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In modern wireless networks deployments, each serving node needs to keep its Neighbour Cell List (NCL) constantly up to date to keep track of network changes. The time needed by each serving node to update its NCL is an important parameter of the network’s reliability and performance. An adequate estimate of such parameter enables a significant improvement of self-configuration functionalities. This paper focuses on the update time of NCLs when an approach of crowdsourced user reports is adopted. In this setting, each user periodically reports to the serving node information about the set of nodes sensed by the user itself. We show that, by mapping the local topological structure of the network onto states of increasing knowledge, a crisp mathematical framework can be obtained, which allows in turn for the use of a variety of user mobility models. Further, using a simplified mobility model we show how to obtain useful upper bounds on the expected time for a serving node to gain Full Knowledge of its local neighbourhood.

Neighbour Cell List Discovery (NCLD) is a core process of modern wireless networks, especially when deployed in an unplanned and decentralised manner like WiFi hotspots and LTE femtocells [

Though related to location discovery, the topic of this manuscript is the discovery of existing neighbours without targeting their actual geographical position. We focus on the process of NCLD via

Example of NCLD. Each user report to the serving node which neighbouring nodes they observe.

Keeping the NCL updated is fundamental for a number of reasons:

Neighbouring cells can be added, removed, or temporarily offline.

The handover to a new cell might be problematic whenever it is not contained in the NCL of the serving cell.

Some cells should not be added to the NCL list because they might reflect spurious measurements, yielding nonreliable handovers.

Neighbours with the same PCI should be handled with specific solutions.

With these reasons in mind, this manuscript studies the time

In the literature, it is usually assumed that neighbourhood information may be instantaneously acquired [

In a framework where the NCL is built via crowdsourced user reports, our main goal is to rigorously characterise

The rest of the paper is organised as follows: in Section

In the field of decentralised algorithm design, it has been shown that local knowledge of network topology is enough to produce a distributed algorithm for resource allocation; such local knowledge also allows the minimisation of scrambling-code collision and confusion in small cell networks; see [

Perhaps the main motivation of this manuscript is the work of [

The user report function is already available in commercial femtocells [

Crowdsourcing approaches have been investigated for different applications, for example, for estimating both density and number of attendees of large events [

With this work, we address the problem of estimating the NCL construction time, which is necessary to assess whether crowdsourcing is effective in a particular network deployment. However, to the best of our knowledge, this problem has not been addressed in the literature yet.

We show in this section some timely use cases where our proposed framework can be applied as a network design tool.

In order to provide seamless mobility and a satisfactory service, the optimisation of the handover function is fundamental in modern 3G cellular networks. To achieve that, the construction of a reliable NCL is one of the most critical tasks. While in the past this was achieved by drive and walk testing, the needs to adapt to changes in the network and to reduce the cost require different solutions [

The so-called Detected Set Reporting (DSR) is an intrafrequency 3GPP functionality that allows users to report cells not defined in the NCL. In this way, whenever a macrocell detects a problem, or when a new cell is deployed [

An important problem that affects the small cells deployment for residential use is code selection. In 3G, base stations have only few scrambling codes available, making the task of selecting the optimal allocation challenging. Moreover, communication with a central controller is discouraged, to avoid signalling overhead. In 4G and 5G, Physical Cell Identity codes and 5G scrambling codes have similar problems.

A fully decentralised algorithm that can converge to the optimal confusion- and collision-free code allocation has been devised in [

Given a set of wireless nodes

Let

Example of tessellation corresponding to scenario of Figure

Whenever a user is in

To keep the model as conservative as possible and to encompass the frequent case of half-duplex nodes, we assume

Let

Given an integer

If

Given a sequence of reports

The characterisation of the first time to FK generally depends on the realisation of a sequence of user reports; this means that

We end the section with a note on the tessellation.

A generic tessellation of

Since

Hypercube representation of the tessellation for

We can now define the main problems of this work.

Given an access point

Obviously, the way the user(s) moves inside the coverage area

There may exist cases where it is only necessary to characterise the first time to attain partial knowledge of the local topology. For example, we may be interested in the first moment when the neighbouring nodes that have been already discovered, that is, the elements of the knowledge set

Let

When

We will hereafter consider the Lebesgue measure

The concept of

The characterisation of

The realisation of

It will prove useful to consider a simplified mobility model in which a single user continuously teleport between tiles, without leaving

A single user moves within

Model

Assuming Model

Let

The following result holds.

