Ad Hoc Networks 11 (2013) 2273–2287

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Ad Hoc Networks

journal homepage: www.elsevier .com/locate /adhoc

Optimum node placement in wireless opportunistic routingnetworks

1570-8705/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.adhoc.2013.05.010

⇑ Corresponding author. Tel.: + 34 3 4016798.E-mail addresses: llorenc@ac.upc.es (L. Cerdà-Alabern), amir@ac.upc.

edu (A. Darehshoorzadeh), vpla@dcom.upv.es (V. Pla).URL: http://personals.ac.upc.edu/llorenc (L. Cerdà-Alabern).

Llorenç Cerdà-Alabern a,⇑, Amir Darehshoorzadeh a, Vicent Pla b

a Univ. Politècnica de Catalunya, Dept. d’Arquitectura de Computadors, c/Jordi Girona, 1-3, Barcelona, Spainb Univ. Politècnica de València, Communications Dep., València, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 November 2012Received in revised form 16 May 2013Accepted 24 May 2013Available online 4 June 2013

Keywords:Wireless networksOpportunistic routingMaximum performanceAnalytical model

In recent years there has been a growing interest in opportunistic routing as a way toincrease the capacity of wireless networks by exploiting its broadcast nature. In contrastto traditional uni-path routing, in opportunistic routing the nodes overhearing neighbor’stransmissions can become candidates to forward the packets towards the destination.

In this paper we address the question: What is the maximum performance that can beobtained using opportunistic routing? To answer this question we use an analytical modelthat allows to compute the optimal position of the nodes, such that the progress towardsthe destination is maximized. We use this model to compute bounds to the minimumexpected number of transmissions that can be achieved in a network using opportunisticrouting.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Multi-hop wireless networks (MWN) [1,2] have becomea very active research field during the last years. Routing inMWN is more challenging than in wired networks becauseof two fundamental differences. The first difference is theheterogeneous characteristics of wireless links. As a conse-quence, there can be significant differences in packet deliv-ery probabilities across the links of a MWN network. Thesecond difference is the broadcast nature of wireless trans-missions [3]. Unlike wired networks, where links are typi-cally point to point, when a node transmits a packet in awireless network the neighbors of the intended destinationnode can overhear it.

Routing protocols in MWN have traditionally managedthe heterogeneous characteristics of wireless links byusing distributed protocols that at each node choose the

best link for every destination (referred to as next-hop).Once all next-hops have been chosen, all packets betweena source and a destination follow the same path. This moti-vates the name of uni-path routing for such type ofprotocols.

Opportunistic Routing (OR) [4–7] has been proposed toincrease the performance of MWNs by taking advantage ofthe broadcast nature of the wireless medium. In OR, in-stead of preselecting a single node to be the next-hop asa forwarder for a packet, an ordered set of nodes (referredto as candidates) is selected as the potential next-hop for-warders. Thus, the source can use multiple potential pathsto deliver the packets to the destination. After the trans-mission of a packet, all the candidates that successfully re-ceive it will coordinate among themselves to determinewhich one will actually forward it; the others will simplydiscard the packet.

Previous research in this field has principally concen-trated on proposing and evaluating different candidateselection mechanisms and routing protocols. Then the per-formance of the proposed mechanism is compared withthat of the baseline scenario of traditional uni-path routing

2274 L. Cerdà-Alabern et al. / Ad Hoc Networks 11 (2013) 2273–2287

or with that of other OR mechanisms. Performance is gen-erally measured in terms of the expected number of trans-missions from the source to the destination (which, like in[8], we refer to as expected any-path transmission, EAX). Tothe best of our knowledge, all studies assume that the net-work topology is given and the evaluations and compari-sons are done over that topology, or a variety of them. Inthis paper we follow a different approach. Here, we inves-tigate the maximum gain that can be obtained by using OR.For this purpose, we study the optimal position of thenodes acting as candidates. The obtained insight is then ap-plied to propose practical design rules for multihop wire-less networks.

In the first part of the paper, we address the question:What is the maximum gain that can be obtained usingOR? We shall refer to gain as the relative difference ofthe expected number of transmissions required betweenOR and the baseline uni-path routing scenario. More spe-cifically, we focus on a scenario where the maximum num-ber of candidates per node is limited. To answer the formerquestion we use a network where the nodes are optimallylocated so that at each transmission the progress towardsthe destination is maximized. To do so, we shall assumethat we have a formula for the delivery probability, p(d),between nodes at a distance d. For the sake of simplicitywe shall assume the same function p(d) for any pair ofnodes. The model, however, could be generalized assuminga different function for every link. In our analysis p(d) willbe given by the radio propagation model. An expression tocompute the expected number of transmissions in OR hasbeen obtained by several authors (e.g., [8–10]). Thatexpression is recursive and has a non-linear dependenceon the delivery probability between the nodes. We shalluse this formula for comparison purposes, and in the restof the paper we shall refer to it as the EAX recursive formula.

Due to the complex form of the EAX recursive formula,even if p(d) is known, it may not give a feasible way to de-rive the optimal position, and thus, the maximum OR gain.We solve this problem by computing the optimal positionof the nodes maximizing the progress towards the destina-tion when OR is used. The same principle was in used in[11] in the context of uni-path routing. We show that thisapproach allows deriving a set of equations that can besolved numerically in order to compute the optimal dis-tances between a node and its candidates. We refer tothese distances as the maximum progress distances (MPD).Additionally, by maximizing the progress towards the des-tination we establish a tight lower bound on the expectednumber of transmissions in OR, and thus, an upper boundon the OR gain.

Studying the optimal position of the candidates is theo-retically insightful, and also allow us to assess the maxi-mum performance achievable with OR and to establishlower and upper bounds. Moreover, from a practical per-spective there are also a number of implications that arederived from the results in our study with application,for instance, to network deployment and design of routingprotocols.

Obviously, to apply our results to deploy future wirelessmulti-hop network it is necessary to be able to decide, tosome extent, where the nodes are to be placed. However,

it is not necessary a complete freedom to place nodes(i.e., it is not necessary to place the node at the very precisepositions given by the MPDs) since our results reveal arather low sensitivity of the performance with respect todeviation from the optimal positions. Therefore, althoughnot optimal, a network design based on the MDPs andthe existing node positioning constraints can still providea good design.

Furthermore, in a practical wireless multi-hop networkdesign, the total number of nodes to be deployed is also avery important practical issue. Even if a limited number ofcandidates per node is considered, building a network inwhich each node has neighbors located precisely at theMPD would imply, in general, a network with a very largeor infinite number of nodes. By applying the insight gainedfrom the analysis of the MPDs, we propose practical designrules based on regular one- and two-dimensional topolo-gies. The proposed solutions exhibit a performance thatis very close to the optimal one, whereas the number ofnodes is greatly reduced.

