1. Highway Vehicular Delay Tolerant Networks: Information Propagation Speed Properties Emmanuel Baccelli, Philippe Jacquet, Bernard Mans, and Georgios Rodolakis Presented by: Mayank Shekhar

2. Contents Information Propagation Speed Significance Dependency
Mathematical Model for Delay Tolerant Networks
Asymptotic Estimate of Speeds
Simulation Results
Conclusion

3. Information propagation speed
Information Propagation Speed- Speed with which information from a vehicle or a beacon is transmitted from a source vehicle to the destination vehicle in a bidirectional delay tolerant vehicular networks such as highway or roads.
In US, almost 75% of the total miles covered are typically on a bidirectional networks.
What’s the advantage over Radio Propagation? `

5. significance
One of the main reasons for this type of the ad-hoc network is that the dependency on the end to end path is eliminated and communication routes can be made dynamic which can be available based on time and mobility.
Communication where nodes carry packets but also advance themselves towards the destination is one way to define Intermittently Connected Networks(ICNs) or Delay Tolerant Networks(DTNs).
Based on these ideas appropriate ad-hoc network routing protocols for a vehicular network can be designed as opposed to the idea of a fixed ad-hoc network.

6. dependency
Under a certain threshold Information Propagation happens at the average speed of the vehicle but can be expedited exponentially when the number of vehicles are increased.
The distance of propagation is dependent on the vehicles moving in the opposite directions.
This speed is widely dependent on the densities of the vehicles present in either direction on the highway and tends to increase when the number of vehicles increase, this is when a phase transition happens.
An information beacon on any particular lane, which for our case is Eastbound lane is transmitted in a zigzag fashion as the next car in the same lane may be out of the radio range.

7. Mathematical Model for d.t.n.
The information Propagation Speed model is hypothesized on the idea that most of the vehicles are distributed exponentially on the highways or roads.
Earlier studies on packet propagation delay have modelled a Delay Tolerant Network as Erdos-Renyi random graphs.
The authors in this paper have used the Poisson Point Process model in a continuous space that allows for both finite and infinite Radio speed thus making the results of speed propagation more accurate.
Erdos-Renyi random graphs which uses fluid limit techniques to derive relationship between buffer space and packet duplication).

8. Poisson process point model
The authors here use Poisson Process Point which gives the distribution for a range R and intensity λ 𝑒 𝑖𝑠 λ 𝑒 𝑅 𝑒 − λ 𝑒 𝑅
In information propagation for phase transition to happen the distribution must be same for both eastbound and west bound vehicles as
λ 𝑤 𝑅 𝑒 − λ 𝑤 𝑅 = λ 𝑒 𝑅 𝑒 − λ 𝑒 𝑅
The authors have chosen the value of R to be 1 which when plugged in corresponds to the equation:
𝑦= 𝑥𝑒 −𝑥
In probability, statistics and related fields, a Poisson point process or Poisson process (also called a Poisson random measure, Poisson random point field or Poisson point field) is a type of random mathematical object that consists of points randomly located on a mathematical space.[1] The process has convenient mathematical properties,[2] which has led to it being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy,[3] biology,[4] ecology,[5] geology,[6] physics,[7] image processing,[8] and telecommunications.[9][10]. The Poisson point process is often defined on the real line. For example, in queueing theory [11] it is used to model random events, such as the arrival of customers at a store or phone calls at an exchange, distributed in time. In the plane, the point process, also known as a spatial Poisson process,[12] may represent scattered objects such as transmitters in a wireless network,[9][13][14][15] particles colliding into a detector, or trees in a forest. For a range (a,b) the distribution is defined as
𝑃 𝑁 𝑎,𝑏 =𝑛 = [λ 𝑏−𝑎 ] 𝑛 𝑛! 𝑒 −λ(𝑏−𝑎)

