Keywords

The second research area considered in this book refers to Wireless Mesh Networks (WMNs) formed by stationary mesh routers organized in a mesh topology [3, 22], providing transportation of flows originating from mesh clients (with little or no mobility). As presented in Fig. 4.1, WMN nodes have the mesh capability meaning that their functioning is not restricted only to transmission of local data. Instead, they are also able to relay in a multi-hop fashion information belonging to flows from other WMN nodes [18, 25]. If equipped with necessary functionality at certain nodes (i.e., gateways), WMNs may be also utilized to provide connectivity with external networks, e.g., Internet [5, 8, 68].

Fig. 4.1
figure 1

Example architecture of a Wireless Mesh Network including wireless mesh routers, mesh clients, and gateway routers

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Most of WMN architectures are based on IEEE 802.11 standard defining how wireless devices can be mutually interconnected to create a mesh network [26]. In general, compared to Wi-Fi solutions, mesh structure of these networks implies a substantial enhancement in terms of the coverage area, connectivity, and scalability improvement, as well as brings about the simplification of deployment and maintenance activities [18, 68]. Additionally, WMN end users are provided with single-domain connectivity, as opposed to switching between Wi-Fi hot-spots. It has been proved that grid organization of WMN nodes provides up to 50 % higher throughput, compared to random node placement [68].

Due to utilization of the 71–86 GHz band [29, 39, 66], as well as highly directional antennas, effective transmission rate can be as high as 1–10 Gb/s per a millimeter-wave link with transmission range of at least several kilometres [64, 72]. Therefore, WMNs can be seen as a promising alternative to wired local or even metropolitan area networks providing last few miles connectivity especially in sparsely populated rural areas [22, 42].

It is also possible to equip each WMN router with MIMO technology (i.e., multiple-input multiple-output) utilizing multiple orthogonal channels [8]. This in turn leads to a further substantial increase of the network capacity [31, 71]. MIMO transmission is especially important in urban areas encountering signal distortions, where such systems help amplify and rebuild signal levels, while directional antenna settings visibly reduce interference between neighboring channels [68]. What is similarly important, WMNs can provide connectivity among users without direct Line of Sight (LOS) links.

WMNs have been also shown to be able to offer low costs of connections in the backhaul area [8]. That is why, utilization of WMN solutions (e.g., instead of applying the fiber optic technology) is well justified for economical, as well as practical reasons. It especially refers to 3G (4G) operators not having their own fiber infrastructure, who otherwise would have to either deploy their own fiber network (which is very expensive in rural areas [20]), or try to lease capacity from other network providers. Deployment of WMNs has been also proposed to obtain affordable access networks for underdeveloped regions [42].

In the last decade, many research teams have been addressing the problems of capacity planning, placement of WMN nodes, as well as routing, channel assignment, power control, topology control, etc. These problems are indeed very closely linked due to the nature of wireless interference. Therefore, when designing a WMN network, a joint consideration of these problems provides much better results in practice than in case of a separate analysis. A comprehensive overview of joint design problems is presented in [42].

A number of WMN installations are already in use in Europe, Australia, and US [17], deployed using equipment provided by, e.g., TerraNet, ArubaNetworks, or Motorola [4, 37, 62]. Example WMN architectures include city-wide (or campus-wide) networks in Las Palmas-Spain and Corpus Cristi [65], Cambridge-Massachusetts US [68], Houston-US [49], Oulu-Finland [59], Madison-US [69], or Dartmouth-US [24] with the number of nodes ranging from tens to hundreds, and the area of coverage measured in tens of square kilometres.

Apart from inheriting the common characteristics of the general ad-hoc networking concept (i.e., decentralized design, distributed communications), WMNs are known to exhibit characteristics that are novel in the wireless context, but rather typical to wired networks, i.e., stationary nodes, no LOS connectivity, high capacity, no limitations referring to node energy consumption [42].

Considering transmission of information itself, we can even say that WMNs possess most of wired networks characteristics with the only clear exception being the time-varying link stability . Therefore, applying the hop-count metric for routing purposes in WMNs is inefficient (as shown in [13]). To respond to dynamic characteristics of WMN links, several routing metrics have been proposed, the most important ones including: expected transmission count (ETX) [12], expected transmission time (ETT) [16], metric of interference and channel switching (MIC) [70], or multi-channel routing (MCR) [32]. They were designed to support WMN routing algorithms, e.g., AODV-ST [46], opportunistic Ex-OR [9], multipath routing [19], geographic routing [33], hierarchical routing [48], or multi-radio routing [32].

However, by incorporating the mentioned metrics into either a single-path or multipath routing [18], the impact of time-varying disruptions leading to partial/full degradation of the effective capacity of WMN links can be reduced only in a reactive way. Proactive protection against failures (commonly known to achieve better performance, e.g., in terms of reduction of the lost traffic after failures) is a rather new research direction for WMNs. The problem is indeed important, since independent of the failure cause (whether the result of an accident, forces of nature, or an intentional attack [63]), data and revenue losses encountered at high transmission rates of several Gb/s may be certainly severe.

In this chapter, we focus on failures of both WMN nodes and links. In particular, failures of WMN links can be covered by failure scenarios of the respective incident nodes (the topic addressed in Sect. 4.1). If referred to WMN links only (as in Sect. 4.2), they are commonly temporal (i.e., not observed after the interval of a negative factor duration).

Although a significant part of research efforts is related to scenarios of isolated random failures of single nodes being result of software errors, or physical faults [1], such an assumption is not proper for WMNs in many realistic scenarios. Example cases comprise natural disasters like earthquakes, volcano eruptions, tornadoes, or malicious human activities, including, e.g., bomb explosions [35] resulting in spatial correlation of failures of WMN nodes. WMN links are in turn very vulnerable to heavy precipitation responsible for remarkable signal attenuation .

In such cases, it is commonly assumed that the extent of negative outcomes depends on characteristics of a particular event, with the major factor being the distance of a network element from the failure epicenter. This in turn gives rise to the region failure scenario [30, 38, 50, 51] addressing simultaneous failures of multiple nodes located close enough to suffer from the results of the event. Following [51], regions of failure can be defined with respect to either network topology or geometry. The latter approach, i.e., geometrical representation of a failure region determined by a circular area of radius \( \overset{\frown }{\overset{\frown }{r}} \), shown in Fig. 4.2, is mostly used due to the predominant role of a node distance from the event epicenter [50, 51].

Fig. 4.2
figure 2

Example of a region failure : dark gray circle centered at the epicenter of disruptions and characterized by a given radius \( \overset{\frown }{\overset{\frown }{r}} \) represents the area of possible failures of WMN nodes

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In particular, to the best of our knowledge, there are no survivability measure s available designed to evaluate the performance of WMNs under region failure s leading to simultaneous failures of multiple WMN nodes (as well as related links). Also, there are very few proposals referring to proactive protection of WMN flows against link failures. To provide the respective solutions, in Sect. 4.1, we introduce the appropriate survivability measures for WMNs, while in Sect. 4.2 – a new approach to proactive protection against weather-based region disruptions based on automatic antenna alignment features. Section 4.3 concludes this chapter.