Another important area of potential application of theresults in this paper is that of routing protocols for wirelessmulti-hop networks. The MPDs can be used for routingprotocols in OR that are based in the geographic routingprinciple. In geographic routing, the position of the nodesand that of the destination is used in the routing decisions.In wireless multi-hop networks, it is considered that rout-ing protocols that are not based on geographic routing arenot scalable [12]. Uni-path routing schemes based on geo-graphic routing often use a greedy approach by forwardingdata packets to the neighbor geographically closest to thedestination [13]. This principle has also been applied inOR [14], where the distance from the candidates that havereceived that packet and the destination is used to decidewhich candidate will forward the packet onto the nexthop. However, a basic premise in this paper is that, in orderto keep within practical limits the overhead due to candi-date coordination, the number of candidates per nodemust be limited and rather small. The MPDs can be usedto select the appropriate candidate for each node.

Finally, from a practical perspective it also worth men-tioning that our results show that instead of needing thevalues of the MDPs, which depend on the propagationcharacteristics, and the distances to the neighboring nodes,a node can select its candidate set by simply estimating thedelivery probabilities to its neighbors.

The remainder of this paper is organized as follows. InSection 2 we study the positions of the candidates thatyield the maximum progress per transmission, and the re-sults are used in Section 3 to derive bounds for the ex-pected number of transmission necessary to reach thedestination. Section 4 introduces the propagation modelthat we have used in the numerical experiments presentedin Section 5. In Section 6 we describe a procedure to con-struct a network using the maximum progress distances.The expected number of transmissions in this networkyields a tight upper bound of the optimal value. Section 7compares the MPD with the optimal distances. In Section 8a practical method is proposed to achieve a performanceclose the optimum one with a much lower number ofnodes. Section 9 investigates this method when applied

L. Cerdà-Alabern et al. / Ad Hoc Networks 11 (2013) 2273–2287 2275

to a two-dimensional grid. Finally, Section 10 surveys therelated work and concluding remarks are given inSection 11.

2. Optimal positions of candidates

We study the position of the candidates in order tomaximize the progress towards the destination. The ingre-dients of our model are: the maximum number of candi-dates per node n; and the formula for the deliveryprobability at a distance d, p(d), which we suppose to bethe same for all the nodes. Assume that the destination isfar from a generic test node for whose candidates are look-ing for. Clearly, the optimum candidates will be locatedover the segment between the test node and the destina-tion (see Fig. 1).

Let {c1, c2, . . . , cn} be the ordered set of candidates of thegeneric test node (cn has the highest priority, and c1 theleast one), and let di be the distance from the test nodeto the candidate ci (see Fig. 1). We assume that a coordina-tion protocol exist among the candidates, such that thehighest priority candidate receiving the packet will for-ward the packet (if it is not the destination), while theother nodes will simply discard it. Assume that p(di) isthe delivery probability from the test node to the candidateci, and let Dn be the random variable equal to the distancereached after one transmission shot. Clearly,

E½Dn� ¼ dn pðdnÞ þ dn�1 pðdn�1Þ ð1� pðdnÞÞ þ . . .

þ d1 pðd1Þ �n

i¼2ð1� pðdiÞÞ: ð1Þ

That is, the packet will progress a distance dn if the highestpriority candidate cn receives it, or a distance di,i = 1, . . . , n � 1 if ci receives it, and no higher priority candi-dates receive the packet. A key observation is that takingthe common factor (1 � p(dn)) in Eq. (1) we get the recur-sive equation:

E½Dn� ¼ dn pðdnÞ þ ð1� pðdnÞÞE½Dn�1�¼ E½Dn�1� þ ðdn � E½Dn�1�Þ pðdnÞ: ð2Þ

We are interested in looking for the value dn that max-imizes Eq. (2). Note that this value also maximizes thefunction

f ðxÞ ¼ ðx� aÞ pðxÞ ð3Þ

where a = E[Dn�1]. Notice that f(a) = 0 and f(x) is increasingin the neighborhood of a. We shall assume that the deliv-ery probability p(x) is differentiable and limx?1x p(x) = 0.Note that this last condition is necessary for the expectedvalue of the distance reachable in a transmission shot to

Fig. 1. Test node and its candidates.

be finite. Since the delivery probability is p(x) P 0, andmust be monotonically decreasing (the higher is the dis-tance, the lower the delivery probability), we conclude that(3) must have a point x⁄ 2 (a,1) for which f is increasingfor x 2 (a,x⁄), and decreasing for x 2 (x⁄,1). In order words,the function f(x) is quasi-concave in x 2 (a,1), having a un-ique critical point equal to its global maximum in thisinterval. Additionally, since di�1 < di (i = 2, . . . , n), it isE[Dn�1] < dn�1 < dn. Thus, we can reduce the optimizationdomain to dn 2 (dn�1,1). Under these conditions we cancompute the distances di(i = 1, . . . , n) that maximize (2)by solving:

@E½Di�@di

¼ 0; di 2 ðdi�1; 1Þ; i ¼ 1; . . . ;n

which gives the set of equations:

pðdiÞ þ ðdi � E½Di�1�Þ p0ðdiÞ ¼ 0; di 2 ðdi�1;1Þ;i ¼ 1; . . . ;n ð4Þ

where E[D0] = 0 and d0 = 0. Note that by using Eq. (4) wecan compute d1 from p(d1) + p0(d1) d1 = 0. Then, substitut-ing in (2) we have E[D1] = d1p(d1), which can be used tocompute d2 using (4) and so on until dn. We shall refer tothese distances as the MPD. In the sequel we shall referto them as d1, . . . , dn, and denote the expected number oftransmissions given by Eq. (2) using these distances asE D�n� �

. Note also that a consequence of Eq. (4) is that theMPD for the already existing candidates do not change ifwe decide to add a new candidate to the candidate set.

Note that the previous model, could be generalized byassuming a different delivery probability for every candi-date of the source s: psi(x). Of course, in this case theMPD to the candidates could be different for every node.

3. Maximum performance of OR

In this section we investigate the performance of OR interms of the expected number of transmissions to send apacket from the source to the destination, i.e. in terms ofthe EAX. To do so, we define sn to be the random variableequal to the number of transmissions required to send apacket from the source to the destination using at most ncandidates per node. We shall use the notation EAXn = E[sn]to refer to the value of EAX in a network with this con-straint. Note that n = 1 correspond to traditional uni-pathrouting. We are thus interested in obtaining bounds toEAXn.

3.1. Infinite number of candidates

We first derive a result that will be useful in the boundsderived afterwards. Assume an infinitely dense networkwhere the nodes can choose an infinite number of candi-dates. Assume further that there is not limitation on theminimum delivery probability that living links can have.Let s1 be the random variable equal to the number oftransmissions required to send a packet from the sourceto the destination in such network. With these assump-tions, some node as close to the destination as we wantcan receive the packet with probability 1 (we can choose

2276 L. Cerdà-Alabern et al. / Ad Hoc Networks 11 (2013) 2273–2287

a region arbitrarily close to the destination that contains aninfinite number of candidates). Therefore, if the destinationdoes not receive the packet after it is firstly transmitted bythe source, by the previous reasoning, one candidate arbi-trarily close to it will have received it, and will relay it tothe destination with just one more transmission. Thus, inthis case, s1 = 2. Let D be the distance between the sourceand the destination. From the previous discussion we con-clude that:

E½s1� ¼ pðDÞ þ 2ð1� pðDÞÞ ¼ 2� pðDÞ: ð5Þ

3.2. A lower bound for the expected number of transmissions

Assume a network with n candidates per node. SinceE½D�n� computed in Section 2 using the MPD given by Eq.(4) is the maximum progress towards the destination afterevery transmission shot, we have that the expected num-ber of transmissions to send a packet from the source tothe destination (E[sn]) is lower bounded as follows

E½sn�PD

E D�n� � ; ð6Þ

where D is the distance between the source and thedestination.

