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Best Signal Quality in Cellular Networks: Asymptotic Properties and Applications to Mobility Management in Small Cell Networks

  • VanMinh Nguyen1Email author,
  • François Baccelli2,
  • Laurent Thomas1 and
  • ChungShue Chen2
EURASIP Journal on Wireless Communications and Networking20102010:690161

https://doi.org/10.1155/2010/690161

Received: 15 October 2009

Accepted: 21 March 2010

Published: 3 May 2010

Abstract

The quickly increasing data traffic and the user demand for a full coverage of mobile services anywhere and anytime are leading mobile networking into a future of small cell networks. However, due to the high-density and randomness of small cell networks, there are several technical challenges. In this paper, we investigate two critical issues: best signal quality and mobility management. Under the assumptions that base stations are uniformly distributed in a ring-shaped region and that shadowings are lognormal, independent, and identically distributed, we prove that when the number of sites in the ring tends to infinity, then (i) the maximum signal strength received at the center of the ring tends in distribution to a Gumbel distribution when properly renormalized, and (ii) it is asymptotically independent of the interference. Using these properties, we derive the distribution of the best signal quality. Furthermore, an optimized random cell scanning scheme is proposed, based on the evaluation of the optimal number of sites to be scanned for maximizing the user data throughput.

Keywords

Signal StrengthPath LossSignal QualityMobility ManagementTail Distribution

1. Introduction

Mobile cellular networks were initially designed for voice service. Nowadays, broadband multimedia services (e.g., video streaming) and data communications have been introduced into mobile wireless networks. These new applications have led to increasing traffic demand. To enhance network capacity and satisfy user demand of broadband services, it is known that reducing the cell size is one of the most effective approaches [14] to improve the spatial reuse of radio resources.

Besides, from the viewpoint of end users, full coverage is particularly desirable. Although today's macro- and micro-cellular systems have provided high service coverage, 100%-coverage is not yet reached because operators often have many constraints when installing large base stations and antennas. This generally results in potential coverage holes and dead zones. A promising architecture to cope with this problem is that of small cell networks [4, 5]. A small cell only needs lightweight antennas. It helps to replace bulky roof top base stations by small boxes set on building facade, on public furniture or indoor. Small cells can even be installed by end users (e.g., femtocells). All these greatly enhance network capacity and facilitate network deployment. Pervasive small cell networks have a great potential. For example, Willcom has deployed small cell systems in Japan [6], and Vodafone has recently launched home 3G femtocell networks in the UK [7].

In principle, high-density and randomness are the two basic characteristics of small cell networks. First, reducing cell size to increase the spatial reuse for supporting dense traffic will induce a large number of cells in the same geographical area. Secondly, end users can set up small cells by their own means [2]. This makes small cell locations and coverage areas more random and unpredictable than traditional mobile cellular networks. The above characteristics have introduced technical challenges that require new studies beyond those for macro- and micro-cellular networks. The main issues concern spectrum sharing and interference mitigation, mobility management, capacity analysis, and network self-organization [3, 4]. Among these, the signal quality, for example, in terms of signal-to-interference-plus-noise ratio (SINR), and mobility management are two critical issues.

In this paper, we first conduct a detailed study on the properties of best signal quality in mobile cellular networks. Here, the best signal quality refers to the maximum SINR received from a number of sites. Connecting the mobile to the best base station is one of the key problems. The best base station here means the base station from which the mobile receives the maximum SINR. As the radio propagation experiences random phenomena such as fading and shadowing, the best signal quality is a random quantity. Investigating its stochastic properties is of primary importance for many studies such as capacity analysis, outage analysis, neighbor cell scanning, and base station association. However, to the best of our knowledge, there is no prior art in this area.

In exploring the properties of best signal quality, we focus on cellular networks in which the propagation attenuation of the radio signal is due to the combination of a distance-dependent path-loss and of lognormal shadowing. Consider a ring of radii and such that . The randomness of site locations is modeled by a uniform distribution of homogeneous density in . Using extreme value theory (c.f., [8, 9]), we prove that the maximum signal strength received at the center of from sites in converges in distribution to a Gumbel distribution when properly renormalized and it is asymptotically independent of the total interference, as . The distribution of the best signal quality can thus be derived.