4.1 Measures of Wireless Mesh Networks Survivability

Due to dependency of region-based failures on multiple characteristics, region failure s need a detailed evaluation concerning their influence on the ratio of WMNs performance degradation (e.g., measured in terms of the fraction of flow surviving failures of WMN nodes located inside a given failure region).

In this section, we present our approach to WMN region failure assessment from [45] based on three introduced measures of WMNs survivability for a circular region failure scenario under random location of failure epicenters, i.e.:

  • region failure survivability function (RFS) being the cumulative probability of all region failure scenarios δ occurrence, for which at least ψ percent of flows are successfully served after failures,

  • p-fractile region failure survivability function (PFRS), providing information on total flow reduction to at most ψ percent after a failure at certain probability p,

  • expected percentage of total flow delivered after a region failure as a function of region radius \( \overset{\frown }{\overset{\frown }{r}} \) (EPFD).

Apart from providing the means of assessment of a given WMN to region disruptions, these measures are also proposed to enable comparisons of characteristics between different WMNs. To the best of our knowledge, besides our methodology from [45], there are currently no other relevant techniques available in the literature appropriate for measuring the vulnerability of WMNs to region failure s of differentiated radiuses \( \overset{\frown }{\overset{\frown }{r}} \) of failure regions.

Methodology of network survivability evaluation is well established with respect to wired networks (see e.g., [21, 47, 53, 55, 61, 67]). Concerning wireless networks, only a few proposals are available focusing, e.g., on connectivity of a network topology as a measure of fault-tolerance [52]. Connectivity can be generally used to provide a binary answer to the question whether the network is k-connected, i.e., able to provide transmission continuity after a simultaneous failure of k-1 nodes. This idea has been extended to cover, e.g., average connectivity [7], distance connectivity [6], or path connectivity [23].

However, majority of existing proposals of WMN evaluation are not suitable in the case of a region failure scenario with faults assumed to occur only in bounded areas. To address this problem, the respective region-based connectivity was proposed (see e.g., [35, 50, 51, 52]). Concerning the scenario of circular failure regions, we can distinguish the models of:

  • deterministic failures (e.g., the single circular model from [51]), where, any node located within the failure region is assumed to always fail with probability 1,

  • probabilistic failures with probability of a node failure due to a disruptive event depending on the node distance from the failure epicenter [35]. This failure probability is assumed to decrease when increasing the node distance from the failure epicenter.

Probabilistic models seem to provide more accurate results due to the common non-deterministic characteristics of natural disasters or attacks, resulting in failures of nodes located within failure regions with a certain probability. It is worth noting that available probabilistic approaches are not limitation-free. For instance in [35], the size of a failure region (given by radius \( \overset{\frown }{\overset{\frown }{r}} \)) is assumed to be constant. Another constraint in [35] is that probability of a node failure (even though decreasing with the increase of a node distance from the failure epicenter) is constant in each i-th area between two consecutive concentric annuluses (see Fig. 4.3a), which results in over- or underestimating the node failure probability values in some areas.

Fig. 4.3
figure 3

Visualization of region failure probabilities: (a) from [35], and (b) the proposed one

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Considering proposals of WMNs characteristics evaluation under region failure s, several approaches have been introduced (e.g., [50, 51, 52]) to determine whether transmission in WMNs is possible between pairs of non-faulty nodes. To the best of our knowledge, our proposal described in this section is the first one to introduce the WMN survivability measure s for the case of varying region radiuses \( \overset{\frown }{\overset{\frown }{r}} \), and using the continuous function of node failure probability (see Fig. 4.3b and Eq. 4.3) that covers the models from [35, 51] as special cases.

It is worth noting that similar survivability measure s have been proposed in the literature so far only for random failure scenarios in wired networks (see e.g., [36]). However, they were designed for failures of network elements assumed to be statistically independent and equally probable, which is completely in contrast to characteristics of WMN region failure s.

In the remaining part of this section, we first present details of the assumed network model (Sect. 4.1.1) followed by introduction of the proposed measures to evaluate the vulnerability of WMNs to region failure s (Sect. 4.1.2). Next, we describe the methodology of WMN survivability evaluation (Sect. 4.1.3) and comment on results of simulations performed for example network topologies (Sect. 4.1.4).

4.1.1 Network Model

In this chapter, we model the WMN topology by graph Γ = (N, A), where N represents the set of WMN stationary nodes (following [42]), while A denotes the set of directed arcs a h  = (i, j). Each WMN link between neighboring nodes i and j is represented by two arcs in opposite directions. Additional information refers to location of each node n defined by coordinates \( \left({\overline{x}}_n,{\overline{y}}_n\right) \). Despite the assumed stationary characteristics of network nodes, methodology of network assessment presented in this section can be also easily adapted to the case of mobile nodes (if performed with respect to the instant topology of a network at time t).

Available capacity of any WMN link is a result of multiple factors the most important ones being: medium access protocol implementation, inter-channel interference implied by the respective link scheduling algorithm [11, 18], or time-varying factors including, e.g., weather-based disruption s caused by heavy rain falls (general propagation conditions) [27]. Since the effective capacity of any WMN link changes over time, it is reasonable to perform evaluations at a given time t, i.e., assuming that capacity of arc a h is equal to c h (t).

The set of demands D consists of demands indexed by r defined by ordered triples (s r , t r , d r ), i.e., described by source and destination nodes s r and t r , and the demanded capacity d r .

Two matrices are used in our model description: A nn and D nn . Node-to-node incidence matrix A nn provides information on connectivity with elements a ij defined by formula (4.1).

$$ {a}_{ij}=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill if\kern0.5em \mathrm{arc}\kern0.5em {a}_h=\left(i,j\right)\in {A}_{nn}\hfill \\ {}0,\hfill & \mathrm{otherwise}\hfill \end{array}\right. $$
(4.1)

Information about aggregate capacities required for flows (commodities) between given pairs of end nodes is stored in elements d st of matrix D nn .

$$ {d}_{st}\equiv \left({s}_r,{t}_r,{d}_r\right) $$
(4.2)

During evaluations, location of a failure epicenter is chosen at random (i.e., following the uniform distribution function of failure epicenter coordinates) within the smallest rectangular area containing the network. We assume a probabilistic failure scenario with a disruptive event affecting nodes localized within a given radius \( \overset{\frown }{\overset{\frown }{r}} \) from the failure epicenter. In particular, in our model:

  • radius \( \overset{\frown }{\overset{\frown }{r}} \) of a failure circular region is uniformly distributed over (0, \( {\overset{\frown }{\overset{\frown }{r}}}_{\max } \)), where \( {\overset{\frown }{\overset{\frown }{r}}}_{\max } \) is equal to half of the largest Euclidean distance between any two nodes in the network,

  • probability P(\( {\overset{\frown }{\overset{\frown }{r}}}_n \)) of node n failure is given by a decreasing continuous function of distance \( {\overset{\frown }{\overset{\frown }{r}}}_n \) between node n and the failure epicenter (see Fig. 4.3b and Eq. 4.3). P(\( {\overset{\frown }{\overset{\frown }{r}}}_n \)) is thus the generalization of the respective formula from [35].