The bound given by Eq. (6) will be tight as long as thedistance D is large compared with dn, and the nodes are lo-cated at the MPD. Clearly, for those nodes that are closerthan dn to the destination, the optimal positions cannotbe given by the MPD. In this case the highest priority can-didate will be the destination. Thus, the distance of thehighest priority candidate will be the distance to the desti-nation, and the optimal position of the other candidatesshould be computed taking the distances that minimizethe EAX. In fact, this ‘‘boundary effect’’ will propagate tothe position of the other nodes between the source andthe destination, and their optimal positions may be slightlydifferent than those obtained using the MPD (we shallinvestigate this in Section 7). Nevertheless, the expecteddistance progress after each transmission could not be ashigh as the one obtained using the MPD, which guaranteesthat (6) is a lower bound.

We can use the result obtained for an infinite number ofcandidates to improve the bound given by (6). First, the ex-pected number of transmissions cannot be less that the va-lue given by Eq. (5). Therefore, we have that:

E½sn�P max 2� pðDÞ; DE D�n� �

!: ð7Þ

The bound given by (7) can still be improved as we ex-plain next. As we said before, when the nodes are closerthan dn to the destination, the position of the nodes cannotbe the MPD. Therefore, using E D�n

� �as the progress in this

region may be a coarse approximation. To estimate theprogress in this region we note that before the packetreaches the destination, at least one node in the interval[D � dn,D) will receive it, because the furthest candidateof any node is at a distance dn. We shall refer to the firstnode in this interval that receives the packet as v(x), wherex is the distance from this node to the destination. Now,

the number of transmissions from the source to v(x) canbe lower bounded by ðD� xÞ=E D�n

� �(i.e. assuming the max-

imum progress), and the number of transmissions fromv(x) to the destination can be lower bounded assumingan infinite number of candidates between v(x) and the des-tination (Eq. (5)). Adding both terms we have E½snjvðxÞ�P ðD� xÞ=E D�n

� �þ 2� pðxÞ ¼ D=E D�n

� �þ 2� pðxÞ � x=E D�n

� �.

Thus, if we want a lower bound we must take x that min-imizes E[snjv(x)] in the interval x 2 (0, dn].

Summing up, we have that:

EAXn¼E½sn�

Pmax 2�pðDÞ; DE D�n� �þ inf

x2ð0;dn �2�pðxÞ� x

E D�n� �

( ) !:

ð8Þ

3.3. An upper bound for the gain

Let us denote by s�n the number of transmissions whenthe candidates are optimally placed. In order to measurethe improvement that can be reached using OR we definethe gain (Gn) as the relative difference of the expected num-ber of transmissions required with the OR with n candidatesðE s�n� �Þ, with respect to the uni-path routing case. Note that

OR with only 1 candidate per node is equivalent to uni-pathrouting. Therefore, we shall refer to the expected number oftransmissions with uni-path routing as E s�n

� �, and thus:

Gn ¼E s�n� �

� E s�n� �

E s�n� � ¼ 1�

E s�n� �

E s�n� � : ð9Þ

Using the same intuition as in (6) we can write

DE D�1� � 6 E s�n

� �6dD=d1ed1

E D�1� � ð10Þ

and using the lower bound for E s�n� �

and the upper boundfor E s�n

� �it follows from (9) that

Gn 6 1� D=d1

dD=d1eE½D�1�E D�n� � : ð11Þ

4. Propagation model

In order to model the delivery probabilities we will as-sume that the channel impairments are characterized by ashadowing propagation model. This model includes deter-ministic path loss and large scale fading. It is a standardpropagation model when considering the network capacity[15]. More specifically, the power received at a distanced(Pr(d)), in terms of the transmitted power (Pt) is given by:

PrðdÞjdB ¼ 10 log10Pt Gt Gr k2

Lð4pÞ2db

!þ XdB ð12Þ

where Gt and Gr are the transmission and reception anten-na gains respectively, L is a system loss, k is the signalwavelength (c/f, with c = 3 � 108 m/s), b is a path loss expo-nent, and XdB is a Gaussian random variable with zeromean and standard deviation rdB.

L. Cerdà-Alabern et al. / Ad Hoc Networks 11 (2013) 2273–2287 2277

Packets are correctly delivered if the received power isgreater than or equal to RXThresh. Note that we shallnot consider collisions in our model. Thus, the deliveryprobability at a distance d(p(d)) is given by:

pðdÞ ¼ ProbðPrðdÞjdB P 10 log10ðRXThreshÞÞ

¼ Q1

rdB10log10

RXThreshLð4pÞ2db

PtGtGrk2

! !ð13Þ

where QðzÞ ¼ 1ffiffiffiffi2pp

R1z e�y2=2dy.

In our numerical experiments we have set the modelparameters to the default values used by the network sim-ulator (ns-2) [16], given in Table 1. Table 2 shows typicalvalues for b and rdB.

Fig. 2 depicts the delivery probability at a varying dis-tance, for three values of the path loss exponent (b) anda standard deviation rdB = 6 dBs. We shall use these valuesin the numerical results presented in later sections.

Distance [m]

Del

iver

y pr

obab

ility

0.0

0.2

0.4

0.6

0.8

1.0

2.73.03.3

0 50 100 150 200

Fig. 2. Delivery probability versus distance for a standard deviationrdB = 6 dBs.

Table 1Default ns values for the shadowing propagation model.

Parameter Value

Pt 0.28183815 WRXThresh 3.652 � 10�10 WGt, Gr, L 1f 914 MHz

Table 2Typical values for b and rdB.

Environment b rdB

Outdoor Free space 2 4–12Urban 2.7–5

Office Line-of-sight 1.6–1.8 7–9.6Obstructed 4–6

5. Numerical results

We now give some numerical examples of the formulasderived in the previous sections. We shall assume that thedelivery probability p(d) is given by Eq. (13). Substitutingp(d) in the Eqs. (4) and solving them numerically we obtainthe maximum progress distances for the candidates shownin Fig. 3. The three curves correspond to three values of theloss exponent of the propagation model: b = 2.7, b = 3 andb = 3.3. Note that the larger the value of b, the lower thetransmission range of the nodes is, and thus, the shorterthe distances to the candidates are.

Fig. 4 shows the delivery probabilities obtained for thecorresponding distances shown in Fig. 3. It is interestingthat the probabilities are very similar for all values of b.This fact could be used as a rule of thumb in the selectionof candidates, or for placing the nodes in the back-haul of amesh network.