The second part of this paper focuses on applying the above results to mobility support in dense small cell networks. Mobility support allows one to maintain service continuity even when users are moving around while keeping efficient use of radio resources. Today's cellular network standards highlight mobile-assisted handover in which the mobile measures the pilot signal quality of neighbor cells and reports the measurement result to the network. If the signal quality from a neighbor cell is better than that of the serving cell by a handover margin, the network will initiate a handover to that cell. The neighbor measurement by mobiles is called neighbor cell scanning. Following mobile cellular technologies, it is known that small cell networking will also use mobile-assisted handover for mobility management.

To conduct cell scanning [1012], today's cellular networks use a neighbor cell list. This list contains information about the pilot signal of selected handover candidates and is sent to mobiles. The mobiles then only need to measure the pilot signal quality of sites included in the neighbor cell list of its serving cell. It is known that the neighbor cell list has a significant impact on the performance of mobility management, and this has been a concern for many years in practical operations [13, 14] as well as in scientific research [1518]. Using neighbor cell list is not effective for the scanning in small cell networks due to the aforementioned characteristics of high density and randomness.

The present paper proposes an optimized random cell scanning for small cell networks. This random cell scanning will simplify the network configuration and operation by avoiding maintaining the conventional neighbor cell list while improving user's quality-of-service (QoS). It is also implementable in wideband technologies such as WiMAX and LTE.

In the following, Section 2 describes the system model. Section 3 derives the asymptotic properties and the distribution of the best signal quality. Section 4 presents the optimized random cell scanning and numerical results. Finally, Section 5 contains some concluding remarks.

2. System Model

The underlying network is composed of cells covered by base stations with omnidirectional antennas. Each base station is also called a site. The set of sites is denoted by . We now construct a model for studying the maximum signal strength, interference, and the best signal quality, after specifying essential parameters of the radio propagation and the spatial distribution of sites in the network.

As mentioned in the introduction, the location of a small cell site is often not exactly known even to the operator. The spatial distribution of sites seen by a mobile station will hence be treated as completely random [19] and will be modeled by an homogeneous Poisson point process [20] with intensity .

In the following, it is assumed that the downlink pilot signal is sent at constant power at all sites. Let be some strictly positive real number. For any mobile user, it is assumed that the distance to his closest site is at least and hence the path loss is in the far field. So, the signal strength of a site received by a mobile at a position is given by
(1)

where is the location of site represents the base station's transmission power and the characteristics of propagation, is the path loss exponent (here, we consider ), and the random variables , which represent the lognormal shadowing, are defined from , an independent and identically distributed (i.i.d.) sequence of Gaussian random variables with zero mean and standard deviation . Typically, is approximately 8 dB [21, 22]. Here, we consider that fast fading is averaged out as it varies much faster than the handover decision process.

Cells sharing a common frequency band interfere one another. Each cell is assumed allocated no more than one frequency band. Denote the set of all the cells sharing frequency band th by , where . So for , and . The SINR received at from site is expressible as
(2)
where is the thermal noise average power which is assumed constant. For notational simplicity, let . Then is given by
(3)

In the following, we will use (3) instead of (2).

3. Best Signal Quality

In this section, we derive the distribution of the best signal quality. Given a set of sites , the best signal quality received from at a position , denoted by , is defined as
(4)

Let us first consider a single-frequency network (i.e., ).

Lemma 1.

In the cell set of single-frequency network, the site which provides a mobile the maximum signal strength will also provide this mobile the best signal quality, namely,
(5)
where
(6)
is the maximum signal strength received at from the cell set , and
(7)

is the total interference received at .

Proof.

Since and , (5) follows from the fact that no matter which cell is considered, is the same and from the fact that with constant is an increasing function of .

Let us now consider the case of multiple-frequency networks. Under the assumption that adjacent-channel interference is negligible compared to cochannel interference, cells of different frequency bands do not interfere one another. Thus, for a given network topology , the SINRs received from cells of different frequency bands are independent. In the context of a random distribution of sites, the SINRs received from cells of different frequency bands are therefore conditionally independent given . Write cell set as
(8)
with the subset of allocated to frequency . Let
(9)
be the best signal quality received at from sites which belong to . The random variables are conditionally independent given . As a result,
(10)

Remark 1.

