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Abstract

This article studies the performance of a wireless sensor network with cognitive radio capabilities to gather information about structural health monitoring of buildings in case of seismic activity. Since the use of the local area network is intensive in office and home environments, we propose the use of empty cellular channels (primary system). As such, the structural health monitoring does not degrade the local communications. Thus, the wireless sensor network for structural health monitoring acts as secondary network. Two discrete-time analytical approaches are proposed and developed to evaluate the system performance in terms of both the average packet delay and average energy consumption. The first one is an approximation suitable for the case when the time slot duration is small relative to the mean call inter-arrival time. The second model is accurate for any time slot duration and inter-arrival times.

Keywords

Wireless sensor network cognitive radio structural health monitoring blocking probability discrete-time digital telephony system

Introduction

Wireless sensor networks (WSNs) are composed of a large number of small sensor nodes which are limited by their computing power, size, and lifetime; usually, they have a single omnidirectional antenna to communicate with each other through wireless transmissions.¹ The organization of their internal software and hardware must be properly configured in order to work effectively and be able to dynamically adapt to new environments, requirements, and applications.² In this work, we focus our study on event-driven WSNs^3,4 where nodes are deployed inside a building or home environments. These nodes are expected to start sending information whenever the structure experiences an important movement due to seismic activity or strong winds among other situations. At this point, the network produces a high number of simultaneous transmissions in a relatively small physical space. This can cause high interference levels in already congested frequencies used primarily by local area networks for Wi-Fi access. Then, it is imperative to make an adequate use of resources to effectively implement the structural health monitoring (SHM) network while maintaining the quality of service of the wireless system used by people inside the building.

To this end, we consider the use of cognitive radio (CR) technology. In CR networks, secondary nodes (nodes that do not own the license to use the spectrum but have an agreement with the system that owns it) identify unoccupied spectrum bands to send their information in an opportunistic manner and vacate the spectrum when it is necessary.⁵ Conversely, wireless users with a specific license to communicate over allocated bands (the primary users (PUs)) have the priority to access the communication channels. CR users, also called secondary users (SUs), can access the channel as long as they do not cause interference to the PUs.⁶ CR capabilities have been proposed to be used in the WSN context, which are traditionally assumed to employ fixed spectrum allocation.⁷

Previous research studies have addressed the issue of SHM by implementing WSNs in the monitored structures.^1,8–14 However, contrary to the present work, none of these research studies considered efficient data transmission by means of a CR system neither average delay nor energy consumption as the main performance metrics. Also, different medium access control (MAC) protocols have been designed for efficient SHM.¹⁵ However, Zheng et al.¹⁵ focused on an alarm-oriented system, while we focus on gathering all the information regarding the event with the purpose of transmitting all the data to a disaster control center in order to detect possible failures in the monitored structure. Hence, our proposed work does not aim at immediately reporting the seism to prevent the people in the building but rather use this information for subsequent damage detection of the building structure. Finally, in our previous works,^16,17 we presented a similar study. However, in the work by Garrido et al.,¹⁶ we presented only simulation results, while in the work by Garrido et al.,¹⁷ we considered a continuous process for the primary network and only the cluster formation (CF) phase was mathematical modeled. We now extend that work to develop an analytical model for the steady-state (SS) phase (data reporting period). To this end, we have discretized the Erlang-B formula, which entails a completely different model than the one proposed in the aforementioned work.

Regarding many practical details in SHM, we focus neither on the characteristics of nodes nor the optimal placement of the nodes. However, we briefly discuss some of these points. To select the type of nodes to be placed in the building for seismic monitoring, the results presented in the work by Zhou and Yi¹⁸ can be used to select the appropriate sensors based on their specific characteristics. Also, the optimal placement of wireless sensors is studied by Zhou and colleagues^19,20 based on firefly algorithms. In real building applications, the optimal placement of nodes is not random but rather in specific points of the structure. Specifically, nodes are placed at the stiffness centroid of each of the floor plans and at the extremes of each floor to measure torsional displacement. Additional nodes in between these main locations can also be placed to enhance the seismic monitoring. However, we consider a random placement of nodes for simplicity and because we assume that nodes have the capacity of transmitting their packets directly to the base station and each floor of the building works independently from the rest of floors. As such, the particular placement of nodes has no impact on the overall performance of the network in terms of both average packet delay and average energy consumption. Indeed, the base station and cluster heads (CHs) receive the same number of packet transmissions per time slot regardless of the placement of nodes. Note that for larger structures, such as bridges, this no longer holds, since packets are relayed in a multi-hop fashion. For such applications, Zhou et al.²¹ designed and implemented an SHM system for a bridge and arch bridge and the work reported by Kurata et al.²² focuses on dense WSNs with a suite of information technologies remotely accessible by the Internet. We mainly focus on the performance of the network and do not consider upper-layer aspects, which are relevant for commercial systems.