The matrix

Let us consider the following partial ordering relation among the states:

The explicit computation of the whole matrix

Let

Even if

Regarding Problem

The following result fully characterises the spectrum of the matrix

For

The matrix

Since each eigenvalue is a sum of positive elements, the second-largest eigenvalue

Using (

Given

Using Lemma

Model

Model

Let us imagine that a single user travels inside the coverage area

As mentioned above, in a general mobility model it is likely that two successive user reports are correlated. These correlations may decay as the inter-report time grows larger and larger. As an example, let us imagine that a single user travels inside the coverage area

Therefore, the formulation and the results developed in Sections

We end this section by briefly mentioning a straightforward application of Model

Regarding the use case of femtocell self-organisation presented in Section

Opposite to the previous example, cells deployed in congested places like a mall have an extremely large basin of potential users. However, in situations where users main interest is other than connecting to the Internet, it is reasonable to expect the single-user reporting-activity to be rather sporadic. Therefore, the Poissonian approximation that we have mentioned at the end of Section

In this section, we offer a preliminary assessment of the possibility of using the machinery developed so far in real applications. To this purpose, we developed a simulation framework in MATLAB and studied a scenario where 8 nodes are positioned on a plane at random according to a uniform (bivariate) probability distribution, that is,

Figure

Empirical probability mass function of the expected time to 0.9-knowledge, and the number of steps to have 0.9-knowledge with 90% confidence, for the teleport model on random positioned nodes. Since the inter-report time is fixed, the simulation time is directly proportional to the number of reports.

Empirical cumulative distribution function of the expected time to 0.9-knowledge, and the number of steps to have 0.9-knowledge with 90% confidence, for the teleport model on random positioned nodes. Since the inter-report time is fixed, the simulation time is directly proportional to the number of reports.

Roughly speaking, a user moving at 0.5 m/s according to a random walk model, and providing

To summarise, simulation on random scenarios show that our proposed bound can be used to estimate the time to

In order to investigate and confirm the ideas of Section

In Figure

Empirical mean 0.9-knowledge time (in hours) of a random walk versus inter-report period, compared with Model

We assume typical femtocell parameters, that is, that coverage radius is 50 m and that the user does a step in a grid of 2.5 m every 5 s. Figure

It is important to notice that the inter-report times used in Figure

To summarise, simulation on random walks corroborate the analysis of Section

A received power map for 4 base stations in the Hynes convention centre have been generated using the Wireless System Engineering (WiSE) [

Figure

Coverage areas at Hynes convention centre. The coloured lines delimit the extension of the coverage areas. Base stations are transmitting at 2.1 GHz with a power of 34 mW. Point

Figure

Expected time to

Figure

Expected time to FK,

To summarise, these results seem to confirm that the values obtained placing random nodes with circular coverage areas in Section

In this paper, we have introduced the problem of user-reports-based Neighbour Cell List Discovery and provided a crisp mathematical formulation of it for a simple mobility model. We have also shown that such mobility model can be effectively used as an upper bound for a wide range of mobility models when the user-reports frequency is lower than the inverse mixing time of the Markov chain of the actual mobility model. Additionally, we have provided a useful method to estimate the time to

Simulations on random scenarios with typical small cells parameters show that the expected number of reports in order to have a high degree of knowledge of the local topology is very small. Roughly speaking, a user moving at 0.5 m/s according to a random walk model, and providing at least one report every hour, can guarantee the serving node will have 0.9-knowledge with high probability in less than 5 h, and in less than two hours if reports are sent at least every 15 minutes. Since we do not expect the network topology to be affected by high network dynamics, these are acceptable times for the problems of interest. We encourage the adoption of the presented framework to assess the possibility of employing crowdsourced user reports in other self-configuration problems, comparing the time to

Simulations in more realistic scenarios show that the bounds obtained are compatible with the ones obtained from statistics on random scenarios with similar parameters. This seems to confirm that the use of statistics obtained from macroscopic parameters, such as densities of deployment and distribution of coverage radii, can be used as a tool to bound the time to

In conclusion, we provide a useful tool to estimate the time to NCL construction, which is fundamental to assess whether a decentralised algorithm can be employed in a given network scenario.

The authors declare that there are no conflicts of interest to disclose regarding the publication of this paper.

The authors would like to thank Anna Zakrzewska, from Bell Laboratories, Alcatel Lucent Ireland, for her insights and useful suggestions.