Appendix E

Derivation of Joint CDF of W

and �

In this appendix, we derive the joint cumulative distribution function (CDF) of the distance W

to the nearest neighbor and the angle � at which the nearest neighbor is from the transmitteron the surface of a torus with length 2R on each edge (see Figure E.1). The distance W

lies in the interval (0,√

2R], and the values of � are in the interval [0, 2π). Due to thetorus assumption, without loss of generality, we can assume that a node is at the origin of theCartesian coordinates. The joint CDF of W and � can be expressed in terms of the conditionalCDF as follows:

F�,W (φ,w) = F�/W (φ/W ≤ w) · FW (w). (E.1)

First, we derive an expression for FW (w), where w ≥ 0. There are two cases to consider.

• Case 1: w ≤ R . As shown in Figure E.1 (a), in this case, the distance W to thenearest neighbor will be greater than w if there is no node within a circle with radius w.In other words, the nearest neighbor cannot be inside the shaded circular area. Hence,the complementary CDF can be written as follows:

Pr(W > w) = Pr(no node in the shaded circular area with radius w). (E.2)

Since the nodes are uniformly distributed over an area of 4R2, the probability that anode falls in the shaded circle is πw2/4R2. Hence, the probability that a node does notfall in the shaded circle is 1 − πw2/4R2. In a network with N nodes, the probabilitythat none of these N nodes falls in the circle is

Pr(no node in the shaded circular area with radius w) =(

1 − πw2

4R2

)N

= Pr(W > w). (E.3)

The CDF in this case is simply

FW (w) = 1 − Pr(W > w) = 1 −(

1 − πw2

4R2

)N

. (E.4)

Ad Hoc Wireless Networks: A Communication-Theoretic Perspective Ozan K. Tonguz and Gianluigi Ferrari© 2006 John Wiley & Sons, Ltd. ISBN: 0-470-09110-X

300 Appendix E. Derivation of Joint CDF of W and �

w

−R R

−R

R

w

√2⋅R

ξ−R R

−R

RR

A2

A1

(a) (b)

Figure E.1 Region where the distance to the nearest neighbor is less than w in the casewhere (a) w ≤ R and (b) R < w ≤ √

2R.

Keeping the node spatial density fixed, in the case where N is large (i.e. when N →∞), it follows that

FW (w) = limN→∞ 1 −

(1 − πw2

4R2

)N

= 1 − e−ρsπw2(E.5)

where ρs = N/4R2 is the node spatial density.

• Case 2: R < w ≤ √2R . In this case, the distance W to the nearest neighbor will be

greater than w if there is no node in the shaded region shown in Figure E.1 (b). We willrefer to this total shaded area as At. The complementary CDF can be written as follows:

Pr(W > w) = Pr(no node in the shaded region At). (E.6)

Note that the area At is the sum of a circular area with radius R and the area of thedark shaded region (portion of an annulus), referred to as Ac. The area of the circle issimply equal to πR2. The total dark shaded area can be computed as follows. FromFigure E.1 (b), note that the total dark shaded area can be constructed from eightsymmetric pieces, where each piece is a combination of the areas A1 and A2. Thus,Ac = 8(A1 +A2). With simple geometric considerations, it can be shown that the areaof the total shaded region At is

At = πR2 +[4(π

4− ξ

)w2 − πR2 + 4R

√w2 − R2

]= 4w2

(π

4− ξ

)+ 4R

√w2 − R2 (E.7)

Appendix E. Derivation of Joint CDF of W and � 301

where ξ is defined as

ξ � cos−1(

R

w

). (E.8)

Given the expression for the total shaded area At, the probability that none of the N

nodes are in the shaded area is

Pr (no node falls in the shaded region) =(

1 − At

4R2

)N

= Pr(W > w). (E.9)

The CDF of W in this case is simply

FW(w) = 1 − Pr(W > w) = 1 −(

1 − At

4R2

)N

. (E.10)

Keeping the node spatial density fixed, in the case where N is large (i.e. when N → ∞),it follows that

FW(w) = limN→∞ 1 −

(1 − At

4R2

)N

= 1 − e−ρsAt . (E.11)

Finally, combining the two cases, the CDF of W can be written as

FW(w) =

0 w < 0

1 −(

1 − πw2

4R2

)N

0 ≤ w < R

1 −(

1 − χ

4R2

)NR ≤ w <

√2R

1 otherwise.

(E.12)

For large N , the CDF can be written as

FW (w) =

0 w < 0

1 − e−ρsπw20 ≤ w < R

1 − e−ρsAt R ≤ w ≤ √2R

1 otherwise,

(E.13)

which establishes (8.3).Next, we will derive an expression for the conditional probability F�/w(φ/W ≤ w).

For this computation as well, two cases have to be considered.

• Case 1: w ≤ R. Given that the distance to the nearest neighbor is less than w, theprobability that the angle � is less than φ is equal to the probability that the nearestneighboring node lies in a sector with angle φ and radius w, which corresponds to thedark shaded region shown in Figure E.2.

302 Appendix E. Derivation of Joint CDF of W and �

w

φ−R R

−R

R

Figure E.2 Region where � ≤ φ when 0 < w ≤ R.

