Mobility Increases the Connectivity of K-hop Clustered Wireless Networks

Mobility Increases the Connectivity of K-hop Mobility Increases the Connectivity of K-hop Clustered Wireless NetworksClustered Wireless Networks

Qingsi Wang, Xinbing WangDepartment of Electronic Engineering

Shanghai Jiao Tong University, China

Xiaojun LinDepartment of Electrical and Computer Engineering

Purdue University, USA

Mobility Increases the Connectivity of K-hop Clustered Wireless Networks 2

OutlineOutline IntroductionIntroduction

BackgroundBackgroundMotivationsMotivationsObjectivesObjectives

K-hop Clustered Network ModelsK-hop Clustered Network Models

Main Results and IntuitionsMain Results and Intuitions

The Impact of Mobility The Impact of Mobility

Concluding RemarksConcluding Remarks

3

Background – I/IIBackground – I/II

Connectivity is a basic concern in designing and Connectivity is a basic concern in designing and implementing wireless networks. implementing wireless networks.

Three main schemes of connecting strategies are Three main schemes of connecting strategies are proposed in the literature.proposed in the literature. Distance-based strategyDistance-based strategy Number-of-neighbor-based strategyNumber-of-neighbor-based strategy Sector-based strategySector-based strategy

Mobility Increases the Connectivity of K-hop Clustered Wireless Networks

4

Background – II/IIBackground – II/II

The connectivity of networks under The connectivity of networks under the distance-based the distance-based connecting strategyconnecting strategy is widely studied: is widely studied:

The critical value of , overall connectivity can The critical value of , overall connectivity can

be established with probability approaching one as if and be established with probability approaching one as if and

only if [1] [2]. only if [1] [2].


log ( )( )

n c nr n

n

n

( )c n

[1] P. Gupta and P.R. Kumar, “Critical Power for Asymptotic Connectivity in Wireless Networks”, [1] P. Gupta and P.R. Kumar, “Critical Power for Asymptotic Connectivity in Wireless Networks”, 1998.1998.

[2] M.D. Penrose, “The Longest Edge of the Random Minimal Spanning Tree”, 1997.[2] M.D. Penrose, “The Longest Edge of the Random Minimal Spanning Tree”, 1997.

5

MotivationMotivation

The network models studied in these prior works The network models studied in these prior works are non-clustered (or flat) and stationary are non-clustered (or flat) and stationary networks.networks.

ClusteringClustering and and mobilitymobility have been found to have been found to improve various aspects of network performance. improve various aspects of network performance.

Studies on the connectivity of Studies on the connectivity of mobilemobile and and clusteredclustered networks are quite networks are quite limitedlimited..--- We don’t even know the definition of the connectivity under such --- We don’t even know the definition of the connectivity under such

circumstancescircumstances..



Objective – I/IIObjective – I/II

Open question:Open question:What is the impact of mobility on What is the impact of mobility on

connectivity of clustered networks connectivity of clustered networks subject to delay constraints?subject to delay constraints?

We studyWe study The critical transmission range for The critical transmission range for

connectivityconnectivityK-hop mobile clustered networks (delay K-hop mobile clustered networks (delay

guarantee)guarantee)Random walk mobility model with non-trivial Random walk mobility model with non-trivial

velocity velocity i.i.d. mobility model (fast mobility).i.i.d. mobility model (fast mobility).


Objective – II/IIObjective – II/II

We compare with the critical transmission range for We compare with the critical transmission range for connectivity in stationary k-hop clustered networks.connectivity in stationary k-hop clustered networks.

Implications on Implications on the power-delay trade-off the power-delay trade-off the energy efficiencythe energy efficiency

Our results show that:Our results show that:

Mobility does improve Mobility does improve connectivityconnectivity in k-hop clustered in k-hop clustered networks, and it also significantly decreases the networks, and it also significantly decreases the energy energy consumptionconsumption and the and the power-delay trade-offpower-delay trade-off..

