Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links Shizhen Zhao, Luoyi Fu, Xinbing Wang Department

Fundamental Relationship between Node Density Fundamental Relationship between Node Density andand Delay in Wireless Ad Hoc Networks with Delay in Wireless Ad Hoc Networks with

Unreliable LinksUnreliable Links

Shizhen Zhao, Luoyi Fu, Xinbing WangDepartment of Electronic Engineering

Shanghai Jiao Tong University, China

Qian ZhangDepartment of Computer Scien Engineering

Hong Kong, China

Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links 2

OutlineOutline BackgroundBackground

Large-scale NetworksLarge-scale Networks Random Connection ModelRandom Connection Model First Passage Percolation ModelFirst Passage Percolation Model

Network Model & ObjectiveNetwork Model & Objective

Main Results and IntuitionsMain Results and Intuitions

Simulation ResultsSimulation Results

Concluding RemarksConcluding Remarks

3

Large-scale Networks Large-scale Networks

Network size is growing.Network size is growing. Number of users is growingNumber of users is growing Need more base stationsNeed more base stations

Unreliable links.Unreliable links. Communication between adjacent nodes is not always availableCommunication between adjacent nodes is not always available

Cause of the unreliability.Cause of the unreliability. Increased interferenceIncreased interference Severe environmentSevere environment Sleep-wake schedulingSleep-wake scheduling

Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

4

Random Connection Model (RCM) Random Connection Model (RCM)

Ramdom Connection ModelRamdom Connection Model Stationary point process (e.x. Poisson point process)Stationary point process (e.x. Poisson point process) Connection functionConnection function

1.1. A non-increasing function h(·) defined on positive realsA non-increasing function h(·) defined on positive reals

2.2. An edge exists between nodes xAn edge exists between nodes x11 and x and x22 with probability h(|x with probability h(|x11-x-x

22|)|)

Phase transition.Phase transition. Condition: Condition:

There exists a critical node density , such that There exists a critical node density , such that If , an infinite large cluster exists.If , an infinite large cluster exists.

1.1. if , all clusters are finite almost surely.if , all clusters are finite almost surely.

20 ( )

Rh r dr

cc

c


5

Random Connection Model (RCM) Random Connection Model (RCM)

Poisson BooleanPoisson Boolean Model Model Special case of RCM withSpecial case of RCM with

Two nodes are connected if and only if their distance is smaller or Two nodes are connected if and only if their distance is smaller or

equal toequal to

0

0

1,( )

0,

r rh r

r r

0r


6

First Passage Percolation ModelFirst Passage Percolation Model

First Passage Percolation Model.First Passage Percolation Model. Assign a random variable to each link .Assign a random variable to each link . Define the passage time for a path :Define the passage time for a path :

Define the first passage time between nodes and :Define the first passage time between nodes and :

Remark.Remark. Use first passage time to model delayUse first passage time to model delay Use the random variable to model the unreliability of links Use the random variable to model the unreliability of links

( )cT e e

( ) ( )p ce

T T e

( , ) inf{ ( ) : }pT x y T is a path from x to y x y

( )cT e


7

OutlineOutline

BackgroundBackground

Network Model & ObjectiveNetwork Model & Objective Network ModelNetwork Model ObjectiveObjective





8

Network Model-ConnectivityNetwork Model-Connectivity

DelayDelay is a basic concern in designing and implementing is a basic concern in designing and implementing larlarge scale ge scale wireless networks. wireless networks. However, the premise of comHowever, the premise of communication is connectivity.munication is connectivity.

In large scale wireless network with unreliable links, it is toIn large scale wireless network with unreliable links, it is too costly to maintain full connectivity. Therefore, we consido costly to maintain full connectivity. Therefore, we consider a slightly weaker connectivity-connectivity in percolation er a slightly weaker connectivity-connectivity in percolation sense.sense. Random Connection Model (RCM)Random Connection Model (RCM) Instantaneous and Long-term ConnectivityInstantaneous and Long-term Connectivity Instantaneous Critical Density ( ) and Long-term Critical Density Instantaneous Critical Density ( ) and Long-term Critical Density

( )( )I

L


9

I I L L

Network Model-ConnectivityNetwork Model-Connectivity

( )g r

( )g r

rO 0r


10

Network Model-DelayNetwork Model-Delay

Usually, the time needed for links to change state is much Usually, the time needed for links to change state is much larger than the time scale used in scheduling, routing, etc. larger than the time scale used in scheduling, routing, etc. Therefore, we assume that it won't take much time for tranTherefore, we assume that it won't take much time for transmission between connected node pairs. In this paper, we smission between connected node pairs. In this paper, we mainly focus on mainly focus on the delay caused by the lack of instantthe delay caused by the lack of instantaneous connectivityaneous connectivity which is closely related to the node which is closely related to the node density in such a netwok.density in such a netwok.

{Cause of Delay

Waiting Delay

Schedul ing and Routing

Network Resources

Propagation Speed

Lack of Instantaneous Connectivi ty

} Propagation Delay(Denoted by )


11

Network Model-DelayNetwork Model-Delay

We assign the delay to each link in the following way, We assign the delay to each link in the following way,

where is the connection function, and is the length owhere is the connection function, and is the length of link .f link .