9. Contd. λe > λ 𝑤 ∗ => 𝑣 𝑝 > 0
The average speed of transmission is 𝑣 𝑝 defined by Elapsed Distance by Elapsed Time. Now the information propagation speed of the Eastbound cars based on the Poisson Intensities are defined as
λe < λ 𝑤 ∗ => 𝑣 𝑝 = 0 if 𝑣 𝑝 < ∞
λe > λ 𝑤 ∗ => 𝑣 𝑝 > 0
Now in order to bridge the road gap between the cluster of the cars in the opposite lanes for a gap length x is for a kth cluster defined is given by the Laplace transform.
there is no westbound cluster so the propagation speed is zero if vp<infinity.
The first cluster greater than x is the kth cluster, in this case the road length Laplace Transform is equal to ( 𝒇 𝒘 (θ,𝒙) λ 𝒘 λ 𝒘 +θ ) 𝒌
From the sum of the k terms of the transforms we arrive at the average road gap

10. Asymptotic estimate of speed
If the vehicle densities are large the propagation speed becomes where λ 𝑤 , λ 𝑒 -> ∞
𝑣 𝑝 ~2𝑣 𝑒 λ 𝑒 + λ 𝑤 λ 𝑤 λ 𝑒 + λ 𝑒 λ 𝑤
Near the threshold the speed of propagation is
𝑣 𝑝 = 𝐸(𝐷) 𝐸(𝑇)
The information propagation speed grows quasi exponentially wrt the total vehicle density(say this for the first point)

11. s = 𝐸 𝑇 𝑤 𝑣+𝐸( 𝐷 𝑒 ) 𝐸( 𝑇 𝑤 )+ 1 𝑣 𝑟 𝐸( 𝐷 𝑒 )
Contd.
If the radio propagation speed including the timings for store and forward becomes finite and constant equal to 𝑣 𝑟 then the average propagation speed becomes
s = 𝐸 𝑇 𝑤 𝑣+𝐸( 𝐷 𝑒 ) 𝐸( 𝑇 𝑤 )+ 1 𝑣 𝑟 𝐸( 𝐷 𝑒 )

12. Simulation results
The authors perform 2 simulations to compare the results obtained from the above calculations. The first one is the Maple Simulations where the authors generate Poisson traffic of eastbound and westbound traffic on two opposite lanes moving at constant speed, which is set to v=1m/s. The radio propagation range is R=1m, and radio transmissions are instantaneous. The length of the highway is sufficiently large to provide a large number of bridging operations (of order at least ) for all considered traffic densities.
The results obtained from this simulation is pretty close to the asymptotic exponential value.

13. Contd.
The second simulation is done using “ONE” Simulation where all the vehicles are uniformly distributed on both the lanes of the roads and they move at a constant unit speed. However the number of vehicles varies from and Range R as 10m.
Here too we get the same results as the earlier simulation which implies that the information propagation speed remains constant below the threshold but increases dramatically beyond that.
The fastest propagation is measured using epidemic broadcast which means the radio propagation is instantaneous with no buffering and congestion delays.
Draw the graph of speed 𝒗 𝒑 𝒗𝒆𝒓𝒔𝒖𝒔 λ 𝒆 + λ 𝒘 where λ 𝒆 = λ 𝒘 𝒘𝒊𝒕𝒉 𝒂𝒔𝒚𝒎𝒑𝒕𝒐𝒕𝒊𝒄 𝒆𝒔𝒕𝒊𝒎𝒂𝒕𝒆𝒔.

14. Conclusion
The authors in this paper have provided a detailed analysis for information propagation in bidirectional vehicular DTNs.
They also analysed the existence of a threshold, concerning vehicle density, above which information speed increases dramatically over vehicle speed, and below which information propagation speed is on average equal to vehicle speed and giving the exact expression of this threshold.

15. Contd.
They computed the exact expression of this threshold, which characterized the information speed near the threshold and showed that, above the threshold, the information propagation speed increases quasi-exponentially with vehicle density.
Combining all these different situations, we obtain a complete image of the way information propagates in vehicular networks on roads and highways, which is useful in determining the performance limits and designing appropriate routing protocols for VANETs. All their theoretical results were validated with simulations in several environments (The One and Maple).

16. Questions?