environments Article
Advancements in the Statistical Study, Modeling, and Simulation of Microwave-Links in Cellular Backhaul Networks Lior Gazit *
ID
and Hagit Messer
ID
School of Electrical Engineering, Tel Aviv University, Tel Aviv 6997801, Israel;
[email protected] * Correspondence:
[email protected]; Tel.: +972-3-640-8764 Received: 3 June 2018; Accepted: 26 June 2018; Published: 28 June 2018
Abstract: While the effect of rainfall and other environmental phenomena on a link budget in microwave wireless communication has been well studied for network design, it has usually been done for each microwave-link separately. Recently, attenuation in multiple microwave-links is being used simultaneously for rainfall mapping over specific areas, and consequently, rain-induced attenuation fields can be constructed. Dedicated algorithms have been designed to relate attenuation in multiple microwave-links to its corresponding rain-field. Their performance depends significantly on the structure of the network. As the topology of a cellular microwave network (CMNs) is region-dependent, general theory for its effect on performance can only be developed statistically. In this paper we study the statistical nature of CMNs and lay the groundwork for such models based on empirical results. Keywords: cellular microwave-links; rainfall mapping; rain-field estimation; network statistics; non-linear regression
1. Introduction In cellular backhaul networks, cellular microwave-links (CMLs) are used as wireless channels to connect two base stations (BS). Figure 1 portrays a single CML. Each BS is equipped with a transmitting-receiving antenna. Giuli et al. describe in their work [1,2] how such microwave channels can be utilized for rainfall mapping. Capitalizing on the use of CMLs in cellular communications, Messer et al. [3] suggested commercial backhaul networks for environmental monitoring. They suggested utilizing the commercial network, already set-up and functional, thus offering a cheap and opportunistic approach to the problem of precipitation monitoring. In that framework, the precipitation field is modeled as the signal to be reconstructed and the CMLs are modeled as random line projections which sample the signal and serve as data observation. As the microwave propagates along the CML, it accumulates attenuation that is attributed greatly to the air’s present moisture, thus providing a form of spatial sampling of the precipitation field. This physical phenomenon is described in [4] and is elaborated on in [1]. In [5] it is shown that in order to yield a reconstruction method for a given sampled field, one must characterize the sampling scheme, i.e., the distribution of CMLs. In order to do so, we studied the design of a backhaul network. When a cellular microwave network (CMN) is being initially architected, the chosen topology of network BS, and thus CMLs, is based on several considerations. One can divide those considerations to two types, micro and macro.
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Figure 1. A single cellular microwave–link (CML) [6].
Micro factors would Figure be those have a minor impact on(CML) the chosen 1. A Athat single cellular microwave–link [6]. position of the BS. For Figure 1. single cellular microwave–link (CML) [6]. instance, after deciding to position a BS on a specific street block, one would consider micro factors for positioning thatwould BS on be a specific building’s rather on than The macro factors, on Micro factors those that have aroof-top minor impact theanother’s. chosen position of the BS. For Micro factors would be those that have what a minor impact onamount the chosen position of the BS. the other hand, will inform the CMN architect should be the of CMLs to deploy in an instance, after deciding to position a BS on a specific street block, one would consider micro factors For instance, after deciding to position a BS on a specific street block, one would consider micro area, how to spatially distribute them, and how their rather lengthsthan (i.e.,another’s. distancesThe between for positioning that BS on a specific building’s roof-top macro linked factors,BS) on factors for positioning that BS on athe specific building’s roof-top rather thanbe another’s.inThe macro factors, should vary. Insights regarding of backhaul networks We suggest the other hand, will inform the CMNdesign architect what should be the can amountfound of CMLs[7]. to deploy in an on the other hand, willfactor inform CMN architect what be the is amount of CMLs to deploy in that in the thethem, spread of CMLs. Thisshould claim, the basis of this paper, relies area,there howistoa random spatially distribute and how their lengthswhich (i.e., distances between linked BS) an area, howmicro to spatially distribute them, and how their lengths (i.e., distances between linked BS) on bothvary. the andregarding macro factors. However, in this paper we focus on found the macro factors as they should Insights the design of backhaul networks can be in [7]. We suggest should vary. Insights regarding the design of backhaul networks can be found in [7]. We suggest that impact the CMLs globally. willwhich be demonstrated, CMLs’ that there is aspatial randomdistribution factor in the of spread of CMLs. ThisAs claim, is the basis of this paper,spatial relies there is a random factor in the spread of CMLs.ofThis claim, which is the basis of this paper, relies on distribution can be divided into subsets distribution categories, each corresponding