2006 - location errors ++

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Geographic routing in the presence of location errors q

Sungoh Kwon, Ness B. Shroff *

Center for Wireless Systems and Applications (CWSA), School of Electrical and Computer Engineering, Purdue University,

1285 EE Building, West Lafayette, IN 47907-1285, United States

Received 2 June 2005; received in revised form 10 October 2005; accepted 17 November 2005Available online 28 December 2005

Responsible Editor: E. Ekici

Abstract

In this paper, we propose a new geographic routing algorithm that alleviates the effect of location errors on routing inwireless ad hoc networks. In most previous work, geographic routing has been studied assuming perfect location informa-tion. However, in practice there could be significant errors in obtaining location estimates, even when nodes use GPS.Hence, existing geographic routing schemes will need to be appropriately modified. We investigate how such locationerrors affect the performance of geographic routing strategies. We incorporate location errors into our objective functionby considering both transmission failures and backward progress. Each node then forwards packets to the node that max-imizes this objective function. We call this strategy Maximum Expectation within transmission Range (MER). Simula-

tion results with MER show that accounting for location errors significantly improves the performance of geographicrouting. Our analysis also shows that our algorithm works well up to a critical threshold of error. We also show thatMER is robust to the location error model and model parameters. Further, via simulations, we show that in a mobile envi-ronment MER performs better than existing approaches. 2005 Elsevier B.V. All rights reserved.

Keywords: Wireless ad hoc networks; Geographic routing; Location errors; Simulations

1. Introduction

Geographic routing for multi-hop wireless net-works has become an active area of study over thelast few years (e.g., see [14] and the referencestherein). Geographic routing is appealing because

of its simplicity and scalability. However, mostwork in this area has implicitly assumed that loca-

tion information available at each node is perfect,while in practice only a rough estimate of this infor-mation is available. In this paper, we will show thatimperfect location information can lead to substan-tial degradation in the performance of geographicrouting. We will also develop a routing scheme thataccounts for location errors, and whose perfor-mance is robust to such errors.

In geographic routing, each node determines itsown location by using either the Global Positioning

1389-1286/$ - see front matter 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.comnet.2005.11.008

q This work has been partially supported by the NSF grantANI-0207728 and a grant from the Indiana 21st Century Fund.* Corresponding author. Tel.: +1 765 494 3471; fax: +1 765 494

3358.E-mail addresses: [email protected] (S. Kwon), shroff@

purdue.edu (N.B. Shroff).

Computer Networks 50 (2006) 29022917

www.elsevier.com/locate/comnet
mailto:[email protected]:shroff@%20purdue.edumailto:shroff@%20purdue.edumailto:shroff@%20purdue.edumailto:shroff@%20purdue.edumailto:[email protected]


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System (GPS) [5,6] or the location sensing tech-niques [710]. It then broadcasts its location infor-mation to other nodes proactively and periodically.Packet forwarding is accomplished based on theneighbors location information stored in each

nodes database (DB) and the destinations locationinformation contained in the packet. Packets aretypically forwarded using what is commonly referredto as thegreedy mode, in which nodes use local infor-mation to forward packets towards their destination.If the greedy mode is not successful (i.e., the destina-tion node is not available in the local databases, or agreedy mode forwarding results in a failure), a spe-cial routine called a recovery mode is initiatedthrough the entire network to find an appropriateroute to the destination. Since the greedy mode useslocal information and most packets are forwarded

in this mode [11], geographic routing is generallyconsidered to be scalable and applicable to largenetworks.

Several forwarding schemes have been proposedfor use in the greedy mode [1215].Fig. 1providesan illustration of such schemes when a node Swithtransmission range R has a packet to send to some

node D. Arcs ab_

and pq_

are centered at node Dand with radii DSand DG, respectively. When thenodes in the wireless network have a fixed transmis-sion range, the Most Forward within Radius

(MFR) scheme [12] and the Greedy RoutingScheme (GRS)[13]have been proposed to minimizethe hop count and the energy consumption. MFRforwards a packet to the neighbor (node M inFig. 1) that is the farthest from the source in the

direction of the destination within the transmissionrange. GRS selects the closest neighbor (node G inFig. 1) to the destination among neighbors. Sincein most cases MFR and GRS provide the same pathto the destination[16], we only consider GRS in this

paper. When nodes have the ability to control thetransmission ranges, the Nearest Forward Progress(NFP) algorithm [14]has been proposed to reduceenergy power consumption. NFP chooses the clos-est neighbor (node Nin Fig. 1) to the sender withinthe forward region. Yet another scheme called com-pass routing [15] is used to select that neighbor(nodeCinFig. 1), which has a minimum angle withrespect to the line between the sender and thedestination.

Most work on geographic routing assumes thatlocation information received from GPS (or other

techniques) is perfect [1,1620]. Effective algorithmsin the recovery mode have been addressed in [1,1619]. Flooding based algorithms have been proposedfor finding alternative paths in[16,19]. Face routing(perimeter routing) has also been studied for routerecovery in [1,17]. In [20], the authors consider theinformation discrepancy due to mobility and pro-pose a mobility prediction scheme based onreported location information. However, the loca-tion information in[20]is assumed to be measuredperfectly by GPS.

In practice, the received location information isnot perfect [5,10,21] and the error in the locationinformation degrades the performance of geo-graphic routing[22]. This inaccuracy is caused evenwhen nodes use GPS because of GPSs inherenterror in location estimation [6]. This location errorcould induce transmission failures and backwardprogress in the greedy mode. A transmission failurehappens when the selected node is out of the trans-mission range. Backward progress occurs when thechosen node is located farther from the destinationthan the sender and could cause looping. Loopsoccur when the selected node is one of the previoussenders in the route. Further, it could also result ina local minimum, i.e., there exist solutions butthere are no nodes in the forwarding direction.Such failures produce unnecessary transitions fromthe greedy mode to the recovery mode, which inturn results in an inefficient routing solution[16,17].