    $$ P\left({\overset{\frown }{\overset{\frown }{r}}}_n\right)=\left\{\begin{array}{cc}\hfill -\frac{{\overset{\frown }{\overset{\frown }{r}}}_n}{\overset{\frown }{\overset{\frown }{r}}}+1=-\frac{\sqrt{{\left({\overline{x}}_n-\overset{\frown }{\overset{\frown }{x}}\right)}^2+{\left({\overline{y}}_n-\overset{\frown }{\overset{\frown }{y}}\right)}^2}}{\overset{\frown }{\overset{\frown }{r}}}+1,\hfill & \hfill if\kern0.5em {\overset{\frown }{\overset{\frown }{r}}}_n\le \overset{\frown }{\overset{\frown }{r}}\hfill \\ {}\hfill 0,\hfill & \hfill \mathrm{otherwise}\hfill \end{array}\right. $$
    (4.3)

    where

    • \( \left({\overline{x}}_n,{\overline{y}}_n\right) \) are coordinates (location) of node n

    • \( \left(\overset{\frown }{\overset{\frown }{x}},\overset{\frown }{\overset{\frown }{y}}\right) \) are coordinates (location) of the failure epicenter

    • \( \overset{\frown }{\overset{\frown }{r}} \) is the radius of a failure region

    • \( {\overset{\frown }{\overset{\frown }{r}}}_n \) is the distance of node n from the failure epicenter

It is reasonable to introduce the WMN node failure probability function as given in Eq. 4.3, since, following [35], the negative impact of real physical attacks (e.g., bomb explosions, or electromagnetic pulse (EMP) attacks), as well as natural disasters (earthquakes, floods, etc.) attenuates gradually with the increase of the distance of WMN nodes from the failure epicenter. As given in [35], the maximum value of node failure probability can be assumed to be equal to 1 for locations of nodes matching exactly the failure epicenter. Its lowest value of 0 is in turn attributed to nodes located at distance \( {\overset{\frown }{\overset{\frown }{r}}}_n \) not smaller than \( \overset{\frown }{\overset{\frown }{r}} \) from the failure epicenter.

It is worth noting that this gradual attenuation of \( P\left({\overset{\frown }{\overset{\frown }{r}}}_n\right) \) values with the increase of the distance \( {\overset{\frown }{\overset{\frown }{r}}}_n \) can be disturbed by several environmental factors including, e.g., topography or node protection characteristics. However, if we neglect them to simplify the analysis (following [35]), the decrease of probability \( P\left({\overset{\frown }{\overset{\frown }{r}}}_n\right) \) of node n failure becomes linear with the increase of node n distance from the epicenter of disruptions, as introduced in Eq. 4.3.

4.1.2 Proposed Measures to Evaluate the Survivability of WMNs

The following notation is used in the remaining part of Sect. 4.1:

  • δ a region failure scenario given by the set of non-operational nodes (after the outage)

  • P(δ) probability of occurrence of a failure scenario δ

  • Ψ(δ) random variable referring to the percentage ψ of flows delivered in scenario δ

  • p Ψ (ψ) probability density function of percentage ψ of flows surviving the region failure , defined by Eq. 4.4

    $$ {p}_{\varPsi}\left(\psi \right)={\displaystyle \sum_{\delta :\varPsi \left(\delta \right)=\psi }P\left(\delta \right)} $$
    (4.4)

We introduce three measures of WMN survivability for a region failure scenario, i.e.:

  1. (a)

    Region failure survivability function (RFS) of the percentage ψ of flows successfully transmitted after region failure s:

    $$ RFS\left(\psi \right)={\displaystyle \sum_{\delta :\varPsi \left(\delta \right)\ge \psi }P\left(\delta \right)}=1-{\displaystyle \sum_{\delta :\varPsi \left(\delta \right)<\psi }P\left(\delta \right)}=1-cdf\left(\varPsi \right) $$
    (4.5)

    As given in Eq. 4.5, RFS(ψ) is defined for any value of ψ as the cumulative probability of all region failure scenarios δ (i.e., for differentiated radiuses \( \overset{\frown }{\overset{\frown }{r}} \) of failure regions), for which at least ψ percent of flows survived the failure. It can be thus expressed as the reverse cumulative distribution function of Ψ. Although Eq. 4.5 shows some similarities with the respective one from [36] for wired networks, calculation of P(δ) values is completely different.

  2. (b)

    p-fractile region survivability (PFRS):

    $$ PFRS(p)= \inf \left\{\psi :{\displaystyle \sum_{\delta :\varPsi \left(\delta \right)<\psi }P\left(\delta \right)=p}\right\} $$
    (4.6)

    Following formula (4.6), the value of p-fractile region survivability refers to the minimum percentage ψ of flows delivered after a region failure , for which the probability of not exceeding this value is equal to p. PFRS thus returns useful information about probability p that the total flow is reduced to at most ψ percent after the failure.

Since RFS and PFRS measures do not depend directly on radius \( \overset{\frown }{\overset{\frown }{r}} \) (i.e., they allow radius \( \overset{\frown }{\overset{\frown }{r}} \) to take any value from (0, \( {\overset{\frown }{\overset{\frown }{r}}}_{\max } \)) interval), they are designed to give a general information on network vulnerability to region failure s. These measures are thus appropriate, if the objective is to analyze the performance of WMNs independent of the failure region size \( \overset{\frown }{\overset{\frown }{r}} \). However, information they provide is of different types.

For instance, if for a given WMN at least ψ percent of traffic should be delivered (e.g., because such a portion of traffic is considered to be critical based on the Service Level Agreement), then RFS is the appropriate one to provide information about probability p of fulfilling this requirement under region failure s independent of size \( \overset{\frown }{\overset{\frown }{r}} \) of the failure region. Naturally, the greater is the value of p, the better performance of a network can be achieved.

PFRS is in turn a suitable measure for a network operator to determine, given the respective probability p, what is the upper bound on the fraction ψ of flow surviving a region failure . It is therefore useful to give information on probability that not all of ψ percent of flows (e.g., referred to as the critical flow) will survive the region failure, i.e., in statements like: “with probability 0.7 the total flow will be reduced to at most 80 % of the traffic served before the region failure”.

The following EPFD function is introduced to obtain a detailed characteristics of a WMN performance related to particular radiuses \( \overset{\frown }{\overset{\frown }{r}} \) of failure regions.

  1. (c)

    Expected percentage of total flow delivered after a failure (EPFD) as a function of region radius \( \overset{\frown }{\overset{\frown }{r}} \):

    $$ EPFD\left(\overset{\frown }{\overset{\frown }{r}}\right)={\displaystyle \sum_{\psi}\psi \cdot {p}_{\varPsi}\left(\psi, \overset{\frown }{\overset{\frown }{r}}\right)} $$
    (4.7)

    where

    • \( \overset{\frown }{\overset{\frown }{r}} \) is the radius of a failure region

    • \( {p}_{\varPsi}\left(\psi, \overset{\frown }{\overset{\frown }{r}}\right) \) is the probability density function of Ψ defined for region failure s of radius \( \overset{\frown }{\overset{\frown }{r}} \)

    $$ {p}_{\varPsi}\left(\psi, \overset{\frown }{\overset{\frown }{r}}\right)={\displaystyle \sum_{\delta :\varPsi \left(\delta \right)=\psi; \overset{\frown }{\overset{\frown }{r}}}P\left(\delta \right)} $$
    (4.8)

EPFD(\( \overset{\frown }{\overset{\frown }{r}} \)) is defined in Eq. 4.7 as the expected value of percentage of flows to survive failures of nodes bounded in circular regions, i.e., derived using the probability density function \( {p}_{\varPsi}\left(\psi, \overset{\frown }{\overset{\frown }{r}}\right) \) obtained for failure regions of a given radius \( \overset{\frown }{\overset{\frown }{r}} \) (see formula (4.8)).