For the coming discussions, we define
(11)

which is the interference from cells in set . In the following, for notational simplicity, the location variable appearing in , and will be omitted in case of no ambiguity. We will simply write , and . Note that since .

Following Lemma 1, the distribution of can be determined by the joint distribution of and , which is given below.

Corollary 1.

The tail distribution of the best signal quality received from cell set is given by
(12)

where is the joint probability density of and .

Proof.

By Lemma 1, we have
(13)

In view of Corollary 1, we need to study the properties of the maximum signal strength as well as the joint distribution of and . As described in the introduction, in dense small cell networks, there could be a large number of neighbor cells and a mobile may thus receive from many sites with strong enough signal strength. This justifies the use of extreme value theory within this context.

For some and such that , let be a ring with inner and outer radii and , respectively. In this section, we will establish the following results.
  1. (1)

    The signal strength received at the center of belongs to the maximum domain of attraction (MDA) of the Gumbel distribution (c.f., Theorem 1 in Section 3.1).

     
  2. (2)

    The maximum signal strength and the interference received at the center of from sites therein are asymptotically independent as (c.f., Corollary 3 in Section 3.1).

     
  3. (3)

    The distribution of the best signal quality is derived (c.f., Theorem 2 in Section 3.3).

     

3.1. Asymptotic Properties

To begin with, some technical details need to be specified. Given a ring as previously defined, we will study metrics (such as e.g., signal strength, interference, etc.) as seen at the center of for a set of sites located in . We will use the notation , and instead of , and , respectively, with
(14)

Lemma 2.

Assume that , that sites are uniformly distributed in , and that the shadowing follows a lognormal distribution of parameters . Then the cdf of the signal strength received at the center of from a site located in is given by
(15)
where , , , , and , , refers to the cdf of a lognormal distribution of parameters , in which
(16)

Proof.

See Appendix .

Under the studied system model, are independent and identically distributed (i.i.d.), and so the cdf and probability density function (pdf) of are directly obtained as follows.

Corollary 2.

Under the conditions of Lemma 2, the cdf and the pdf of are given, respectively, by
(17)
(18)

where is given by (15), and is the pdf of , .

Since is the maximum of i.i.d. random variables, we can also study its asymptotic properties by extreme value theory. Fisher and Tippett [9, Theorem  3.2.3] proved that under appropriate normalization, if the normalized maximum of i.i.d. random variables tends in distribution to a nondegenerate distribution , then must have one of the three known forms: Fréchet, Weibull, or Gumbel distribution. In the following, we prove that belongs to the MDA of a Gumbel distribution. First of all, we establish the following result that is required to identify the limiting distribution of .

Lemma 3.

Under the conditions of Lemma 2, the signal strength received at the center of from a site located in has the following tail equivalent distribution:
(19a)
(19b)

where , and .

Proof.

See Appendix .

Equation (19b) shows that the tail distribution of the signal strength is close to that of , although it decreases more rapidly. The fact that determines the tail behavior of is in fact reasonable, since is the distribution of the signal strength received from the closest possible neighboring site (with and ). The main result is given below.

Theorem 1.

Assume that , that sites are uniformly distributed in , and that shadowings are i.i.d. and follow a lognormal distribution of parameters with . Then there exists constants and such that
(20)
where is the standard Gumbel distribution:
(21)
and represents the convergence in distribution. A possible choice of and is
(22)

with given by Lemma 3.

Proof.

See Appendix .

By Theorem 1, the signal strength belongs to the MDA of the Gumbel distribution, denoted by . From [23, 24], we have the following corollary of Theorem 1.

Corollary 3.

Let be the variance and be the mean of signal strength . Let . Let , where and are given by (22). Under the conditions of Theorem 1,
(23)

where is the Gumbel distribution and the standard Gaussian distribution, and where the coordinates are independent.

Proof.

Conditions and provide . Then the result follows by Theorem 1 and [23, 24].

