Minkowski Distance for Geographical Routing in VANETs

Empirical Analysis of the Minkowski Distance Order in GeographicalRouting Protocols for VANETs

13th International Conference on Wired & Wireless Internet Communications

Luis Urquiza-Aguiar1 Carolina Tripp-Barba2 Jos Estrada-Jimnez3

Mnica Aguilar Igartua1

1Department of Network Engineering, Universitat Politcnica de Catalunya, Barcelona, SpainEmail: [ luis.urquiza, monica.aguilar]@entel.upc.edu

2Faculty of Informatics, Autonomic University of Sinaloa, Mazatlan, MexicoEmail: [email protected]

3Department of Electronics and Telecommunications, Escuela Politcnica Naciona, Quito, EcuadorEmail: [email protected]

Mlaga, Spain, May 25-27th.

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Introduction

Minkowskidis-tanceinge-o-graph-i-caldis-tancerout-ingmet-ric

EmpiricalAnal-y-sisofMinkowskior-derr

Conclusions

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Agenda

Introduction

Minkowski distance in geographical distance routing metric

Empirical Analysis of Minkowski order r

Conclusions and Future work

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Introduction

Vehicular ad hoc networks (VANETs)VANETs are seen as a special case of mobile ad hoc networks (MANETs), wherenodes are vehicles.I Faster topology changes.I Short link lifetime.I Greater number of nodes. (Non-uniformly distributed)I Nodes (vehicles) follow roads and respect traffic signals

Geographical routing protocolsA routing paradigm based only on local information.I Typically based on distance between nodes.I Position of destination and neighbors have to be known .

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Introduction

Greedy Perimeter Stateless Routing (GPSR)

(a) Greedy forwarding. (b) Perimeter mode.

Greedy Buffer Stateless Routing (GBSR)

I It uses a buffer instead of perimeter mode. (Delay Tolerant applications)I It uses more information to improve the position estimation. (e.g. speed, time)I GBSR improves packet delivery ratio but introduces delay.

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Introduction

ObjectiveTo test if the Euclidean distance is the most suitable function for VANET routingpurposes.

Why not using other distances?

(a) Manhattan distanced = x + y .

(b) Euclidean distance.d =

x2 + y2(c) Dominant distance.d = max(x , y)

All these cases are particular case from Minkowski distance function.

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5 Minkowskidis-tanceinge-o-graph-i-caldis-tancerout-ingmet-ric


Conclusions

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Minkowski Distance in Geographical VANET routing

Distance functionA distance function (x , y) for two n-dimensional points x and y satisfies:

(x , y) = (y , x) (1a)

(x , y) 0 (1b)(x , x) = 0 (1c)

Minkowski distanceThe Minkowski distance [1] of order r between the points x and y is:

r (x , y) =

(n

i=1

|xi yi |r)1/r

(2)

When r < 0, the Minkowski distance function (2) can be seen as a similaritymeasure

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Minkowski circles of radius = 1

1.0 0.5 0.0 0.5 1.0

1.0

0.5

0.0

0.5

1.0

r = 0.5

x

y O

1.0 0.5 0.0 0.5 1.0

1.0

0.5

0.0

0.5

1.0

r = 1

x

y O

1.0 0.5 0.0 0.5 1.0

1.0

0.5

0.0

0.5

1.0

r = 1.5

x

y O

1.0 0.5 0.0 0.5 1.0

1.0

0.5

0.0

0.5

1.0

r = 2

x

y O

1.0 0.5 0.0 0.5 1.0

1.0

0.5

0.0

0.5

1.0

r = 4

x

y O

1.0 0.5 0.0 0.5 1.0

1.0

0.5

0.0

0.5

1.0

r = infinite

x

y O

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Effects of Minkowski order r in routing decision

1. The size and form of the searching area to find a the next forwarding node.

2. The decision of which neighbor is the closest to destination.

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8 EmpiricalAnal-y-sisofMinkowskior-derr

Conclusions

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Empirical Analysis of Minkowski order r in VANET geo routing

Simulation SettingsI The mobility of vehicles was obtained with SUMO [4]/C4R [3]I 100 and 150 vehicles, 1 Access PointI IEEE 802.11p. Estinet simulator [2].I GBSR [5] in the routing layer.I Inter-packet time TU(2,6) s E(T ) = 4 s. Packets of 1000 bytes

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Percentage of packet losses vs r

Packet losses increase when r < 2 and almost constant with r > 2.