In this paper, we study the impact of these loca-tion errors on the performance of geographic rout-ing. We further propose a new routing scheme to

improve the performance of geographic routing.

D

C

N

G

M

S

R

p

a

b

q

Fig. 1. The examples of forwarding schemes when node Swithtransmission range R has a packet to send to node D: compass

routing(C), GRS(G), MFR(M) and NFP(N).

S. Kwon, N.B. Shroff / Computer Networks 50 (2006) 29022917 2903


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We focus on the case when the transmission rangesof the nodes are fixed. Using numerical simulationswe verify the performance of the proposed algo-rithm and the robustness of the location error mod-els and model parameters. Unless otherwise stated,

the term location error means location errordue to measurement through this paper.The rest of the paper is organized as follows. In

Section 2, we study how location errors in geo-graphic routing arise and evaluate the impact oflocation errors on the performance of the geo-graphic routing. In Section 3, we propose a newalgorithm for geographic routing in an environmentwith location errors and analyze properties of thealgorithm. In Section4, we use simulation to com-pare the performance of our scheme with that ofknown schemes. Section5 concludes this paper.

2. The impact of location error on geographic

routing performance

In this section, we develop a location errormodel and investigate the impact of location errorson the performance of geographic routing. Asmentioned in the introduction, errors in the loca-tion information affect the forwarding scheme inthe greedy mode, which could cause unnecessarytransitions into the recovery mode of the algorithm

in order to find an alternative route to the destina-tion. The goal of the greedy mode is to succeed intransmitting packets to a neighbor with forwardprogress (i.e., the neighbor is closer to the destina-tion). Thus, it is important to account for bothtransmission failures and backward progress to ana-lyze how the location error affects the performanceof geographic routing.

2.1. Error modeling

Location errors occur during the process of esti-mating the location (via GPS or other techniques)[5,6,10,21]. In GPS, the performance of an estimatedlocation depends on the geometry of satellites, loca-tion sensing techniques, radio environments, etc.[5,23,24]. For example, when GPS uses a single fre-quency, the root mean square measurement error istypically 6 m[6]. On the other hand, the typical mea-surement error is about 3 m for a dual-frequencyreceiver[6]. The geometry of satellites and locationsensing techniques are known parameters to theGPS receiver. Other factors are also estimated and

adjusted by the GPS receiver.

In this paper we make the following assumptions.All nodes are equipped with GPS1 to measure theirown locations. These locations are proactivelybroadcasted. The location errors at different nodesare independent. The location error at each nodeis modeled by a Gaussian distribution2 with zeromean and finite standard deviation. The zero-meanassumption implies that, for an given environment,the average of location errors over all nodes is equalto zero, i.e., it follows from the strong law of large

numbers[25]that limn!11nPn

k1Wk is equal to zeroalmost surely, where Wirepresents a measurementerror of node i. However, note that for a given envi-ronment, the time average of location errors at a sin-gle node could be non-zero.Therefore, a scheme thatuses simple averaging of samples in time at eachnode will not be able to overcome the errors.

LetXibe the real position of nodeiand letX0ibe

its measured position. Then Xican be expressed asXiX0iWi , where Wi is a Gaussian randomvector with zero mean and standard deviation ri.

For convenience, we assign node i to be at theorigin and destination node d to be on the x-axis,as in Fig. 2. Let node j be a neighbor of node i,and letz be the real distance between the two nodes.Since Xiand Xjare independent and Gaussian, theprobability density function fj(z,h) that node j islocated at (z,h) is

t

z

i

i

j

i

j

Node dDestination

d

r

X

X

X

X

X

measured location

real location

Fig. 2. Location error modeling when node iwith the transmis-sion range r ihas a packet to send the destination node d.

1 Our analysis and algorithm are applicable to other locationestimation technologies, however in this paper we focus onnetworks where the nodes are equipped with GPS.2 In Section4, we will also study cases when the errors are not

Gaussian and their impact on the routing performance.

2904 S. Kwon, N.B. Shroff / Computer Networks 50 (2006) 29022917


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fjz; h z2pr2ij

exp z2 g2ij2r2ij

!exp

zgij

r2ijcos h

!;

whererijis the standard deviation ofXi Xj,his anangle of Xjwith respect to x-axis, and gij kX0iX0jk as in Fig. 2. Hence the probability densityfunction f(z) that the distance between two nodesis zis

fjz Z 2p

0

fjz; hdh

zr2ij

exp z2 g2ij2r2ij

!I0

zgij

r2ij

! for zP 0;

where I0(x) is the modified Bessel function of thefirst kind and zero order, which is defined byI0

x

12p R2p0 expx cos hdh.

2.2. Transmission failure probability

A packet transmission failure occurs when a cho-sen node is out of the transmission range of a sen-der, as shown in Fig. 3. Most work assumes thatthe transmission range of each node is perfectly cir-cular and identical so that a neighbor is within thetransmission range of a node if and only if the nodeis located within the transmission range of theneighbor. In practice, nodes have imperfect circular

transmission patterns [26] and the transmissionranges deviate from the ideal case [20]. Moreover,a network may be composed of heterogeneous

nodes that have different transmission ranges. Inthese cases, even though a node is located in thetransmission range of a neighbor, the neighbormay be out of the transmission range of the node.Such a link is an asymmetric communication link.

