## Coverage of Wireless Sensor Networks

Coverage of Wireless Sensor Networks Guo-Lie Lan 1,2 and Zhi-Ming Ma1 and Su-Yong Sun 1,2 1 Institute of Applied Mathematics Academy of Mathematic...
Author: Cody Griffith
Coverage of Wireless Sensor Networks Guo-Lie Lan

1,2

and Zhi-Ming Ma1 and Su-Yong Sun

1,2

1 Institute of Applied Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences, Beijing 100080, China. 2 Graduate School of the Chinese Academy of Sciences

Emails: [email protected]; [email protected]; [email protected] Abstract. In this paper we study the limiting achievable coverage problems of sensor network. For the sensor networks with uniform distributions we obtain a complete characterization of the coverage probability. For the sensor networks with non-uniform distributions, we derive two different necessary and sufficient conditions respectively in the situations that the density function taking its minimum value on a set with positive Lebesgue measure or at finitely many points. We propose also an economical scheme for the coverage of sensor networks with empirical distributions. Keywords. coverage, sensor network, large-scale, wireless, uniform distribution.

1

Introduction

Recently, there has been a growing interest in studying large-scale wireless sensor networks. Such a network consists of a large number of sensors which are densely deployed in a certain area. For some reasons (reducing radio interference, limited battery capacity, etc.), these sensors are small in size and they communicate with each other in short distance. There are many fundamental problems that arise in the research of wireless sensor networks. Among them one important issue is that of limiting achievable coverage. A point in an area can be detected by a sensor provided the point is within the distant r of the sensor, where r is the sensing radius of the sensor. The area is said to be covered if every point in the area can be detected by a sensor. In the literature there have been several discussions concerning the minimum sensing radius, depending on the numbers of (active) sensors per unit area, which guarantees that the area is covered in a limiting performance. In [12], the authors considered the problem of covering a square of area A with randomly located circles whose centers are generated by a two-dimensional Poisson point process of density D points per unit area. Suppose that each Poisson point represents a sensor with sensing radius R pwhich may depend on D and A. They proved that, for any ε > 0, if R = (1 + pε) ln A/πD, then limA→∞ Pr[square covered] = 1. On the other hand, if R = (1 − ε) ln A/πD, then limA→∞ Pr[square covered] = 0. Therefore the authors observed that, to guarantee that the area is covered, a node must have π[(1 + ε) ln A/πD]D or a little more than ln A nearest neighbors (Poisson point that lie at a distant of R or less from it) on the average. In the paper [14], the authors studied the coverage of a grid-based unreliable sensor network. They derived necessary and sufficient conditions for the random grid network to cover a unit square area. Their result shows that the random grid network asymptotically cover a unit square area if and only if pn rn2 is of the order (ln n)/n,

2

G.L. Lan, Z.M. Ma and S.S. Sun

where rn is the sensing radius and pn is the probability that a sensor is “active” (not failed). In connection with the above two results we mention that Hall[3] has considered the coverage problem of the following model: Circles of radius r are placed in a unit-area disc D at a Poisson intensity of λ. Let V (λ; r) denote the vacancy within D, i.e., V (λ; r) is the region of D not covered by the circles. It has been shown ([3], Theorem 3.11) that 2 1 min{1, (1 + πr2 λ2 )e−πr λ } < Pr{|V (λ; r)| > 0} 20 2 < min{1, 3(1 + πr2 λ2 )e−πr λ },

