Critical Transmission Range for Mobile Clustered Wireless Networks

1. Critical Transmission Range for Mobile Clustered Wireless Networks Qi Wang, Liang Liu, Xinbing Wang Department of Electronic Engineering Shanghai Jiao Tong University, China

2. Outline Preliminary knowledge Wireless Network Structure I.I.D. model K-Hop in mobile network Network Deployment Network model Main results Discussion 2

3. Wireless Network Structure BS BS MN MN MN MN

4. Wireless Network Structure cluster head cluster member r transmission range Wireless Network Structure

5. Mobility Model - I.I.D. Model At the beginning of each time slot, each cluster member node randomly and uniformly choose a position within the region and remain static for the rest of the time slot. [1] Wang Q, Wang X, Lin X. Mobility increases the connectivity of k-hop clustered wireless networks[C]//Proceedings of the 15th annual international conference on Mobile computing and networking. ACM, 2009: 121-132.

6. Explanation of -hop in mobile network • Data transmission is divided into time slots. • For each cluster member node, if it can move within the transmission region of a certain cluster head in time slots, then it is connected. Otherwise it is disconnected. • If all the member nodes are connected, then the network is fully connected.

7. Explanation of -hop in mobile network disconnected node connected node Connectivity illustration

8. Network Deployment • Cluster head nodes are initially distributed randomly and uniformly within the unit square and always remain static. There is a path that connects all the cluster heads in the initial graph . • Cluster member nodes are initially distributed randomly and uniformly within the unit square and move according to the I.I.D. model. If at time slot ,member node is within the transmission range of head , an edge is added to . • The number of cluster member nodes is . • The number of cluster heads is , where is the cluster head exponent, and .

9. Previous result:

10. Outline Preliminary knowledge Network model Network Parameters Correlated Mobility Model Cluster Scalability Main results Discussion 10

11. Network Parameters • : the number of clusters. • : the radius of each cluster. • : the number of nodes in each cluster. [2] Ciullo D, Martina V, Garetto M, et al. Impact of correlated mobility on delay-throughput performance in mobile ad hoc networks[J]. IEEE/ACM Transactions on Networking (TON), 2011, 19(6): 1745-1758.

12. Correlated Mobility Model • After deploying the initial network architecture, the cluster heads will remain stationary while the cluster members will move. • The movement of a cluster-member node consists of two steps. • The movement of home point. At the beginning of each time slot, each home point will randomly and independently choose a position within the unit square. • The relative movement of cluster member.Each cluster member will uniformly and independently choose its location within its corresponding cluster region.

13. Correlated Mobility Model Correlated Mobility illustration

14. Cluster Scalability • : cluster-sparse state. • : cluster-dense state. • : cluster-inferior dense state.

15. Cluster Scalability Correlated Mobility and Cluster Scalability

16. Outline Preliminary knowledge Network Model Main Results Critical Transmission Range Main Results Discussion 16

17. Critical Transmission Range • Connectivity with probability (w.p.): Let denote the event that the network with cluster member nodes is fully connected, then is the critical transmission range with which the network will be connected with probability (w.p.) if • if and ; • if and .

18. Critical Transmission Range • Connectivity almost surely (a.s.): Let denote the event that the network with cluster member nodes is fully connected, then is the critical transmission range with which the network will be connected almost surely (a.s.) if • if and ; • if and . • “lim inf” denotes limit inferior of a sequence of events and • “lim sup” denotes limit superior of a sequence of events and

19. Main Results • Cluster-sparse state: • Cluster-dense state: • Cluster-inferior dense state: • Cluster-mixed state (w.p.): • Cluster-mixed state (a.s.):

20. Main Results • Cluster-Sparse State (, ) Proposition 1: Let denote the probability that the network is disconnected. If , where , , and , we have

21. Main Results • Cluster-Sparse State (, ) • Clustering effect is dominant in this state. • () is due to the principle of inclusion-exclusion. • denotes the probability that the th cluster is disconnected. • , .

22. Main Results • Cluster-Dense State (, ) • Cluster members behave more like independent nodes. • , .

23. Main Results • Cluster-Inferior Dense State (, ) • Cut out non-overlapping circular areas (sub-areas), with radius . • Considering time slots, we obtain sub-clusters. • , , .

24. Main Results • Cluster-Inferior Dense State (, ) • We use a sequence to denote a sub-cluster, where , . • A node if and only if is in sub-area during time slot . sub-area during time slot , sub-area during time slot , sub-area during time slot , sub-area during time slot . E.g.

25. Main Results • Cluster-Mixed State (w.p.) • This state is a generalization of the three separate states. • In this state, we denote the network as , the th cluster has members and radius . • , , . There is no linear relation between and . • : number of clusters in the cluster-sparse state (). • : number of clusters in the cluster-inferior dense state (). • : number of clusters in the cluster-dense state ().

26. Main Results • Cluster-Mixed State (w.p.) denotes the number of node groups in the whole network.

27. Main Results • Cluster-Mixed State (a.s.) • A stronger connectivity condition than w.p. connectivity. • Major approach: Borel-Cantelli lemma. • Price from w.p. connectivity to a.s. connectivity: .

28. Outline Preliminary knowledge Network Model Main Results Discussion 28

29. Discussion

30. Discussion Price from w.p. connectivity to a.s. connectivity:

31. Thank you!