In the case of asymmetric communication links,transmission failures can happen in the presence oflocation errors even though each node preciselyknows its own transmission range and pattern. Ina mobile environment, displacements of nodesinduced by mobility cause transmission failures thatdisplacement prediction may reduce [20,27]. How-ever, this is still vulnerable to the transmissionfailure due to location errors in measurement eventhough the prediction is perfect and the communi-cation links are symmetric. In the case when thetransmission range is controllable, the adjusted

transmission range of a sender can also fail to trans-mit a packet to its neighbor that is still within themaximum transmission range of the sender. Here,we study failures caused only by inaccurate locationinformation.

Assume that node ihas a packet to transmit andnode jis chosen as the next node. We assume thatnode icalculates the distance between nodes iandjas gijsuch that gij kX0iX0jk, and sets its trans-mission range to ri, which may be fixed or control-lable, in order to forward data to node j. The

probability that a packet transmission from node ito node jfails is

Prftransmission failure at nodejg PrfZ> rig

Z 1

ri

fzdz

Z 1

ri

z

r2ijexp z

2 g2ij2r2ij

!I0

zgij

r2ij

!dz

Z 1

ririj

rexp 12

r2 g2ij

r2ij

! !I0 r

gij

rij dr Q1

gij

rij;ri

rij

; 1

whereZ= kXi Xjk,rijis the standard deviation ofXi Xj, andQ1(a, b) is a MarcumsQ function withm= 1 defined as in[28].

It follows from MarcumsQ function withm = 1that the transmission failure probability increaseswhen the standard deviation of location errors rijincreases. When rij is fixed and the chosen node iscloser to the edge of the transmission range, the

transmission failure probability increases. In other

Measured location

Real location

Real (or controlled) Tx range

Ideal (or max.) Tx rangeNode displacement

Heterogeneity

DisplacementIrregular Tx pattern

Controlled Tx range

Tx failure

Fig. 3. Examples for transmission failure.



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words,given an error environment, a longer transmis-sion range reduces the transmission failure probability.

2.3. Backward progress probability

Backward progress occurs when a chosen node j

is located farther from a destination than the send-ing node i. Note that there may exist a route to thedestination even though there is no neighbor in theforward region. This case is typically called a localminimum in the geographic routing literature andcannot be solved by using only the greedy mode,so recovery mode is needed. Assume that node ihas a packet to transmit and node j is chosen asthe next node. For simplicity, in this subsection,we assume that the destination location Xd in thepacket has no error.3

The probability that the chosen node j, such thatkX0iXdkP kX0jXdk, is located behind (isfurther away from the destination than node i) thesender iis

Prfbackward progress at node jgPrfz; hjkXiXdk 6 kXjXdkg

Z

fz;hjkXiXdk6kXjXdkgfjz; hdzdh

Zfz;hjkXiXdk6kXjXdkgz

2pr

2

ij

exp z2 g2ij2r2ij

!exp

zgij

r2ijcos h

!dzdh; 2

where gij kX0iX0jkand rijis the standard devia-tion ofXi Xj.

In general, the integral above does not reduce toa closed form and must be numerically evaluated.However, if ri is much smaller than the distancebetween node iand the destination, h is close to 0,we can approximate Eq.(2)as follows:

Prfbackward progress at node jg Q gijrij

;

where gij kX0iX0jk and Qx 1ffiffiffiffi2pp R1x exp x2

2

dx.

It follows from the Q function thatthe backwardprogress probability increases when the standard devi-

ation of location errorsrijincreases. Whenrijis fixedand the chosen node is closer to the sender, thebackward progress probability increases.

Figs. 4 and 5 show these two probabilities. InFig. 4, we fix the location of node jand show the

relationship between the failure probabilities andthe standard deviation of the location error. InFig. 5, we fix the location error and show the rela-tionship between the failure probabilities and thedistance between nodeiand node j. Given the loca-tion error, when the chosen node is closer to the

3 The case where the destination location is in error can besimilarly treated, although the equations become more notation-ally complex. For the simulations, we assume that the location

information of all nodes, including the destination, has errors.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized standard deviation of location errors

Failureprobability

Transmission failure probability

Backward progress probability

Fig. 4. Failure probabilities versus the standard deviation oflocation errors when gij= 0.8ri. The standard deviation oflocation errors is normalized by the transmission range r i.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized location from a sender

Failureprobab

ility

Transmission failure probability

Backward progress probability

Fig. 5. Failure probabilities versus the location of node j whenthe standard deviation of location errors is 0.1ri. The location of

the chosen node is normalized by the transmission range r i.



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sender or the edge of the transmission range, thefailure probability increases.

2.4. Impact of location error on geographic routing

All forwarding schemes in geographic routingsuffer from the above mentioned failures. GRS (orMFR) selects the closest neighbor to a destination(or the farthest neighbor from the source in thedirection of the destination), so the node is morelikely close to an edge of a transmission range thanany other neighbors. They are susceptible to a trans-mission failure. NFP chooses a neighbor which isclosest to the sender. This scheme is susceptible tobackward progress. Compass routing does not con-sider the distance between the sender and the inter-mediate node, but cares for only the angle with

respect to the line between the sender and the desti-nation. Hence, compass routing is vulnerable toboth factors described above. When the number ofnodes increases in a given area, a chosen node is clo-ser to the sender or the edge of the transmissionrange. Therefore, denser nodes can potentiallyworsen the performance (as will also be shown vianumerical studies in Section4).

