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In this paper, we study the connectivity of multihop wireless networks
under the log-normal shadowing model by investigating the precise distribution
of the number of isolated nodes. Under such a realistic shadowing model, all
previous known works on the distribution of the number of isolated nodes were obtained
only based on simulation studies or by ignoring the important boundary effect
to avoid the challenging technical analysis, and thus cannot be applied to any
practical wireless networks. It is extremely challenging to take the
complicated boundary effect into consideration under such a realistic model
because the transmission area of each node is an irregular region other than a
circular area. Assume that the wireless nodes are represented by a Poisson
point process with densitynover a
unit-area disk, and that the transmission power is properly chosen so that the
expected node degree of the network equals ln*n* + *ξ* (*n*), where *ξ* (*n*) approaches to a constant *ξ *as n → ∞. Under such a shadowing model with the boundary effect taken into
consideration, we proved that the total number of isolated nodes is
asymptotically Poisson with mean e$ {-*ξ*}. The Brun’s sieve is utilized to derive the precise asymptotic distribution.
Our results can be used as design guidelines for any practical multihop
wireless network where both the shadowing and boundary effects must be taken
into consideration.

Connectivity is one of the most fundamental properties of multi-hop wireless networks. It is the premise for enabling a network with proper functions. The path-loss model (also known as the unit-disk communication model) of wireless networks assumes that the received signal strength at a receiving node from a transmitting node is only determined by a deterministic function of the Euclidean distance between the two nodes. Under such a simple communication model, two nodes are directly connected if and only if their Euclidean distance is no more than a given threshold, and network connectivity has been well studied in the literature (e.g., [

The study of multihop wireless networks with the log-normal shadowing model can date back to the early of 1980s [

In this paper, we assume that the wireless networking nodes are represented by a Poisson point process with density

The vanishing of isolated nodes is not only a prerequisite but also a good indication of network connectivity. Under the path-loss model, it is well-known that the probability of having a connected network equals the probability of having no isolated nodes in the network as the node density

In what follows,

The remaining of this paper is organized as follows. In Section 2, we give a literature review for related work of our paper. The log-normal shadowing model is introduced and explained in Section 3. In Section 4, we give some definitions and geometric results that will be used to prove the main result of this paper. In Section 5, we derive the precise asymptotic distribution of the number of isolated nodes in the network under the log-normal shadowing model. Finally, we conclude our paper in Section 6.

Under the unit-disk communication model, network connectivity has been extensively studied, and a huge number of existing research work are available in the literature [

where

The log-normal shadowing model is a much more realistic radio propagation model and has been widely used by many researchers for network connectivity [

Most of the results in these known works were obtained only based on simulation studies or ignoring the important boundary effect to avoid the rigorous analysis by assuming the toroidal metric as done in the literature. To the best of our knowledge, there are no theoretical results on asymptotic distribution of the number of isolated nodes in the network obtained by rigorous analytical studies with the realistic log-normal shadowing model when the complicated boundary effect is taken into consideration.

With the path-loss model, the received power levels decrease as the distance between the transmitter and the receiver increases. Attenuation of radio signals due to path-loss effect has been modelled by averaging the measured signal power over long times and distances around the transmitter. The averaged power at any given distance

where

But the path-loss model could be inaccurate because in reality the received power levels may show significant variations around the area mean power value. Due to these variations, short links could disappear while long links could merge. The log-normal shadowing model allows for random power variations around the area mean power. With the log-normal shadowing model, the received mean power taken over all possible locations that are at distance

Assume that links are symmetric and the received power at node

where

For any two nodes separated by the Euclidean distance

And we say that any two nodes are directly connected if and only if there exists a link between them.

Define

If both sides of Equation (1) minus

Then Equation (2) is equivalent to

Thus for any two nodes separatedy the Euclidean distance

When

Thus, any two nodes are directly connected if and only if their Euclidean distance is at most

When

The following lemma demonstrates how the probability

Lemma 1. When

Proof. According to Equation (5),

In this section, we shall give some definitions that will be used to prove our main result of this paper. The results in this section are purely geometric, with no probabilistic content. Let

For the given maximum transmission radius r, the unit-area disk

Then we have

(a) (b)

(a) Unit-disk communication model; (b) Log-normal shadowing model

Partition of the unit-area disk

In this section, we assume that all the nodes transmit at a uniform power

We use the same notations as in Section 3. Recall that

Let

Refer to the discussions in [

Based on our assumptions,

Let

Then, when

In this paper, we make the following two assumptions:

1)

2) the transmission power

The main theorem of this paper is stated below:

Theorem 2. Under the two assumptions given above, the total number of the isolated nodes in the network is asymptotically Poisson with mean

Remarks. If the probability

where

If the probability

where

Theorem 2 will be proved by using the Brun’s sieve in the form described, for example, in [

Theorem 3. (Brun’s sieve) Let

If there is a constant

then

To apply Theorem 3, let

Therefore, in order to prove Theorem 2, it is sufficient to show that for any fixed positive integer

The proof of this asymptotic equation will use the following lemmas.

Lemma 4. Assume the conditions of Theorem 2 hold. Then there exist a sufficiently large constant

for all

Proof. We prove the lemma by contradiction and assume the contrary is true. Then for any arbitrarily large

The above inequality holds for any arbitrarily large

The following lemma shows that

Lemma 5. Assume the conditions of Theorem 2 hold. Then we have

where

Proof. Note that

By Lemma 4, there exist a sufficiently large constant

(both independent of

Thus, the lemma is proved.

Next we introduce a lemma that has only one event involved and has been proved in [

Lemma 6. For any

If

Lemma 7. For any

Proof. First we prove the lemma holds when

Case 1.

It remains to show that there is a constant

For any

Let

Apply the same approach in deriving the probability for

Case 2.

center

Note that the inequality still holds for annuli not fully contained in

For any

Then

Thus,

The lemma holds for the constant

When

centers is at most

as the case for

Next we assume

Since at least one

Thus, the lemma is proved.

Now we are ready to prove the asymptotic Equation (12). The proof of this asymptotic equation is divided into three lemmas. The case for

Lemma 8.

Proof.

We will prove

For the integral over

For the integral over

Next we calculate the integral over

Therefore, by Equation (7) and Equation (17)

To complete the proof of the lemma, it is sufficient to prove that this integral over

Case 1.

Thus,

The last equation holds since

Case 2.

Thus,

Therefore, the lemma is proved.

Lemma 9. For any

Proof. By Equations (15) and (16), it is straightforward to verify that the lemma holds when

We first prove the case when

where the second last equation holds from Lemma 5.

Next we assume

Then we have

where the last equation holds by following the similar arguments as the case

This completes the proof of the lemma.

Lemma 10. For any

Proof. For any

Next we calculate the two integrals

the last equation holds by following the same steps as we used in the proof of Lemma 8.

where the last equation holds from Lemma 9.

This completes the proof of the lemma.

In this paper, we assume that the wireless nodes are represented by a Poisson point process with density n over a unit-area disk, and that the transmission power is properly chosen so that the expected node degree of the network equals

The work of Dr. Lixin Wang in this paper is supported in part by the NSF grant HRD-1238704 of USA.