Note that the total interference can be written as where denotes the complement of in . Under the assumptions that the locations of sites are independent and that shadowings are also independent, and are independent. The asymptotic independence between and thus induces the asymptotic independence between and . This observation is stated in the following corollary.

Corollary 4.

Under the conditions of Theorem 1, and are asymptotically independent as .

This asymptotic independence facilitates a wide range of studies involving the total interference and the maximum signal strength. This result will be used in the coming sub-section to derive the distribution of the best signal quality.

Remark 2.

The asymptotic properties given by Theorem 1 and Corollaries 3 and 4 hold when the number of sites in a bounded area tends to infinity. This corresponds to a network densification process in which more sites are deployed in a given geographical area in order to satisfy the need for capacity, which is precisely the small cell setting.

3.2. Convergence Speed of Asymptotic Limits

Theorem 1 and Corollaries 3 and 4 provide asymptotic properties when . In practice, is the number of cells to be scanned, and so it can only take moderate values. Thus, it is important to evaluate the convergence speed of (20) and (23). We will do this based on simulations and will measure the discrepancy using a symmetrized version of the Kullback-Leibler divergence (the so-called Jensen-Shannon divergence (JSdiv)).

Let us start with some numerical evaluations of the convergence of to its limiting distribution. Figure 1(a) shows for different and compares to empirical simulation results. As expected the analytical distributions obtained by (17) of Corollary 2 match with the empirical distributions for all . Figure 1(b) plots the analytical distribution and its limiting distribution, that is, the Gumbel distribution . There is a discrepancy in the negative regime (see the circled region in Figure 1(b)). It is worth pointing out that as a maximum of signal strengths, and thus since . This means that , , whereas , . This explains the gap observed for small . This dissimilarity should have limited impact as long as we only deal with positive values of (resp., ).
Figure 1
Figure 1

CDF of under different : . Comparison between analytical and empirical CDFsComparison between analytical and limiting CDFs

We now study the symmetrized divergence between the analytical and limiting distributions of for some moderate values of and under different and . The convergence is best for around 10 dB and around two to four. For practical systems, is approx. 8 dB and . We compute the Jensen-Shannon divergence for and and plot the results in Figures 2(a) and 2(b), respectively. For these (and other) values (within the range given above) of and , , and have low divergence.
Figure 2
Figure 2

Jensen-Shannon divergence between and . Under different : Under different :

Let us now measure the (dis)similarity between the empirical joint distribution, , and the product of the empirical marginal distributions, . Figure 3 shows an example with , and . We see that these two density functions are very similar. Figure 4 compares these two density functions for different values of and . Within the range defined above, the divergence between the two distributions is again small. Comparing Figure 2 and Figure 4, one can conclude that even if the convergence of remains slow, and quickly become uncorrelated. Thus, the independence between and holds for moderate values of , that is,
(24)
and so the same conclusion holds for the independence between and .
Figure 3
Figure 3

Example of joint densities of and : . Here, norm. refers to , while norm. refers to .

Figure 4
Figure 4

Jensen-Shannon divergence between and . Under different : Under different :

3.3. Distribution of the Best Signal Quality

From the above results, we have the distribution of and the asymptotic independence between and . In order to derive the distribution of the best signal quality, we also need the distribution of the total interference.

Lemma 4.

Assume that shadowings are i.i.d. and follow a lognormal distribution of parameters , and that sites are distributed according to a Poisson point process with intensity outside the disk of radius . Let be the interference received at the disk center, and be the characteristic function of . Then:
  1. (1)
    (25)

    where , and is the characteristic function of ;

     
  2. (2)

    for all , where is the space of absolutely integrable functions;

     
  3. (3)
    If is large, then admits the following approximation:
    (26)

    where , with denoting the gamma function.

     

Proof.

See Appendix .

Theorem 2.

Under the assumptions of Lemma 4, let be the ring of inner and outer radii and , respectively. Denote the best signal quality received at the center of from sites in by . Assume that , that is large, and that , with high probability, where is some positive integer. Then the tail distribution of can be approximated by
(27)

Proof.

See Appendix .

The approximation proposed in Theorem 2 will be used in Section 4 below. It will be validated by simulation in the context considered there.

