IndependentLinks Heuristic.

Upon the arrival of a new node (nnew) on the network,ILH assumes that this node acquires a local knowledgeby the reception of spatial reuse frames, called R, whichare transmitted at full power. This knowledge entails itsneighborhood, namely its neighbors identities, their

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Fig. 7. Multi-hop scenarios.

equivalent distances and the communication power con-cerning itself.

Let us consider a neighbor of nnew, called nn, upon thereceipt of a frame R sent by nnew. At this time, nn verifieswhether it is the master of link (nn,nnew), that is, the nodethat should run the heuristic. Otherwise, nn simply replieswith a frame R, which will result in nnew being chosen asmaster and running the heuristic. The master selectionshould be undertaken through operations on the identitiesof both nodes involved that guarantee a statistical equilib-rium in the choice of each node as master. The link underconsideration is referred to as the working link, that is,the one which undertakes an optimization attempt at thetime.

The heuristic itself corresponds basically to the follow-ing steps:

1. Discover which neighbor links belong to sets Ip or Dp.The union of these sets represents the set T.

2. Calculate a decision factor cd for each link in T.3. Sort the links in T in decreasing order of decision

factors.4. Try to optimize for all links in T.5. For every successful optimization, send a frame to the

master node of the neighbor link requiring transmissionpower reconfigurations.

6. Both nodes of the link adjust their CSth according to thenew set of dependent links (Dd [ DP).

The first step is performed based on the local knowl-edge that each node has about the topology, especiallywith respect to equivalent distances to each of itsneighbors. This information allows the node to obtainknowledge about existing neighbor links, and their corre-sponding distances (a,b,c,d,e, f) to the working link. Thus,the master node of the working link can test whether theconditions of independence are met, and identify whichneighbor links are in the set T (i.e., those that are neitherin Id nor in Dd).

The decision factor mentioned in Item 2 is used as anordering criterion of links in T in which attempts at optimi-zation are made. The importance of the ordering sequenceof attempts lies in its impact on the total amount of suc-cessful optimizations.

Before executing the heuristic, the nodes that composethe working link may operate with any pair of discretepower levels provided that they are sufficient to maintainthe communication between them. Thus, the initial localsolution set consists of all those pairs of power levels.

After the first successful optimization with a neighborlink, the local solution power set of the working link willbe reduced, only the pairs of power levels that maintainthe condition of independence with the neighbor linkremaining. The next attempt at optimization will take thisinto consideration by testing only the power levels that arepart of the local solution set. New successful optimizationstend to further reduce the local solution set, often tendingto a single pair of power levels to be used in the working link.

Thus, it is clear that if the first optimizations are veryrestrictive, the local solution set can be quickly reduced,preventing more links becoming independent of the

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8 12 16 20 24

Heu

ristic

effi

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cy (%

)


Preliminary heuristicILH

Fig. 8. Comparison of heuristics’ performance.


working link. On the other hand, finding the ordering se-quence that allows the largest number of optimizations re-quires the covering of all possibilities, and totaling theremaining local solution set for each one of them. Thiswould make the mechanism become very complex.

So, the decision factor was defined in a way that linksare sorted according to a greater likelihood of maximizingthe resulting local solution set at the end of eachoptimization.

As previously exposed, the regions of potential solu-tions in each of the Cartesian planes (Ps1 � Ps2 , Pr1 � Ps2 ,Ps1 � Pr2 and Pr1 � Pr2 ) are the areas between two lines, lo-cated in each plane. More specifically, they correspond tothe operating points (i.e. pairs of power levels) that arewithin the defined areas.

Thus, given one of the Cartesian planes, as the angle be-tween these lines becomes larger, the number of operatingpoints in the solution region tends to be bigger. It is note-worthy that it is just a tendency, since, depending on therelative position between the lines and the possible oper-ating points, one can envision situations in which largerangles lead to smaller sets of solutions.

Another point to be stressed is that the useful angleshould be considered, that is, the one which defines an areawithin the polygon of solutions imposed by the limits ofpower. So, the angle to be considered is only the one be-tween the limiting lines. These lines are y ¼ Pmin

Pmaxx and

y ¼ PmaxPmin

x.

Finally, for each neighbor link candidate for optimiza-tion, there are four Cartesian planes, each one with twolines, totaling four useful angles. Thus, the decision factoris defined as the smallest one of the four calculated angles,which represents a major constraint.