to on both the micro and macro factors. However, in this paper we focus on the macro factors as they both the micro and macro factors. However, in this paper we focus on the macro factors as they impact characteristic population density and impact the spatial distribution of topography. CMLs globally. As will be demonstrated, CMLs’ spatial the spatial distribution of CMLs globally. As distributed will be demonstrated, CMLs’ spatial distribution be As pointed out in [8], CMLs tend to be in clusters. This means not only iscan their distribution can be divided into subsets of distribution categories, eachthat corresponding to divided into subsets of distribution categories, each corresponding to characteristic population density spatial distribution observed to beand non-uniform, characteristic population density topography.it also leaves regions with little or no coverage. We and will topography. make the case these clusters in their to population this As pointed outthat in [8], CMLs tend correspond to be distributed in locations clusters. This means thatdensities. not only As is their As pointed outsupported, in [8], CMLs tend to bean distributed in clusters. This means that not only is their hypothesis will be it provides intuition for the macro factors of CMLs’ distribution spatial distribution observed to be non-uniform, it also leaves regions with little or no coverage. We spatial distribution observed to be non-uniform, it also leaves regions with little or no coverage. and will volume. make the case that these clusters correspond in their locations to population densities. As this We will the CMLs case that clustersofcorrespond in their locations to population densities. dense), As this Wemake analyze in these the context distribution namely being hypothesis will be supported, it provides an intuitioncategories, for the macro factors ofurban CMLs’(most distribution hypothesis will be supported, it provides an intuition for the macro factors of CMLs’ distribution suburban, and rural (least dense) [9,10]. In [11] one may also find a connection between population and volume. and volume. density perspective BSthe service capacity. Population volumesnamely may bebeing high urban so as to saturate BS Weand analyze CMLs in context of distribution categories, (most dense), Wecapacity, analyze CMLs in the for context of distribution categories, namely being urbanTable (most1 dense), service thus calling the allocation of additional BS to share the load. shows suburban, and rural (least dense) [9,10]. In [11] one may also find a connection between population suburban, and rural (least dense) [9,10]. In [11] one may also find a connection between population results BS density studies. densityofand perspective BS service capacity. Population volumes may be high so as to saturate BS density and perspective BS capacity.To Populationthis volumes may be high so as to saturate BS BS capacity, densities project on service CML densities. relationship, 2 presents four service thus calling for the allocationwitness of additional BS to shareFigure the load. Table 1the shows service capacity, thus calling for the allocation of additional BS to share the load. Table 1 shows results common topologies. One can see that for each of these, the number of edges is similar to the results of CML BS density studies. of BS density studies. number of nodes, project as they on symbolize the CMLs BS, respectively. BS densities CML densities. Toand witness this relationship, Figure 2 presents the four common CML topologies. One can see that for each of these, the number of edges is similar to the Table 1. Statistical base station (BS) densities [10]. BS densities depend on population densities. Since Table Statistical basesymbolize station (BS) the densities densities depend on population densities. Since number of 1.nodes, as they CMLs[10]. andBSBS, respectively. cellular microwave-links (CMLs) topologies are such that the number of CMLs is nearly identical to cellular microwave-links (CMLs) topologies are such that the number of CMLs is nearly identical to the number of BS, this table also reflects spatial densities of CMLs. the number of BS, this also(BS) reflects spatial densities of CMLs. Table 1. Statistical basetable station densities [10]. BS densities depend on population densities. Since cellular microwave-links (CMLs) topologies are such that the number of CMLs is(1/km nearly2)identical to 2 ) 2) 2) Area BS Amount RegionRegion BS AmountBS Density Area (km(km BS Density (1/km the number of BS, this table also reflects spatial densities of CMLs. Most Dense × 40 62516251 2.604 2.604 Most Dense City City 60 60 × 40 2 Second Densest City 30 × 50 1911 1.2742) Second Densest City 30 × 50 1911 1.274(1/km Region Area (km ) BS Amount BS Density Third Third Densest City 40 × 40 977 City 40 977 0.611 MostDensest Dense City 60 × 40 6251 2.604 0.611 Rural 200 × 200 12,691 0.317
Second Rural Densest City Third Densest City Rural
200 30 × 200 50 40 × 40 200 × 200
12,691 1911 977 12,691
0.317 1.274 0.611 0.317
Figure 2. Typical cellular microwave-link (CML) topologies. All four are such that the number of Figure 2. Typical cellular microwave-link (CML) topologies. All four are such that the number of CMLs CMLs is approximately identical to the number of BS [6]. is approximately identical to the number of BS [6]. Figure 2. Typical cellular microwave-link (CML) topologies. All four are such that the number of CMLs is approximately identical to the number of BS [6].