where |V (λ;pr)| is the area of V (λ; r). Note that by Hall’s result, if we set λ = n and rn = (ln n + ln ln n + an )/πn, then limn→∞ Pr[square covered] = 1 for an → +∞, and limn→∞ Pr[square covered] < 19/20 for an → −∞. However, it was not clear whether the limit limn→∞ Pr[square covered] = 0 for an → −∞. In the literature there are also various other discussions about the coverage problems of wireless sensor networks, see [4 - 11, 13, 15, 16] and reference therein. In this paper, we employ some results in Aldous[1] to study the coverage problems. Among other results, we p show that in the above mentioned Hall’s model, if we set λ = n and rn = (ln n + ln ln n + an )/πn with an = o(ln n), then Pr[square covered] = exp[− exp(−an )] + o(1), provided the limit of {an } exists in [−∞, +∞] (see Theorem 2.1 and Remark 2.2 below). In particular, if an → −∞, then limn→∞ Pr[square covered] = 0. Thus we clarify the above mentioned question. Moreover, the above equality tells us something more, that is, if an → a for a ∈ (−∞, +∞), then limn→∞ Pr[square covered] = exp(−e−a ) ∈ (0, 1). The structure of this paper is as follows. In section 2 we investigate the coverage of sensor networks with uniform distributions, which is asymptotically the same as the sensor networks with sensors located according to a Poisson distribution. In section 3 we investigate the coverage of sensor networks with non-uniform distributions. Our research shows that in non-uniform case the minimum sensing radius in the coverage problem relies mainly on the behavior of the distribution density function around its minimum point. We give first a sufficient condition in Theorem 3.1. We then derive two different necessary and sufficient conditions in Theorem 3.2 and Theorem 3.3, respectively in the situations that the density function taking its minimum value on a set with positive Lebesgue measure or at finitely many points. Finally in Section 4 we propose an economical scheme for the coverage of sensor networks with empirical distributions.

Coverage of Wireless Sensor Networks

2

3

Coverage of Sensor Networks with Uniform Distributions

Let A = [0, 1]2 be a unit square. Suppose {X1 , · · · , Xn } are n independent random points with the uniform distribution on A and rn is a positive number. We consider {rn ; X1 , · · · , XS n } as a sensor network on the region A with sensing n radius rn . Let M(n, rn ) = k=1 B(Xk , rn ) be the union of n circles with centers X1 , · · · , Xn and common radius rn . Then the region is covered by the sensor network if and only if A ⊂ M(n, rn ). The question that we investigate in this section is to find the smallest rn that guarantees lim Pr[A ⊂ M(n, rn )] = 1.

n→∞

(2.1)

We have the following complete result. Theorem 2.1 Let {X1 , · · · , Xn } be n independent p random points with the uniform distribution on A. Suppose that rn = (ln n + ln ln n + an )/πn, where {an } is a sequence of real numbers such that an = o(ln n) and has a limit in Sn [−∞, +∞]. Let M(n, rn ) = k=1 B(Xk , rn ) be as above. Then we have Pr[A ⊂ M(n, rn )] = exp[− exp(−an )] + o(1).

(2.2)

In particular, limn→∞ Pr[A ⊂ M(n, rn )] = 1 if and only if an → +∞, and limn→∞ Pr[A ⊂ M(n, rn )] = 0 if and only if an → −∞. Proof. Define a random variable Ln as the radius of the largest circle that lies inside A containing no points of {X1 , · · · , Xn }. Then A is covered by M(n, rn ) if and only if Ln < rn . Applying the approximation method used by Aldous in the study of stochastic geometry, we can get ([1], p150, H1d)   Pr[Ln < rn ] ≈ exp −n2 πrn2 exp(−nπrn2 )(1 − 2rn )2 , (2.3) where the relation “≈” is understood as   2 Pr[Ln < rn ] = exp −n2 πrn2 exp(−nπrn2 )(1 − 2rn )2 + O(e−nπrn ). By the assumption of rn , we have nπrn2 → ∞. Therefore, Pr[A ⊂ M(n, rn )] = Pr[Ln < rn ] = exp[−I(n)] + o(1), where I(n) = n2 πrn2 (1 − 2rn )2 exp(−nπrn2 ). Replacing nπrn2 by ln n + ln ln n + an in the expression of I(n), we get (1 − 2rn )−2 I(n)

= n(ln n + ln ln n + an ) exp(− ln n − ln ln n − an ) = n(ln n + ln ln n + an )(n ln n)−1 exp(−an ) = exp(−an )[1 + ln ln n(ln n)−1 + an (ln n)−1 ]

(2.4)

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G.L. Lan, Z.M. Ma and S.S. Sun

By our assumption an = o(ln n), which implies an (ln n)−1 = o(1) and rn = o(1). Therefore, from the above formula we get I(n) = exp(−an ) + o(1). Thus by (2.4) lim Pr[A ⊂ M(n, rn )] = lim exp[−I(n)] = lim exp[− exp(−an )],

n→∞

n→∞

n→∞

verifying (2.2).  Remark 2.2 The conclusion of Theorem 2.1 is equally available in the model that the sensors put in the unit square according to a Poisson point process with intensity n points per unit area. Because the difference of these two schemes is negligible in the limiting procedure. In fact, Aldous’s result [1] (H1d) (see (2.3) in this paper) was obtained via the approximation of large number of uniformly distributed points by Poisson point processes with large intensity.

