Capacity Scaling in Delay Tolerant Networks with Heterogeneous Mobile Nodes

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 notations • Main results • Some hints on the derivation of results • Conclusions

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 constant • 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 R D • Drawback: large delays(minutes/hours) • Delay-tolerant networking

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)

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 notations • Main results • Some hints on the derivation of results • Conclusions

9. l l l l l l Assumptions • 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 l (n) bits/sec • all transmissions employ the same nominal range or power • all transmissions occur at common rate r • single channel, omni-directional antennas destination node source node

10. Protocol Model • Let dijdenote the distance between node i and nodej, andRT the common transmission range • A transmission from i to j at rater is successful if: for every other node k simultaneously transmitting RT (1+Δ)RT k j i

11. Realistic mobility models for DTNs • characterized by : Restricted mobility of individual nodes: Non-uniform density due to concentration points From: J.H.Kang, W.Welbourne, B. Stewart, G.Borriello, Extracting Places from Traces of Locations, ACM Mobile Computing and Communications Review, July 2005. From: Sarafijanovic-Djukic, M. Piorkowski, and M. Grossglauser, Island Hopping: Efficient Mobility-Assisted Forwarding in Partitioned Networks,, IEEE SECON 2006

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

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

14. Anisotropic node density (clustering) • Achieved through the distribution of home-points 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 Uniform model: home-points randomly placed over the area according to uniform distribution n = 10000

15. constant size increasing density increasing size constant density We assume that: Scaling the network size 10 nodes……100 nodes…..1000 nodes Moreover: node mobility process does not depend on network size

16. Asymptotic capacity • We say that the per-node capacity isif 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 notations • Main results • Some hints on the derivation of results • Conclusions

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

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

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

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:

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)

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

25. Mapping over Generalized Random Geometric Graph (GRGG) • Link capacities can be evaluated in terms of contact probabilities: which depend only on the distance dijbetween the homepoints of i and j • Wecan 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

26. Upper bound : network cut 1 0 1/2 1

27. Average/random flow through the cut • The “average” flow through the cut is computed as • fundamental question: Answer: YES ! 1 Proof’s idea: Consider regular tessellation where squarelets have area γ(n) Take upperandlowerbounds for number of homepoints falling in each squarelets, combined, respectively, with lower and upper bounds of distances between homepoints belonging to different squarelets 0 1/2 1

28. An optimal routing scheme • The above routing strategy sustains per-node traffic Routing strategy: Consider a regular tessellation where squarelets have area s Create a logical route along sequence of horizontal/vertical squarelets, choosing any node whose home-point lie inside traversed squarelet as relay d

29. Sub-critical regime • Network is not uniformly dense • Transmission range may fail even to guarantee network connectivity • When s(d) has finite support: • Nodes have to use • System behaves as network of m static node

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

31. Questions ? Comments ?