Concerning scenarios of EPFD measure utilization, it can be useful in any performance analysis/comparison of WMNs under region failure s being result of, e.g., natural disasters (like floods, or volcano eruptions), for which the failure region is commonly expected to have a circular shape defined by a given radius \( \overset{\frown }{\overset{\frown }{r}} \). Another application of EPFD measure would be, e.g., when expecting failures confined to a given region characterized by radius \( \overset{\frown }{\overset{\frown }{r}} \) (e.g., incoming flood), to predict their impact on WMN performance being helpful to take preventive actions.

The later part of this section provides information on how to utilize the three introduced measures to evaluate vulnerability of WMNs to region failure s, as well as how to use them to provide comparisons of performance characteristics of different topologies.

4.1.3 Method of a WMN Survivability Evaluation

In this section, we explain our methodology of WMN survivability characteristics evaluation under region failure s. In particular, we focus on how to determine the introduced RFS, PFRS, and EPFD characteristics for example WMNs.

Proposed measures are derived from the auxiliary function F[ψ] providing information on the frequency of a given percentage ψ of flows (ψ∈{0,1,…, 100}) is successfully delivered after region failure s. F[ψ] values can be collected for a given WMN based on network performance observations after consecutive occurrences of disruptive events implying failures of WMN nodes confined to given regions. However, due to rather long inter-failure time intervals (typically measured in terms of months/years), deriving any characteristics based on real-life experiments is rather time-consuming and practically impossible.

In this section, an iterative procedure is presented to simulate consecutive region failure s in a way to eliminate the inter-failure time. In this way, it is possible to analyze not only the performance of existing networks, but also to predict the survivability characteristics of planned (i.e., non-deployed) WMNs using information related to the abstract WMN topology and estimated demand volumes.

The 13-steps procedure to determine F[ψ] values for a single set of demands is given in Fig. 4.4. The most important input information is related to:

Fig. 4.4
figure 4

Method of determining F[ψ] values

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  • topology of existing/planned WMN defined by graph Γ with sets N and A of nodes and directed arcs, representing network nodes and links, accordingly,

  • location of network nodes defined by coordinates (\( {\overline{x}}_n,{\overline{y}}_n \)),

  • demands r, given by the requested throughput d r , as well as source and destination nodes s r / t r .

After initialization Steps 1–2, the purpose of each iteration given by Steps 3–13 is to obtain the percentage ψ of flows delivered after failures of WMN nodes occurring in a given failure region. Coordinates of each failure epicenter and radius \( \overset{\frown }{\overset{\frown }{r}} \) of a failure region are defined as random values by the continuous uniform distribution function (following [35]).

In particular, it implies that in each iteration of the analyzed procedure:

  • location of a failure epicenter is chosen at random within the smallest rectangular area containing the WMN topology, using the continuous uniform distribution function,

  • radius \( \overset{\frown }{\overset{\frown }{r}} \) of a failure circular region is uniformly distributed over \( \left(0,{\overset{\frown }{\overset{\frown }{r}}}_{\max}\right) \), with \( {\overset{\frown }{\overset{\frown }{r}}}_{\max } \) equal to half of the largest Euclidean distance between any two nodes in the network.

After the iteration initialization Steps 3–5, Step 6 is to identify the set of failed nodes (based on formula (4.3)). To evaluate the percentage ψ of flows delivered in a given region failure scenario, for each flow with both end nodes being non-faulty, our method tries to find an alternate path of capacity d r (Steps 7–9). If the new path is found, but, due to link capacity limitations, it cannot be assigned the demanded capacity d r , multipath routing is then applied to increase as much as possible the capacity assigned to demand r after a region failure (Step 10).

The percentage ψ of flows successfully delivered after a failure is calculated in Step 11 based on the ratio of the aggregate flow \( \overset{\frown }{\overset{\frown }{f}} \) restored after the failure to the total flow f being transported before the failure (i.e., after finding the alternate paths for all demands in a given region failure scenario). Following Steps 12–13, analysis is repeated until the number FR of failure regions are evaluated.

All three introduced functions (RFS, PFRS, and EPFD) are next derived based on F[ψ] values. In particular:

  • RFS(ψ) is calculated based on empirical probabilities of restoring ψ percent of flows after failures (each such probability is obtained by dividing the respective value of F[ψ] by FR, i.e., by the total number of analyzed region failure s). According to formula (4.5), RFS(ψ) is determined as the reverse cumulative distribution function of Ψ,

  • PFRS(p) is obtained based on the cumulative distribution function of Ψ (formula (4.6)),

  • EPFD(\( \overset{\frown }{\overset{\frown }{r}} \)) is calculated based on probability density functions \( {p}_{\varPsi}\left(\psi, \overset{\frown }{\overset{\frown }{r}}\right) \) found separately for each radius \( \overset{\frown }{\overset{\frown }{r}} \) of a failure region using Eq. 4.7.

In order to find the optimal solution to the problem of determining a new set of paths in a capacity-constrained network after failures with the objective to maximize the amount of restored flows, the respective linear programming formulation of the problem (LP) is necessary [40]. However, due to its NP-completeness (see e.g., [43]), the optimal solution can be found in reasonable time using offline approaches only for small problem instances (e.g., for networks up to 12–15 nodes). Therefore, in the proposed method, calculating the alternate paths (Steps 9.2 and 10 in Fig. 4.4) is done using the heuristic approach based on Dijkstra’s algorithm [15] that is proved to have the polynomial computational complexity bounded in above by O(|N|2), where |N| is the number of WMN nodes.

4.1.4 Analysis of Modeling Results and Conclusions

In this section, we present evaluation of vulnerability of five example WMNs to region failure s (i.e., N29, N29_2, N29_3, N44, and N59 networks from Fig. 4.5), utilizing the proposed survivability measure s. First three networks (presented in Figs. 4.5a–c) are formed by 29 nodes located in 4000 × 10,000 m2, 6000 × 6000 m2, and 8000 × 8000 m2 fields, accordingly, connected by 68, 68, and 57 wireless links, respectively. The other two networks shown in Figs. 4.5d, e consist of 44 and 59 nodes (located in fields of 10,000 × 10,000 m2), respectively, connected by 97 and 150 wireless links, accordingly.

Fig. 4.5
figure 5

Evaluated topologies of: (a) N29, (b) N29_2, (c) N_29_3, (d) N44, and (e) N59 networks

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It is worth noting that for N29 network, due to visible differences between horizontal and vertical sizes of the rectangular area (4000 m and 10,000 m, accordingly), this network is likely to obtain the worst results concerning the portion of flows surviving the region failure s (since for each network, the analyzed radiuses \( \overset{\frown }{\overset{\frown }{r}} \) of failure regions were up to half of the largest Euclidean distance between any two nodes in the network).

