Wireless Ad Hoc and Sensor Networks

308 

Wireless Ad Hoc and Sensor Networks

s

u

s

v

s

a

s

b

s

c

s

d

(a)

s

u

s

v

s

a

s

b

s

c

s

d

(b)

Figure 12.5: (a) Interference analysis in Phase I.

more, from Section 12.3, we know that the distribution of both the primary transmit-

ters and the primary receivers satisfies a two-dimensional Poisson point process with

density

λ

. Therefore, we have

p(E1)=

e

−

λπ

r

2

I

(−

λπ

r

2

I

)

0

0!

(12.1)

= e

−

λπ

r

2

I

. (12.2)

Similarly, p(E2)=e

−

λπ

R

2

I

. Therefore, the probability that a spectrum opportunity

exists from s

u

to s

v

during a time slot is

p = p(E1) ·p(E2) (12.3)

= exp(−

πλ

(r

2

I

+ R

2

I

)). (12.4)

2

Since r

I

= r and R

I

= R under the UDG model, let p = exp(−

πλ

(r

2

+R

2

)) in this

section. In Phase I, the task is to broadcast ...