Regarding the primary network considered in this work, that is, a cellular system, it is common in the literature to consider a digital telephone system based on arrivals and departures as continuous time processes. However, in this work, we require a discretized system to model the possible events that can occur in a time slot–based structure. As such, we develop a time discretization of the well-known Erlang-B formula in order to mathematically model the complete CR system. This time discretization is suitable for the case when the time slot duration is small compared to the mean inter-arrival time (in this work, the average channel holding time is considered constant and considerably larger than the time slot duration^23,24). However, we also propose and develop a second model that provides an exact description of the system for any value of the time slot duration. Due to the complexity of this exact model, we based our CR analysis on the approximated discrete model. A similar discrete system for the telephonic process is proposed by Veretennikov.²⁵ Our contribution relative to the previously related published work can be summarized as follows. An approximation when the time slot is sufficiently small compared to the average inter-arrival time is developed that provides closed expressions on the blocking probability. Moreover, a discrete-time model that provides an exact description of the Erlang-B model is proposed and developed. Transition probabilities are obtained through the discretization of the arrival rate and holding times from the continuous time Erlang-B model. The effect of the time discretization proposed by Nyiyato and Hossain²⁶ is quantified. Moreover, the conditions under which this discretization renders accurate results are provided.

We believe that the presented method performs better than the other methods for the following reasons. From the system design point of view, structural monitoring systems are widely adopted to monitor the behavior of structures during forced vibration testing or natural excitation (e.g. earthquakes, winds, live loading). The monitoring system is primarily responsible for collecting data from all sensors installed in specific points of the structure and storing the measured data in a central data repository. To guarantee reliable collection of measurement data, structural monitoring systems employ coaxial wires for communication between the sensors and the repository. While coaxial wires provide a very reliable communication link, their installation in structures can be expensive and labor-intensive. To address the limitations of current sensing technologies placed on both local- and global-based damage detection methods, the research community is actively exploring new technologies that can advance the current state of practice in SHM. In particular, wireless sensors represent one potential sensing technology that can help advance the structural engineering field’s ability to economically implement SHM. Interest in wireless sensors was initially motivated by their low-cost attributes. The eradication of extensive lengths of coaxial wires in a structure results in wireless systems having low installation costs. These low costs promise wireless monitoring systems to be defined by greater nodal densities as compared to traditional tethered monitoring systems. With potentially hundreds of wireless sensors installed in a single structure, the wireless monitoring system is also better equipped to screen for structural damage by monitoring the behavior of critical structural components, thereby implementing local-based damage detection.²⁷ From the analytical method point of view, the proposed/developed discrete model allows to accurately track both the primary system and the secondary system at the time slot level, while the previous closely related methods present only an approximation based on the timescale decomposition to approximately evaluate the performance of the considered systems which is accurate only for certain range of values of the system parameters (when the timescale of the primary network is two or more orders of magnitude higher or lower than the secondary network timescale.) As such, this proposed model could be used for any parameter settings providing accurate numerical results.

It is important to note that our work is focused on the MAC layer by studying the effect of the number of nodes and transmission probability on the system performance for different conditions of the primary network. As such, nodes transmit in a two-hop fashion to the sink node, first to the CH and then to the sink node. Hence, there is no need to consider a routing protocol. As such, the network layer is not considered. In addition, the cluster selection is based on the arrival order of the CF packets. Specifically, the first $N_{CH}$ nodes that successfully transmit their control packets in the CF phase become CHs, while the rest of the nodes become cluster members (CMs). Although this is not the optimal CH selection, in the literature, it has been shown that this selection strategy achieves similar results than other intelligent schemes in certain conditions.²⁸

The rest of the article is organized as follows. First, section “System operation” describes the system operation of the CR system, then section “Erlang-B discretization” develops the time discretization of the Erlang-B formula including both proposals. Following this, the teletraffic analysis is presented in section “Teletraffic analysis for the CR network.” The article concludes with relevant numerical results in section “Numerical results” and conclusions.