Applying first passage percolation model here, we could dApplying first passage percolation model here, we could define the first passage time for node and :efine the first passage time for node and :

e

( ( ) ) (1 ( )) ( )kcP T e k g r g r

( )g re

r

( , )T x y x y

( , ) inf{ ( ) : }pT x y T is a path from x to y


12

ObjectiveObjective

Previous result:Previous result:

We studyWe study:: Detailed relationship between and node density . Detailed relationship between and node density .

Two steps:Two steps: Ignore the propagation delay ( ).Ignore the propagation delay ( ). Consider the propagation delay ( ).Consider the propagation delay ( ).

( , )lim ( )

( , )d

T x y

d x y

0 0


13

OutlineOutlineBackgroundBackground

Network Model & ObjectiveNetwork Model & Objective

Main Results and IntuitionsMain Results and Intuitions Main resultsMain results IntuitIntuitionsions Impact of Propagation DelayImpact of Propagation Delay


Concluding RemarksConcluding RemarksFundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

14

Main ResultsMain Results

Properties of Properties of , , There existsThere exists , such that , such that For any For any , , is a monotone decreasing functionis a monotone decreasing function

( ) M ( ) M

[ , )L

I ( ) 0 ( )

( )

IL

M

0


15

Main ResultsMain Results

Upper bound and lower bound of Upper bound and lower bound of

is a constantis a constant g(·) is the connection functiong(·) is the connection function is the size of a cluster is the size of a cluster E(·) is the expectationE(·) is the expectation

( )

[ , )0

0

1 1( ) inf 1

[ ( ) ] Lg L LE S r

g r

( )gS


16

IntuitiIntuitiononss-Upper Bound-Upper Bound

For two nodes and , the first passage time is For two nodes and , the first passage time is bounded by the the passage time along any path bounded by the the passage time along any path connecting and .connecting and .

x y ( , )T x y

x y

0

1( , ) ( , ) 1 , [ , )L

L L

T x y d x y

g r

least number of hops

average delay of one hop( )N d

distantce of one hop

Basic idea: and are connected by a sequence of hops with

distance smaller or equal to

x y

0 Lr


17


Existance of such pathsExistance of such paths The long-term critical density can be viewed as the critical The long-term critical density can be viewed as the critical

density of the Poisson Boolean Model with transmission rangedensity of the Poisson Boolean Model with transmission range

(Proposition 2 in [14]) Let be the critical density for the (Proposition 2 in [14]) Let be the critical density for the Poisson Boolean Model when the transmission range is , thenPoisson Boolean Model when the transmission range is , then

The critical density for The critical density for is , and is , and . Therefore, . Therefore, such a path exists. such a path exists.

L

0r

( )c rr

2 21 1 2 2( ) ( )c cr r r r

0 Lr r


18


Least Number of hops of such pathsLeast Number of hops of such paths Scale the network up by Scale the network up by , then each hop of such , then each hop of such

paths becomes , and paths becomes , and

We can show that We can show that

Then, we can see that Then, we can see that

( )N d

L 0r

( ( , )) ( ( , ) )L LN d x y N d x y

( )lim L

d

N d

d

( )limd

L

N d

d


IntuitiIntuitiononss--LowLower Bounder Bound

Waiting occurs at the boundry of a clusterWaiting occurs at the boundry of a cluster

19

x

y

t1t 2t 3t

MMM

MM

M

M

MM

Cluster to Cluster transmission


20

IntuitiIntuitiononss--LowLower Bounder Bound

Delay between and Delay between and

Motivation: The minimum number of clusters between Motivation: The minimum number of clusters between and gives a lower bound of delay and gives a lower bound of delay

Let the size of a cluster be , Let the size of a cluster be ,

then the minimum number of then the minimum number of

clusters is lower bounded byclusters is lower bounded by

x y

3 1( , ) 2T x y t t number of clusters-1

xy

( )gS

( )gS

( , )

[ ( )]g

d x y

E S


21

Impact of Propagation Delay Impact of Propagation Delay

Upper Bound Upper Bound The average delay of one hop changes The average delay of one hop changes

Lower BoundLower Bound

The minimum number of The minimum number of

clusters is lower boundedclusters is lower bounded

byby

0

1( , ) ( , ) , [ , )L

L L

T x y d x y

g r

average delay of one hop changes

( )gS

0

1r

0

( , )

[min{ ( ), }]g

d x y

E S r


22

OutlineOutline

BackgroundBackground

Network ModelsNetwork Models & Objective & Objective


Simulation ResultsSimulation Results without Propagation Delaywithout Propagation Delay with propagation delaywith propagation delay



23

Without Propagation DelayWithout Propagation Delay


24

With Propagation DelayWith Propagation Delay


25

OutlineOutlineIntroductionIntroduction

K-hop Clustered Network ModelsK-hop Clustered Network Models


The Impact of Mobility The Impact of Mobility



26

Concluding Remarks Concluding Remarks

We have studied the first passage delay of large scale We have studied the first passage delay of large scale network with unreliable links.network with unreliable links.

In two senarios (with propagation delay and without In two senarios (with propagation delay and without propagation delay), we propagation delay), we sketch by excavating its properties;sketch by excavating its properties; provide upper bound and lower bound to .provide upper bound and lower bound to .

( ) ( )


Thank you !Thank you !

Documents

Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links Shizhen Zhao, Luoyi Fu, Xinbing Wang Department