Shizhen Zhao, Luoyi Fu , Xinbing Wang Department of Electronic Engineering

1. Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links Shizhen Zhao, Luoyi Fu, Xinbing Wang Department of Electronic Engineering Shanghai Jiao Tong University, China Qian Zhang Department of Computer Scien Engineering Hong Kong, China

2. Outline Background Large-scale Networks Random Connection Model First Passage Percolation Model Network Model & Objective Main Results and Intuitions Simulation Results Concluding Remarks Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 2

3. Large-scale Networks • Network size is growing. • Number of users is growing • Need more base stations • Unreliable links. • Communication between adjacent nodes is not always available • Cause of the unreliability. • Increased interference • Severe environment • Sleep-wake scheduling Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 3

4. Random Connection Model (RCM) • Ramdom Connection Model • Stationary point process (e.x. Poisson point process) • Connection function • A non-increasing function h(·) defined on positive reals • An edge exists between nodes x1 and x2 with probability h(|x1-x2|) • Phase transition. • Condition: • There exists a critical node density , such that • If , an infinite large cluster exists. • if , all clusters are finite almost surely. Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 4

5. Random Connection Model (RCM) • Poisson Boolean Model • Special case of RCM with • Two nodes are connected if and only if their distance is smaller or equal to Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 5

6. First Passage Percolation Model • First Passage Percolation Model. • Assign a random variable to each link . • Define the passage time for a path : • Define the first passage time between nodes and : • Remark. • Use first passage time to model delay • Use the random variable to model the unreliability of links Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 6

7. Outline Background Network Model & Objective Network Model Objective Main Results and Intuitions Simulation Results Concluding Remarks Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 7

8. Network Model-Connectivity • Delay is a basic concern in designing and implementing large scale wireless networks. However, the premise of communication is connectivity. • In large scale wireless network with unreliable links, it is too costly to maintain full connectivity. Therefore, we consider a slightly weaker connectivity-connectivity in percolation sense. • Random Connection Model (RCM) • Instantaneous and Long-term Connectivity • Instantaneous Critical Density ( ) and Long-term Critical Density ( ) Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 8

9. Network Model-Connectivity Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 9

10. Network Model-Delay • Usually, the time needed for links to change state is much larger than the time scale used in scheduling, routing, etc. Therefore, we assume that it won't take much time for transmission between connected node pairs. In this paper, we mainly focus on the delay caused by the lack of instantaneous connectivity which is closely related to the node density in such a netwok. Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 10

11. Network Model-Delay • We assign the delay to each link in the following way, where is the connection function, and is the length of link . • Applying first passage percolation model here, we could define the first passage time for node and : Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 11

12. Objective Previous result: We study: Detailed relationship between and node density . Two steps: Ignore the propagation delay ( ). Consider the propagation delay ( ). Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 12

13. Outline Background Network Model & Objective Main Results and Intuitions Main results Intuitions Impact of Propagation Delay Simulation Results Concluding Remarks Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 13

14. Main Results • Properties of , • There exists , such that • For any , • is a monotone decreasing function Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 14

15. Main Results • Upper bound and lower bound of • is a constant • g(·) is the connection function • is the size of a cluster • E(·) is the expectation Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 15

16. Intuitions-Upper Bound • For two nodes and , the first passage time is bounded by the the passage time along any path connecting and . distantce of one hop average delay of one hop least number of hops Basic idea: and are connected by a sequence of hops with distance smaller or equal to Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 16

17. Intuitions-Upper Bound • Existance of such paths • The long-term critical density can be viewed as the critical density of the Poisson Boolean Model with transmission range • (Proposition 2 in [14]) Let be the critical density for the Poisson Boolean Model when the transmission range is , then • The critical density for is , and . Therefore, such a path exists. Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 17

18. Intuitions-Upper Bound • Least Number of hops of such paths • Scale the network up by , then each hop of such paths becomes , and • We can show that • Then, we can see that Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 18

19. Intuitions-Lower Bound • Waiting occurs at the boundry of a cluster M M M M M M M M M Cluster to Cluster transmission Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 19

20. Intuitions-Lower Bound • Delay between and • Motivation: The minimum number of clusters between and gives a lower bound of delay • Let the size of a cluster be , then the minimum number of clusters is lower bounded by number of clusters-1 Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 20

21. Impact of Propagation Delay • Upper Bound • The average delay of one hop changes • Lower Bound • The minimum number of clusters is lower bounded by average delay of one hop changes Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 21

22. Outline Background Network Models & Objective Main Results and Intuitions Simulation Results without Propagation Delay with propagation delay Concluding Remarks Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 22

23. Without Propagation Delay Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 23

24. With Propagation Delay Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 24

25. Outline Introduction K-hop Clustered Network Models Main Results and Intuitions The Impact of Mobility Concluding Remarks Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 25

26. Concluding Remarks • We have studied the first passage delay of large scale network with unreliable links. • In two senarios (with propagation delay and without propagation delay), we • sketch by excavating its properties; • provide upper bound and lower bound to . Fundamental Relationship between Node Density andDelay in Wireless Ad Hoc Networks with Unreliable Links 26

27. Thank you !