Analytical Study of P cov and P com

In this section, we derive the lower bounds on Pcovand Pcomunder the random sensing radius model

and the random communication link model described in Sections 3.2.1.1 and 3.2.1.2, respectively. Similar to [14, 86], we apply the coverage process techniques introduced in [85] and the results in the percolation theorem [77] for our study.

3.3.1 Asymptotic Lower Bound (PL

cov) on Pcov

LEMMA4. Let n points distributed independently and uniformly in a unit-area convex field D within R2_{, then as n goes to +∞, these points form a stationary Poisson point process with density n.}

Lemma 4 is a well-known result and its proof is given in [85] (Chapter 1.7, Page 39). Let P ≡ {ξi, i > 1} denote the set of active sensors. It is shown in Lemma 5 that P is also a stationary Poisson

point process with density np for infinitely large n. Note that, we consider the point coverage process in fixed deployment field D in this chapter, which might look a little different from that in [85], where the radius of covered disk is fixed and the size of D goes to infinity. However, those two point coverage processes are essentially equivalent after scaling.

LEMMA5. Let n points distributed independently and uniformly in a unit-area convex field D within R2_{. Each point is marked independently as an active point with probability p, where 0 < p 6 1. Then}

the set of active points, P = {ξ_i, i > 1}, form a stationary Poisson point process with density np as n goes to +∞.

Let Si denote a random disk with radius rs,i centered at the origin of R2, which is defined as

Si ≡ {x ∈ R2 : |x| 6 rs,i}, where rs,i is the sensing radius of the i-th active sensor ξi. Here, we

assume that all sensing radii are i.i.d. random variables following an unknown distribution F (r), with known mean r0 and variance σ2sr20, i.e., all Si’s are distributed as S ≡ {x ∈ R2 : |x| 6 r, r ∼

F (r)}. Then, the sensing disk (abbreviated as disk) centered at active sensor ξi can be defined as

D_i ≡ ξ_i+ S_i = {ξ_i+ y : y ∈ S_i}. The set of {D_i, i > 1} forms a stationary coverage process. For such a coverage process, Lemma 6 gives the distribution of the number of disks with certain properties.

LEMMA6. Let Q = {ξ_i+ S_i, i > 1} denote a stationary coverage process, where {ξ_i} is a stationary Poisson point process with density λ within D, and Si’s are distributed as S defined above. For a given

deterministic condition C, let Y denote the number of disks in Q that satisfy the condition C. Then, Y is Poisson-distributed with mean µ = λ · E

h

k{x : I_C(x + S) = 1}k i

, where I_C(·) is the indicator function of whether a disk satisfies the condition C, and k · k denotes the area.

Proofs of Lemmas 5 and 6 are included in [87]. Using Lemmas 4, 5, and 6, we can derive a lower bound (P_covL ) on Pcov. A similar bound has been given in [14] for the case of deterministic sensing

radius model and non-sleeping sensor networks. Theorem 3 is a generalization of the results in [14] for the random sensing radius model.

THEOREM3. For 0 < p 6 1 and as ≡ E

£

kSk¤= πr2

0(1 + σs2) < 1, as n goes to +∞, we have

Pcov > PcovL ≡ 1 − 2e−npas

à 1 +¡n2p2a0_s+ 2npr0 ¢Xk−1 i=0 (npas)i i! ! , (3.1) where a0 s≡ πr20(1 + σs2/2).

The proof of Theorem 3 is included in [80].

3.3.2 Asymptotic Lower Bound (PL

com) on Pcom

LEMMA7. Given the network described in Section 3.2.1, the number of a sensor’s communication neighbors (i.e., a communication link exists between two nearby sensors) is Poisson-distributed with mean µ = nacwhere ac ≡ π ¯L2(σc2+ 1).

The proof of Lemma 7 is included in [87]. THEOREM4. For 0 < p 6 1, as ≡ E

£

kSk¤= πr2

0(1 + σ2s) < 1, and ac ≡ π ¯L2(σ2c + 1) < 1, as n

goes to +∞, we have Pcom> PcomL ≡ 1 − npe−npac· h(n, ¯L), where

h(n, ¯L) =        min n 1, 1₂ · (8npa0 s+ 4n2p2a0sas) · e−λc P_k−1 j=0 (λc) j j! o if rL> LU, 1 if r_L6 L_U, (3.2)

and

λc = np(as− L2Uπ)(1 − (np)2e−2npac). (3.3)

Proof: According to Theorem 6.3 and Propositions 6.4-6.6 in Chapter 6.5 of [77], in a Poisson point process under the random-connection model (the same as the random communication link model described in Section 3.2.1.2), there is at most one unbounded connected component and the size of any other finite component converges to one, as the number of points goes to infinity. This implies that, in a randomly-deployed sensor network, there exists a dominant connected component (DCC) consisting of most sensors while the rest of the sensors are individually isolated (i.e., not connected to any other sensor), as n goes to infinity. Therefore, Pcomis equal to the probability that there does not exist an

isolated sensor whose sensing disk is not k-covered by sensors belonging to the DCC. Recall that Pcom

is the probability that the sensing disks of sensors isolated from the DCC are all k-covered by sensors belonging to the DCC in an infinite-size sensor network.

Pcom= 1 − P (∃ an isolated single active sensor s whose sensing disk is not k-covered

by sensors belonging to the DCC)

> 1 − np · P (s is isolated and its sensing disk is not k-covered by sensors belonging to the DCC) = 1 − np · P (s is isolated) · Pv = 1 − np · e−npac· Pv,

(3.4) where Pvdenotes the conditional probability that the sensing disk of s (denoted by Φs) is not k-covered

by sensors belonging to the DCC, given that s is isolated. The key step in obtaining PL

comis to find an appropriate upper bound for Pv. Applying the similar

coverage process techniques as those used in the proof of Theorem 3 (see [80] for details), suppose M0

belonging to the DCC, we have Pv 6 P (Mk0 > 2|s is isolated) 6 1 2· E(M 0 k|s is isolated) = 1

2· E(number of crossings within Φs|s is isolated)

· P (a crossing within Φsis not k-covered by sensors belonging to the DCC|s is isolated)

< 1

2· E(number of crossings within Φs)

· P (a crossing within Φsis not k-covered by sensors belonging to the DCC|s is isolated).

There are two types of crossings within Φs: the crossings on Φs’s boundary and the crossings inside

Φs’s boundary. Applying the similar calculation as that in the proof of Theorem 3 (see [80] for details),

we can get the expression of the expected number of crossings within Φsas below:

E(number of crossings within Φs) = 1₂(8npa0s+ 4n2p2a0sas). (3.5)

Now, combining (3.5) with the results in [87] about the conditional probability that a crossing within Φsis not k-covered by sensors belonging to the DCC given that sensor s is isolated, we can derive the

upper bound of Pvas follows:

P_v 6        min n 1, 1 2 · (8npa0s+ 4n2p2a0sas) · e−λc P_k−1 j=0 (λc) j j! o if r_L> L_U, 1 if rL6 LU, (3.6)

where λc = np(as − L2_Uπ)(1 − (np)2e−2npac). Recall that rL is the lower bound of the sensing

radius rs. By inserting Pv’s upper bound expression into (3.4), we complete the proof. ¤

Note that when r_L6 L_U, it is very difficult to derive an effective upper bound for P_vbased only on the mean and variance of rsand L. So we simply use 1 as Pv’s upper bound and hence have the lower

bound of Pcomequal to (1 − np · e−npac). Further study and analysis will be included in the future work