Finally, Fig. 5 depicts the lower bound of the expectednumber of transmissions (Eq. (8)) for a distanceD = 300 m between the source and the destination, andFig. 6 shows the corresponding upper bound to the gain(Eq. (11)). As we shall see in Section 6, the lower bound gi-ven by Eq. (8) is very tight. Consequently, the gains thatcan be obtained using OR are close to the upper bound de-picted in Fig. 6. These figures show that the highest gain in-crease occurs when we move from 1 to 2 candidates(approximately 30% of gain). After that, the gain increasesup to approximately 60% with 10 candidates. However,implementing an OR protocol with a high number of can-didates is difficult, and possibly will introduce large signal-ing overhead and duplicated transmissions that wouldprevent to reach such large gains. This motivates thatselecting a maximum number of candidates per node equalto 2, or maybe 3, is possibly a sensible choice.

6. Quasi Optimal OR network

In this section we compute an upper bound for the ex-pected number of transmissions, EAX. This bound is ob-tained by building a network where the candidates are

Candidate

Max

imum

pro

gres

s di

stan

ce [

m]

50

100

150

200

250

300

2.73

3.3

1 2 3 4 5 6 7 8 9 10

Fig. 3. MPD for the candidates.

Del

iver

y pr

obab

ility

0.1

0.2

0.3

0.4

0.5

0.6

0.7 2.73

3.3

Candidate

1 2 3 4 5 6 7 8 9 10

Fig. 4. Delivery probability to each candidate located at the maximumprogress distances.

Number of candidates per node

Gai

n up

per

boun

d (%

)

20

30

40

50

60

2 3 4 5 6 7 8 9 10

2.73

3.3

Fig. 6. Gain (upper bound).

Number of candidates per node

Exp

ecte

d nu

mbe

r of

tran

s. (

low

erbo

und)

3

4

5

6

7

8

9 2.73

3.3

1 2 3 4 5 6 7 8 9 10

Fig. 5. Expected number of transmissions, EAX (lower bound).

Fig. 7. Quasi Optimal OR network with a maximum of 3 candidates pernode.

2278 L. Cerdà-Alabern et al. / Ad Hoc Networks 11 (2013) 2273–2287

positioned whenever possible using the MPD computed asin Section 2. Once the network is built, its exact EAX can becomputed by using the EAX recursive formula (see e.g.[10]). Note that not all the candidates can be located usingthe MPD distances, since for some nodes the distance tothe destination can be shorter than the distance to the can-didate. For these nodes we will use as candidates the des-tination and its closest neighbors located between thenode and the destination. Since these candidates, at least,are not located at the optimum positions, the expectednumber of transmissions computed for such network willbe an upper bound to the minimum expected number oftransmissions that can be achieved using OR. We shall re-fer to such network as Quasi Optimal OR (QOO) network.

Fig. 7 depicts an example of a network with 3 candi-dates per node build using these rules. The source is vs

and the destination is vd. Nodes 2, 3 and 4 are located atthe MPD from vs: d1, d2 and d3 respectively. Nodes 5 and6 are located at the MPD from node 2: d1 and d2 respec-tively. Since vd is closer from node 2 than d3, vd is takenas the third candidate of node 2. Since node 6 is at a dis-

tance d1 from node 3, and vd is closer from this node thand2 and d3, the candidates of node 3 are nodes 5, 6 and vd.Likewise it is done for the other nodes.

Fig. 8 shows the expected number of transmissionsvarying the distance D between the source and the destina-tion for a QOO network build as explained before. Thecurves shown in the figure have been obtained using amaximum number of candidates per node equal to 1, 2, 3and 5 (cfr. the numbers in the legend). Fig. 9 shows thenumber of nodes that resulted in the QOO networks usedto obtain the corresponding values of Fig. 8. In Fig. 8 wehave also added the lower bounds obtained using Eq. (8)(thin lines), and the lower bound for an infinite numberof candidates given by Eq. (5) (dashed line).

The delivery probability of the links (p(d)) has been ob-tained using the shadowing model (Eq. (13)) with a pathloss exponent b = 2.7. The expected number of transmis-sions has been obtained using the Markov chain that wehave proposed in [10].

Fig. 8 confirms that the lower bounds of the expectednumber of transmissions obtained with Eq. (8), especiallywhen n > 1, are very tight, since they are very close tothe upper bound obtained with the QOO network.

D [m]

Opt

imum

can

dida

te d

ista

nce

[m]

0

50

100

150

200

0

50

100

150

200

0 100 200 300 400 500

cand.12

Fig. 10. Optimum distances of the candidates ðd�i Þ with a maximum ofn = 1 and n = 2 candidates per node (top and bottom respectively). Thedashed lines are the MPD (di).

Exp

ecte

d nu

mbe

r of

tran

smis

sion

s

1

2

3

4

5

6

71235

D [m]

0 100 200 300 400 500

Fig. 8. Lower and upper bounds (thin and thick lines respectively) of theminimum expected number of transmissions achievable with OR forn = 1, 2, 3 and 5 maximum number of candidates per node. The dashedline corresponds to infinite number of candidates.

L. Cerdà-Alabern et al. / Ad Hoc Networks 11 (2013) 2273–2287 2279

Furthermore, this result seems to indicate that the MPD arevery close to the optimum distances. We shall investigatethis in the next section. Note that the discontinuities ofthe upper bound occur at the distances where a new nodeis added to the QOO network. For instance in the scenariowith 1 candidate, which occurs when the distance betweenthe source and the destination (D) is a multiple of d1.

7. Optimal distance to candidates in a finite network

In the previous sections the MPD have been obtainedand used to derive bounds, which are rather accurateapproximations as well, of the performance of OR mea-sured by the mean number of transmission required toreach the destination.

For a network of finite length (D <1), the optimal dis-tances of the candidates—in the sense of minimizing themean number of transmissions required to reach the desti-nation—are more complex to obtain and in general maynot coincide with the MPD.

Fig. 9. Number of nodes of the Quasi Optimal OR networks used in Fig. 8.

In this section, we use a numerical approximation tofind the optimal distances in a finite length network. Thisis used to confirm some of the intuitions that have beenapplied previously, and provide a further insight into theoptimal distances problem.

Let Vn(x) be the minimum mean number of transmis-sions required to reach the destination that is at distancex from the source node, when a maximum of n candidatesper node is used. We can write

VnðxÞ¼ minx1<���<xn

1

1�Yn

i¼1

qðxiÞ� 1þpðxnÞVnðx�xnÞþ . . .þ

Yn

i¼2qðxiÞpðx1ÞVnðx�x1Þ

� �8>>>><>>>>:

9>>>>=>>>>;

ð14Þ

where q(d) = 1 � p(d) and Vn(0) = 0. If the number of nodesbetween the source and the destination is less than n, thenthe destination and the intermediate nodes are taken ascandidates. We shall refer as d�i to the optimal distancesxi that minimize Eq. (14).

Fig. 11. Expected number of transmissions obtained with the optimumdistances of Fig. 10, and its upper and lower bounds.

Fig. 12. MPD of the candidates (thin lines) and its d1/4 (dashed line) andd1/2 (dot-dashed line) approximations.

