1 Capacity Scaling in Delay Tolerant Networks with Heterogeneous Mobile Nodes Michele Garetto – Università di Torino Paolo Giaccone - Politecnico di Torino.

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Capacity Scaling in Delay Tolerant Networks with Heterogeneous Mobile

Nodes Michele Garetto – Università di Torino

Paolo Giaccone - Politecnico di Torino

Emilio Leonardi – Politecnico di Torino

MobiHoc 2007

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Outline

Introduction and motivation Assumptions and notationsMain resultsSome hints on the derivation of

resultsConclusions

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Introduction The sad Gupta-Kumar result:

In static ad hoc wireless networks with n nodes, the

per-node throughput behaves as

P. Gupta, P.R. Kumar, The capacity of wireless networks, IEEE Trans. on Information Theory, March 2000   

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Introduction

The happy Grossglauser-Tse result: In mobile ad hoc wireless networks with n nodes,

the per-node throughput remains constanto assumption: uniform distribution of each node presence

over the network area

M. Grossglauser and D. Tse, Mobility Increases the Capacity of Ad Hoc Wireless Networks, IEEE/ACM Trans. on Networking, August 2002

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Introduction

Node mobility can be exploited to carry data across the network Store-carry-forward communication

scheme

S DR

Drawback: large delays (minutes/hours)

Delay-tolerant networking

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Mobile Ad Hoc (Delay Tolerant) Networks

Have recently attracted a lot of attention

Examplespocket switched networks (e.g., iMotes)vehicular networks (e.g., cars, buses, taxi)sensor networks (e.g., disaster-relief

networks, wildlife tracking)Internet access to remote villages (e.g., IP

over usb over motorbike)

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The general (unanswered) problem

Key issue: how does network capacity depend on the nodes mobility pattern?

Are there intermediate cases in between extremes of static nodes (Gupta-Kumar’00) and fully mobile nodes (Grossglauser-Tse’01)?

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Outline

Introduction and motivation Assumptions and notationsMain resultsSome hints on the derivation of

resultsConclusions

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n nodes moving over closed connected region independent, stationary and ergodic mobility processes uniform permutation traffic matrix: each node is origin

and destination of a single traffic flow with rate (n) bits/sec

all transmissions employ the same nominal range or power

all transmissions occur at common rate r single channel, omni-directional antennas

Assumptions

s

ou

rce

nod

e

destination node

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Protocol Model

Let dij denote the distance between node i and node j, and RT the common transmission range

A transmission from i to j at rate r is successful if:

for every other node k simultaneously transmitting

RT (1+Δ)RT

ij k

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Realistic mobility models for DTNs

characterized by : Restricted mobility of individual nodes:

Non-uniform density due to concentration points

From: Sarafijanovic-Djukic, M. Piorkowski, and M. Grossglauser, Island Hopping: Efficient Mobility-Assisted Forwarding in Partitioned Networks,, IEEE SECON 2006

From: J.H.Kang, W.Welbourne, B. Stewart, G.Borriello, Extracting Places from Traces of Locations, ACM Mobile Computing andCommunications Review, July 2005.

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Home-point based mobility

Each node has a “home-point”

… and a spatial distribution around the home-point

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Home-point based mobilityThe shape of the spatial distribution of each

node is according to a generic, decreasing function s(d) of the distance from the home-point

s(d)

d

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Anisotropic node density (clustering)

Achieved through the distribution of home-points

0

1

0 1

Uniform model: home-points randomly placed over the area according to uniform distribution

n = 10000

0

1

0 1

Clustered model: nodes randomly assigned to m = nν clusters uniformly placed over the area. Home-points within disk of radius r from the cluster middle point

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Scaling the network size

10 nodes……100 nodes…..1000 nodes

We assume that:

Moreover: node mobility process does not depend on network size

increasing sizeconstant density

constant sizeincreasing density

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Asymptotic capacity We say that the per-node capacity is if

there exist two constants c and c’ such that

sustainable means that the network backlog remains finite

Equivalently, we say that the network capacity in this case is

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Outline

Introduction and motivation Assumptions and notationsMain resultsSome hints on the derivation of

resultsConclusions

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Asymptotic capacity results

Recall:

0 1/2

per-

nod

e c

ap

aci

ty

0

-1/2

-1

logn [(n)] Uniform Model

Independently of the shape of s(d) !

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Asymptotic capacity results

Recall: #clusters

0 1/2

per-

nod

e c

ap

aci

ty

0

-1/2

-1

logn [(n)] Clustered Model

“Super-critical regime”: mobility helps

“Sub-critical regime”: mobility does not help

?Lower bound : in case s(d) has finite support

Lower bound : in case s(d) has finite support

?

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Outline

Introduction and motivation Assumptions and notationsMain resultsSome hints on the derivation of

resultsConclusions

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A notational note

In the analysis we have fixed the network size, L=1

and let the spatial distribution of nodes s(d) to scale with n, i.e., s(f(n)d)

f(n)=1 f(n)=2 f(n)=3

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Uniformly dense networks We define the local asymptotic node density ρ(XO) at

point XO as:

Where is the disk centered in XO , of radius

A network is uniformly dense if:

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Properties of uniformly dense networks

Theorem: the maximum network capacity is achieved by scheduling policies forcing the transmission range to be

Corollary: simple scheduling policies leading to link capacities

are asymptotically optimal, i.e., allow to achieve the maximum network capacity (in order sense)

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Super-critical regime

Let (m = n in the Uniform Model)

When we are in the super-critical regime

Theorem: in super-critical regime a random network realization is uniformely dense w.h.p.Transmission range is

optimalScheduling policy S* is optimal

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Mapping over Generalized Random Geometric Graph

(GRGG) Link capacities can be evaluated in terms of contact probabilities:

which depend only on the distance dij between the homepoints of i and j

We can construct a random geometric graph in which vertices stand for homepoints of the nodesedges are weighted by

Network capacity is obtained by solving the maximum concurrent flow problem over the constructed graph

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Upper bound : network cut

0 1

1

1/2

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Average/random flow through the cut

0 1

1

1/2

The “average” flow through the cut is computed as

fundamental question:

Proof’s idea:

Consider regular tessellation where squarelets have area γ(n)

Take upper and lower bounds for number of homepoints falling in each squarelets, combined, respectively, with lower and upper bounds of distances between homepoints belonging to different squarelets

Answer: YES !

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An optimal routing scheme

The above routing strategy sustains per-node traffic

s

d

Routing strategy:

Consider a regular tessellation where squarelets have area

Create a logical route along sequence of horizontal/vertical squarelets, choosing any node whose home-point lie inside traversed squarelet as relay

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Network is not uniformly dense

Transmission range may fail even to guarantee network connectivity

When s(d) has finite support:

Sub-critical regime

Nodes have to use

System behaves as network of m static node

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ConclusionsWe analyzed the capacity of mobile ad-

hoc networks under heterogeneous nodes Our study has shown the existence of two

different regimes:superctriticalSubcritical

In this paper we have mainly focused on the supercritical regime

Subcritical behavior must be better explored

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Comments ?

Questions ?