Surveillance coverage of sensor networks under a

Surveillance coverage of sensor networks under a random mobility strategy George Kesidis

Takis Konstantopoulos

Shashi Phoha

EE & CSE Depts, Penn State University University Park, PA, USA [email protected]

ECE Dept, University of Texas Austin, TX, USA [email protected]

ARL, Penn State University University Park, PA, USA [email protected]


 

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º \ ¿ Æ Á »U between two points ¿XÁ$¼ÈÇ ÉPÊ ]V» Ê Â8¿ Ê _ ¬ . ­ is the Euclidean distance We assume in the following that the maximum coverage ³´ area ®GË ¬ (maximum assuming no overlapping disks) is Á Á 3$ % T:. of significantly less than the area ­ of ­ . Thus, the nodes is necessary to maintain surveillance coverage of the whole region ­ to some degree of confidence as described below.

in the given region, i.e., ¼ ® be
Á ­ theÁ ormean¼density  ®
Á ­ ofÁ ifnodes ® is random. The objective of surveillance is to gather information about certain target objects that move through the region ­ . These objects exist in a so-called time-space] neighborhood charac_ terized by a time-varying function  ° for times ° in some º finite interval of (continuous) time  °  ° ¸ where  takes values in the subsets of ­ . For convenience of notation, we can ] _`¼  for ° Æ  °  º ° ¸ . In the following, we simply define  ° assume that if ] ± ² ] ° _[º ³ ´ _   Ê ] ° _  ¼  ¹  >[

then sensor node ¯ object  at time ° . :

5  = &   %  & . in the sense that, for a fixed Certain targets may be » and ³ ,  ] ° _¥¼ ]^»…º

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­

®

­

­

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±  ] ° _ & ¼(' º)º+*,*+* .° - 

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The three-dimensional environment In this section, each stochastic process

;

±  ] ° _¥¼ ] ±  ¸ ] ° _º ±  ¬ ] ° _º ±   ] ° _ _

is a 3-dimensional Brownian motion with independent Cartesian coordinates and variance coefficient (measured in m /s). Without loss of generality, suppose that a target is located at the origin. Let be the ball of radius (the surveillance radius) centered at the origin. At each point of time, the area that can be monitored by the sensor network is represented by the set

¬

¹

«…¬

] º ³=_

 _° ¼ ¶ ] ±  ] ° _ ] ºQ >/XSWXY e[$Z Â Q ´]  > S H$R \ d KE - ³"*  HUT ¬LH$V R ´ ¬ ]% ^ _

The random variable
H _[Å c ) eX] _ where the expression for < G > H is given as a ratio of 

modified Bessel functions in 2.0.1, p. 297, of [2]. This # lower bound could ) then be substituted into (2) to obtain a lower bound on and thus obtain an upper bound on the tail of the target detection time distribution (1).

#$$Y] ] ° _ _

Random mobility design A design objective here could be as follows: Suppose that a point target is required to be detected within a prespecified time with an error probability less than  R , where  is a positive number. Our design variable is the variance of the node mobility. When is very small, the target detection goal can be achieved if we choose roughly larger than  :

°

³

«X¬ «¬



«…¬

* ³ Ë ° J 


]J Ë ³ °_



º



R  R T  

¼ ! ] ° _ ° f  )  ¼ b 'OÂ  Ë )    A | ]%
) _ c |]V»‘_ where  is the tail of the standard normal distribution

with the P obvious )   ' choice of the constants. The integration f be performed f d can to give Y

function. Using the inequality

f

Y

| ]V»‘_Ã  » R = ) A¬ Ë

f g together, we get Collecting the calculations #

° _¼



bounds can be obtained. Finally, by substituting the values f of the constants, we / / obtain an exact g expression:

) R¼

'

/

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R   

> b

Ë J

' Â ³



³

Ë ¬



>





b

) ³  c,c *

This is a rapidly decreasing function of (exponential decay). SUMMARY We considered the problem of surveillance of a potentially large region undertaken by a potentially large group of mobile sensors. Under a random mobility strategy for the sensor grid, the distribution of the contact time between two nodes and the distribution of the time-until-detection of slowly moving (point) targets were studied. Both two and three dimensional environments were considered. Finally, design issues pertaining to the single parameter of mobility, the variance , were discussed.

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