8

OutlineOutlineIntroductionIntroduction

K-hop Clustered Network ModelsK-hop Clustered Network ModelsAn overview of flat networksAn overview of flat networksK-hop clustered network modelsK-hop clustered network models





9

An Overview of Flat NetworksAn Overview of Flat Networks

Defining Connectivity in Flat NetworksDefining Connectivity in Flat Networks LetLet denotedenote a unit area in Ra unit area in R22, and , and ((nn)) bebe thethe graphgraph formedformed when when

nn nodes nodes areare placedplaced uniformlyuniformly andand independentlyindependently inin . . An edge An edge eeijij exists between two nodes exists between two nodes ii and and jj, if the distance , if the distance

between them is less than between them is less than rr((nn) under the distance-based strategy.) under the distance-based strategy.

Flat networks under Flat networks under the distance-based the distance-based connecting strategyconnecting strategy


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Clustered networksClustered networks nn normal nodes normal nodes and and nndd cluster-head nodes cluster-head nodesStatic or mobileStatic or mobile

Mobility ModelMobility ModelRandom Walk Mobility Model with Non-Random Walk Mobility Model with Non-

Trivial VelocityTrivial Velocity– Uniformly chosen direction Uniformly chosen direction – Constant velocity (continuous path)Constant velocity (continuous path)

I.I.D. Mobility ModelI.I.D. Mobility Model– IndependentlyIndependently and uniformly reshuffledand uniformly reshuffled– Static within a single time slotStatic within a single time slot


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Mobile Networks: Transmission SchemeMobile Networks: Transmission Scheme

TTLTTL (time to live): the number of hops that the packet has (time to live): the number of hops that the packet has been forwarded.been forwarded.

SYNSYN (synchronize): preamble for data-flows synchroni- (synchronize): preamble for data-flows synchroni-zationzation


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Mobile Networks: Routing StrategyMobile Networks: Routing Strategy

Direct delivery to the cluster head without relayDirect delivery to the cluster head without relay


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Clustered Network ModelsClustered Network Models

For stationary k-hop clustered networks, we say For stationary k-hop clustered networks, we say that a cluster member is that a cluster member is connectedconnected if it can reach if it can reach a cluster head within a cluster head within k hopsk hops. .

For mobile clustered networks, a cluster member For mobile clustered networks, a cluster member is is connectedconnected if it can reach a cluster head within if it can reach a cluster head within k k slotsslots. .

If all the cluster members in a network are If all the cluster members in a network are connected, we define that the network has connected, we define that the network has full full connectivityconnectivity..


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Main Results and IntuitionsMain Results and IntuitionsDefinition of critical transmission rangeDefinition of critical transmission rangeMain resultsMain results Intuitive explanationsIntuitive explanations




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Critical Transmission RangeCritical Transmission Range

Definition: For stationary or mobile k-hop clustered Definition: For stationary or mobile k-hop clustered networks, networks, rr((nn)) is the critical transmission range if is the critical transmission range if

E:E: the event that all cluster members are connected the event that all cluster members are connected

lim ( ) 1, if ( ) and 1;

lim ( ) 1, if ( ) and 1.n

n

P E r cr n c

P E r c r n c


16

Main ResultsMain Results

Under the random walk mobility pattern, the critical Under the random walk mobility pattern, the critical

transmission range is , where d is the cluster transmission range is , where d is the cluster

head exponent, head exponent, 0 < 0 < dd ≤ 1 ≤ 1, and v is the velocity of all , and v is the velocity of all

member nodes.member nodes.

Under the i.i.d. mobility pattern, the critical transmission Under the i.i.d. mobility pattern, the critical transmission

range is , where range is , where 1/1/kk < < dd ≤ 1 ≤ 1..