3. Geographic routing scheme with location errors

In this section we propose a new geographic rout-ing scheme that can mitigate the impact of locationerrors. Since MFR is similar to GRS in most cases[4], we focus on improving GRS. For ease of illus-tration, from here on we assume that the transmis-sion range is fixed. However, it should be readilyapparent that the methodology can be extended tothe case when the transmission range is controllable.Since each node measures its location and estimatesits own error characteristic, we attach an errorinformation field in a message for geographic rout-ing and announce the statistical characteristics ofthe location error to neighbors with locationinformation.

3.1. Objective function

Let nodesiand jbe located at Xiand Xj, respec-tively. We assume that X0i and X

0j are the measured

locations of nodes i and j, respectively. As before,these are expressed as XiX0i Wi and XjX0jWj , where Wi and Wj are Gaussian randomvectors with zero means and standard deviations

ri and rj, respectively. Then the real position of

node jwith respect to node iis a Gaussian randomvector with mean X0jX0i and standard deviationrij

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2i r2j

q . Hence, the probability that node j

is located within uj from X0jX0i is PrfkX

X0iX0jk 6 ujg 1 exp u2

j

2r2ij

, where Xis the

real position of node jwith respect to node i.

Fix a sender i and the destination d. We definemeasured progress to node j to be tj kXdkkXd X0jX0ik and the measured margin fromthe boundary to be sjri kX0iX0jk, where ri isthe transmission range of nodeiand Xdis the desti-nation position with respect to node i. Then we canexpress GRS in this (location) error-free environ-ment as follows. IfkX0iXdk 6 ri, choose node d.If

kX0i

Xd

k> ri, choose node ksuch that

karg maxj2Ni

tj 3subject tokX0iX0jk 6 ri 8j2Ni;where Ni is the set of neighbors of node i.

When we have location errors, instead of usingthe measured progress tj in(3), we propose using adifferent metric to determine which neighbor toforward the packets.

We define true progress to node j to be sjkXdk kXd XjX0ik when node j is actuallylocated at position Xj. Then, sj(Xj) is a random

variable with probability density function fj(Xj).Since the probability density function, fj(Xj),that node j is located at Xj is circularly symmetricwith respect to the point X0jX0i, we consider areaAj such that Aj fX2 R2jkX X0jX0ik 6 ujgand the expected progress of node j over Aj asfollows.

EfsjXj1Ajg ZAj

sjXjfjXjdX. 4

If uj=

1, (4) becomes the expected progress over

the entire domain of X. However, if uj>sj, node jmay be out of the transmission range of node i. Ifuj>tj, nodejmay be located behind node i. In orderto find a neighbor to be able to successfully transmitto and result in forward progress, we let uj=min{sj, tj}.

For simplicity, we use an approximation of (4).For realistically sized wireless networks, in most

caseskXdk is much greater than ri. Then, arc AB_

and arc MN_

in Fig. 6 are nearly straight. Sincefj(z) is circularly symmetric, we can simplify (4) as

follows:



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EfsjXj1Ajg tjZAj

fjXjdXj;

where Aj fX2 R2jkX X0jX0ik 6 ujg for u=min{sj, tj}.

Hence, we define Ej as the revenue of node jasfollows.

Ej,tjFjuj; 5where

Fj

uj

Prfk

X

X0j

X0ik

6 ujg

1 exp u2j

2r2ij

!

for uj= min{sj, tj}.Based on the calculated revenue of each node,

node iselects the next node which has a MaximumExpectation within transmission Range ri (MER).Our MER algorithm for forwarding packets is givenas follows. IfkX0iXdk 6 ri di, where 0 6 di ri di, choose nodeksuch thatkarg max

j2NEj

subject tokX0iX0jk 6 ri 8j2Ni;whereNiis the set of nodeis neighbor nodes storedin the nodeis DB. Here,diis a function of the loca-tion errors and could be a tuning parameter for spe-cific implementations.d ican be simply the standarddeviation of location errors or the distance from themaximizer of the objective function to the transmis-sion range edge. In [20], the authors proposed a

scheme to improve the forwarding performance as

follows: if the destination node exists in the neigh-bor list, the sender forwards the packet to the desti-nation without any effort. However, since in thelocation error environment the destination may belocated out of the transmission range, it is impera-

tive to consider the parameter di to increase thetransmission success rate. In Fig. 7, we summarizeour algorithm. For the simulations in Section 4,we also add functionalities to the protocol to detectloops and local minima.

3.2. Properties of the algorithm

We now analyze the MER algorithm. For sim-plicity, we assume that a selected node is locatedon the line between the sender and the destinationsince the node on the line (OD in Fig. 6) is the

most likely chosen among neighbors that have thesame measured progress. Let the transmission rangeof the sender ribe 1. We let r denote the standarddeviation for the overall location error betweenthe sender and the intermediate node. Then, (5)becomes

Et tFu t 1 exp u2

2r2

; 6

where t 2 (0,1) is the position of the selected nodeand u= min{t, 1

t}.

We define an effective search range to be an areafrom the source to the most likely position chosenby the algorithm and tmax to be a maximizer ofE(t) such that E(tmax)P E(t) for t 2 (0,1).Property 1. Let r 1, then MER is identical toGRS within an effective search range for a next node.

The range is reduced by d such that E0(1 d) = 0,where E0(t) is the derivative of E(t) and the maximizerof E(t), t

max, is a decreasing function ofr.