When assessing the vulnerability of network flows to region disruptions, all transmission paths (both before and after failures) were calculated as the cheapest ones using the standard metric of distance [34, 41]. After failures, reactive approach was utilized to redirect flows with survived end nodes. To provide the appropriate statistical analysis related to RFS, PFRS, and EPFD functions, the original values of F[ψ] were obtained as the aggregate ones including all 100 investigated demand sets of a certain size. For each set of demands, failures related to FR = 9000 random regions were simulated.

Three simulation scenarios were considered. The first two, referred to as Scenarios A and B, were prepared to use the proposed measures to evaluate characteristics of different WMNs under a similar network load. To achieve this, the sets of unicast transmission demands included 25 % of randomly chosen node pairs. Scenario A was to verify characteristics of WMNs of the same size in terms of the number of nodes (i.e., N29, N29_2, and N29_3 networks consisting of 29 nodes), while Scenario B was aimed at evaluating networks of a similar area they covered (i.e., not necessarily comparable in terms of the number of nodes). Therefore, topologies analyzed in Scenario B included: N29, N44, and N59.

Additional Scenario C was to verify the properties of our measures under differentiated loads of N59 network. In particular, four sizes of demand sets (i.e., consisting of randomly chosen 25, 50, 75, and 100 % node pairs) were examined. Capacity d r of each unicast demand r was assumed to be unitary.

Each network link offered 160 units of unitary capacity in each direction. Considering failure scenarios, radiuses \( \overset{\frown }{\overset{\frown }{r}} \) of failure regions were uniformly distributed in range \( \left(0,{\overset{\frown }{\overset{\frown }{r}}}_{\max}\right) \), where \( {\overset{\frown }{\overset{\frown }{r}}}_{\max } \) was equal to half of the largest Euclidean distance between any two network nodes. Statistical analysis of results was based on 95 % confidence intervals. However, since sizes of obtained intervals did not exceed 1 % of the original values, due to low visibility they are not shown in Figs. 4.6–4.12.

Fig. 4.6
figure 6

RFS(ψ) function (Scenario A)

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4.1.4.1 Region Failure Survivability (RFS)

Evaluation of vulnerability of WMN topologies to region failure s using the RFS measure under the assumptions of Scenario A is presented in Figs. 4.6 and 4.7. Recall that RFS measure, defined in Eq. 4.5, was introduced to evaluate the probability that at least ψ percent of flows survives after a region failure.

Fig. 4.7
figure 7

RFS(ψ) function (Scenario B)

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As presented in Figs. 4.6 and 4.7, with the increase of ψ, RFS starts decaying from the value of 1 (since independent of the network topology, probability of reducing the total flow to at least 0 % is equal to 1). When comparing RFS characteristics for any two network topologies, greater values of RFS for any value of ψ imply a better performance of a network after a failure (since they reflect a greater chance of total flow reduction to at least ψ percent after a failure).

The general conclusion following from Figs. 4.6 and 4.7 is that better results concerning network survivability characteristics under region failure s are attributed to WMN networks with RFS functions driven by a slower decay with the increase of ψ (i.e., for which independent of ψ parameter, RFS values are higher). For instance, as shown in Fig. 4.6, N29 network (for which its horizontal and vertical sizes are remarkably different) is outperformed by N29_2 and N29_3 networks (located inside a square area) in Scenario A. In the same way, N44 and N59 networks turned out to outperform the N29 network in Scenario B (Fig. 4.7).

4.1.4.2 p-Fractile Region Survivability (PFRS)

Figures 4.8 and 4.9 show the evaluation of WMN survivability characteristics using the p-fractile region survivability (PFRS) measure for Scenarios A and B. Recall that PFRS (Eq. 4.6) is to provide information on probability p that the fraction of total flow delivered after region failure s will not exceed ψ (Y axis on Figs. 4.8 and 4.9).

Fig. 4.8
figure 8

PFRS(p) function (Scenario A)

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Fig. 4.9
figure 9

PFRS(p) function (Scenario B)

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For any WMN, it is thus better if, for any value of p, the upper bound on the portion ψ of flow surviving the failure is higher. As shown in Figs. 4.8 and 4.9, independent of the network topology, PFRS values are always positively correlated with p. In general, the lower the values of PFRS, the network is more vulnerable to region failure s. Similar to results for RFS measure, PFRS also showed that N29 network has the worst properties among all analyzed WMNs in Scenarios A–B.

4.1.4.3 EPFD Function

Figures 4.10 and 4.11 show values of EPFD function obtained in Scenarios A and B. Recall that EPFD function is defined by formula (4.7) as the expected percentage of the total flow delivered after failures occurring in circular areas of a certain radius \( \overset{\frown }{\overset{\frown }{r}} \). For any radius \( \overset{\frown }{\overset{\frown }{r}} \), greater values of EPFD function imply more network flows surviving the failures. As shown in Figs. 4.10 and 4.11, N29 network obtained the worst characteristics also with respect to EPFD measure (which is compliant with the respective RFS and PFRS characteristics from Figs. 4.6–4.9, accordingly).

Fig. 4.10
figure 10

EPFD(\( \overset{\frown }{\overset{\frown }{r}} \)) function (Scenario A)

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Fig. 4.11
figure 11

EPFD(\( \overset{\frown }{\overset{\frown }{r}} \)) function (Scenario B)

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It is worth mentioning that all three measures do not depend on the network load (as shown in Fig. 4.12 for Scenario C). Therefore, they can be used to compare characteristics of different WMN topologies.

Fig. 4.12
figure 12

Characteristics of: (a) RFS(ψ), (b) PFRS(p), and (c) EPFD(r) functions for Scenario C (N59 network)

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In this section, we focused on the evaluation of vulnerability of WMNs to region failure s occurring in circular areas and introduced three measures for evaluation of WMN survivability . The first two measures, i.e., region failure survivability function – RFS, and p-fractile region survivability function – PFRS, were proposed to provide assessment of WMN vulnerability to region failures independent of the radius \( \overset{\frown }{\overset{\frown }{r}} \) of the failure region. The third measure – the expected percentage of total flow delivered after a region failure as a function of region radius \( \overset{\frown }{\overset{\frown }{r}} \) (EPFD) – was in turn designed for the evaluation of WMN performance depending on radius \( \overset{\frown }{\overset{\frown }{r}} \) of a circular failure region.

Proposed measures were later utilized to evaluate the properties of three example topologies of WMNs. Simulation analysis confirmed that these measures give adequate and consistent information on WMN networks vulnerability to region failure s. Since for all introduced measures, achieved characteristics did not depend on the network load, they can be thus utilized in comparisons of different WMNs.

4.2 A New Approach to Design of Weather Disruption-Tolerant Wireless Mesh Networks

As discussed in the former part of this chapter, failures of WMN nodes/links may imply severe data losses. In this section, we focus on link failures and present the respective approach to survivable routing to improve the WMN performance under link failures. As stated in [57], WMN links are very susceptible to weather disruptions in particular related to precipitation. Heavy rain storms may cause high signal attenuation remarkably reducing the available link capacity or implying a link failure leading to instability problems of routing (i.e., route flapping).