2280 L. Cerdà-Alabern et al. / Ad Hoc Networks 11 (2013) 2273–2287

We have solved the optimization problem of (14) in anapproximate fashion by considering a discrete network (afinite number of nodes are evenly distributed betweensource and destination) and then performing an exhaustiveoptimum search. The network density, i.e., the number ofnodes, has been increased until the minimum does notvary significantly. Obviously, the exhaustive search be-comes unfeasible as the maximum number of candidates,n, or the network size, D, grow. For this reason, we havelimited this method to a maximum number of candidatesequal to n = 1 and n = 2. Nevertheless, as we will see inthe following, these two scenarios are enough for ourpurposes.

Fig. 10 compares the optimal distances d�i� �

and theMPD (di) as functions of D. The optimum mean numberof transmission obtained for the optimal distances isshown in Fig. 11 along with its corresponding lower andupper bounds.

Number of candidates per node

Exp

ecte

d nu

mbe

r of

tran

smis

sion

s

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10

2.73

3.3

Fig. 13. Expected number of transmissions: (i) lower bound (thin lines),(ii) using the QOO network of Section 6 (solid lines), (iii and iv) using,respectively, its d1/4 (dashed line) and d1/2 (dot-dashed line)approximations.

We observe that the optimal distances converge to themaximum progress distances (d1 � 102 m andd2 � 150 m) when D grows. It is also observed that whilethe MPD of the first candidate (d1) are the same for differ-ent values of D and n, the optimum distances d�1 are differ-ent for n = 1 and for n = 2, although they converge to thesame value (that of the maximum progress distance, d1,as we said before).

Notice that, as expected, when n = 1 and D is a multipleof d1, the optimal distance equals the maximum progressdistance d�1 ¼ d1

� �and the lower bound of V1(D) turns

out to yield an exact value. Also, as it has been predicted,the lower bound for V1(D) is tighter than that for V2(D).On the other hand, in both cases (n = 1,2) when D growsthe shape of Vn(D) tends to be a straight line whose slopeis matched by that of the lower bound, i.e., by 1=E D�n

� �.

Moreover, the shape of V2(D) gets smooth more rapidlythan V1(D) does. A similar observation can be made aboutthe rate of convergence of the optimal distances to theMPD.

8. Node position in linear OR networks

The MPD computed in Section 2 can be of practicalinterest in the design of a static network using OR. Forexample, the back-haul of a mesh network, or the positionof the nodes in a sensor network. A first approach could bethe Quasi Optimal OR Network described in the previoussection. However, for such network the number of nodesincreases nearly exponentially with the distance betweenthe source and the destination, D, as shown in Fig. 9. In thissection we look for positions of the nodes that, being closeto their optimal values, allow reducing the number ofnodes of the network.

Looking at the MPD obtained for different parameters ofthe propagation model (Fig. 3), we can observe thatd2 � d1 + d1/2 and d3 � d1 + d1/2 + d1/4. This suggest that agood compromise is positioning the nodes equally spacedat a distance d1/4, choosing d1 for the first candidate,d̂2 ¼ d1 þ d1=2 for the second, andd̂i ¼ d1 þ d1=2þ ði� 2Þ � d1=4 for the candidates i > 2.Doing this way, a distance D would require a number ofnodes N 6 4 � dD/d1e. If only 2 candidates are going to beused, or if we wish to reduce further the number of nodes,a coarser approach would be positioning the nodes equallyspaced at a distance d1/2, choosing d1 for the first candi-date and d̂i ¼ d1 þ ði� 1Þ � d1=2 for the candidates i > 2.Doing this way, the required number of nodes would beN 6 2 � dD/d1e. We shall refer to these approximations asd1/4 and d1/2 respectively. Fig. 12 shows the MPD com-puted as in Section 2 and its d1/4 and d1/2 approximations.

Figs. 13 and 14 show the sensitivity of the expectednumber of transmissions to the d1/4 and d1/2 approxima-tions. Here we have used the same settings as in Section 5:D = 300 m; b = 2.7, 3 and 3.3. For each value of b Fig. 13shows four curves of the expected number of transmis-sions: (i) the lower bound computed as in Section 3 (notethat these curves are the same than those shown inFig. 5); (ii) using the QOO network of Section 6 (solidlines); and (iii and iv) using its d1/4 (dashed line) and

L. Cerdà-Alabern et al. / Ad Hoc Networks 11 (2013) 2273–2287 2281

d1/2 (dot-dashed line) approximations. Fig. 14 shows thenumber of nodes of the networks that where used to com-pute the expected number of transmissions for the corre-sponding cases (ii–iv) of Fig. 13.

As shown in Fig. 13 the expected number of transmis-sions obtained for the QOO network is very close to thelower bound. Nevertheless, Fig. 14 shows that buildingthe QOO network requires a high number of nodes. Themaximum value is 665 nodes, obtained for b = 3.3 (wherethe nodes’ coverage is the shortest) and 10 candidatesper node. Fig. 13 also shows that the expected number oftransmissions obtained for the d1/4 and d1/2 approxima-tions it is very close to the lower bound too. Only forb = 3.3 and more than 5 candidates per node the differenceis noticeable. However, in Fig. 14 we can see that the num-ber of nodes using the d1/4 and d1/2 approximations isenormously reduced (e.g. it is 27 and 15 nodes respectivelyfor the d1/4 and d1/2 approximations in the same scenariofor which 665 nodes are used with the QOO network).

We conclude that choosing the position of the 2 candi-dates closest to the sender near to their optimal positions,is the most critical in order to minimize the expected num-ber of transmissions. Consequently, what we have calledd1/2 approximation may be a sensible rule of thumb inthe design of the node positions in a linear static networkusing OR routing. In summary, the d1/2 approximationyields a good trade-off between performance and numberof nodes when we want to deploy a linear network.

9. Node positioning in two-dimensional multihopnetworks

In this section we investigate whether the d1/2 approx-imation proposed in previous sections can also be used in atwo-dimensional network. We consider a square grid withN = k � k nodes, and denote by d the distance between twoadjacent nodes on the square side (see Fig. 15). Clearly, thebest positions for those candidates to reach destinationslocated in verticals or horizontals lines departing from

Fig. 14. Number of nodes of the networks used in Fig. 13 (i) using theQOO network of Section 6 (solid lines), (ii and iii) using its d1/4 (dashedline) and d1/2 (dot-dashed line) approximations.

the source will be the MPD. Thus, setting d = d1/2 wouldbe a good rule of thumb for these destinations. However,most destinations are not located in these lines. Thus, itis not clear whether another value for d may be a betterchoice.

In order to investigate an appropriate value for d wehave varied its value, while maintaining the number ofnodes of the grid, N. We have assumed that the source Vs

and the destination Vd are located at positions (1,1) and(i,k) where i 2 {1, . . . , k}, respectively (see Fig. 15). Fur-thermore, we refer to D as the distance between Vs andVd. We have used a semi-optimum candidate selectionalgorithm, DPOR [17], to select the candidates of the nodestowards the destination. DPOR is a fast and efficient candi-date selection algorithm whose performance is very closeto the optimum algorithms proposed in the literature,whereas it runs much faster (see [17]). The link deliveryprobabilities between two nodes have been assigned withthe shadowing model with b = 2.7 and rdb = 6.0. We haveassumed that a link between any two nodes exists only ifthe delivery probability between them is greater (or equal)than min.dp = 0.1.