Proof. Since the exponential term in (6),exp u22r2

, goes to zero faster than r2 whenr goes

to zero, for r 1, we haveEt t or2. 7Note that(7) is equivalent to the objective functionof GRS(3). Since E0(t) = tF(t) is always an increas-ing function of t 2 (0,0.5), we need to considert 2 (0.5,1) to check the maximum ofE(t). The deriv-ative of the objective functionE0(t) fort 2 (0.5,1) is

E0t

F1

t

tf1

t< 1

1

2f

1

t.

i

tji

u

sj

ijj

j

Dd

Destination

r

ONode

X

X'X'

A

M

N

B

X

Fig. 6. The expected progress of nodejwith respect to node ithathas the transmission range r iand is located at O.



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Sincer 1 from the assumption above andf(t) hasa maximum 1

rexp 1

2

at t=r, E0(t) 1. The

values of E(t) precipitates from a maximum valueto zero, and the effective search range reduces totmax such that E

0(tmax) = 0. In other words, theeffective search range is reduced by d such thatd= 1 tmax. Since r2 > 0 andE0t F1 t tf1 t

F1 t 1 tf1 t f1 t

1 tmax2

2r2 1 tmax

4

8r4 1 tmax

2

r2

1 tmax4

2r4 1 tmax 1 tmax

3

2r2

o

1 tmax4

r4 !;

we have

r2 1 tmax

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4 tmax2 101 tmax

q 4

1 tmax4 tmax4

o1 tmax3; 8

where tmax is the maximizer of(6). From(8) above,ris a monotonically decreasing function oftmaxfortmax 2 (0.5,1). Hencetmaxis a monotonic decreasingfunction ofr. h

Property 2. There exists a threshold rth such that

the decrement of r results in an increment of tmaxfor rP rth, b u t tmax does not depends on r for

r 6 rth. The thresholdrth is numerically 0.315.

A packet arrives at node i

||Xi'-Xd|| > ri- i

Choose node dCalculate t

j= ||X

d-X

i||-||X

d-X

j||

for all neighboring nodes

Calculate Ej

= tj

Fj

for all neighoring nodes

Choose k = arg max Ej

Succeedin transmiting?

Fail to deliver the

packet to the destination

Succeedin transmiting?

Succeed in delivering the

packet to the destination

Yes

No

No YesYes

Set i = k.

Fig. 7. Block diagram for MER.



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Proof. tF(t) is an increasing function for t 2[0,1]. tF(1 t) has a maximizer tmax 2 [0, 1] anda minimum value 0 at t = 0 and 1. tF(1 t) in-creases for t 2 [0, tmax] and decreases for t 2 [tmax, 1]monotonically. It follows from (6) that if tmax 2 [0,0.5],

E(t) has a maximum at

t= 0.5 andif tmax 2 [0.5,1), E(t) has a maximum at t=tmax.

rth is r such that E0 1

2

0. The numerical valueofrth is 0.315. h

Figs. 810 depict how the objective function ofthe proposed algorithm (MER) works. For thesefigures, we assume that the selected node is located

onOD inFig. 6.Property 1shows that MER worksidentically to GRS except in the outskirts of thetransmission range, as shown in Fig. 8. However,MER may require more hops to route packets fromsource to destination compared with the case of per-fect location environment, since the position of themost likely chosen node,tmax, decreases due to loca-tion error, as inFigs. 8 and 10.

Property 2gives us some insights on geographic

routing in an environment with location errors. Ifthe standard deviation of the location error isgreater than some threshold, i.e. rP rth, the maxi-mizer of (6), tmax, decreases when r increases. Ifr 6 rth, tmax does not depend on r, as in Fig. 10and only the maximum revenue decreases as inFig. 9. tmax is fixed at 0.5 after r= 0.315. In thisregion, increasing r does not affect the selectednode, but decreases the revenue. This means thatfor a standard deviation of error larger than rth,taking error estimates into account does not helpin improving the performance of geographic rout-ing. The algorithm is simply reduced to selectingthe node that is closest to the middle point of theforward region. Hence, rth becomes the criticalpoint that determines the utility of the algorithm.For example, when each node is equipped with aGPS receiver that has a standard deviation of 3 mof the location error, Bluetooth [29] which has atransmission range of 10 m cannot use geographicrouting. However, a geographic routing scheme thatincorporates error information (such as MER) isapplicable in the case of IEEE 802.11 with a nomi-

nal transmission range 250 m.

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized distance from a sender

Revenue

GRS

MER with =0.01

MER with =0.03

MER with =0.05

Fig. 8. The revenue versus the distance from the sender when thetransmission range is 1.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Maximum

Revenue

Fig. 9. The maximum revenue versus the standard deviation

normalized by the transmission range.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Maximize

r

Fig. 10. The maximizer versus the standard deviation when thetransmission range is 1.



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4. Simulation results

In this section, we use numerical simulations toverify the performance of the proposed algorithm,MER, in a mobile environment as in [11,18,20].

We use the following random way point (RWP)mobility model. Each node chooses a destinationin a given area and moves at a constant speed,which is uniformly chosen between 0 and 50 m/s.The node stays for a pause time, which is uniformlydistributed between 0 s and 30 s. Each node broad-casts its own location periodically and proactively.To avoid collisions, the interval of these broadcastsis uniformly chosen from 1 s to 3 s, as in [18]. Wealso set the neighbor timeout interval at 9 s. Sincelocation information errors in the mobile environ-ment can be categorized into two classes, displace-

ment due to mobility and location errors inmeasurement, we use two cases of simulations inthe mobile environment in order to study the impactof location errors on geographic routing. First, weassume that location information errors exist onlyin measurement. The displacements of the nodesare assumed to be perfectly estimated in the mobileenvironment. In the case when there is no locationerror due to measurement, the authors in[20]studythe performance of geographic routing due to errorsin displacement using extensive simulations with

several mobility models. In our simulation environ-ment, we compare the performance of MER versusGRS when the parameters of the location errormodel are known in Section4.1. We further demon-strate the robustness of the algorithm to changingparameters in the error model in Section 4.2.Finally, we study the performance of MER in anoisy environment due to inaccurate measurementand mobility in Section 4.3. In order to reduce theimpact of mobility on the routing performance, weadopt a displacement estimation method used in[20,27]. Through our simulations we assume thatthere is no time delay to route data from a sourceto destination.