After determining the decision factor of each neighborlink candidate for optimization, attempts will be made indescending order of their factors. Optimizing first thoselinks whose angles that define the solution region are lar-ger tends to result in the desired effect, that is, the largestnumber of independent links at the end of the optimizingprocess.

Another point that needs to be noted concerns theselection of the solution to be adopted in the case of a suc-cessful optimization. At the end of an attempt, it will oftenbe possible to achieve independence between links withmultiple solutions, each one of the form ðPs1 ; Pr1 ; Ps2 ; Pr2 Þ.

Again, the choice should be the solution that allowsgreater flexibility, so new optimizations may also besuccessful. The objective remains to maximize thelocal solution power levels set. Each local solution hasthe form ðPs1 ; Pr1 Þ. We choose, therefore, the solutionðPs1 ; Pr1 ; Ps2 ; Pr2 Þ, such that the pair of remote power levelsðPs2 ; Pr2 Þ makes valid solutions with the greatest possiblenumber of pairs of local power levels ðPs1 ; Pr1 Þ.

6. Simulation results

As in the preliminary evaluation, we used a modifiedversion of ns-2 version 2.31 to compute the accumulatedinterference. The simulation parameters were kept thesame, except when explicitly highlighted.

We generated a set of 300 random topologies for eachone of the maximum neighborhood degrees: 8, 12, 16, 20and 24. Each plotted point is an average of the results inthese 300 scenarios, with confidence intervals of 95%.Two propagation models were used, log-distance andtwo-ray ground.

Initially, we evaluated the performance of the new heu-ristic (ILH) in relation to the preliminary one. The metricused was the ability to convert power-based dependentlinks in power-based independent links. In the beginningof each simulation, we counted the number of elementsof the set Dp of each one of the links. After the executionof the heuristic, we counted the number of links that wereconverted to independent, expressing the result as a per-centage of that initial number.

Fig. 8 shows the performance comparison of the twoheuristics. We can see that ILH is almost three times moreefficient than the preliminary method. The superiority ismaintained even in the situation of increased density ofthe network, when it becomes more and more difficult tomake links independent of each other.

Fig. 9 shows that ILH is able to offer higher aggregatedthroughput in relation to the use of fixed Pt and CSth.The log-distance propagation model was used in thesesimulations.

In Fig. 10, one can see the performance of the evaluatedmechanisms in relation to two factors: the path loss expo-nent a and the used frame size.

As a increases, the power received by a node, at thesame distance from a neighbor, will be lower, leading toa reduced level of accumulated interference. As a result,all methods show an improvement in aggregatedthroughput, and the ILH mechanism presents the bestperformance.

Frame size also significantly impacts the throughput, aspresented in Fig. 10b.

We also evaluated the performance of the mechanismswhen channel propagation conditions follow the two-rayground propagation model. This model basically dividesthe communication range into two zones, in which the re-ceived power is calculated in different ways. Up to a cer-tain distance from the transmitter, normally known as

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Fig. 9. Independent links heuristics.

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Fig. 10. Path loss exponent and frame size.

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Fig. 11. Fading channel – two-ray ground propagation model.


cross-over distance, it is assumed that the propagation oc-curs in free space, with the received power being calcu-lated with a path loss exponent equal to 2.

If the receiver is beyond the cross-over distance, the re-ceived power is calculated with a path loss exponent of 4,

since the model considers the influence of a reflected waveon the ground.

This is an important evaluation, since ILH is based onthe log-distance propagation model, for which the pathloss exponent is held constant for any distance from thetransmitter.

Fig. 11a shows the results obtained with the two-rayground model. We have also added one more recently pro-posed mechanism to the comparison, the one presented in[14]. This method is based on the two-ray ground model,presenting a different behavior in relation to neighborsthat are located within the cross-over distance. For thisreason, it was used only in this specific evaluation.

We can see that the ILH presents the best aggregatedthroughput. Fig. 11b shows the same evaluation, but con-sidering a fading channel, in which instantaneous varia-tions on the received power are calculated by the Ricedistribution.

We used the same evaluation shown in Fig. 11a to mea-sure two key factors for spatial reuse mechanisms: thenumber of collisions in the receivers and the time forwhich transmitters receive power above their carrier sens-ing threshold CSth.

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As previously stated, the balance between these factorsis essential for a spatial reuse scheme to be successful. Wemeasured the average number of collisions per second ateach receiver and the average percentage of the total sim-

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Fig. 13. Performance comparison w

ulation time that the transmitters were unable to transmit,because they were sensing a carrier in the channel.