Note that varying d, the value of D and possibly thenodes chosen by the candidate selection algorithm alsochange. Thus, in order to compare the goodness of the dif-ferent values of d, we have defined the metric that we callAverage Distance Progress (ADP) as:

ADPnðd;Vs;VdÞ ¼Dðd;Vs;VdÞ

EAXnðd;Vs;VdÞð15Þ

where n is maximum number of candidates per node,D(d, Vs, Vd) is the distance from the source Vs to the destina-tion Vd, and EAXn(d, Vs, Vd) is its EAXn using the candidateschosen for a given value of d. Note that ADPn(d, Vs, Vd) rep-resents the average number of meters that a packet pro-gress towards Vd at each transmission shot. Therefore,the optimum value of d would maximize Eq. (15).

Fig. 16 shows the value of ADPn(d, Vs, Vd) for N = 400nodes varying the distance between adjacent nodes onthe square side in the range 1 6 d 6 200 m. The value ofEAXn has been computed using the EAX recursive formula.The curves have been obtained for different maximumnumber of candidates: n = {2, 3, 5}. In Fig. 16 we have fixedthe position of the source and the destination at the diag-onal end points, i.e. Vs = (1,1) and Vd = (20,20). For the sakeof comparison, we have included the value of d1/2 (see ver-tical dotted line in Fig. 16). For a shadowing propagationmodel with the parameters used in Fig. 16 it isd1/2 � 51 m (see Fig. 3).

Fig. 15. Grid topology.

Fig. 17. Mean value of ADP using DPOR for Vs = (1, 1) and Vd = (i,20), i 2 {1, 2, . . . , 20}, varying the distance between nodes on the squareside.

20

40

60

80

100

120

140

1 20 40 60 80 100 120 140 160 180 200

532

Fig. 16. ADP using DPOR for Vs = (1, 1) and Vd = (20, 20), varying thedistance between two adjacent nodes on the square side.

2282 L. Cerdà-Alabern et al. / Ad Hoc Networks 11 (2013) 2273–2287

For each n, the thick and solid vertical lines in Fig. 16show the values of d that maximize ADPn. Fig. 16 alsoshows an upper bound of ADPn (dashed line) obtainedusing the value of EAXn computed by means of Eq. (8). Notethat in the grid topology the nodes are not located at theiroptimal positions. Therefore, the fact that the maximumvalue obtained for ADPn is close to the upper bound showsthat the performance that can be obtained with OR is notvery sensitive to deviations in the candidates placementwith respect to their optimal positions. This fact was al-ready observed in Section 8 for a linear setting.

Fig. 16 shows that the value of ADP increases withincreasing the maximum number of cadidates (n). Addi-tionally, Fig. 16 shows that the curves of ADP is rather flataround their maximum point. For instance, with n = 2,ADP2 reaches its maximum (101 m) at d = 36 m. However,using d = d1/2 � 51 m it is obtained that ADP2 = 97 m,which is only 5% smaller than the maximum.

Fig. 16 shows that the optimum d is smaller than d =d1/2 � 51 m for n = 2, while it is higher for n = {3,5}. Thereason is the following. Recall that the optimum positionof the first candidate is d1, and the second is very close tod1/2. Therefore, choosing d ¼ ðd1=2Þ=

ffiffiffi2p� 36 m we obtain

the best position for the candidates in the diagonal. If morethan 2 candidates are used, the best position of the othercandidates is not well fitted by nodes in the diagonal. Thus,the candidate selection algorithm also chooses nodes outof the diagonal, and closer to the sending node than theirMPDs (since they are better than nodes located farther).Being the candidates closer to Vs than their optimal posi-tions motivates that a slightly higher performance can beachieved using a higher value for d, as shown in Fig. 16.

Fig. 17 shows the average value of ADP over all destina-tions in one side of the square opposite to the source, i.e.Vd = (i,20),i 2 {1, 2, . . . , 20}. As before, the vertical linesare located at d = d1/2 � 51 m (dotted) and the values ofd that maximize ADP, and the horizontal dashed lines arethe upper bounds of ADPn, obtained using the value of EAXn

computed by means of Eq. (8). Interestingly, Fig. 17 showsthat taking the average all vertical lines almost overlap,coinciding with d = d1/2.

We can conclude that in a grid topology, ADP is practi-cally insensitive to d over a wide interval. Additionally, set-ting d = d1/2 is a good rule of thumb, yielding an ADP closeto the optimum for all possible destinations.

To further study the d1/2 approximation we have runanother experiment with a grid of N = 10 � 10 nodes. Wehave used the same conditions as before, thus d =d1/2 � 51 m. Like in the previous scenario, we have fixedthe position of the source to Vs = (1,1). The position ofthe destination is varied by choosing the nodes on thediagonal, i.e. Vd = (i, i), 2 6 i 6 10. Due to the reduced sizeof the network, for the candidate selection now we haverun an optimum algorithm (MTS [18]). The maximumnumber of candidates has been set equal to n = 5.

Fig. 18 shows the candidates sets that are chosen foreach destination. This is done by showing the distancefrom the source node Vs to each of the 5 selected candi-dates versus the distance between source and the destina-tion. It turns out that all candidates are chosen among thenodes located in the three central diagonals of the grid (i.e,(i, j)ji � jj 6 1). In order to identify these nodes, their dis-tance to Vs is indicated with the labels (i, j) on the right sideof the plot (note that the distance of nodes (i, j) and (j, i) toVs is the same). For the sake of comparison, the figure alsoshows the MPD (dashed lines), with the labels di on theright. For each destination in Fig. 18, which shows theADP5 (solid line), and its upper bound (dashed line), ob-tained as in Figs. 16 and 17.

We can see in Fig. 18 that for different positions of thedestination close to the source (Vd = (i, i), 2 6 i 6 7), differ-ent candidates sets are chosen in the neighborhood of thesource. For further distances (Vd = (i, i), i > 7), the candi-dates set remains the same. The geographical positions ofthe candidates in this set are depicted in Fig. 20. Fig. 20 alsoshows the MPDs (dashed lines), with labels di. Note thatcandidates located at the MPDs would be in the intersec-tion of these dashed lines with the segment connectingthe source and the destination (dotted line in Fig. 20 for

L. Cerdà-Alabern et al. / Ad Hoc Networks 11 (2013) 2273–2287 2283

the destinations located in the diagonal of the grid). Fig. 20shows that the candidates chosen by the optimal algo-rithm, MTS, are indeed those closest to the MPDs. Fig. 20also gives a pictorial view about how far are the chosencandidates from their optimal position. Note that choosingthe destination in the diagonal is a worst case, since thegrid is built applying the d1/2 rule of thumb for the nodeslocated in verticals or horizontals lines departing from thesource.