In our simulations we compare two differentschemes: GRS and MER. We use the performanceof GRS with perfect location information as anupper bound on the routing performance of allschemes in the presence of location errors.

To compare the fundamental performance inSection 4.1, we use two measures: delivery successrate and the number of attempts. In the other sec-tions we compare delivery success rates. For the

delivery success rate, we do not use retransmissions

or algorithms to find alternative routes when packetforwarding failures occur. Packet delivery is said tosucceed only if the packet is delivered to the destina-tion by the schemes. For the number of attempts, weemploy an algorithm to find another node when

packet forwarding fails. When the chosen node isnot available, a sender immediately finds the nextavailable node without retransmitting to the samenode. We define this as a reattempt. When there isno available node in the forward area, the routinestops finding an alternative node. In the case whenthere is no reattempt, the total number of attemptsis equivalent to the number of hops. The reattemptsrequire extra energy as well as time delay so that thethroughput will be degraded. Since the time delayper reattempt is protocol-specific (such as the num-ber of retransmission before declaring the chosen

node to be unavailable), hence we focus only onthe number of reattempts. We count the numberof attempts and reattempts until a hundred packetsare successfully delivered to their own destinations.

In MER, we use the distance from the maximizerof the objective function to the transmission rangeedge as a tuning parameter di. In order to studyhow system parameters affect the performance ofgeographic routing in the presence of locationerrors, for our numerical results, we focus on thecase when the wireless environment is homogeneous

across all nodes. In practice, the wireless environ-ment at each node in the network may be different.For instance, in the case of GPS, each node couldreceive a different number of satellite signals dueto obstacles. Our algorithm described in Section 3can handle such a heterogeneous environment,and we show that our algorithm works well in theheterogeneous environment by simulations (Section4.1).

We run 20 simulations with different randomseeds for each scenario and average the results.

4.1. The performance of MER when the distribution

of the location error is known

We investigate three scenarios, as illustrated inTable 1. In the first scenario, we deploy 100 nodesin an area of 1000 1000 m2. The standard devia-tion of the location error for each node is 10 m.We vary the transmission range from 25 m to500 m.Fig. 11shows that the transmission successrate of MER is close to that of GRS with perfectlocation information. However, the performance

of GRS degrades severely in the presence of location



11/16

errors. Table 2 provides a comparison betweenMER and GRS in terms of the total number ofattempts required for 100 successful transmissions.Table 2shows that MER with location errors needsadditional attempts (or hops) to reach its destina-tion when compared with GRS with perfect locationinformation. The reason is that the location errorscause MER to reduce its effective search range whenforwarding packets to its neighbors, as shown byProperty 1. Hence, MER requires additionalattempts to deliver packets from source to destina-

tion. When we compare the total number ofattempts of MER and GRS in the presence of loca-tion errors, they are similar so that the reattemptroutine seems to dilute the benefits of MER. How-ever, GRS requires many more reattempts than

our algorithm. The reattempts may induce timedelay in the system. When the transmission rangesincrease, the number of attempts of GRS with loca-tion errors decreases since the failure probabilitydecreases, as in(1).

In the second scenario, the transmission range isfixed at 250 m, which is the nominal transmissionrange of IEEE 802.11, and we vary the standarddeviation of the location error from 3 m (1.2%) to50 m (20%). As expected, Fig. 12shows that MERperforms much better than GRS when there arelocation errors. The performance of GRS starts to

degrade when the standard deviation is above 3 m(1.2%). However, the performance of MER doesnot decrease significantly. As the error increases,

Table 2The comparison of the number of attempts in Scenario 1

ri(m) GRS MER

No error Error Error

(a) (b) (c) (a) (b) (c) (a) (b) (c) (d)

300 238.15 0.0 0.0 260.00 17.85 0.0 255.70 1.20 0.0 107.37350 208.50 0.0 0.0 225.65 14.75 0.0 219.05 0.60 0.0 105.06400 184.95 0.0 0.0 195.95 9.40 0.0 193.45 0.50 0.0 104.60500 155.45 0.0 0.0 162.05 5.70 0.0 159.85 0.10 0.0 102.83

(a) The average number of total attempts to deliver packets, (b) the average number of reattempts to find alternative nodes, (c) the average

number of times backward progress is made, (d) a

of MER with error

a of GRS with no error %.

0 100 200 300 400 5000

10

20

30

40

50

60

70

80

90

100

Transmission range

Succe

ssrate[%]

GRS without error

MER with error

GRS with error

Fig. 11. The performance comparison of the forwarding schemesin Scenario 1.

Table 1Scenarios for simulations: A, N, R and r represent a deployedarea, the number of deployed nodes, the transmission range ofnodes and the standard deviation of location errors, respectively

Scenario A (m2) N R(m) r (m)

1 1000 1000 100 25500 10 (240%)

2 1000 1000 100 250 350 (1.220%)3 1000 1000 25500 250 5 (2%)

0 5 10 15 2040

50

60

70

80

90

100

Standard deviation of location errors[%]

Successra

te[%]

MER with error

GRS with error

Fig. 12. The performance comparison of the forwarding schemesin Scenario 2.