4. Random Cell Scanning for Data Services

In this section, the theoretical results developed in Section 3 are applied to random cell scanning.

4.1. Random Cell Scanning

Wideband technologies such as WiMAX, WCDMA, and LTE use a predefined set of codes for the identification of cells at the air interface. For example, 114 pseudonoise sequences are used in WiMAX [25], while 504 physical cell identifiers are used in LTE [26]. When the mobile knows the identification code of a cell, it can synchronize over the air interface and then measure the pilot signal quality of the cell. Therefore, by using a predefined set of codes, these wideband technologies can have more autonomous cell measurement conducted by the mobile. In this paper, this identification code is referred to as cell synchronization identifier (CSID).

In a dense small cell network where a large number of cells are deployed in the same geographical area, the mobile can scan any cell as long as the set of CSIDs used in the network is provided. This capability motivates us to propose random cell scanning which is easy to implement and has only very few operation requirements. The scheme is detailed below.
  1. (1)

    When a mobile gets admitted to the network, its (first) serving cell provides him/her the whole set of CSIDs used in the network. The mobile then keeps this information in its memory.

     
  2. (2)

    To find a handover target, the mobile randomly selects a set of CSIDs from its memory and conducts the standardized scanning procedure of the underlying cellular technology, for example, scanning specified in IEEE 802.16 [25], or neighbor measurement procedure specified in 3G [27] and LTE [12].

     
  3. (3)

    The mobile finally selects the cell with the best received signal quality as the handover target.

     

In the following, we determine the number of cells to be scanned which maximizes the data throughput.

4.2. Problem Formulation

The optimization problem has to take into account the two contrary effects due to the number of cells to be scanned. On one hand, the larger the set of scanned cells, the better the signal quality of the chosen site, and hence the larger the data throughput obtained by the mobile. On the other hand, scanning can have a linear cost in the number of scanned cells, which is detrimental to the throughput obtained by the mobile.

Let us quantify this using the tools of the previous sections.

Let be the average cell bandwidth available per mobile and assume that it is a constant. Under the assumption of additive white Gaussian noise, the maximum capacity that the mobile can have by selecting the best among randomly scanned cells is
(28)
Hence
(29)
where is the pdf of . By an integration by parts of and , this becomes
(30)

Note that is the expected throughput from the best cell. Since is the maximum signal quality of the cells, increases with and so does . Hence, the mobile should scan as many cells as possible. However, on the other hand, if scanning many cells, the mobile will consume much time in scanning and thus have less time for data transmission with the serving cell. A typical situation is that where the scanning time increases proportionally with the number of cells scanned and where the data transmission is suspended. This for instance happens if the underlying cellular technology uses a compressed mode scanning, like for example, in IEEE 802.16 [25] and also inter-frequency cell measurements defined by 3GPP [12, 27]. In this mode, scanning intervals, where the mobile temporarily suspends data transmission for scanning neighbor cells, are interleaved with intervals where data transmission with the serving cell is resumed.

Another scenario is that of parallel scanning-transmission: here scanning can be performed in parallel to data transmission so that no transmission gap occurs; this is the case in, for example, intrafrequency cell measurements in WCDMA [27] and LTE [12].

Let be the average time during which the mobile stays in the tagged cell and receives data from it. Let be the time needed to scan one cell (e.g., in WCDMA, the mobile needs if the cell is in the neighbor cell list and if not [28], whereas in WiMAX, , i.e., two 5-ms frames). Let be the duration of the suspension of data transmission due to the scanning of the cells:
(31)

Finally, let be the average throughput received from the serving cell when no scanning at all is performed (this would be the case if the mobile would pick as serving site one of the sites of set at random).

The gain of scanning cells can be quantified by the following metric, that we will call the acceleration
(32)

In this definition, (resp., is the expected amount of data transmitted when scanning cells (resp., doing no scanning at all). We aim at finding the value of that maximizes the acceleration .

It is clear that when (i) , that is, the mobile stays in and receives data from the tagged cell forever, or (ii) , that is, parallel scanning-transmission is enabled. In these cases, increases with and the mobile "should" scan as many cells as possible. However, is often concave and the reward of scanning then decreases. To characterize this, we introduce a growth factor defined as follows:
(33)

Special cases as those considered above can be cast within a general framework which consists in finding the value of that maximizes under the constraint that , where is a threshold.