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BS densities project on CML densities. To witness this relationship, Figure 2 presents the four common CML topologies. One can see that for each of these, the number of edges is similar to the number of nodes, as they symbolize the CMLs and BS, respectively. Figure 2 and Table 1 suggest that the spatial distribution of CMLs is not, and cannot be homogenous. However, our study is based on the assumption that any given region can be partitioned into sub-regions, each homogeneous in the sense of CMLs density, meaning that all the CMLs in a sub-region have their positions drawn from the same uniform distribution. Here and throughout this paper, when referring to a CML’s position, it is the position of its midpoint that is considered. Based on [12], we suggest describing CML distribution in any homogeneous region by three characteristics: 1. 2. 3.
Spatial density of the CMLs; Orientations of the CMLs; Lengths of the CMLs.
The rest of the paper is organized as follows: in Sections 2 and 3 we study the CMLs’ statistical characteristics as listed above and suggest statistical models. Section 2 addresses the CMLs’ spatial density, and Section 3 addresses their orientations and lengths. Section 4 suggests a mathematical model for the relationship between CMLs’ lengths and their spatial density. Section 5 then combines the statistical derivations from previous sections to suggest a novel computational method for simulating sets of synthetic CMLs. Section 6 then concludes and discusses the results. The geographical regions analyzed here are described in the appendix. Note that as this paper is an extension of [13], its novelty is presented in Sections 4 and 5. 2. Spatial Distribution In order to be able to characterize the distributions of CMLs in a given region, we suggest partitioning the region into sub-regions, each with a spatially uniform CML density. Such sub-regions are expected to correspond to the common environmental terms: urban, suburban, and rural. If such a partition to homogeneous regions was not performed and the CMLs were clustered together, their spatial distribution would need to be addressed more meticulously in order to evaluate reconstruction potential, as it would not have been simply uniform. When partitioning a region, it is critical to maintain a nominal area size which is appropriate for capturing relevant rain phenomena. Typical rain clouds over Israel tend to stretch over an area of up to 10 × 10 (km2 ) [14]. It is recommended to maintain a minimum of such region size. Accordingly, in this paper we examined regions that are indeed 10 × 10 (km2 ). 3. Length and Orientation The locations and volume of CMLs that were discussed in Section 2 characterize CMLs collectively. Individual CML attributes present a significant attenuation measurement factor as well. These characteristics are the orientation and length of the CML. The understanding of all three factors allows for 2-D modeling of CMLs, as this paper is concerned with. It should be mentioned that if one were to examine the 3-D modeling of CMLs, the CMLs’ height variability would be a factor as well. The study presented here regards CMLs in the state of Israel belonging to a single cellular provider, Cellcom (see Appendix A for further details). All CMLs are operating in the same frequency range, the K-band. Results show that in any type of region studied, the CMLs’ orientation takes on any angle with equal probability. This means the direction of CMLs is distributed uniformly and is not at all correlated with the type of population density. Figure 3 portrays this conclusion. Moreover, the orientation is found to be statistically independent of the other factors studied here, the CML’s length and density.