3

Coverage of Sensor Networks with Non-uniform Distributions

In this section we investigate the coverage of sensor networks which are deployed non-uniformly according to a general distribution density function g(x) on the unit square A. We assume that g is a bounded smooth function. Our results show that the minimum sensing radius in the coverage problem relies mainly on the minimum value of g and the behavior of g around its minimum value. We derive first a sufficient condition as follows. Theorem 3.1 Suppose {X1 , · · · , Xn } are n independent random points in the unit square A according to a smooth distribution density function g(x), where p g(x) satisfies 0 < β ≤ g(x) ≤ γ < ∞. Let rn = (ln n + ln ln n + an )/βπn, where {an } is a sequence in R tending to +∞. Then we have lim Pr[A ⊂ M(n, rn )] = 1.

n→∞

(3.1)

(Here and henceforth M(n, rn ) is the same as defined in the beginning of Section 2.) Proof. Analogously, we define a random variable Ln as the radius of the largest circle that lies inside A containing no points. Following (H9c) in Aldous [1] (p160) and its argument, we can get  Z  2 Pr[Ln < rn ] = exp − Jn (x)dx + O(e−nπrn ), A

where Jn (x) = n2 πrn2 g(x)2 exp(−ng(x)πrn2 ).

(3.2)

Coverage of Wireless Sensor Networks

5

Since nβπrn2 = ln n + ln ln n + an , we have Jn (x)

= ng(x)2 β −1 (ln n + ln ln n + an ) · exp[−g(x)β −1 (ln n + ln ln n + an )] ≤ nγ 2 β −1 (ln n + ln ln n + an ) exp[−(ln n + ln ln n + an )] = nγ 2 β −1 (ln n + ln ln n + an )(n ln n)−1 exp(−an ) = γ 2 β −1 [exp(−an ) + ln ln n(ln n)−1 exp(−an ) + an (ln n)−1 exp(−an )].

This yields that Jn (x) tend to 0 as an → ∞. By the dominated convergence theorem, Z Z lim Jn (x)dx = lim Jn (x)dx = 0. n→∞

A n→∞

A

Thus,  Z  lim Pr[A ⊂ M(n, rn )] = lim Pr[Ln < rn ] = lim exp − Jn (x)dx = 1. 

n→∞

n→∞

n→∞

A

In the remainder of this section, we explore the necessary and sufficient condition that guarantees (3.1) holds. It turns out that the behavior of g around its minimum point will effect the coverage rate of rn . See Theorem 3.2 and Theorem 3.3 for details. Theorem 3.2 Let {X1 , · · · , Xn } be n independent random points in the unit square A according to a smooth distribution density function g(x). Suppose that g(x) satisfies 0 < β ≤ g(x) ≤ γ 0, where λ is the Lebesgue measure. Let rn = (ln n + ln ln n + an )/βπn, where {an } is a sequence in R such that an = o(ln n). Then, lim Pr[A ⊂ M(n, rn )] = 1

(3.3)

n→∞

if and only if an tends to +∞ as n → ∞. Moreover, lim Pr[A ⊂ M(n, rn )] = 0

(3.4)

n→∞

if and only if an tends to −∞ as n → ∞. Proof. By the approximation formula (3.2), 2

Pr[A ⊂ M(n, rn )] = exp [−I(n)] + O(e−nπrn ), where

Z I(n) =

(3.5)

n2 πrn2 g(x)2 exp(−ng(x)πrn2 )dx.