4.3. Numerical Result

In the following, we show how to apply the above results to find the optimal . We adopt WCDMA as the underlying cellular network technology. 100 omnidirectional small cell base stations are deployed in a square domain of . The network density is thus equal to
(34)
It is assumed that any cell synchronization identifier can be found in a radius . We take equal to 2 meters. The propagation path loss is modeled by the picocell path loss model [29]:
(35)

where is the distance from the base station in meters, the number of penetrated floors in the propagation path. For indoor office environments, is the default value [22]; however, here, the small cell network is assumed to be deployed in a general domain including outdoor urban areas where there are less penetrated walls and floors. So, we use in our numerical study.

It is assumed that the total transmission power including the antenna gain of each small cell base station is . Shadowing is modeled as a random variable with lognormal distribution with an underlying Gaussian distribution of zero mean and standard deviation. The signal strength received at any distance from a base station is expressible as
(36)
By (35), we have
(37)
The parameters and appearing in can be identified from (37) after converting the received signal strength from the dBm scale to the linear scale:
(38)
The received noise power is given by
(39)

where the effective bandwidth  Hz, is the Boltzmann constant, and is the temperature in Kelvin,  W/Hz and is equal to 7 dB.

It is assumed that the mobile is capable of scanning eight identified cells within 200 ms [28]. So, the average time needed to scan one cell is given by .

In order to check the accuracy of the approximations used in the analysis, a simulation was built with the above parameter setting. The interference field was generated according to a Poisson point process of intensity in a region between and . For a number , the maximum of SINR received from base stations which are randomly selected from the disk between radii and was computed. After that the expectation of the maximal capacity received from the selected BSs was evaluated.

In Figure 5(a), the expectation of the maximal throughput for different is plotted, as obtained through the analytical model and simulation. The agreement between model and simulation is quite evident. As shown in Figure 5(a), increases with , though the increasing rate slows down as increases. Note that in Figure 5(a), is plotted after normalization by .
Figure 5
Figure 5

Numerical results in the random cell scanning optimization. Plot of Plot of secondOptimal number of cells to be scannedGrowth factor under different

Figure 5(b) gives an example of acceleration for second and . In the plot, is normalized by its maximum. Here, an agreement between model and simulation is also obtained. We see that first increases rapidly with , attains its maximum at by simulation and by model, and then decays.

Next, using the model we compute the optimal number of cells to be scanned and the growth factor for different . Note that in (32), the factor can be rewritten as
(40)

It is clear that this factor also depends on the ratio . Figure 5(c) plots the optimal for different values of . Larger will drive the optimal towards larger values. Since can be roughly estimated as the mobile residence time in a cell, which is proportional to the cell diameter divided by the user speed, this can be rephrased by stating that the faster the mobile, the smaller and thus the fewer cells the mobile should scan.

Finally, Figure 5(d) plots the growth factor with different . In Figure 5(d), the "limiting case" corresponds to the case when or . We see that is quite stable w.r.t. the variation of . Besides, flattens out at about 30 cells for a wide range of . Therefore, in practice this value can be taken as a recommended number of cells to be scanned in the system.

5. Concluding Remarks

In this paper, we firstly develop asymptotic properties of the signal strength in cellular networks. We have shown that the signal strength received at the center of a ring shaped domain from a base station located in belongs to the maximum domain of attraction of a Gumbel distribution. Moreover, the maximum signal strength and the interference received from cells in are asymptotically independent as . The above properties are proved under the assumption that sites are uniformly distributed in and that shadowing is lognormal. Secondly, the distribution of the best signal quality is derived. These results are then used to optimize scanning in small cell networks. We determine the number of cells to be scanned for maximizing the mean user throughput within this setting.

Appendices

A. Proof of Lemma 2

Let be the distance from a site located at to a position . Under the assumption that site locations are uniformly distributed in , the distance from a site located in , that is, , to the center of has the following distribution:
(A1)
Let , for , its distribution is equal to
(A2)

where , , and . The density of is given by .