Issue of survivable routing is well researched with respect to wired networks (see e.g., [47, 55, 58, 61, 67]), in particular concerning protection of WDM network flows ([47, 55, 56, 60]). Among few proposals related to resilience of routing in wireless networks, we can mention reference [10] addressing shared medium problems and node mobility issues. However, these solutions cannot be directly applied to WMNs due to remarkably different characteristics. In particular, WMNs are commonly non-mobile and do not encounter contention problems (if equipped with directional antennas). Therefore, except for link stability issues, WMNs seem to share the most important characteristics with wired networks [27].

In order to provide protection of flows against weather-based disruption s of WMN links, it seems reasonable to use information related to expected incoming rain storms (e.g., achieved from radar echo measurements) to predict the real shapes of signal attenuation regions. Based on this idea, two approaches were introduced in [27], called XL-OSPF and P-WARP, to modify the link-state OSPF routing based on weather predictions. Both techniques utilize formulas (4.9–4.10) from [14] defining the dependency of signal attenuation on the rain rate.

$$ \Omega \left({R}_p,\varTheta \right)=\alpha {R}_p^{\beta}\left[\frac{e^{u\beta \vartheta }-1}{u\beta }-\frac{b^{\beta }{e}^{\iota \beta \vartheta }}{\iota \beta }+\frac{b^{\beta }{e}^{\iota \beta \varTheta }}{\iota \beta}\right],\kern1em \vartheta \le \varTheta \le 22.5\kern0.5em \mathrm{km} $$
(4.9)
$$ \Omega \left({R}_p,\varTheta \right)=\alpha {R}_p^{\beta}\left[\frac{e^{u\beta \varTheta }-1}{u\beta}\right],\kern1em 0\le \varTheta \le \vartheta $$
(4.10)

where

  • Ω is the signal attenuation in dB

  • Θ is the length of the path over which the rain is observed

  • R p is the rain rate in mm/h

  • α, β are the numerical constants from [14]

    $$ \begin{array}{cc}\hfill u=\frac{ \ln \left(b{e}^{\iota \vartheta}\right)}{\vartheta },\hfill & \hfill b=2.3{R}_p{{}^{-}}^{0.17},\hfill \\ {}\hfill \iota = 0.026-0.03 \ln {R}_p,\hfill & \hfill \vartheta = 3.8-0.6\ \ln {R}_p.\hfill \end{array} $$

In particular, XL-OSPF utilizes a special metric of link cost being proportional to the observed bit error rate (BER) of the link (which is justifiable due to the clear impact of signal attenuation on the effective BER, as well as on packet error rate – PER). This metric is utilized in a reactive manner to update the OSPF routing characteristics. However, such an approach is not easy to deploy, since in the Media Access Control (MAC) layer there is no information on the actual BER between network nodes (it can be estimated using signal-to-noise ratio – SNR).

P-WARP in turn estimates the costs of WMN links using weather-based predictions of future conditions of links. This can be done either at one dedicated node or at a subset of nodes capable of collecting the weather-related radar data.

In this section, we focus on the issue of reducing the level of signal attenuation along millimeter-wave links in the presence of rain storms. In particular, in Sect. 4.2.1, we present in detail our method from [44] to perform in advance the periodic updates of a WMN topology following forecasts of heavy rain storms, using the functionality of a dynamic antenna alignment offered by a number of equipment vendors (see e.g., [54]). Next, in Sect. 4.2.2, we describe the ILP model proposed by us to obtain the optimal routing solution in accordance with the forecasted levels of signal attenuation at WMN links (that also returns the proper assignment of non-interfering channels to intersecting links). After that, in Sect. 4.2.3, we present the analysis of computational complexity of the problem followed by evaluation of our approach characteristics (Sect. 4.2.4).

To the best of our knowledge, protection of WMN links against weather-based region failure s has not been sufficiently researched so far. In particular, there is no other proactive approach that is based on periodic updates of a WMN topology.

4.2.1 Proposed Approach

The technique to provide protection of WMN links against weather-based disruption s described here does not impose any modifications of the routing algorithm. Therefore, it can be used in conjunction with practically any routing scheme, which makes our solution easily deployable. In particular, transmission paths are established based on conventional metric of link costs (e.g., the number of hops).

The main idea of our approach is to prepare the network to changing weather conditions by applying the periodic updates of WMN topology to improve the throughput during rain storms. We propose to perform the consecutive updates of a WMN topology by means of dynamic antenna alignment features (offered by a number of equipment vendors) utilizing predictions related to future conditions of WMN links based on rain storm forecasts obtained from real echo rain maps. This in turn implies periodic creation (or deletion) of WMN links, if low (or high) values of signal attenuation are expected for them, accordingly.

The network is modeled in this section by graph Γ = (N, A) in a similar way as in Sect. 4.1.1. In particular, any link between two neighboring nodes i and j is represented by two directed arcs a h  = (i, j) and a h′  = (j, i), accordingly, and is assigned a given transmission channel from the set of available transmission channels. In order to focus on time-varying characteristics of WMN links, definition of graph Γ is extended by:

  • \( \overset{\smile }{T} \) denoting the lifetime of a network

  • \( \vartheta \left(\overset{\smile }{T}\right) \): \( A\times \overset{\smile }{T}\ \to \left\{0,\ 1\right\} \) function determining existence of links at time \( t\in \overset{\smile }{T} \)

  • \( \gamma \left(\overset{\smile }{T}\right) \): \( A\times \overset{\smile }{T}\ \to \boldsymbol{R} \) link cost function based on signal attenuation ratio at time \( t\in \overset{\smile }{T} \) (formulas (4.9–4.10))

We assume the existence of a dedicated core node responsible for the alignment of antennas of all network nodes that has access to:

  • the set of active network nodes and their locations,

  • radar echo rain measurements (received periodically),

  • demands to provide transmission between WMN end nodes.

The role of this core node is also to execute the procedure shown in Fig. 4.13. In particular, in Step 1 of this scheme, the estimated signal attenuation ω h at each potential arc a h  = (i, j) is determined using formulas (4.9–4.10). Actions of Step 2 are to return a new configuration of WMN links. In particular, in the proposed scheme ω h values are used as link costs to obtain the set of the cheapest (in terms of signal attenuation) potential paths. If in Step 2, a given link is not used by any path, it will not be present in the updated WMN topology.

Fig. 4.13
figure 13

Proposed methodology of periodic updates of alignment of WMN antennas

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In the method from Fig. 4.13, we propose to utilize the heuristic approach to proceed with Step 2, since the problem to determine the optimal alignment of WMN antennas with the objective to minimize the aggregate signal attenuation over all transmission paths, defined in Sect. 4.2.2, is NP-complete (as proved in Sect. 4.2.3). New alignment of antennas (Step 3) is expected every τ time units (as defined in Step 4).

It is worth to recall that metric ω h is used in our approach only to update the alignment of antennas at WMN nodes. Routing is in turn performed by means of a conventional protocol with all its characteristics unchanged. This implies that the original metric of link costs (i.e., the one normally used by the routing algorithm) is utilized instead of ω h values to obtain the real transmission paths.

4.2.2 ILP Formulation of Weather-Resistant Links Formation Problem (WRLFP)

The problem to determine the optimal alignment of WMN antennas (Step 2 from Fig. 4.13) to minimize the aggregate signal attenuation over all transmission paths at time t can be solved by determining the solution to the following ILP model.