A

Letting Jn (x) = n2 πrn2 g(x)2 exp(−ng(x)πrn2 ) and S = {x : g(x) = β}, we have I(n) = I1 (n) + I2 (n)

(3.6)

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G.L. Lan, Z.M. Ma and S.S. Sun

with

Z

Jn (x)dx = λ(S)n2 πrn2 β 2 exp(−nβπrn2 )

I1 (n) :=

(3.7)

S

and

Z

Jn (x)dx ≤ λ(A \ S)n2 πrn2 γ 2 exp(−nβπrn2 ).

I2 (n) :=

(3.8)

A\S

Since λ(S) > 0, we have I2 (n) ≤ αI1 (n) for some α ∈ (0, ∞). This implies that I1 (n) ≤ I(n) ≤ (1 + α)I1 (n).

(3.9)

Since nβπrn2 = ln n + ln ln n + an , hence we obtain I1 (n) = λ(S)β exp(−an ) + o(1).

(3.10)

By (3.5), (3.9) and (3.10), we have lim Pr[A ⊂ M(n, rn )] = 1 ⇔ lim I(n) = 0 ⇔ lim I1 (n) = 0 ⇔ an → +∞.

n→∞

n→∞

n→∞

Similarly, we can check that lim Pr[A ⊂ M(n, rn )] = 0 ⇔ lim I1 (n) = +∞ ⇔ an → −∞. 

n→∞

n→∞

In the following we assume that the distribution density function g(x) has finitely many minimum points at which some regular conditions hold. For a real function g(x) = g(x1 , x2 ) on R2 which is secondly differentiable at x∗ , we define 2 g (x∗ ), i, j = 1, 2. B(x∗ ) as the 2 × 2 matrix with entries bi,j = ∂x∂i ∂x j Theorem 3.3 Let {X1 , · · · , Xn } be n independent random points according to a smooth distribution density function g(x) on A. Suppose that g(x) has finitely ∗ many minimum points {x∗i } ⊂ A, 0 < β = g(x∗i ) ≤ g(x) p ≤ γ < ∞, and B(xi ) are well defined and strictly positive definite. Let rn = (ln n + an )/βπn, where {an } is a sequence in R such that an = o(ln n). Then lim Pr[A ⊂ M(n, rn )] = 1

n→∞

(3.11)

if and only if an tends to +∞ as n → ∞. Moreover, lim Pr[A ⊂ M(n, rn )] = 0

n→∞

(3.12)

if and only if an tends to −∞ as n → ∞. Before proving Theorem 3.3, let us prepare several lemmas. Lemma 3.4 Suppose B is a d-dimensional matrix which is symmetric and strictly positive definite. Then Z 1 exp(− xT B −1 x)dx = (2π)d/2 |B|1/2 , (3.13) 2 R where |B| is the determinant of the matrix B.

Coverage of Wireless Sensor Networks

7

Proof. To prove (3.13), it is enough to note that 1 (2π)−d/2 |B|−1/2 exp(− xT B −1 x) 2 is the density function of a normal distribution. Lemma 3.5 Suppose B is a d × d strictly positive definite matrix. Let Z n Dn = exp(− xT B x)dx 2 d R and

Z Dn (r) =

n exp(− xT B x)dx. 2 |x|≤r

Then for r > 0, lim Dn (r)/Dn = 1.

n→∞

Proof. By Lemma 3.4, it can be calculated that Dn = (2π)d/2 n−d/2 |B|−1/2 .

(3.14)

Since B is a strictly positive definite matrix, there exists δ > 0 such that xT B x ≥ δ|x|2 , where | · | is the Euclidean norm in Rd . Thus, Z 1 Dn − Dn (r) = exp(− nxT B x)dx 2 |x|>r Z n n ≤ exp(− δr2 − xT B x)dx 4 4 |x|>r Z ≤ exp(−nδr2 /4) exp(−n/4 · xT B x)dx Rd

