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1
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
2
Outline
Introduction and motivation Assumptions and notationsMain resultsSome hints on the derivation of
resultsConclusions
3
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
4
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
5
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
6
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)
7
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)?
8
Outline
Introduction and motivation Assumptions and notationsMain resultsSome hints on the derivation of
resultsConclusions
9
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
14
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
15
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
17
Outline
Introduction and motivation Assumptions and notationsMain resultsSome hints on the derivation of
resultsConclusions
18
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) !
19
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
?
20
Outline
Introduction and motivation Assumptions and notationsMain resultsSome hints on the derivation of
resultsConclusions
21
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
22
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
25
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
26
Upper bound : network cut
0 1
1
1/2
27
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 !
28
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
29
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
30
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
31
Comments ?
Questions ?