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the effective search range in MER is furtherreduced. This reduction in the effective search rangedecreases the number of neighbor nodes to beselected and degrades the performance. Similarly

to Scenario 1, MER with location errors requiresadditional attempts to route packets when com-pared to the case with perfect information, as shownin Table 3. When we compare GRS with MER,GRS needs more reattempts than our algorithm inthe erroneous environment. Since MER and GRStend to choose the closest node to the transmissionrange, they are less susceptible to backward pro-gress. As analyzed in Section 2, backward progressrarely happens, however, backward progressincreases with larger location errors.

In the third scenario, we vary the number ofdeployed nodes from 25 to 500. We fix the transmis-sion range at 250 m and the standard deviation ofthe location error at 5 m. Fig. 13 shows that theperformance MER is not affected by the number

of nodes while the performance of GRS is. The lar-ger density reduces the distance between two adja-cent nodes. This reduction in distance means thatthe selected node is closer to the edge of the trans-

mission range. Hence, transmission failure is morelikely to happen in GRS. Similarly to Scenario 1,MER with location errors needs additionalattempts (Table 4) compared to the case withoutlocation errors, however the success rate does notdecrease when the node density of the network isincreased. Note that inTable 4, unlike previous sce-narios, the ratio of the additional attempts forMER does not change significantly as the numberof nodes increases, since the maximizer of the objec-tive function depends only on the ratio of the loca-

tion error to the transmission range. In the case ofGRS with errors, the total number of attemptsdecreases as the number of nodes, N, increasesbetween 100 and 200. However, after that region,the total number of attempts increases due to theincrement of the failure probability as the numberof nodes increases.

Depending on RF environments, the locationerrors in the same system may be different. We usethe same settings as Scenario 3 except that nowthere exits a shadowing area (250 250 m2) in whichnodes have twice the location errors of the otherareas.Fig. 14shows that our algorithm still workswell in such a heterogeneous environment.

4.2. Robustness to estimation error

In Section2, we modeled the location error by aGaussian error distribution. In practice, the errormay not follow a Gaussian distribution and/or theparameters of the model may be incorrect. In thissubsection, we study the robustness of MER withrespect to these two kinds of modeling errors: the

distribution function and parameter error.


r (m) GRS MER


(a) (b) (c) (a) (b) (c) (a) (b) (c) (d)

3 289.75 0.0 0.0 295.95 5.30 0.00 295.45 0.10 0.00 101.975 289.75 0.0 0.0 300.35 8.55 0.00 300.10 0.65 0.00 103.577 289.75 0.0 0.0 303.05 11.15 0.05 304.35 1.20 0.05 105.04

10 289.75 0.0 0.0 310.60 17.05 0.10 310.35 1.35 0.10 107.11


number of times backward progress is made, (d) a of MER with errora of GRS with no error %.

0 100 200 300 400 50050

55

60

65

70

75

80

85

90

95

100

The number of deployed nodes

Successrate[%]

GRS without error

MER with error

GRS with error

Fig. 13. The performance comparison of the forwarding schemes

in Scenario 3.



13/16

In Fig. 15, we simulate three different locationerror models: uniformly distributed error, exponen-tially distributed error, and Gaussian error. How-ever, the MER algorithm always assumes aGaussian model. The transmission range of eachnode is 250 m and the standard deviation of thelocation error is 5 m. The simulation results showthat MER is quite robust to difference in error dis-tribution functions.

In Fig. 16, the underlying error model is alsoGaussian. However the parameter used by MERis different from the true parameter of the underly-ing model. In the simulation the transmission rangeof each node is 250 m and the standard deviation ofthe location error is assumed to be 5 m. However,the actual standard deviation of the location erroris varied from 0 m to 10 m. Fig. 16 illustrates thatMER is robust to the estimation error and outper-

forms GRS.

0 100 200 300 400 50050

55

60

65

70

75

80

85

90

95

100


Successrate[%]

GRS without error

MER with Gaussain error

MER with uniform error

MER with exponential error

GRS with Gaussain error

GRS with uniform error

GRS with exponential error

Fig. 15. The performance of forwarding schemes versus thenumber of nodes with different error distributions and the fixedtransmission range.

0 2 4 6 8 1080

82

84

86

88

90

92

94

96

98

100

Real Location Error

Successrate[

%]

MER

GRS

Fig. 16. The performance of forwarding schemes versus theparameter error in estimation with the fixed number of nodes and

the fixed transmission range.


N GRS MER


(a) (b) (c) (a) (b) (c) (a) (b) (c) (d)

100 289.75 0.0 0.0 300.35 8.55 0.0 300.10 0.65 0.0 103.57200 275.95 0.0 0.0 291.55 13.75 0.0 285.90 0.45 0.0 103.61400 271.05 0.0 0.0 293.15 19.75 0.0 281.65 0.45 0.0 103.91500 268.95 0.0 0.0 294.60 23.45 0.0 279.85 0.25 0.0 104.05


number of times backward progress is made, (d) a of MER with errora of GRS with no error %.

0 100 200 300 400 50050

55

60

65

70

75

80

85

90

95

100


Successrate[%]

GRS without error

MER with error

GRS with error

Fig. 14. The performance comparison of the forwarding schemesin Scenario 3 with a heterogeneous RF environment.