= exp(−nδr /4)(2π)d/2 (n/2)−d/2 |B|−1/2 = exp(−nδr2 /4) · 2d/2 Dn . 2

Therefore, Dn (r)/Dn = 1 − exp(−nδr2 /4) · 2d/2 = 1 − o(1).  In what follows for two sequences {xn } and {yn } in R+ , we write xn ∼ yn to mean that a ≤ lim inf n→∞ xn /yn ≤ lim supn→∞ xn /yn ≤ b for some a, b ∈ (0, +∞). It is clear that “∼” is a equivalence relation. Lemma 3.6 Suppose that a real function g(x) on R2 has a unique minimum points x = 0 and B(0) is well defined and strictly positive definite. Define Z En (Ω) = exp[−ng(x)]dx Ω

for Ω ⊆ R . Let Fn (ε) = En ({x : |x| ≤ ε}) and Gn (α) = En ({x : |x| ≤ n−α }) for ε, α > 0. Then (a) for any α ∈ (0, 1/2] and ε > 0, Fn (ε) ∼ Gn (α); (b) for any bounded set Ω containing a neighborhood of x = 0, En (Ω) ∼ Fn (ε). 2

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G.L. Lan, Z.M. Ma and S.S. Sun

Proof. We prove only the assertion (a). The proof of the assertion (b) is similar. Since B(0) is a strictly positive definite matrix, there exits δ1 , δ2 > 0 such that δ1 |x|2 ≤ 1/2 · xT B(0)x ≤ δ2 |x|2 . By the Taylor’s formula, g(x) = g(0) + 1/2 · xT B(0)x + o(|x|2 ). Moreover, we can choose δ∗ and δ ∗ such that for |x| ≤ ε, δ∗ |x|2 ≤ 1/2 · xT B(0)x + o(|x|2 ) ≤ δ ∗ |x|2 . Therefore, Z Fn (ε)

=

exp[−ng(x)]dx |x|≤ε

Z

exp[−n(g(0) + 1/2 · xT B(0)x + o(|x|2 ))]dx

= |x|≤ε

Z ≤

exp[−n(δ∗ |x|2 + g(0))]dx Z exp[−ng(0)] exp(−nδ∗ |x|2 )dx. |x|≤ε

=

|x|≤ε

Now we calculate the integral in the above right hand side . Let x = (ρcosθ, ρsinθ), where θ ∈ [0, 2π] and ρ ∈ [0, ε]. Then the Jacobi determinant |J| = ρ. Thus, Z exp(−nδ∗ |x|2 )dx |x|≤ε ε Z 2π

Z =

0

= Therefore, Fn (ε) ≤

exp(−nδ∗ ρ2 )ρdρdθ

0

π [1 − exp (−nδ∗ ε2 )]. nδ∗

π [1 − exp (−nδ∗ ε2 )] exp[−ng(0)]. nδ∗

Analogously, for Gn (α) we have Z Gn (α) = exp[−ng(x)]dx |x|≤n−α Z 1 = exp[−n(g(0) + xT B(0)x + o(|x|2 ))]dx 2 |x|≤n−α Z ≥ exp[−n(δ ∗ |x|2 + g(0))]dx |x|≤n−α Z = exp[−ng(0)] exp(−nδ ∗ |x|2 )dx. |x|≤n−α

(3.15)

Coverage of Wireless Sensor Networks

9

A similar calculation leads to  π  Gn (α) ≥ ∗ 1 − exp(−n1−2α δ ∗ ) exp[−ng(0)]. nδ

(3.16)

Comparing (3.15) and (3.16), one obtains δ∗ 1 − exp(−nδ∗ ε2 ) Fn (ε) ≤ · . Gn (α) δ∗ 1 − exp(−n1−2α δ ∗ ) Since α ≤ 1/2, letting n → ∞, we obtain 1 ≤ lim inf Fn (ε)/Gn (α) ≤ lim sup Fn (ε)/Gn (α) ≤ n→∞

n→∞

δ∗ 1 < ∞.  · δ∗ 1 − e−δ∗

Proof of Theorem 3.3. Without loss of generality, we assume that g(x) has a unique minimum point x∗ . By the approximation formula (3.2), 2

Pr[A ⊂ M(n, rn )] = exp [−I(n)] + O(e−nπrn ), where

Z I(n) =

(3.17)

n2 πrn2 g(x)2 exp[−ng(x)πrn2 ]dx.

A

Since 0 < β ≤ g(x) ≤ γ < +∞, we have Z I(n) ∼ n2 πrn2 exp[−ng(x)πrn2 ]dx.

