Subject: [manet] A question about an integral

From: "Hang Su" <[email-masked]>
Date: Thu, 22 Apr 2004 10:46:15 +0800

hi all,

I have read the paper [1] recently. I cannot understand the integral deduced by the paper.

There is a square area whose side is 2a. There are sensors placed in the area. The number of sensors in the 
square area is a Poisson random variable, N with mean \lambda * A, where A=4*a*a.
Let D_i be a random variable that denotes the length of the segment from a sensor located at (x_i,y_i), i=1,2,...n to
the processing center. The position of the processing center is (a,a). Then,
E[D_i|N=n]=\int_A{(x_i^2+y_i^2)^(1/2)*(1/4/a/a)}dA=0.765a
                                                  ~~how to get it. The detailed procedure.
My question is how to deduce the third "=".

Any help would be appreciated. 

[1] S. Bandyopadhyay and E. J. Coyle, An Energy Efficient Hierarchical Clustering Algorithm for Wireless Sensor Networks,
IEEE Infocom2003

Hang

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From: "John Mullen" <[email-masked]>
Date: Thu, 22 Apr 2004 06:50:22 -0600

Hi Hang,

I think I can answer this, assuming I decoded your question properly.

If sensors are distributed within a region according to a Poisson
process, the distribution of those sensors, given that there are N in
the region, is uniform.  Hence, the density of the placement
distribution is 1/(4a^2) = (1/4/a/a).  Given Sensor i located at (x_i,
y_i), the straight line distance, d_i, to the processing center located
at (0,0) is (x_i^2 + y_i^2)(1/2).  Hence the integral is of the form:

    \int_A d_i f_(XY)(x_i,y_i) dA.

This would be the expectation of D_i, given that N sensors are in the
region you described.

The computation of this integral is a bit difficult.  One way, of
course, is to use Maple or something like that.

:-)

If you want to do it the old fashioned way, note first that because of
symmetry, you need only evaluate the integral over the first half of the
first quadrant and multiply the result by eight.  Also, I usually work
in polar coordinates, so the integral would be over theta from zero to
pi/4 radians. Because we are in a square region, r will range from zero
to a/cos theta for that range of theta.  (It helps to draw a picture).
The change of variable technique involved here is a common engineering
trick.  It is described on p. 519 of _Advanced Engineering Mathematics_
by Kreyszig.  At this point, you would have:

	8/(4a^2) int_0^(pi/4) int_0^(a/cos theta) r^2 dr d theta

Hope this helps,

John Mullen

-----Original Message-----
From: [email-masked] [mailto:[email-masked]] On Behalf Of
Hang Su
Sent: Wednesday, April 21, 2004 7:46 PM
To: manet
Cc: shzju
Subject: [manet] A question about an integral

hi all,

I have read the paper [1] recently. I cannot understand the integral
deduced by the paper.

There is a square area whose side is 2a. There are sensors placed in the
area. The number of sensors in the 
square area is a Poisson random variable, N with mean \lambda * A, where
A=4*a*a. Let D_i be a random variable that denotes the length of the
segment from a sensor located at (x_i,y_i), i=1,2,...n to the processing
center. The position of the processing center is (a,a). Then,
E[D_i|N=n]=\int_A{(x_i^2+y_i^2)^(1/2)*(1/4/a/a)}dA=0.765a
                                                  ~~how to get it. The
detailed procedure. My question is how to deduce the third "=".

[1] S. Bandyopadhyay and E. J. Coyle, An Energy Efficient Hierarchical
Clustering Algorithm for Wireless Sensor Networks, IEEE Infocom2003

Hang