A Uniform Continuum Model for the Scaling of Ad-Hoc Networks

1. A Uniform Continuum Model for the Scaling of Ad-Hoc Networks Ernst W GrundkeA Nur Zincir-Heywood Faculty of Computer ScienceDalhousie UniversityHalifax NS www.cs.dal.ca/~grundke/ants/continuumModel.html

2. Acknowledgements The Ant Colony: Nur Zincir-Heywood Allan Jost Owen Yue Donald Morrison Nick Pilon Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

3. In a mobile ad-hoc network, ... • … nodes move • … nodes enter & leave • … communication is short-range wireless • … hosts = routers • … node power is scarce What happens when such networks get large? Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

4. A Continuum Model • Only the node density is known Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

5. A Continuum Model • Only the node density is known • Think of nodes as “smeared out” Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

6. A Uniform Model • Assume similar conditions everywhere: no edge effects X X Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

7. Model • Continuum of nodes • Uniform: no edge effects • Simple, optimistic assumptions • Mathematically tractable: no simulation • Dimensionless parameters • Analytical results • Approximate at best • J. B. S. Haldane, 1928, “On being the right size” Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

8. Network Geometry • Nearest neighbour distance d1 • Node density = 1/(r12) Node Voronoi cell r1 = d1/2 Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

9. Network Geometry • N nodes • Max separation D • D = d1(N-1) D D+d1 Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

10. Node Behaviour • Generate user data randomly: pT packets per unit time • Finite transmission range R:d1 R D • Forward user data packets B C A R Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

11. Nodes Behaviour (cont.) • Motion: pW link events per unit time  = pW/pT (the walk/talk ratio) Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

12. Nodes Behaviour (cont.) • Group of g nodes competing for a single channel:g =(R+r1)2/r12 • Finite transmission bandwidth:pW<b, pT<b in isolation, pW<b/g, pT<b/g in network Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

13. Forwarding User Data Packets • Define forwarding overhead:  = mean hops per packet • Uniformity: data rate = pT(per node) • If packets travel D/2 in hops of R,  = (r1/R)(N-1) • Example: <5 requires N<121. Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

14. Ceiling (Forwarding) pT < b/g pT/b< R/(r1(N1))  r12 /(R+r1)2= r1R/((R+r1)2(N1)) • Small range (R 2r1) is best; then pT/b< 0.22/(N1) • Example: If N=100 then pT/b < 2.5% • Applies regardless of routing & mobility Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

15. Ceiling (Forwarding) pT/b< 0.22/(N1): N Max pT/b 10 10.3% 100 2.47% 1,000 0.73% 10,000 0.22% Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

16. Routing Traffic • Define routing overhead:  = packet transmissions caused by one link event • Uniformity: data rate = pW (per node) • Assume proactive routing; flat. • If each node broadcasts to (g1)/2, N = (g1)/2. • For R 2r1: = (N/4)1 Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

17. Ceiling (Routing) pW < b/g ...pW/b < 1/(2N) approx. • Example: If N=100, pW/b< 0.5%.For b~1Mbps and 1000 bits/packet, pW < 5 link events/second • Numerically not serious for modest N. Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

18. Ceiling (Routing) Routing:  (1/N)Forwarding:  (1/N) N Max pw/b Max pT/b 10 5% 10.3% 100 0.5% 2.47% 1,000 0.05% 0.73% 10,000 0.005% 0.22% Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

19. Total Traffic: Data + Routing pT + pW<b/g ( + ) pT<b/g ( = pW/pT) • Forwarding traffic dominates:  < / • Routing traffic dominates:  > / Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

20. Power • Average antenna power per node:kR2(pT + pW) ~ (N) Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

21. Other Dimensions • See paper! Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

22. Conclusions • For flat ad-hoc networks: Keep N,R small; use hierarchies • Dimensionless parameters:  = pW/pT = the walk/talk ratio  = forwarding overhead  = routing overhead • In 2D:  is (N/R),  is (N/R2) • / characterizes traffic type • Map simulations into a common space? Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling

23. ? Comments?Questions? www.cs.dal.ca/~grundke/ants/continuumModel.html Grundke & Zincir-Heywood, Dalhousie University: Uniform Continuum Model for Scaling