Thus, the distribution of the power is equal to
(A3)
Substituting with lognormal distribution of parameters and given above into (A.3), after changing the variable such that , we have
(A4)
where the first integral is straightforward. By doing an integration by parts of and for the second integral, we get
(A5)
After some elementary simplifications, we can obtain
(A6)

where , , , , and . Let , be the lognormal distribution of parameters , can be rewritten as (15).

B. Proof of Lemma 3

Let and note that , we have from (15)
(B1)
This yields the tail distribution :
(B2)
For (B.2), we have . An asymptotic expansion of for large [30, 7.1.23] gives us:
(B3)
in which after a Taylor expansion of the last term on the right-hand side, we can have
(B4)
This implies that
(B5)
In the same manner, we have
(B6)
A substitution of (B.5) and (B.6) into (B.2) results in
(B7)
Moreover, yields . Then, we have the following result for large :
(B8)
Taking this into account in (B.7), finally we have
(B9)

C. Proof of Theorem 1

We will use Lemma 3 and the following two lemmas to prove Theorem 1.

Lemma 5 (Embrechts et al. [9]).

Let be i.i.d. random variables having distribution , and . Let be an increasing real function, denote , and . If with normalizing constant and , then
(C1)

Lemma 6 (Takahashi [31]).

Let be a distribution function. Suppose that there exists constants , , , and such that
(C2)
For and , let . Then, with normalizing constants and , where
(C3)
Let be a real function defined on , is increasing with . Let be the random variable such that . By (19a) of Lemma 3, the tail distribution is given by
(C4)
By (C.4), satisfies Lemma 6 with constants , , , and . So, with the following normalizing constants:
(C5)
Then, by Lemma 5, we have
(C6)
By a Taylor expansion of , we have
(C7)
Since when , we have
(C8)

Substituting and from (C.5) into (C.8), we obtain and for (22). The conditions , and provide . This leads to , and consequently, .

D. Proof of Lemma 4

Under the assumptions of the lemma, the interference field can be modeled as a shot noise defined on excluding the inner disk of radius . Hence, using Proposition  2.2.4 in [20], the Laplace transform of is given by
(D1)
Noting that
(D2)
we have from (D.1) that
(D3)
Using the change of variable , we obtain
(D4)

where . So, substituting this into (D.3), we get the first part of the Lemma 4.

From (25), for all , we have
(D5)
Since , we have
(D6)
Therefore
(D7)

where is some positive constant, and hence the right hand-side of this is an absolutely integrable function. This proves the second assertion of Lemma 4.

Under the assumption that , can be approximated by
(D8)
For , we have
(D9)
Since , we can write . Taking expectations on both sides, we get
(D10)
Substituting this into (D.8) and noting that
(D11)

for lognormally distributed, we obtain (26).

E. Proof of Theorem 2

Under the assumption that sites are distributed as a homogeneous Poisson point process of intensity in , the expected number of cells in is . We assume that is much larger than , which ensures that there are cells in with high probability, so that is well defined.

Under the conditions of Theorem 2, and are asymptotically independent according to Corollary 4. So, by substituting (24) into (12), we have
(E1)
where
(E2)
It is easily seen that is square-integrable with respect to , and its Fourier transform w.r.t. is given by
(E3)
which yields:
(E4)
Besides, according to Lemma 4 we have that and , where is the space of square integrable functions. And so, by Theorem  3 in [32, page 509], is bounded continuous and square integrable. Hence, by applying the Plancherel-Parseval theorem to the inner integral of (E.1), we have
(E5)
where is the Fourier transform of . Take (E.4) into account for (E.5) and (E.1), we have
(E6)
where we further have
(E7)
Note that
(E8)
And under the assumption that , is approximated by (26). Thus, by (26) and (E.4), we have for :
(E9)
By (E.9), we get
(E10)

Substitute the above into (E.7) and then into (E.6), we have (27).

Authors’ Affiliations

(1)
Network and Networking Department, Bell Labs, Alcatel-Lucent, Nozay, France
(2)
TREC (Research Group on Network Theory and Communications), INRIA-ENS, (École Normale Supérieure), Paris, France

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Copyright

© Van Minh Nguyen et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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