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(a) (a)
(b) (b)
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Figure 3.3.The Thedistribution distribution of CML angles. (a)Israel, all Israel, top of northern as Hasharon rural, (c) Figure of CML angles. (a) all (b) top(b) of northern Israel as Israel rural, (c) Figure 3. as The distribution of Aviv CMLasangles. (a) all Israel, (b) top of northern Israel as rural, (c) Hasharon suburban, (d) Tel urban [6]. as suburban, (d) Tel Aviv as urban [6]. Hasharon as suburban, (d) Tel Aviv as urban [6].
Sendik [12] studied the distribution of CMLs as well. To do so, a map of Israel was partitioned Sendik [12] studied the distribution of CMLs do aa map of Israel was partitioned Sendik [12]based studied distribution CMLs as as well. well. To do so, so,29.5° map was into four parts on the latitude (as Israel stretches from latitude N of to Israel 33.29° N). partitioned These four ◦ N). These four parts into (as(as Israel stretches from latitude 29.5◦29.5° N toN 33.29 into four fourparts partsbased basedononlatitude latitude Israel stretches from latitude to 33.29° N). These four parts were used as sub-regions to study CMLs. However, these four regions were heterogeneous in were sub-regions to study CMLs.CMLs. However, these these four regions were heterogeneous in their parts used wereas used as sub-regions toby study However, four regions were heterogeneous in their environmental types. Here, isolating sub-regions of homogeneous environment types and environmental types.types. Here,Here, by isolating sub-regions of homogeneous environment typestypes and then their environmental by isolating sub-regions of homogeneous environment and then characterizing CML statistics by such types, we suggest a contribution to Sendik’s work on characterizing CML statistics by suchbytypes, we suggest a contribution to Sendik’s work on statistical then characterizing CML statistics statistical modeling of CMLs in Israel. such types, we suggest a contribution to Sendik’s work on modeling of CMLs in Israel. statistical modeling of CMLs in Israel. The study of CMLs’ lengths yielded much different results than the orientations. Unlike The of of CMLs’ lengths yielded much much different results than the orientations. Unlike orientations, Thestudy study CMLs’ lengths yielded different the orientations. Unlike orientations, CML lengths are distributed non-uniformly. CMLresults lengthsthan are distributed exponentially. CML lengthsCML are distributed non-uniformly. CML lengths are distributed exponentially. Moreover, orientations, lengths are distributed non-uniformly. CML lengths are distributed exponentially. Moreover, the lengths’ statistical characteristics depend on the type of environment. Figure 4 the lengths’the statistical depend on depend the typeonof the environment. Figure 4 illustrates Moreover, lengths’characteristics statistical characteristics type of environment. Figure 4 illustrates both findings. both findings. illustrates both findings.
(a) (a)
(b) (b)
(c) (c)
(d) (d)
Figure 4. Lengths distributions for various environments. (a) all Israel, (b) top of northern Israel as Figure(c) 4.Hasharon Lengths environments. top of of northern Israel as Figure 4. Lengths distributions distributions for various environments. Israel,(b)(b) top northern Israel rural, as suburban,for (d)various Tel Aviv as urban [6]. (a)(a)allallIsrael, rural, (c) Hasharon as suburban, (d) Tel Aviv as urban [6]. as rural, (c) Hasharon as suburban, (d) Tel Aviv as urban [6].
The sample-means specified in Figure 4 are used for the fitting of an exponential distribution. The sample-means specified in Figure are used for the fitting of density an exponential distribution. Expused (1/ ) for For an exponential random variable, e.g., T44~are , thethe probability functiondistribution. is: The sample-means specified in Figure fitting of an exponential For an exponential random variable, e.g., T ~ Exp (1/ ) , the probability density function is: For an exponential random variable, e.g., T ∼ Exp(1/θ ), the probability density function is: t
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