Recall that the upper bound shown in Fig. 19 is obtainedusing Eq. (8). For the first two points (Vd = {(2,2), (3,3)}),the destination is close to the source, and the first compo-nent of Eq. (8) applies (EAX � 2 � p(D)). For the otherpoints (Vd = (i, i), i > 3), the second component of Eq. (8),is used. Recall that in this case Eq. (8) assumes the candi-dates located at the MPD. The MPD are computed for a des-tination far from the source. In fact, we observe in Fig. 19that for the points corresponding to Vd = (i, i), i > 3 the dif-ference between the bound and the measured ADP is re-duced slightly when the distance between the source anddestination increases. However, the fact that this differ-ence is small shows that the MPDs are also a good estima-tion when the destination is relatively close to the source.Additionally, Fig. 19 shows that for Vd = (10,10) it is ob-tained ADP � 129 with an upper bound �138, thus, aboutonly 6.7% relative error. As explained above, choosing thedestination in the diagonal is a worst case. Therefore, thissmall error shows again that the performance is not verysensitive to deviations from the optimal position of thecandidates. Summing up, we conclude that choosingd = d1/2 is a good rule of thumb for building the grid.

Fig. 19. ADP and upper bound (dashed line) obtained for each candidateset depicted in Fig. 18.

10. Related work

The majority of previous studies that evaluate the per-formance of opportunistic routing do not use analyticalmethods, instead they resort to simulations or empirical

Fig. 18. Geographic distance of the optimum candidates in the grid to the

measurements [5,19–23]. On the other hand, most of theworks are devoted to the selection of the candidates, theway of acknowledging packet reception and how to pre-vent, or at least reduce, duplicate transmissions.

Biswas and Morris [4,5] designed and implemented Ex-tremely Opportunistic Routing (ExOR) for wireless multi-hop networks. It selects the candidates set according toroute costs based on Expected Transmission Count (ETX)[24]. In [8,25] Zhong et al. proposed the expected anypathtransmission (EAX) as a new metric for OR that generalizesthe ETX, and proposed a candidate selection and prioritiza-tion algorithm based on it. In addition to ETX and EAX,some OR schemes such as Geographic Opportunistic Rout-ing (GOR) [26] and Contention-Based Forwarding (CBF)[27] considers geographic distance as metric. Under somescenarios, relay nodes are assumed to be aware of theirpositions. Therefore, routing can be done by selecting the

source for Vs = (1, 1), varying the destination to Vd = (i, i), 2 6 i 6 10.

Fig. 20. Candidates of Fig. 18 chosen for Vd = (i, i), i > 7 (circles with thicklines). Dashed lines with labels di show the maximum progress distancesfrom the source, Vs.

2284 L. Cerdà-Alabern et al. / Ad Hoc Networks 11 (2013) 2273–2287

candidate, which is geographically closest to thedestination.

Dubois-Ferrière et al. [28] introduced a shortest any-path algorithm that finds optimal candidates sets. Theauthors generalized the well-known Bellman–Ford algo-rithm for anypath routing and proved its optimality, butthe resulting algorithm has exponential computationalrunning time. They provided another optimal polyno-mial-time algorithm in [29] to jointly select the optimalcandidates set. They also extended their model proposedin [28,9] to a multi-rate scenario. In [18] the key problemof how to optimally select the candidates sets is addressed,and an optimal algorithm that minimizes the expected to-tal number of transmissions is developed. The authors in[30] provided an analytical framework to model the prob-lem of selecting the optimal candidates set for both theconstrained and unconstrained candidates set selection.They proposed two algorithms for optimal candidates setselection, one for the constrained and one for the uncon-strained case.

In the recent work from [31], the authors presented aBandwidth-aware Opportunistic Routing (BOR) withadmission control protocol. They proposed a new metricto determine the priority of candidates in the candidatesset. The new metric analyzes the expected available band-width (EAB) and the expected transmission cost (ETC)using OR. The candidate with a high bandwidth and lowtransmission cost will relay the packet with a higher prior-ity. They devised an admission control mechanism to rejecttraffic flows with a high bandwidth requirement that can-not be satisfied by the discovered opportunistic path. Thismechanism guarantees that the QoS of ongoing traffic ismaintained. Authors in [32] analyzed the opportunisticrouting gain under the presence of link correlation consid-ering the loss of data and acknowledgment packets. Theydefined a new metric that captures the expected numberof anypath transmissions under the effect of link correla-tion. Based on the new metric they proposed a candidateselection algorithm. Their results show that consideringthe correlation between the links improved the expectednumber of transmissions from the source to thedestination.

Using multiple next-hop candidates however faces thechallenge of avoiding duplicate forwarding, which occurswhen more than a candidate relays the same packet. Thisissue is usually called as candidate coordination which isone of the important issues of OR. There are different algo-rithms with different performances for the candidate coor-dination. In [7] authors surveyed opportunistic routingissues with emphasize on the candidate coordinationalgorithms.

ExOR [4] uses a modified version of the 802.11 MAC forcandidate coordination. It reserves multiple slots of timefor the receiving candidates to return ACKs. Each ACK indi-cates that the packet is successfully received by the candi-date. In addition, it contains the IDs of the successfulrecipients with the higher priority known to the ACKs sen-der. All candidates listen to all ACK slots before decidingwhether to forward; this is done in case a low-priority can-didates ACK reports a higher-priority candidate’s ID whoseACK was not correctly received.

Authors in [33] showed how in the ACK-based methodif the highest priority candidate fails to receive the datapacket, the channel will then be idle for a period of timelonger than the DIFS. It may happen that some other nodes,from another flow, not hearing the data packet, would sendtheir packets; this would cause a collision with subsequentACKs, if any, from the lower priority candidates. Theauthors proposed FSA [33] to provide the coordination be-tween the candidates with lower delay than ACK-basedmethod. FSA is only based on a single ACK which is sentthrough the highest priority candidate that has receivedthe packet successfully.

Combining the idea of OR with Network Coding (NC)[34] can provide an elegant method for preventing dupli-cate transmissions without coordinating the candidatesby coding the packets [35–37]. MAC-independent Oppor-tunistic Routing & Encoding (MORE) [35] was proposedby Chachulski et al. The data flow is divided in somebatches which contain a certain amount of packets. MOREadds the candidates set to the packet’s header and broad-casts the coded packet. The receiving node checks whetherits ID is in the candidates set. Furthermore, the node checkswhether the packet is linearly independent of the packetsit has received before. If so, it creates a new coded packetby linearly combining the received packets and rebroad-casting it. The simulations results show that MORE can sig-nificantly improve the network throughput in comparisonto ExOR; and, it can also eliminate the requirement of glo-bal coordination among candidates. A practical NC mecha-nism that enables the supporting of efficient unicastcommunication in wireless mesh networks is COPE [38].By using OR, COPE enables each node to learn about localstate information. The CORE [39] is a coding-aware ORmechanism that combines OR and localized inter-flow NCfor improvement of the throughput performance in a wire-less mesh network. Through OR, CORE allows the next-hopnode with the most coding gain to continue packet for-warding. Through localized NC, CORE attempts to maxi-mize the number of packets that can be carried in asingle transmission.

