1. Analytical Model for Connectivity in Vehicular Ad Hoc Networks Published in IEEE TVT vol. 57, no. 6, November 2008 Authors: Saleh Yousefi, Eitan Altman, Rachid El-Azouzi, Mahmood Fathy (INRIA, Sophia Antipolis, France) Presenter: Hojin Lee

2. Connectivity Cluster Platoon size –The number of vehicles in each spatial connected cluster –How many vehicles can hear or exchange data Information sharing … Connectivity distance –The length of the connected path –How far are packets delivered by wireless multi-hop (forwarding) Announcement area Accessibility to roadside equipment (e.g., Internet gateway) …

3. Factors for connectivity Traffic pattern –Traffic flow –Average speed –Density –… –Cf> Traffic flow = average speed x density

4. Assumptions Vehicles arrival follows a Poisson process –Poisson process? N discrete levels of constant speed –This assumption will be relaxed to truncated Gaussian distribution

5. Basic Idea M/D/∞ queueing system –Queueing system? Arrival Service queued –Borrow the concept of busy period Shift from time domain to space domain –Inter-arrival time  inter-vehicle distance –Service time  transmission range –Busy period length  connectivity distance

6. Inter-vehicle distance (1/3) Arrival rate of cars with speed v i : λ i –Poisson process (assumption) Total arrival rate: λ = ∑ i λ i –Poisson process (merging) Occurrence probability of cars with speed v i – P i = λ i /λ

7. Inter-vehicle distance (2/3) Inter-vehicle distance between cars with speed v i –Inter-arrival time sequence of cars with speed v i (T n i :=t n i - t n-1 i ) Exponentially distributed with a parameter λ i –E[T i ]=1/λ i –Inter-vehicle distance sequence of cars with speed v i S n i := T n i v i E[S i ]= v i /λ i –Exponentially distributed with a parameter λ i /v i = λ P i /v i

8. Inter-vehicle distance (3/3) Poisson distribution: merge and split Inter-vehicle distance between any cars –Inter-arrival time: exponential distribution ≡ the number of arrival: Poisson distribution –Merging inter-vehicle distance of cars with speed v i Exponentially distributed with a parameter λ L = ∑ i λ i /v i = λ∑ i P i /v i –CDF: F L (x)=Pr(L<x)=1-exp(-λ L x) –Pr(L>x)=exp(-λ L x)

9. Speed levels (1/2) The speeds in the free-flow traffic: normally distributed – –μ: the average speed, σ: the standard deviation

10. Speed levels (2/2) Use a truncated version –to avoid dealing with negative speeds or even getting close to zero speed –v min, v max (μ -3σ, μ + 3σ) – about 99.7% of the whole area – where v min < v < v max, and

11. Queueing system Inter-arrival time distribution –Inter-vehicle distance: exponential distribution with a parameter λ d Service time distribution –Transmission range: deterministic R The number of queues –∞ M/D/∞ queueing system

12. Platoon Size The number of customers served during busy period  platoon size Probability generating function of – – PMF of platoon size – Expectation of platoon size –

13. Connectivity distance Pdf of busy period  pdf of connectivity distance d: random variable for connectivity distance f d (s) = Laplace transform (moment generating function) of the pdf – Expectation of connectivity distance –

14. Conclusions and Future directions Busy period in Queueing system –Time domain  space domain –This framework is easily adaptable for other fields Also consider speed levels –But, in free-flow traffic state (normal distribution - independence) –Not for car-following model Platoon size –Shared data storage like p2p –Unit of DTN routing Platoon size/connectivity length –Could be used as the congestion metric Forwarding or carrying –Performance Evaluation of a DTN as a City-wide Infrastructure Network - CFI '09 –Delay Bounded Routing in Vehicular Ad-hoc Networks – ACM MobiHoc ‘08

15. Appendix

16. Distributions (1/2) Bernoulli trial –Success/failure Binomial distribution  Poisson distribution –Independent event (arrival) –Success rate  arrival rate –Discrete time  continuous time Geometric distribution  Exponential distribution

17. Distributions (2/2) Normal distribution –PDF: –CDF (standard):

18. Poisson XXX Poisson distribution Poisson process Stochastic process Markov process

19. Generating function Probability generating function – – The PMF of X is recovered by taking derivatives of G – The expectation of X is given by Moment generating function ((Laplace) transform) –