For stationary k-hop clustered networks, the critical For stationary k-hop clustered networks, the critical

transmission range is , where transmission range is , where 0 < 0 < dd < 1 < 1..

log( )

2 d

nr n

kvn

log( )

d

nr n

k n

1 log( )

d

d nr n

k n


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Intuitive Explanations – I/IIIntuitive Explanations – I/II

Suppose there are Suppose there are nn cluster members and cluster members and nndd cluster heads cluster heads uniformly distributed in a unit square. Thus, roughly uniformly distributed in a unit square. Thus, roughly speaking, there is one cluster head within an area of speaking, there is one cluster head within an area of 1/1/nndd . .

Area argument for random walk mobility:Area argument for random walk mobility:


1( ) , or

1( )

d

d

kvr nn

r nkvn

k·v

r(n)

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Intuitive Explanations – I/IIIntuitive Explanations – I/II

Area argument for i.i.d mobility:Area argument for i.i.d mobility:


1( ) , or

1( )

d

d

kr nn

r nkn

２

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Intuitive Explanations – II/IIIntuitive Explanations – II/II

Area argument for static case:Area argument for static case:


2 1( ) , or

1 ( )

d

d

kr nn

r nk n

１…

k·r(n)

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The Impact of Mobility The Impact of Mobility Transmission powerTransmission powerEnergy consumption per flowEnergy consumption per flowDiscussionDiscussion



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Transmission Power – I/IITransmission Power – I/II

We assume the free space propagation model, i.e.,We assume the free space propagation model, i.e.,

PPtt = transmission power of an isotropic source, = transmission power of an isotropic source,

GGtt = transmitting antenna gain, = transmitting antenna gain,

GGrr = receiving antenna gain, = receiving antenna gain,

ll = propagation distance between antennas, = propagation distance between antennas,

λλ = = carrier wavelength.carrier wavelength.

Replace Pr with PrReplace Pr with Prthth and replace the propagation distance l and replace the propagation distance l

by the transmission range r. We then haveby the transmission range r. We then have


2

,4r t t rP PG G

l

2tP r

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Energy ConsumptionEnergy Consumption

Let Let ĒĒ denote the energy consumption per flow. denote the energy consumption per flow.

where is the average number of hops per flow.where is the average number of hops per flow.

PPtt affects a single node in energy-constrained networks affects a single node in energy-constrained networks

like wireless sensor networks. like wireless sensor networks.

ĒĒ provides a picture of the life-time expectation both of provides a picture of the life-time expectation both of each single node and of the entire network.each single node and of the entire network.


2 ,tE HP Hr H

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Discussion – I/VIDiscussion – I/VI

Note that in these calculations, we have ignored the Note that in these calculations, we have ignored the energy consumption due to mobility. Hence, these results energy consumption due to mobility. Hence, these results should not be interpreted as a reason to introduce mobility should not be interpreted as a reason to introduce mobility to an otherwise static network, but rather represent to an otherwise static network, but rather represent an an inherent advantage of having mobility in the systeminherent advantage of having mobility in the system. .

Similarly, the comparison with the flat network is not Similarly, the comparison with the flat network is not entirely fair, since in a clustered network, a packet only entirely fair, since in a clustered network, a packet only needs to reach a cluster head. Hence, our following results needs to reach a cluster head. Hence, our following results should be viewed as should be viewed as an inherent advantage of clustered an inherent advantage of clustered network due to the availability of infrastructure supportnetwork due to the availability of infrastructure support..


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Discussion – II/VIDiscussion – II/VI

Using the previous results of the critical transmission range Using the previous results of the critical transmission range rr((nn)), we can compute the order of , we can compute the order of PPtt and and ĒĒ. All the results . All the results

in this paper are reported in the following table.in this paper are reported in the following table.


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Discussion – III/VIDiscussion – III/VI

We haveWe have33

random walk mobility with clustering can increase the number of random walk mobility with clustering can increase the number of

transmission that a node can undertake and transmission that a node can undertake and extend the life-timeextend the life-time both of each single node and of the entire network.both of each single node and of the entire network.