The results are shown in Fig. 12. Thus, we can see thatILH gets a higher aggregated throughput by being moreaggressive than the other methods, allowing a higher fre-quency of collisions, which is balanced by a smaller aver-age back-off time on the transmitters.

Finally, we compared the performance of ILH withthe brute force solution for the problem of maximizationof independent links. In order to make the solution com-putationally possible, we used 10 random scenarios com-posed each by 8 TX–RX pairs, totaling 16 terminals, andthe number of allowed power levels was fixed in three,i.e. 13, 16 and 19 dBm. Each TX terminal is randomlydistributed in a squared area of size 1 km � 1 km andits corresponding RX is randomly positioned within itscommunication range given by the lowest power level.This way, the TX terminal can make use of any of thethree possible power levels.

For each of the 10 sets, every possible combination ofpower levels with respect to the 16 terminals was tested.This way, it was possible to determine, for each scenario,a combination of power levels that generates the highernumber of pairs of independent links. This exhaustivesearch was performed in approximately 24 h on a comput-ing cluster composed by 72 cores. With such predeter-mined combinations, it was possible to set each terminalwith its respective transmission power level and adjustits CSth similarly to the way described in the ILH heuristic.The only difference is that actual distances between termi-nals are used instead of estimated distances. Thus, a net-work configuration for each scenario is obtained foroptimal operation according to the criterion of maximizingthe number of independent links pairs based on a globalknowledge. In some scenarios, the brute force encountereddifferent combinations that provide the same maximumnumber of independent links, thus a tie-breaking criterionwas defined in order to select the final solution. The crite-rion was the sum of the power levels of all terminals, i.e.the combination with the minimum sum of powers wasselected.

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ith the global optimization.


The results of the comparison are shown in Fig. 13a andb in scenarios without and with fading, respectively. Thecurves show the aggregate throughput as function of thetraffic rate at each TX terminal for the Global Optimiza-tion, ILH, Fixed PtMin and CSth, Fixed PtMax and

CSth, Gurses and Boutaba (2009) solutions. Each pointin the curves corresponds to the average aggregatethroughput calculated from 10 simulation runs, and theconfidence intervals are of 95%. For the Fixed PtMin

and CSth and Fixed PtMax and CSth solutions, the CSth

was set to �99 dBm. One can notice that the ILH has a per-formance that is very close or even better for some inter-mediate traffic rate than the one obtained with the globaladjustment. The main reason for the ILH outperforms theglobal optimization solution is related to the fact that max-imizing the number of independent links is not always thesolution that leads to the optimal capacity. Furthermore, itshould be highlighted that the region in which the ILH

outperforms the Global Optimization is not in asymp-totic load conditions, for which the mechanism was de-signed. Despite of suffering larger performancedegradation under fading than the other solutions, ILH stilloutperforms them.

7. Conclusions

In this work, the problem of increasing spatial reuse inCSMA-based networks is explored through the concept ofmutually independent links. We derived the conditionsthat define independence between links and demonstratethe possibility of increasing the number of independentlinks through power control and the use of asymmetriclinks. It is also shown that the adjustment of CSth is a cru-cial component in converting independent links into effec-tive spatial reuse.

Aiming to increase the number of independent linkpairs, we presented an original heuristic for Pt and CSth

adjustments. By extensive evaluation, we demonstratedthat the performance achieved by the proposed mecha-nism achieves a higher aggregate throughput when com-pared to fixed Pt and CSth and other methods in therecent literature.

References

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Alexandre A. Pires received the Ph.D. degreein Electrical Engineering from UniversidadeFederal do Rio de Janeiro (UFRJ) in 2012. Hereceived his B.Sc. degree in Electronics Engi-neering from Instituto Militar de Engenhariaand his M.Sc. in Electrical Engineering fromUFRJ. His research interests focus on wirelessnetworks and distributed algorithms.

José F. de Rezende received the B.Sc. andM.Sc. degrees in Electronics Engineering fromUniversidade Federal do Rio de Janeiro (UFRJ)in 1988 and 1991, respectively. He receivedthe Ph.D. degree in Computer Science fromUniversité Pierre et Marie Curie in 1997,where he was an associate researcher duringthat year. Since 1998 he has been an associateprofessor at UFRJ. His research interests are indistributed multimedia applications, multi-peer communication, high speed and mobilenetworks, and quality of service in the Inter-

net. He has served in the editorial board of Ad Hoc Networks from Else-vier.

independent links: a new approach to increase spatial reuse in wireless networks

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