System operation

In this section, we present the design of the CR system where the cellular system is used as a primary system and the secondary network is the WSN for seism detection and monitoring.

Secondary network: WSN

WSN is a clustered-based and event-driven system. Clustering has been used to reduce energy consumption.^29,30 Nodes that belong to a specific cluster (CMs) only transmit their data to a specific node inside the cluster (CH) instead of transmitting throughout the entire network. Then, CHs transmit the collected data from all their CMs to the sink node. It is important to note that nodes do not transmit directly to the cellular system, that is, nodes transmit to the CHs and CHs transmit to the sink using the cellular system’s frequency bands, but no transmission is performed directly to the base station.

In the CF phase, nodes transmit a control packet to the entire network using the slotted NP-CSMA (non-persistent carrier sense multiple access) protocol.^31,32 The shared medium is considered to be slotted and nodes (re)transmit according to a geometric backoff with parameter $τ$ . Once the clusters are formed, the CMs transmit in an orderly fashion to their respective CH by means of a TDMA (time division multiple access) protocol. The network activity ends when the last node of the network depletes its energy. The selection of the value of $τ$ is of major importance for the adequate performance of the system as described by Bouabdallah et al.³³

To analyze the performance of the SS phase, we consider an average of $N_{CM}$ CMs per cluster, each transmitting packets of $P$ bits. If the total transmission rate is $R (bits / s)$ , then the transmission time of the packet $T$ is $P / R$ , which is also the time slot duration. Hence, the frame duration is $(N_{CM} + 1) T$ .³² It is important to notice that the CH consumes much more energy than a CM simply because in the SS, CHs remain active at all times receiving data from the CMs and transmitting these data to the sink node.

Energy consumption in the network is assumed to be as in the work by Mercier et al.:⁷ control packets transmitted in the CF phase consume $E_{tx}^{CF} = 1 energy units$ , control packets received in the CF phase consume $E_{rx}^{CF} = 0.1 energy units$ , data packets transmitted in the SS phase consume $E_{tx}^{S} = 0.5 energy units$ , data packets received in the SS phase consume $E_{rx}^{S} = 0.1 energy units$ , and data packets transmitted to the sink node consume $E_{tx}^{sink} = 1.5 energy units$ .

Indeed, it is considered that the packet reception procedures consume small amounts of energy that basically depends on the packet length. As such, a packet received at the CF phase consumes less energy than a packet received in SS (using the TDMA technique) since CF packets are basically control packets with the information of the node (mainly the node’s ID) but no data are transmitted. On the one hand, packet transmission power has to be sufficiently high in order to reach the destination target. As such, the energy needed to reach the sink node is much higher than the energy needed to transmit within the network area or the nodes inside the cluster. On the other hand, it is assumed that a data packet transmitted inside the cluster consumes more energy than a control packet transmitted in the CF phase due to the length of the packet. Note that these values strongly depend on the particular commercial nodes used in the system. We use these values for illustrative purposes. As such, these values only represent a case of study in order to analyze the system’s performance. In addition, it is straightforward to produce numerical results for different energy consumption values to the ones considered in this work.

Primary network: cellular system

A discrete-time blocked customers cleared (BCC) system is considered. Hence, nodes that arrive to the system and find no available resources are blocked. We assume that the system can have a maximum of $S$ clients with an active call. In the continuous time system, users arrive to the system according to a Poisson process. This process is time discretized in order to consider a geometric arrival process.

In particular, we consider a TDMA-based system such as global system for mobile communications (GSM), general packet radio service (GPRS), or enhanced data rates for GSM evolution (EDGE), which is composed of $M$ slots per frame to serve users. In a GSM system, the frame is divided into eight time slots; the duration of a frame is 120 ms and the size of the slot is approximately 15 ms.