{r.w.} ( {flat}) and {r.w.} ( {flat})t tP o P E o E

33 Note: By the implication from [27], we know that when d < 1/2, bottleneck of capacity may appear, Note: By the implication from [27], we know that when d < 1/2, bottleneck of capacity may appear, and thus we assume d > 1/2 in our following discussion.and thus we assume d > 1/2 in our following discussion.[27] S. Toumpis, “Capacity Bounds for Three Classes of Wireless Networks: Asymmetric, Cluster, [27] S. Toumpis, “Capacity Bounds for Three Classes of Wireless Networks: Asymmetric, Cluster, and Hybrid”, 2004.and Hybrid”, 2004.

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Discussion – IV/VIDiscussion – IV/VI

We haveWe have To identify the contribution of mobility and k-hop clustering on the To identify the contribution of mobility and k-hop clustering on the

improvement of network performance, we haveimprovement of network performance, we have


1/ 2

due to due tor.w. mobility, -hop

non-trivial clustering

11/ 2

loglog{r.w.} {flat}

log log{r.w.} {stat} {r.w.} {stat}

log l{stat} {flat} {stat} {flat} {r.w.}

d d

t td d k

dt t td

nnE E

n nn nE E P P

n n

nE E P n P P

n

；

； 1

due to-hop due to

clusteringr.w. mobility,non-trivial

og{flat}d

td

k

nn P

n

27

Discussion – V/VIDiscussion – V/VI

We haveWe have From the perspective of From the perspective of energy consumptionenergy consumption per flow, clustered per flow, clustered

networks have an inherent advantage in terms of energy-efficiency networks have an inherent advantage in terms of energy-efficiency due to the availability of infrastructure support.due to the availability of infrastructure support.

Mobile k-hop clustered networks under the Mobile k-hop clustered networks under the i.i.d mobility modeli.i.d mobility model and and stationary clustered networksstationary clustered networks may have comparable performance may have comparable performance and this can be understood intuitively since nodes under the i.i.d. and this can be understood intuitively since nodes under the i.i.d. mobility model actually remain static during the time-slot.mobility model actually remain static during the time-slot.


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Discussion – VI/VIDiscussion – VI/VI

In conclusion, In conclusion, random walk mobilityrandom walk mobility with non-trivial with non-trivial velocity velocity increases connectivityincreases connectivity in k-hop clustered in k-hop clustered networks, and thus significantly networks, and thus significantly improves the energy improves the energy efficiencyefficiency and the and the power-delay trade-offpower-delay trade-off of the of the network.network.


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30

Concluding Remarks – I/IIConcluding Remarks – I/II

We have studied the effects of mobility on the critical We have studied the effects of mobility on the critical transmission range for asymptotic connectivity in k-hop transmission range for asymptotic connectivity in k-hop clustered networks.clustered networks.

Our contributions are twofold.Our contributions are twofold. developed the developed the critical transmission rangecritical transmission range for the mobile k-hop for the mobile k-hop

clustered network under the random walk mobility model with non-clustered network under the random walk mobility model with non-trivial velocity and the i.i.d. mobility model, and for the stationary k-trivial velocity and the i.i.d. mobility model, and for the stationary k-hop clustered network, respectively.hop clustered network, respectively.

These formulations not only provide an asymptotic description of These formulations not only provide an asymptotic description of the critical power needed to maintain the connectivity of the the critical power needed to maintain the connectivity of the network, but also help to identify the network, but also help to identify the contribution of mobilitycontribution of mobility in the in the improvement of network performance.improvement of network performance.