(3.18)

A

Note that cn := nπrn2 → ∞. By Lemma 3.6 we get, Z I(n) ∼ ncn exp[−cn g(x)]dx. −1/2

(3.19)

|x−x∗ |≤cn

By the Taylor’s formula, g(x) = g(x∗ ) + 1/2 · (x − x∗ )T B(x∗ )(x − x∗ ) + o(|x|2 ). Then the right hand side of (3.19) can be written as Z h c i n ∗ ncn exp[−cn g(x )] exp − xT B(x∗ )x − cn o(|x|2 ) dx. (3.20) −1/2 2 |x|≤cn Note that g(x∗ ) = β. Applying Lemma 3.4-3.6, one obtains Z h c i n I(n) ∼ ncn exp(−cn β) exp − xT B(x∗ )x dx −1/2 2 |x|≤cn Z h c i n ∼ ncn exp(−cn β) exp − xT B(x∗ )x dx 2 |x|≤ε Z h c i n ∼ ncn exp(−cn β) exp − xT B(x∗ )x dx 2 R2 ∗ −1/2 = ncn exp(−cn β) · 2πc−1 n |B(x )| ∼ n exp(−cn β).

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G.L. Lan, Z.M. Ma and S.S. Sun

Since cn β = πβnrn2 = ln n + an , we have n exp(−cn β) = exp(−an ). Hence, I(n) ∼ exp(−an ).

(3.21)

Therefore, lim Pr[A ⊂ M(n, rn )] = 1 ⇔ exp [−I(n)] → 1 ⇔ I(n) → 0 ⇔ an → +∞.

n→∞

Thus the first assertion is verified. The second assertion can be checked similarly. 

4

Coverage of Sensor Networks with Empirical Distributions

From the discussion of the above section, we see that if sensors are deployed non-uniformly, then the minimum sensing radius in the coverage problem relies mainly on the minimum value of g(x), which is certainly not a economical scheme. In this section we propose a more economical scheme for non-uniformly distributed sensor networks. Note that in practice, the density function g(x) can always be approximated by empirical functions. PKThus we assume that g is an empirical distribution functions, that is, g(x) = i=1 ci IAi , where {A1 , · · · , AK } is SK T a partition of A, i.e. i=1 Ai = A and Ai Aj = φ for i 6= j. In this situation, we can choose different sensing radius ri (n) for different Ai . We deploy sensors by the following procedure. (see Fig. 1). (a) Divide n sensors into K groups. The number ni of sensors in each group is proportional to ci Si , where Si is the area of the region Ai . (b) Deploy the sensors of each group into the corresponding area Ai independently and according to the uniform distribution on Ai .

A2  A1 . '\$  . r2  . r1 &% '\$ r. 3

. m

. A3&%m

r4 . m

. m A4

Fig. 1. Sensors with different sensing radii in different regions

Coverage of Wireless Sensor Networks

11

With the above scheme we have the following result. SK T Proposition 4.1 Let A = [0, 1]2 be divided as A = i=1 Ai with Ai Aj = φ for i 6= j, i, j = 1, · · · , K, where each Ai is a rectangle with area µ(Ai ). For n large enough, we put mi = dnci µ(Ai )e sensors in Ai , which are located as mi independent random points with uniform distribution on Ai . Suppose that p each sensor in Ai has a sensing radius ri (n) = (ln n + ln ln n + ai (n))/ci πn. Suppose further that ai (n) = o(ln n) and limn→∞ ai (n) exists in [−∞, +∞]. Then we have " K # X Pr[A is covered] = exp − exp(−ai (n)) + o(1). (4.1) i=1

Proof. In the limiting procedure we may ignore the boundary effects. Thus, P [A is covered] =

K Y

P [Ai is covered] + o(1).

i=1

It is enough to check that P [A1 is covered] = exp [− exp(−a1 (n))] + o(1).