Indices

Γ(N, A):

directed network

N :

set of network nodes; |N| is the number of network nodes

A :

set of directed arcs; |A| is the number of arcs

h :

arc index; h = 1, 2, …, |A|

D :

set of demands; |D| is the number of demands

r :

demand index; r = 1, 2, …, |D|

L h :

set of transmission channels available at arc a h  = (i, j)

1…Λ h :

indices of transmission channels at arc a h  = (i, j); ∀ h Λ h  = Λ

Constants

s r (t r ):

source (destination) node of r-th demand

d r :

capacity of r-th demand

c h (t):

estimated total capacity of arc a h  = (i, j) at time t

ω h (t):

estimated signal attenuation due to rain falls for arc a h  = (i, j) at time t

Variables

x l r;h :

equals 1, if l-th channel is assigned for r-th demand path at arc a h  = (i, j); 0 otherwise

Objective

It is to find the end-to-end transmission paths for all demands minimizing the cost defined by formula (4.11):

$$ \varphi \left(x,t\right)={\displaystyle \sum_{r\in D}{\displaystyle \sum_{l\in {L}_h}{\displaystyle \sum_{h\in A}{\omega}_h(t)\cdot {x}_{r,h}^l}}} $$
(4.11)

where ω h (t) is the cost of arc a h  = (i, j) based on signal attenuation ratio at time t.

Constraints

  1. 1.

    Flow conservation rules (based on Kirchhoff’s law) for end-to-end paths:

    $$ {\displaystyle \sum_{l\in {L}_h}{\displaystyle \sum_{\begin{array}{l}h\in \Big\{h:{a}_h\equiv \left(n,j\right)\in A;\\ {}\kern1.5em j\in N;j\ne n\Big\}\end{array}}\kern-3.5em {x}_{{}_{r,h}}^l}}\kern2em -\kern1em {\displaystyle \sum_{l\in {L}_h}{\displaystyle \sum_{\begin{array}{l}h\in \Big\{h:{a}_h\equiv \left(i,n\right)\in A;\\ {}\kern1.5em i\in N;i\ne n\Big\}\end{array}}\kern-3.5em {x}_{{}_{r,h}}^l}}\kern2em =\kern1em \left\{\begin{array}{c}\hfill 1,\hfill \\ {}\hfill -1,\hfill \\ {}\hfill 0,\hfill \end{array}\right.\kern1em \begin{array}{c}\hfill if\kern0.5em n={s}_r\hfill \\ {}\hfill if\kern0.5em n={t}_r\hfill \\ {}\hfill \mathrm{otherwise}\hfill \end{array} $$
    (4.12)

    where a h  = (n, j) arc incident out of node n; a h  = (i, n) arc incident into node n; r ∈ D; n ∈ N

  2. 2.

    On finite capacity of arcs a h (i.e., to assure that the total flow assigned to arc a h will not exceed the maximum available capacity):

    $$ {\displaystyle \sum_{l\in {L}_h}{\displaystyle \sum_{r\in D}{x}_{r,h}^l\cdot {d}_r}}\le {c}_h(t);\kern2em h\in A $$
    (4.13)
  3. 3.

    On selection of different channels to interfering links (at most one link from the set of interfering links can be assigned a given channel l):

    $$ {\displaystyle \sum_{r\in D}{x}_{r,h}^l}+{\displaystyle \sum_{r\in D}{x}_{r,h\prime}^l}\le 1 $$
    (4.14)

    for each pair of intersecting arcs a h and a h′ ; l ∈ L h

4.2.3 Computational Complexity of WRLFP Problem

In this section, we discuss the complexity of the considered optimization problem (4.11–4.14). In particular, by proving that it belongs to the class of NP-complete problems (by showing that one of its subproblems being the channel assignment problem, referred to as WR_CAP, is NP-complete), we explain that there is no efficient algorithm proposed so far to find the optimal solution in polynomial time.

Since assignment of channels to links is confined to the set of Λ available channels (where Λ can be any arbitrarily chosen small integer value), optimization version of WR_CAP channel allocation subproblem can be defined as follows.

WR_CAPopt(A′)

Given the set of network arcs A′ utilized by paths in Step 2 from Fig. 4.13 , find the optimal assignment of transmission channels to arcs a h minimizing the number of used channels, providing that none of intersecting arcs receives the same channel.

To show the NP-completeness of WR_CAP, it is sufficient to analyze its recognition version (i.e., a problem with “yes/no” answer) [28] shown below.

WR_CAPrec(A′, k)

Given a set of arcs A′ utilized by paths in Step 2 from Fig. 4.13 , is it possible to find the optimal assignment of channels to arcs a h in the network that requires k different channels, providing that none of intersecting arcs receives the same channel?

If recognition version of the problem is NP-complete, so is its optimization version [2].

Theorem

WR_CAP problem is NP-complete.

Proof

Following [2], when proving the NP-completeness of WR_CAP problem, it is sufficient to show that:

  1. (a)

    WR_CAPrec(A′, k) belongs to the class of NP problems

  2. (b)

    A known NP-complete problem polynomially reduces to WR_CAPrec(A′, k)

  • Regarding (a): WR_CAP problem belongs to complexity class NP, since it can be determined in polynomial time whether a given assignment of transmission channels to arcs a h is valid (i.e., whether it requires exactly k channels from the set {1,…, |Λ|}). In particular, checking the assignment of channels can be done in at most O(|A′|) ≤ O(|n 2|) operations, while verifying whether different channels are assigned to intersecting links requires at most O(|n 2|) steps.

  • Regarding (b): To provide the second part of the proof, we will show that the known NP-complete problem of determining the optimal vertex-coloring of a graph of conflicts G [28], here referred to as VCGC, can be transformed in polynomial time to WR_CAP problem. As shown in [28], recognition version of VCGC problem can be defined in the following way.

VCGCrec(G, k)

Given a graph of conflicts G = (V, E), where V is the set of vertices, and E is the set of edges e h  = (i, j) representing conflicts between the respective vertices i and j, is it possible to find the optimal assignment of colors to vertices from V requiring exactly k colors in a way that any two conflicting vertices i and j (i.e., connected by an edge in G) receive different colors?

Assume that:

  • {G = (V, E), k} is the input to the VCGC recognition instance of the problem

  • G also represents the graph of conflicts for links to be installed in the network after executing Step 2 of the method from Fig. 4.13. In this graph:

    •  vertices from V represent links to be installed in the network,

    •  there exists edge e h  = (j, k) in G, if the respective network arcs a j and a k in Γ intersect with each other, i.e., if they have to be assigned different channels

(⇒):

Let us assume that it is feasible to color vertices from G using k different colors. In this case, any valid coloring of G by k different colors in VCGCrec(G, k) automatically returns a proper assignment of k different channels to interfering links in WR_CAPrec(A′, k).

(⇐):

Assume that k channels are sufficient to determine the solution to WR_CAPrec(A′, k) problem. Then, after creating the respective graph of conflicts G for interfering WMN links, we automatically have a valid coloring of G vertices that requires k different colors.

▪

If we relax the problem by disregarding the requirement on allocation of different channels to intersecting links, the simplified problem remains NP-complete as a basic task to determine transmission paths between |D| pairs of nodes in capacity-constrained networks (classified as NP-complete in [43]). Therefore, to perform Step 2 from Fig. 4.13, heuristic Dijkstra’s algorithm from [15] is used.

