Critical Transmission Range for Mobile Clustered Wireless Networks

Qi Wang, Liang Liu, Xinbing WangDepartment of Electronic EngineeringShanghai Jiao Tong University, China

2

OutlinePreliminary knowledge

Wireless Network Structure I.I.D. modelK-Hop in mobile networkNetwork Deployment

Network model

Main results

Discussion

Wireless Network Structure

BS

BS

MN

MN

MN

MN

Wireless Network Structure

cluster headcluster member transmission range

r

Wireless Network Structure

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.

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.

Explanation of -hop in mobile network

Connectivity illustration

connected node

disconnected node

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 .

log 1, 1cnr

k n k

Previous result:

10

Outline

Preliminary knowledge

Network modelNetwork ParametersCorrelated Mobility ModelCluster Scalability

Main results

Discussion

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.

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.

Correlated Mobility Model

Correlated Mobility illustration

Cluster Scalability

: cluster-sparse state.

: cluster-dense state.

: cluster-inferior dense state.

Cluster Scalability

Correlated Mobility and Cluster Scalability

16

Outline

Preliminary knowledge

Network Model

Main ResultsCritical Transmission RangeMain Results

Discussion

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 .

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

Main Results

Cluster-sparse state:

Cluster-dense state:

Cluster-inferior dense state:

Cluster-mixed state (w.p.):

Cluster-mixed state (a.s.):

Main Results

Cluster-Sparse State (, )

Proposition 1: Let denote the probability that the network is disconnected. If , where , , and , we have

. ._inf ( , , ,l ) (1 )im , w pf css cn

n r e e

P

Main Results

Cluster-Sparse State (, )

.

1_

1

. ( )( , , , , ) ( ) ( )w pf c

m m

i i ji i j i

ss c F F Fn r

P P P

Clustering effect is dominant in this state.

() is due to the principle of inclusion-exclusion.

denotes the probability that the th cluster is disconnected.

, .

. . logw pc

nrk n

Main Results

Cluster-Dense State (, )

Cluster members behave more like independent nodes.

, .

. . logw pc

nrk n

Main Results

Cluster-Inferior Dense State (, )

Cut out non-overlapping circular areas (sub-areas), with radius .

Considering time slots, we obtain sub-clusters.

, , .

. . [ ( 2 ) ]logw pc

k nrk n

Main Results

Cluster-Inferior Dense State (, )

E.g.

sub-area during time slot ,

sub-area during time slot ,

sub-area during time slot ,

sub-area during time slot .

We use a sequence to denote a sub-cluster, where , .

A node if and only if is in sub-area during time slot .

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 .

32( 2 )

11 1

i

mmk

ii i

m m n

: 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 ().

Main Results

. . logw pc

mrk n

Cluster-Mixed State (w.p.)

denotes the number of node groups in the whole network.

Main Results

Cluster-Mixed State (a.s.)

. . . . 2log2a s w pc c

mr rk n

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

Discussion

Comparisonbetween correlatedmobility∧ i . i . d .mobility

𝐂𝐨𝐫𝐫𝐞𝐥𝐚𝐭𝐞𝐝 𝐦𝐨𝐛𝐢𝐥𝐢𝐭𝐲𝐚𝐧𝐝 𝐜𝐥𝐮𝐬𝐭𝐞𝐫 𝐬𝐜𝐚𝐥𝐚𝐛𝐢𝐥𝐢𝐭𝐲 𝐜𝐚𝐧𝐡𝐞𝐥𝐩𝐬𝐚𝐯𝐞𝐭𝐫𝐚𝐧𝐬𝐦𝐢𝐬𝐬𝐢𝐨𝐧 𝐩𝐨𝐰𝐞𝐫 .

Discussion

Price from w.p. connectivity to a.s.

connectivity:

Thank you!