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Concluding Remarks – II/IIConcluding Remarks – II/II

For future work:For future work: Extend the results for the random walk mobility model to account Extend the results for the random walk mobility model to account

for the case where each node moves with for the case where each node moves with different speeddifferent speed;; In our current model for random walk, each node changes In our current model for random walk, each node changes

direction after one time-slot. An interesting extension is to study direction after one time-slot. An interesting extension is to study the case where the the case where the change of directions occurs at random timeschange of directions occurs at random times (e.g., a node may move a random distance before it changes (e.g., a node may move a random distance before it changes direction).direction).

Account for Account for wireless interferencewireless interference in the system. in the system. It would be interesting to study the case where It would be interesting to study the case where cluster-heads may cluster-heads may

movemove as well. as well.


Thank you !Thank you !

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Reference – I/IIReference – I/II [1] P. Gupta and P.R. Kumar, “Critical Power for Asymptotic Connectivity in Wireless [1] P. Gupta and P.R. Kumar, “Critical Power for Asymptotic Connectivity in Wireless

Networks,” Stochastic Analysis, Control, Optimization and Applications: A Volume in Networks,” Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, W.M. McEneaney, G. Yin, and Q. Zhang, Boston: Birkhauser, Honor of W.H. Fleming, W.M. McEneaney, G. Yin, and Q. Zhang, Boston: Birkhauser, 1998.1998.

[2] M.D. Penrose, “The Longest Edge of the Random Minimal Spanning Tree,” Annals of [2] M.D. Penrose, “The Longest Edge of the Random Minimal Spanning Tree,” Annals of Applied Probability, vol. 7, pp. 340-361, 1997.Applied Probability, vol. 7, pp. 340-361, 1997.

[8] P. Gupta and P.R. Kumar, “The Capacity of Wireless Networks,” IEEE Transactions [8] P. Gupta and P.R. Kumar, “The Capacity of Wireless Networks,” IEEE Transactions on Information Theory, vol. 46, pp. 388-404, March 2000.on Information Theory, vol. 46, pp. 388-404, March 2000.

[9] P. Gupta, R. Gray, and P.R. Kumar, “An Experimental Scaling Law for Ad Hoc [9] P. Gupta, R. Gray, and P.R. Kumar, “An Experimental Scaling Law for Ad Hoc Networks,” Univ. of Illinois at Urbana-Champaign, May 2001.Networks,” Univ. of Illinois at Urbana-Champaign, May 2001.

[10] W. Heinzelman, A. Chandrakasan and H. Balakrishnan, “Energy-efficient [10] W. Heinzelman, A. Chandrakasan and H. Balakrishnan, “Energy-efficient Communication Protocol for Wireless Micro Sensor Networks,” in Proc. the 33rd Annual Communication Protocol for Wireless Micro Sensor Networks,” in Proc. the 33rd Annual Hawaii International Conference on System Sciences, pp. 3005-3014, 2000.Hawaii International Conference on System Sciences, pp. 3005-3014, 2000.

[11] Qiangfeng Jiang and D. Manivannan, “Routing Protocols for Sensor Networks,” in [11] Qiangfeng Jiang and D. Manivannan, “Routing Protocols for Sensor Networks,” in Consumer Communications and Networking Conference (CCNC 2004), pp. 93-98, 2004.Consumer Communications and Networking Conference (CCNC 2004), pp. 93-98, 2004.

[19] U. Kozat and L. Tassiulas, “Throughput capacity of random ad hoc networks with [19] U. Kozat and L. Tassiulas, “Throughput capacity of random ad hoc networks with infrastructure support,” in Proc. ACM MobiCom 2003, Annapolis, MD, USA, June 2003.infrastructure support,” in Proc. ACM MobiCom 2003, Annapolis, MD, USA, June 2003.

[20] M. Grossglauser and D. Tse, “Mobility Increases the Capacity of Ad Hoc Wireless [20] M. Grossglauser and D. Tse, “Mobility Increases the Capacity of Ad Hoc Wireless Networks,” IEEE/ACM Transactions on Networking, vol. 10, no. 4, pp. 477-486, August Networks,” IEEE/ACM Transactions on Networking, vol. 10, no. 4, pp. 477-486, August 2002.2002.