(4.2)

For convenience we assume that c1 µ(A1 )n = m1 is an integer. Scaling the area and the sensing radius simultaneously, we can apply Theorem 2.1 to obtain P [A1 is covered] = exp[−I(m1 )] + o(1), where I(m1 ) =

  1 1 m21 πr1 (n)2 exp −m1 πr1 (n)2 . µ(A1 ) µ(A1 )

(4.3)

Under our assumption on r1 (n), we have πr1 (n)2

= = =

1 (ln n + ln ln n + a1 (n)) c1 n µ(A1 ) [ln(c1 µ(A1 )n) + ln ln(c1 µ(A1 )n) + a1 (n) + O(1)] c1 nµ(A1 ) µ(A1 ) [ln m1 + ln ln m1 + a1 (n) + O(1)]. (4.4) m1

Combining (4.3) and (4.4) we obtain I(m1 ) = exp[−a1 (n)][1 + ln ln m1 (ln m1 )−1 + a1 (n)(ln m1 )−1 ]. Thus I(m1 ) = exp[−a1 (n)] + o(1). Hence (4.2) follows. 

(4.5)

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G.L. Lan, Z.M. Ma and S.S. Sun

Acknowledgment

We thank Yibing Cai for his talk on wireless sensor networks at our seminar. We thank also Qingyang Guan, Xue Cheng, Shaojun Guo, Yongsheng Song for helpful discussions. Zhi-Ming Ma is grateful to the organizers of the Conference CJCDGCGT2005 for inviting him to give a talk. The work is partly supported by NSFC and 973 Project.

References [1] David Aldous, Probability Approximations via The Poisson Clumping Heuristic, SpringVerlag, New York, 1989. [2] P. Gupta and P. R. Kumar, Critical Power for Asymptotic Connectivity in Wireless Networks, in Stochastic Analysis, Control, Optimization and Applications, pp. 547-566, 1998. [3] P. Hall, Introduction to the Theory of Coverage Processes, John Wiley & Sons, New York, 1988. [4] C.-F. Huang and Y.-C. Tseng, The Coverage Problem in a Wireless Sensor Network, in ACM Intl. Workshop on Wireless Sensor Networks and Applications, pp. 115-121, 2003. [5] K. Kar and S. Banerjee, Node Placement for Connected Coverage in Sensor Networks.in Proc. of Wiopt, Sophia Antipolis, France, 2003. [6] H. Koskinen, On the Coverage of a Random Sensor Network in a Bounded Domain, in Proceedings of 16th ITC Specialist Seminar, pp. 11-18, 2004. [7] B. Liu, P. Brass, O. Dousse, P. Nain, D. Towsley, Mobility Improves Coverage of Sensor Networks,submitted to the ACM MobiHoc 2005 Symposium, 2004. [8] B. Liu and D. Towsley, A Study of the Coverage of Large-scale Sensor Networks, in IEEE International Conference on Mobile Ad-hoc and Sensor Systems, 2004. [9] X. Y. Li, P. J. Wan, O. Frieder. Coverage in Wireless Ad Hoc Sensor Networks, IEEE Trans. on Computers, vol. 52, pp. 753-763, 2003. [10] S. Meguerdichian, F. Koushanfar, M. Potkonjak, and M. B. Srivastava, Coverage Problems in Wireless Ad-hoc Sensor Networks.in Proc. IEEE Infocom, pp. 1380C1387, 2001, [11] D. Miorandi and Eitan Altman, Coverage and Connectivity of Ad-Hoc Networks in Presence of Channel Randomness, Proc. of IEEE INFOCOM 2005, Miami, USA, Mars 13-17, 2005. [12] T. K. Philips, S. S. Panwar, and A. N. Tantawi, Connectivity Properties of a Packet Radio Network Model. IEEE Transactions on Information Theory, 35(5), September 1989. [13] S. Poduri, G. S. Sukhatme. Constrained Coverage for Mobile Sensor Networks, IEEE International Conference on Robotics and Automation, pp. 165-172, May 2004, LA, USA. [14] S. Shakkottai, R. Srikant and N. Shroff, Unreliable Sensor grids: Coverage, Connectivity and Diameter, in Proc. IEEE Infocom, 2003. [15] D. Tian and N. D. Georganas, A coverage-preserving Node Scheduling Scheme for Large Wireless Sensor Networks, in First ACM International Workshop on Wireless Sensor Networks and Applications, pp. 32-41, 2002. [16] H.h. Zhang and J. C. Hou, Maintaining Sensing Coverage and Connectivity in Large Sensor Networks, Ad Hoc & Sensor Wireless Networks, Vol. 1, pp. 89C124, March 3, 2005.