Authors in [40] developed an anycast mechanism at thelink layer for wireless ad hoc networks. They used explicit

L. Cerdà-Alabern et al. / Ad Hoc Networks 11 (2013) 2273–2287 2285

control packet(s) exchanged immediately before sending adata packet. In this approach the sender multicasts the Re-quest-To-Send (RTS) to the its candidates set. The RTS con-tains all the candidates addresses which are orderedaccording to a metric. When an intended candidate re-ceives the RTS packet, it responds by a Clear-To-Send(CTS). These CTS transmissions are sent in decreasing orderof candidate priority. When the sender receives a CTS, ittransmits the DATA packet to the sender of this CTS aftera Short Inter-frame Space (SIFS) interval. All such receiversthen set their Network Allocation Vector (NAV) until theend of ACK period. This mechanism is guaranteed to havea single winner and it can avoid duplicate transmissions.In [41] a method for avoiding duplicate forwarding inopportunistic routing is proposed which does not dependon any form of information exchange between next-hopcandidates. The proposed technique enables forwardingnodes to control relaying at their neighbors on a per-pack-et basis using a small amount of information piggybackedon packets. The authors compared their proposal withExOR and demonstrated that their proposal improvesthroughput of the network by reducing unnecessarytransmissions.

There are some papers which propose analytical modelsto study the performance of OR. In [42] Shah et al. pre-sented a framework to model OR for low loaded sensornetworks. They also explored the performance of opportu-nistic routing for different node densities, channel qualityand traffic rates, and compared it with geographic routing.They also identified optimal points for the duty cycle ofnodes that minimize the power consumption. Baccelliet al. [43] aimed at quantifying and optimizing the poten-tial performance gains of opportunistic routing strategiescompared with classical routing schemes. Their analysiswas under the assumptions of Aloha-based MAC layer. Zu-bow et al. in [22] claimed that shadow fading losses forspatially close candidates are not independent from eachother, unlike commonly assumed. They presented mea-surements obtained from an indoor testbed and concludedthat correlations cannot be neglected if nodes are sepa-rated by less than 2 m. In [44] an analytical approach forstudying OR in wireless multi-hop networks have beenproposed. They used lognormal shadowing and Rayleighfading models for packet reception. In their model they as-sume that the nodes are uniformly distributed over theplane. The authors did not consider any specific candidateselection algorithm, but simply compute the expected pro-gress of the packet transmissions based on the probabilityof any node in the progressing region successfully receivesthe packet. They extended their work by using directionalantennas and different radio propagation models and spa-tial node distributions in [45]. In [46] an algebraic ap-proach was applied to study the interaction of OR routingalgorithms and routing metrics. They showed that OR incombination with ETX could degrade the performance ofnetwork. Authors in [26] analyzed the trade-off amongthe packet advancement, reliability and MAC coordinationtime cost in geographic opportunistic routing. They pro-posed a local metric, expected one-hop throughput (EOT),to balance the trade-off between the packet advancementand the cost of candidate coordination. They showed that

although having more candidates brings more chancesfor the packet to get closer to the destination and be deliv-ered, the gained benefit is marginal.

11. Conclusions

In this paper we have derived the equations that yieldthe distances of the candidates in Opportunistic Routing(OR) such that the per transmission progress towards thedestination is maximized. We have called them as the max-imum progress distances (MPD). The only ingredient to ob-tain these distances is the law for the delivery probabilitybetween nodes as a function of distance. An important con-sequence of our derivation is that the MPD for the alreadyexisting candidates do not change if we decide to add anew candidate to the candidate set. We have also observedthat, while the MPD vary when the propagation conditionschange, the delivery probabilities to the candidates locatedat the MPD remain approximately constant. Thus, to selectthe candidates, or to place the nodes of the back-haul of amesh network during the network design, we may beimplement a method based on the measurement of thedelivery probabilities.

Based on these MPD, we have proposed a lower bound tothe expected number of transmissions needed to send a pack-et using OR. The lower bound has proven to be very tight.

By modeling the delivery probabilities with a shadow-ing propagation model we obtained numerical resultsshowing that the expected number of transmissions canbe reduced up to a 30% with only 2 candidates, whereasin order to reduce it another 30% the number of candidateshas to be increased up to 10.

We have constructed a quasi optimum OR networklocating the nodes and their candidates at the MPD when-ever possible. This quasi optimum OR network is used toobtain an upper bound for the expected number of trans-missions. This upper bound happens to be very close toour lower bound, which confirms that it is indeed verytight. We have further validated these results by buildinga dense network and computing the optimal distances ofthe candidates by an exhaustive optimum search. We haveseen that the optimal distances of the candidates convergerapidly to the MPD as the length of the network increases.

We have also investigated the sensitivity of the perfor-mance to the position of the candidates. Our results al-lowed us to conclude that choosing the distance of thefirst two candidates near to their optimal positions is themost critical aspect in order to minimize the expectednumber of transmissions. Based on this result, we haveused the MPD to provide a rule of thumb for placing thenodes in a static network using OR. Compared to the opti-mal layout, this method will slightly increase the averagenumber of transmissions while the total number of nodesrequired is reduced enormously. This can be of practicalinterest in the design of the back-haul of a mesh network,or in the positioning of the nodes in a sensor network.

Finally, we have investigated the maximum perfor-mance of OR in two-dimensional scenarios. Our results re-vealed that a network design with grid layout yields aperformance that, for all practical purposes, is very close

2286 L. Cerdà-Alabern et al. / Ad Hoc Networks 11 (2013) 2273–2287

the optimum. Furthermore, in the grid design, adjustingthe shortest distance between two adjacent nodes to itsoptimal value is not a critical aspect. In our experimentsthe sensibility of the performance with respect to this de-sign parameter has proven to be rather low in a wide re-gion around its optimal value.

Acknowledgments

This work was supported by the Spanish Ministerio deCiencia e Innovación through the Projects TIN2010–21378-C02-01 and TIN2010-21378-C02-02 and by theGeneralitat de Catalunya through Project 2009-SGR-1167.

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c Networks 11 (2013) 2273–2287 2287

Llorenç Cerdà-Alabern is associate professorat the Computer Architecture Department,

Universitat Politècnica de Catalunya (UPC). Heobtained his PhD in TelecommunicationEngineering in 2000 from the UPC. His areasof interest include protocols for wireless adhoc and mesh networks (MAC and routinglayers). He has published papers in interna-tional conferences and journals and partici-pated in National and EU funded researchprojects.

L. Cerdà-Alabern et al. / Ad Ho

Amir Darehshoorzadeh obtained his PhD atthe Computer Architecture Department,Universitat Politècnica de Catalunya (UPC) in2012. He received his BSc in computer engi-neering from Shahid Bahonar University ofKerman (SBUK), Kerman, Iran in 2003, and hisMSc from Iran University of Science andTechnology (IUST), Tehran, Iran in 2006. Hisresearch interests include opportunistic rout-ing, wireless networks, Quality of serviceprovisioning and multicast routing.

Vicent Pla received the M.E. and PhD degreesin Telecommunication Engineering from theUniversitat Politècnica de València (UPV),València, Spain. He is currently an AssociateProfessor with the Department of Communi-cations at the UPV. He has published numer-ous papers in refereed journals andconference proceedings. His research interestsinclude teletraffic and the modeling and per-formance evaluation of communication net-works. During the past few years, most of hisresearch activity has focused on resource

management in wireless networks.