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Reference – II/IIReference – II/II [21] S. Čapkun, J. Hubaux and L. Buttyán, “Mobility Helps Security in Ad Hoc Networks,” [21] S. Čapkun, J. Hubaux and L. Buttyán, “Mobility Helps Security in Ad Hoc Networks,”

in Proc. ACM MobiHoc 2003, June 2003.in Proc. ACM MobiHoc 2003, June 2003. [27] S. Toumpis, “Capacity Bounds for Three Classes of Wireless Networks: [27] S. Toumpis, “Capacity Bounds for Three Classes of Wireless Networks:

Asymmetric, Cluster, and Hybrid,” in Proc. ACM MobiHoc 2004, pp. 133-144, Roppongi, Asymmetric, Cluster, and Hybrid,” in Proc. ACM MobiHoc 2004, pp. 133-144, Roppongi, Japan, May 24-26, 2004.Japan, May 24-26, 2004.


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Mobility Pattern – II/IIMobility Pattern – II/II

Due to the assumption that Due to the assumption that vv = = Θ(1)Θ(1), , the mixing time under the mixing time under the r.w. mobility model is the r.w. mobility model is on the same orderon the same order as the mixing as the mixing time under the i.i.d. mobility model. time under the i.i.d. mobility model.

However, since under the r.w. mobility model nodes can However, since under the r.w. mobility model nodes can communicate during the course of movement, they will communicate during the course of movement, they will have a higher chance to connect to the cluster head have a higher chance to connect to the cluster head compared to nodes under the i.i.d. mobility model.compared to nodes under the i.i.d. mobility model. 1 1

11 Note : This advantage will become clear after we define the transmission scheme. Note : This advantage will become clear after we define the transmission scheme.


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Motivation – II/IIMotivation – II/II

In a clustered network, a packet only needs to reach one In a clustered network, a packet only needs to reach one of the cluster heads. of the cluster heads. In a stationary k-hop clustered network, a packet must reach a In a stationary k-hop clustered network, a packet must reach a

cluster head within k hops. cluster head within k hops. In a mobile k-hop clustered network, a packet must reach a cluster In a mobile k-hop clustered network, a packet must reach a cluster

head directly in k time-slots. head directly in k time-slots.

Clearly, clustering has an inherent advantage compared to Clearly, clustering has an inherent advantage compared to flat networks, and it can alter the energy efficiency and flat networks, and it can alter the energy efficiency and delay of the system.delay of the system. a different critical transmission range for connectivitya different critical transmission range for connectivity different delaydifferent delay different energy consumption of the networkdifferent energy consumption of the network


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Routing Strategy – II/IIRouting Strategy – II/II

Such an assumption would be valid when Such an assumption would be valid when The cluster heads are static and the cluster member has The cluster heads are static and the cluster member has

knowledge of its own position and the positions of cluster heads;knowledge of its own position and the positions of cluster heads; The cluster heads broadcast a pilot signal that covers the The cluster heads broadcast a pilot signal that covers the

transmission range of a cluster membertransmission range of a cluster member.. We do not actually use multi-hop transmissions in mobile We do not actually use multi-hop transmissions in mobile

k-hop clustered networks because multi-hop transmissions k-hop clustered networks because multi-hop transmissions require require significantly higher overhead significantly higher overhead due to the need to due to the need to discover cluster-heads at a further distance away and to discover cluster-heads at a further distance away and to establish multi-hop paths on demand.establish multi-hop paths on demand.

We proposed a simplified routing strategy to avoid the We proposed a simplified routing strategy to avoid the technicalities of a more complicated one which may technicalities of a more complicated one which may obscure our main target.obscure our main target.


Mobility Increases the Connectivity of K-hop Clustered Wireless Networks

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