Since the position of a node is uniformly distributed, the probability that a node liesin a sector can be obtained from the ratio of the sector area and the total area where anode can be, while satisfying the condition that W ≤ w. In this case, the area where anode could possibly be is a circle with radius w. Consequently, F�/W (φ/W ≤ w) willbe given as

F�/W (φ/w ≤ W) =12φw2

πw2 = φ

2π. (E.14)

The CDF of W in a torus in the case where w ≤ R is given as

FW (w) = 1 − e−ρsπw2. (E.15)

Therefore, the joint CDF F�,W (φ,w) in this case, i.e. for w ≤ R, can be written as

F�,W (φ,w) = φ

2π(1 − e−ρsπw2

). (E.16)

• Case 2: R < w ≤ √2R . Similarly, the conditional CDF of the angle � given W ≤ w

can be obtained by computing the ratio of the area where the nearest neighboring nodecould be, constrained to the fact that � ≤ φ, and W ≤ w and the total area where thecondition W ≤ w is satisfied. The total possible area where the nearest neighboringnode could be is the same as the area At computed earlier. Next, we compute the area ofthe region where the nearest neighboring node could be, while satisfying the condition� ≤ φ. There are sixteen possible regions to be considered for the angle φ.

– Region 1: φ ≤ ξ . The region at which the angle � will be lower than φ is thedark shaded region shown in Figure E.3 (a). The dark shaded region is a triangleand its area is

As1 = 1

2R(R tan φ) = R2

2tan φ. (E.17)

Appendix E. Derivation of Joint CDF of W and � 303

w√2⋅R

φ ξ−R R

−R

R

(a) 0 < φ ≤ ξ

w

√2⋅R

φ ξ−R R

−R

R

(b) ξ < φ ≤ π

4

w

√2⋅R

φξ

−R R

−R

R

(c)π

4< φ ≤ π

2− ξ

w

√2⋅R

φξ

−R R

−R

R

(d)π

2− ξ < φ ≤ π

2

Figure E.3 Regions where � ≤ φ, when R < w ≤ √2R.

The conditional CDF can then be written as

F�/W (φ/W ≤ w) = As1

At= R2

2Attan φ. (E.18)

– Region 2: ξ < φ ≤ π/4 . The region at which the angle � will be lower than φ

is the dark shaded region shown in Figure E.3 (b). The dark shaded area can beobtained by subtracting the area of a sector with angle (π/4 − φ) and radius w

from one-eighth of the total area At. The area of the dark shaded region is

As2 = At

8− w2

2

(π

4− φ

)(E.19)

304 Appendix E. Derivation of Joint CDF of W and �

Table E.1 Conditional CDF F�/W (φ/W ≤ w)

Range of φ F�/W (φ/W ≤ w)

0 < φ ≤ ξ1

At

[R2

2tan φ

]

ξ < φ ≤ π

4

1

At

[At

8− w2

2

(π

4− φ

)]π

4< φ ≤ π

2− ξ

1

At

[At

8+ w2

2

(φ − π

4

)]π

2− ξ < φ ≤ π

2

1

At

[At

4− R2

2tan

(π

2− φ

)]π

2< ξ ≤ π

2+ ξ

1

At

[At

4+ R2

2tan

(φ − π

2

)]π

2+ ξ < φ ≤ 3π

4

1

At

[At

4+ At

8− w2

2

(3π

4− φ

)]3π

4< φ ≤ π − ξ

1

At

[At

4+ At

8+ w2

2

(φ − 3π

4

)]

π − ξ < φ ≤ π1

At

[At

2− R2

2tan(π − φ)

]

π < φ ≤ π + ξ1

At

[At

2+ R2

2tan(φ − π)

]

π + ξ < φ ≤ 5π

4

1

At

[At

2+ At

8− w2

2

(5π

4− φ

)]5π

4< φ ≤ 3π

2− ξ

1

At

[At

2+ At

8+ w2

2

(φ − 5π

4

)]3π

2− ξ < φ ≤ 3π

2

1

At

[3At

4− R2

2tan

(3π

2− φ

)]3π

2< φ ≤ 3π

2+ ξ

1

At

[3At

4+ R2

2tan

(φ − 3π

2

)]3π

2+ ξ < φ ≤ 7π

4

1

At

[3At

4+ At

8− w2

2

(7π

4− φ

)]7π

4< φ ≤ 2π − ξ

1

At

[3At

4+ At

8+ w2

2

(φ − 7π

4

)]

2π − ξ < φ ≤ 2π1

At

[At − R2

2tan(2π − φ)

]

Appendix E. Derivation of Joint CDF of W and � 305

and the conditional CDF can then be written as

F�/W (φ/W ≤ w) = As2

At

= 1

8− w2

2At

(π

4− φ

). (E.20)

– Region 3: π/4 < φ ≤ π/2 − ξ . In this case, the region at which the angle � willbe less than φ is the dark shaded area shown in Figure E.3 (c). The dark shadedarea corresponds to the sum of the area of a sector with angle (φ − π/4) andradius w and one-eighth of the total area At. The area can be written as

As3 = At

8+ w2

2

(φ − π

4

)(E.21)

and the corresponding conditional CDF is

F�/W (φ/W ≤ w) = As3

At

= 1

8− w2

2At

(φ − π

4

). (E.22)

– Region 4: π/2 − ξ < φ ≤ π/2 . In this case, the region at which the angle � willbe lower than φ is the shaded area shown in Figure E.3 (d). The area of the darkshaded region can be obtained by subtracting the area of the light shaded trianglein the first quadrant from the area of one fourth of the total area At. The area canbe written as

As4 = At

4− 1

2R[R tan

(π

2− φ

)]= At

4− R2

2tan

(π

2− φ

). (E.23)

The corresponding conditional CDF becomes

F�/W (φ/W ≤ w) = As4

At

= 1

4− R2

2Attan

(π

2− φ

). (E.24)

Using the same approach for the regions in the other three quadrants, the conditionalCDF (in the case where R < w ≤ √

2R) can be obtained. The final results are listed inTable E.1.