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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

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2 Outline Introduction and motivation Assumptions and notations Main results Some hints on the derivation of results Conclusions

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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

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4 Introduction The happy Grossglauser-Tse result: In mobile ad hoc wireless networks with n nodes, the per-node throughput remains constant oassumption: 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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5 Introduction Node mobility can be exploited to carry data across the network Store-carry-forward communication scheme S D R Drawback: large delays (minutes/hours) Delay-tolerant networking

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6 Mobile Ad Hoc (Delay Tolerant) Networks Have recently attracted a lot of attention Examples pocket 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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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)?

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8 Outline Introduction and motivation Assumptions and notations Main results Some hints on the derivation of results Conclusions

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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 source node destination node

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10 Protocol Model Let d ij denote the distance between node i and node j, and R T the common transmission range A transmission from i to j at rate r is successful if: for every other node k simultaneously transmitting RTRT (1+Δ)R T i j k

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11 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 and Communications Review, July 2005.

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12 Home-point based mobility Each node has a “home-point” … and a spatial distribution around the home-point

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13 Home-point based mobility The 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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14 Anisotropic node density (clustering) Achieved through the distribution of home-points Uniform model: home-points randomly placed over the area according to uniform distribution n = Clustered model: nodes randomly assigned to m = n ν clusters uniformly placed over the area. Home-points within disk of radius r from the cluster middle point

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15 Scaling the network size 10 nodes……100 nodes… nodes We assume that: Moreover: node mobility process does not depend on network size increasing size constant density constant size increasing density

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16 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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17 Outline Introduction and motivation Assumptions and notations Main results Some hints on the derivation of results Conclusions

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18 Asymptotic capacity results Recall: 01/2 per-node capacity 0 -1/2 log n [ (n)] Uniform Model Independently of the shape of s(d) !

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19 Asymptotic capacity results Recall: #clusters 01/2 per-node capacity 0 -1/2 log n [ (n)] Clustered Model “Super-critical regime”: mobility helps “Sub-critical regime”: mobility does not help ? Lower bound : in case s(d) has finite support ?

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20 Outline Introduction and motivation Assumptions and notations Main results Some hints on the derivation of results Conclusions

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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

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22 Uniformly dense networks We define the local asymptotic node density ρ(X O ) at point X O as: Where is the disk centered in X O, of radius A network is uniformly dense if:

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23 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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24 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 optimal Scheduling policy S * is optimal

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25 Mapping over Generalized Random Geometric Graph (GRGG) Link capacities can be evaluated in terms of contact probabilities: which depend only on the distance d ij between the homepoints of i and j We can construct a random geometric graph in which vertices stand for homepoints of the nodes edges are weighted by Network capacity is obtained by solving the maximum concurrent flow problem over the constructed graph

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26 Upper bound : network cut /2

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27 Average/random flow through the cut /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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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

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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

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30 Conclusions We analyzed the capacity of mobile ad-hoc networks under heterogeneous nodes Our study has shown the existence of two different regimes: superctritical Subcritical In this paper we have mainly focused on the supercritical regime Subcritical behavior must be better explored

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31 Comments ? Questions ?

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