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Average end-to-end packet delay vs r

Notice that average delay for r < 2 is similar to the obtained r = 2, but thepercentage of packet losses are different.

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Average number of hops vs r

Manhattan distance (r = 1) has the worst performance.When r > 2 the number of hops decrease.

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Statistical test results

Vehicle Pairwise Standardized p-ValueIs the Difference Median of

Density (r,2) Test Statistic 1 SideSignificant Differences(p-Value < 0.025)?

Percentage packet losses

150 2.5 2.091 0.018 Yes 2.549%+ 2.24 0.012 Yes 3.096 %

Average end-to-end delay

2.5 2.427 0.007 Yes 0.500 s150 3 2.763 0.002 Yes 0.584 s

4 2.203 0.013 Yes 0.409 s+ 1.269 0.108 No 0.186 s

Average number of hops

100 2.5 2.837 0.002 Yes 0.367 hops+ 3.173 0.0005 Yes 0.515 hops

3 2.165 0.015 Yes 0.0136 hops150 4 1.979 0.024 Yes 0.66 hops

+ 3.323 0.0005 Yes 0.11 hops

Table: p-values of Wilcoxon signed rank test for a pairwise comparison of the effect of theMinkowski distance order r

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ConclusionsOur results in a realistic grid urban scenario, indicate:I The use of the Minkowski order (r < 2) is not a good idea. (higher packet

losses)I The use of the dominant distance (r +) in the routing decision leads to

better performance than the one obtained Euclidean distance (r = 2). (shorterpaths, lower packet losses, same delay)

I The performance differences between euclidean distance are not far from thebest ones obtained by other Minkowski r value. Euclidean distance is always agood choice.

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Future work include:I To this same comparison in other city topologiesI To develop a geographical routing protocol that combines some Minkowski

distances.I A distance function to select candidates nodes and other distance function to

compute the best forwarding node.I A linear combination of distances in the selection of next forwarding nodes.

Thanks for your attentionif Questions then

if Time WWIC15_limit thenPlease ask

elseemail to: [email protected]

return Answer & Thanks

Luis Urquiza Aguiarwww.lfurquiza.com

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References

[1] Borg, I., Groenen, P.: Modern Multidimensional Scaling - Theory andApplications. Springer New York, New York, second edn. (2005)

[2] Estinet-Technologies: EstiNet 7 Network Simulator and Emulator (2015),http://www.estinet.com/products.php?lv1=13&sn=15

[3] Fogue, M., Garrido, P., Martinez, F.J., Cano, J.C., Calafate, C.T., Manzoni, P.: Arealistic simulation framework for vehicular networks. In: 5th International ICSTConference on Simulation Tools and Techniques. pp. 3746. ACM, Brussels,Belgium (2012), http://dl.acm.org/citation.cfm?id=2263019.2263025

[4] Krajzewicz, D., Erdmann, J., Behrisch, M., Bieker, L.: Recent development andapplications of SUMO - Simulation of Urban MObility. International Journal OnAdvances in Systems and Measurements 5(3&4), 128138 (2012)

[5] Tripp Barba, C., Urquiza Aguiar, L., Aguilar Igartua, M.: Design and evaluationof GBSR-B, an improvement of GPSR for VANETs. IEEE Latin AmericaTransactions 11(4), 1083 1089 (2013)

IntroductionVANET routingWork objective

Minkowski distance in geographical distance routing metricMinkowski distanceEffects of Minkowski order ``r''

Empirical Analysis of Minkowski order ``r''Simulation SettingsSimulation Results

Conclusions and Future workConclusionsFuture work

Documents

Minkowski Distance for Geographical Routing in VANETs