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4.3. Noisy mobile environment

In this subsection, we compare the performanceof MER and GRS in the presence of location errorsdue to inaccurate measurement and mobility. When

each node chooses the next node, we use the pre-dicted positions of neighbors at the transmissiontime similar to[20,27]in order to improve the rout-ing performance. The authors in [20,27]predict theneighbor positions at the transmission time by usingtwo positions reported at two recent times asfollows:

X0X1X1X2t1 t2 t

0 t1; 9

where X(0) is the predicted position at the currenttime t(0), X(1) is the reported location at the first re-

cent timet(1), andX(2) is the reported location at thesecond recent time t(2). The method improves therouting performance by reducing the mobility errorwhen the reported location information is assumedto be perfect. However, this method will accumulatemeasurement and prediction errors when reportedlocation information is noisy. Hence, we use the in-stant velocity, which is available to GPS equippednodes, since the velocity measured by GPS is con-sidered to be very accurate[6]. The instant velocityis announced with location information. Each node

predicts neighbors positions when forwarding apacket as follows.

X0X1 v1 t0 t1; 10where X(0) is the predicted position at current timet(0), X(1) is the reported location at the first recenttime t(1), and v(1) is the reported velocity at t(1).

Fig. 17 shows the simulation results when thelocation error at each node has a Gaussian distribu-tion with standard deviation 5 m and the transmis-sion range is fixed at 250 m. GRS with thedisplacement prediction performs better than GRSwithout the displacement prediction, as shown in[20]. MER is slightly affected by mobility but out-performs GRS in all cases. As shown in [11], theperformance degradation from mobility can be fur-ther improved by reducing time discrepancy. How-ever, the measurement error of X(1) in (10) doesnot decrease, as the time discrepancy reduces. Thereduction of the time discrepancy also does notaffect the location error of a sender. Hence, anyeffort to reduce the time discrepancy from mobilitycannot help mitigate the impact of measurement

errors on geographic routing.

4.4. Discussion

In order to solve the fundamental problem ofgeographic routing with location errors, in thispaper we do not use a protocol specific solution tohelp mitigate errors. However, such a protocol-spe-cific solution could in certain cases help combatlocation errors [30]. For example, in the case when

nodes are static and have fixed transmission ranges,the transmission failures caused by asymmetriccommunication links can be reduced by a three-way handshake protocol when nodes join the net-work. However, the three-way communication doesnot alleviate backward progress. Even in a staticwireless network, such a protocol cannot avoidtransmission failures when the transmission rangeis controllable in the presence of location errors.Further, the solution is not suitable in a densemobile wireless network, where frequent topologychanges may take place. Moreover, in contrast toour proactive announcement (one-way communica-tion) of location information, the three-way com-munication results in 2n(n 1) overhead messagesper time interval in a transmission range, where nrepresents the number of nodes within a transmis-sion range. The excessive overhead messages mayin fact worsen network performance such asthroughput. For the above mentioned reasons, inthis paper, we do not focus on protocol-specificsolutions to alleviate location errors. However, suchapproaches can be potentially used in conjunction

with our method on a case by case basis.

100 150 200 250 300 350 400 450 50060

65

70

75

80

85

90

95

100

The number of nodes in 1000 by 1000 square area

Successrate

[%]

MER with perfect prediction

MER with prediction

MER without prediction

GRS with perfect prediction

GRS with prediction

GRS without prediction

Fig. 17. The performance of forwarding schemes versus thenumber of nodes with Gaussian location errors and the RWP

mobility model.



15/16

Another feature of our approach is to provide anunderstanding of the intrinsic performance achiev-able using geographic routing with location errors.This serves to provide design guidelines for imple-menters to choose equipment that provides an

appropriate level of accuracy.

5. Conclusions

In this paper, we consider the impact of locationerrors on geographic routing in multi-hop wirelessnetworks. We have shown that location errors cansignificantly impact the performance of geographicrouting. The degradation in the routing performancedepends on the transmission range of the sender,error characteristics of the sender and its neighbors,and the deployed density of nodes. We have proposed

a new algorithm called MER in order to mitigate theeffect of noisy location information by explicitly con-sidering the error probability when making routingdecisions. In doing so we find that our algorithm per-forms quite well, in many cases, even approximatingthe performance of geographic routing without loca-tion errors. We have also used simulations to showthat MER is robust to different location error modelsand errors in model parameters.

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Sungoh Kwonreceived his B.S. and M.S.degrees in electrical engineering fromKAIST, Daejeon, Korea, in 1994 and1996, respectively. From 1996 to 2001,he had worked as a research staff inShinsegi Telecomm Inc., Seoul, Korea.Since 2001, he has worked for the Ph.D.in the School of Electrical and ComputerEngineering from Purdue University. Hiscurrent interests are in wireless commu-nication networks.

Ness B. Shroffis a Professor of Electricaland Computer Engineering at Purdueand director of the Center for WirelessSystems and Applications (CWSA), auniversity-wide center on wireless sys-tems and applications. His researchinterests span the areas of wireless andwireline communication networks, wherehe investigates fundamental problems inthe design, performance, pricing, andsecurity of these networks.

He is an active member in the networking research communityand has been involved in the leadership of various conferencesand journals. He currently serves on the editorial boards ofIEEE/ACM Transactions on Networking and the ComputerNetworks Journal. He was the technical program co-chair ofIEEE INFOCOM03, the premier conference in communicationnetworking. He was also the conference chair of the 14th AnnualIEEE Computer Communications Workshop (CCW99), theprogram co-chair for the symposium on high-speed networks,IEEE Globecom 2001, and the panel co-chair for ACM Mobi-com02. He was also a co-organizer of the NSF workshop onFundamental Research in Networking, held in Arlie HouseVirginia, in 2003.

He has received numerous awards for his networking research,including the NSF CAREER award, the best paper of the yearaward for the Computer Networks journal, and the best paper ofthe year award for the Journal of Communications and Networks(JCN).


2006 - location errors ++

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