4.2.3.1 Example Execution Steps of the Proposed Method

Results of a single iteration of the proposed method execution are presented in Fig. 4.14. Initial alignment of antennas is shown in Fig. 4.14a. Based on actual information related to the predicted rain intensity from Fig. 4.14b, a single iteration of our procedure is to provide the update of the network topology necessary to prepare the network for the forthcoming rain.

Fig. 4.14
figure 14

Example execution steps of the proposed procedure to modify the network topology (here the artificial Irish Network) according to the current rain storm forecasts including: (a) initial topology of the network, (b) extended topology including all possible links, and (c) results of the algorithm execution

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For this purpose, the WMN topology is first extended by the respective core node (responsible for determining the updates of a network topology) in a way to include all possible links between neighboring nodes (see Fig 4.14b). Based on the forecasted attenuation of a signal along each potential link, a new alignment of antennas is next determined (see Fig. 4.14c). As a result, the updated topology from Fig. 4.14c does not include links located within heavy rain storm areas (e.g., links (3, 4), (10, 11), (14, 15), and (15, 16)).

4.2.4 Analysis of Modeling Results and Conclusions

Simulations were performed to verify characteristics of our approach for two example artificial WMN topologies from Fig. 4.15, located in the area of Southern England and Ireland, respectively. Topology of each network included 42 nodes and formed a grid structure, with link lengths equal to 15 km. Characteristics of our technique (here referred to as “with protection”) were compared with the common one implying no changes in the alignment of antennas (further referred to as the “no protection” case).

Fig. 4.15
figure 15

Example topologies of WMNs used in simulations. (a) Southern English Network (b) Irish Network

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In the proposed technique, the initial set of WMN links included the ones marked with solid red lines in Fig. 4.15. Dashed blue lines are in turn used in Fig. 4.15 to indicate the extension of the set of links for possible utilization by the proposed technique. In the reference “no protection” approach, the set of links did not change over time (i.e., it was determined only by red lines from Fig. 4.15). In each network, nodes 1 and 42 were configured as gateways connecting the other ones to the Internet. Traffic outgoing the network via one of these gateways was assumed to be generated by each WMN node at a rate of 3 Mb/s.

Simulations were focused on measuring the average signal attenuation ratio due to rain storms along transmission paths, as well as the average path hop count for three real scenarios of rain storms that occurred in November 2011:

  • Scenario A: Southern England, Nov. 25, 2011, from 3:00 AM till 10:00 AM

  • Scenario B: Ireland, November 26–27, 2011, from 8:00 PM till PM 7:00 AM

  • Scenario C: Ireland, November 24, 2011, from 10:00 AM till 12:00 PM

Radar rain maps utilized in simulations were recorded every 15 min. Duration of the analyzed rain storms varied from 7 to 14 h. A limited set of investigated rain maps (one map per hour) is shown in the Appendix (Sect. 4.2.5).

4.2.4.1 Signal Attenuation

As shown in Fig. 4.16, during heavy rain intervals, the level of signal attenuation increased remarkably. However, due to periodic updates of antenna alignment according to the forecasted signal attenuation ratio, our approach was able to prepare the WMN topology in advance for the forthcoming rain, and, as a result, to significantly decrease the signal attenuation ratio (up to 90 %, as shown in Fig. 4.16). A general conclusion is that the greatest improvement was observed for periods of heavy rain (which is a very desired feature). On the contrary, in the case of light rains, updating the alignment of antennas implied only a slight reduction of the analyzed signal attenuation ratio.

Fig. 4.16
figure 16

Obtained results concerning reduction of signal attenuation

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4.2.4.2 Number of Path Links

Considering the average hop count of end-to-end transmission paths, for the common “no protection” method (for which the costs of links were independent of signal attenuation ratio), the average number of path links was equal to 5.6.

Our technique, due to operations of WMN links creation/deletion being implication of changing attenuation conditions, resulted in establishing WMN links in a more elastic way. In particular, this often implied forming diagonal links (e.g., between nodes 1 and 8), which in general resulted in shorter paths. As presented in Fig. 4.17, the average end-to-end hop count for our technique was often visibly lower than for the reference approach. However, during heavy rain periods (Scenario B, 10:00 PM–1:00 AM; Scenario C, 4:00 PM–10:00 PM), the average hop count for our approach was higher due to the need to provide detours over heavy rain areas.

Fig. 4.17
figure 17

Obtained results concerning the average hop count

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In this section, we addressed the problem of signal attenuation in WMNs due to heavy rain storms. In order to improve the performance of the network during rainy intervals, we presented a method to apply in advance the periodic updates of a WMN topology that utilizes information from radar echo rain measurements. Our approach can be easily implemented in practice, as functionality of dynamic antenna alignment is available in a number of commercial products. Another advantage is that our approach does not imply any changes of a routing algorithm.

It was verified by simulations performed for real radar rain maps that the proposed technique can bring about a significant decrease (up to 90 %) of signal attenuation , compared to the results of the reference “no protection” approach of not applying any changes to WMN topology. This improvement was observed for heavy rain periods (which is indeed a very desired feature).

4.2.5 Appendix – Rain Radar Maps Used in Simulations

Radar rain maps used in Sect. 4.2 are presented in this Appendix in 1 h interval (during simulations, rain maps were, however, collected every 15 min).

Each map presented here was created based on the following rain intensity scale provided by www.weatheronline.com service:

figure afigure a

4.3 Summary

As shown in this chapter, resilience of WMNs is an emerging issue. In terms of resilient routing , WMNs seem to exhibit most of characteristics commonly attributed to wired networks (e.g., stationary nodes, high capacity, or no limits on energy consumption), however, with a clear exception referring to the time-varying link stability . Due to high frequency communications, vulnerability of WMN links to weather-based disruption s is even more challenging than in conventional 802.11 architectures. That is why, direct application of resilience mechanisms originally designed for pure wired or ad hoc (wireless) networks is not proper.

As shown in this chapter, the number of proposals addressing the resilient routing issue in WMNs is limited. They include, e.g., updates of routing metrics to keep changing in a reactive way the communication paths as a response to time-varying characteristics of WMN links. However, a general observation (following from research results on wired networks resilience) is that considering the extent of losses after failures, better results would be achieved when applying the proactive approach (implying preparation of an alternate transmission solution in advance – before occurrence of a failure). Additionally, there have been no survivability measure s proposed so far to evaluate the WMN performance for a common scenario of region failure s (implied e.g., by weather-based region disruptions).

To address these issues, the respective survivability measure s have been proposed in this chapter to allow for evaluation of a WMN performance under region failure s leading to massive failure s of WMN nodes/links. Unique characteristics of WMN links also made us propose the transmission scheme able to prepare the network in advance for the forthcoming heavy rain by means of automatic antenna alignment features. As a result, due to information from radar echo rain maps, settings of WMN antennas could be proactively updated to create links omitting areas of predicted heavy rain (which reduced the signal attenuation ratio up to 90 %).

It seems that other resilience approaches proposed for wired networks, e.g., based on multiple alternate paths could be also applied to WMNs after adapting them to characteristics of WMN links. This is a wide area for future research.