CR network

The CR network works as follows. In the CF phase of the secondary network, all nodes transmit their control packet to establish the role of each node by first scanning each time slot of the primary network. When nodes detect an empty slot, they transmit the aforementioned packet according to the random access protocol. In the steady phase, where nodes have their time slot assigned inside the cluster, each node that is scheduled to transmit first scans the channel in order to detect if it is unused by a PU, in which case it transmits its data packet. On the contrary, if the channel is used by a PU, that is, a cellular node, then the sensor nodes remain listening to the channel until an empty slot is found and then they transmit their information packets. It is important to notice that the occupation of the primary network is of major importance for the appropriate operation of the WSN. Specifically, when the primary network is saturated and almost all channels are being used by PUs, the CF time increases drastically making the use of the WSN unpractical for the seismic detection and reporting. Also, in these conditions, secondary nodes in the SS are practically unable to transmit the information to the sink node causing a high number of packet losses, rendering the WSN useless.

Cellular networks have been modeled using Markov chains, deriving in the widely known Erlang-B formula.³⁴ More complex systems based on wireless and wired systems have also used this mathematical tool to model the system, such as CR,³⁵ peer-to-peer,³⁶ and WSNs,³³ among others. Throughout these studies, this tool has proven its accuracy by providing numerical results very close to simulation and even experimental results. Building on this, the use of Markov chains seems to be a straightforward option to model the proposed system. In the following section, we describe in detail such Markovian models.

Erlang-B discretization

In this work, the BCC policy is considered for a system with $S$ servers. We consider that during the active periods, the voice packets are deterministically generated every $T_{slot}$ seconds. A typical value used for $T_{slot}$ is 20 ms.²³ Also, packet spacing of 10 and 20 ms has been used by Karapantazis and Pavlidou.²⁴ Call arrivals are assumed to follow a Poisson process with rate $1 / λ$ . This continuous time process can also be described by a discrete-time process as in the work by Nyiyato and Hossain.²⁶ Indeed, the probability that $n$ packets arrive in a time slot interval $(t - T_{slot}, t)$ is given as follows

$P_{A} (n) = P {N (t - T_{slot}, t) = n} = \frac{{(λ T_{slot})}^{n} e^{- λ T_{slot}}}{n!}$ (1)

On the other hand, call duration is considered to be exponentially distributed with mean $1 / μ$ . It is straightforward to show that the probability that a user finishes its session in the $j th$ time slot after its arrival is given by $P_{j} = [1 - e^{(- μ T_{slot})}] e^{- (j - 1) μ T_{slot}}$ for $j = 1, 2, \dots$ . Hence, the probability that a particular user finishes its session in a given time slot can be approximated by a geometric distribution with parameter

$P_{s} = 1 - e^{- μ T_{slot}}$ (2)

Building on this, let us define $U_{n}$ as a binomial random variable that represents the number of users that end their session in the $n th$ time slot. Hence

$P (U_{n} = j) = (\begin{matrix} n \\ j \end{matrix}) P_{s}^{j} (1 - P_{s})^{n - j}$ (3)

Based on this description, we now develop two discretized models for the Erlang-B formula. In the first model, we simplify the birth and death process by considering that, during a time slot, only one user can arrive or leave the system. As such, it corresponds to an approximation. In subsequent sections, we validate the accurateness of such approximation. In the second model, we relax this assumption and consider all possible events in a time slot. Due to the complexity of this second discretization model, we only use the first approximation for reasons of simplicity. Also, as shown in the “Numerical Results” section, the value of the time slot is sufficiently small to obtain accurate results with such approximation. For the sake of completeness, we provide the exact discrete Erlang-B model but the CR network is studied under the approximated approach.

Bernoulli-geometric model

This model, is developed for systems where the time slot duration is significantly smaller than the mean inter-arrival time. As such, in these conditions $(T_{slot} << 1 / λ)$ , it is reasonable to consider that in a single slot, the probability of having two or more arrivals is very small as described by Veretennikov.²⁵ Hence, $P {N (t - T_{slot}, t) \geq 2} \approx 0$ . Also note that, given the small value of $T_{slot}$ , the following supposition is in order

$P {N (t - T_{slot}, t) = 1} = P_{a} = (λ T_{slot}) e^{- (λ T_{slot})}$ (4)

Therefore, the arrival process is modeled as a Bernoulli process with parameter $P_{a}$ . For the case of the system’s departures, when $T_{slot} << 1 / μ$ , it can be seen in the work by Bi et al.²³ that the probability that two or more users leave the system in the same time slot is almost zero. Hence, to simplify the mathematical analysis and as an approximation, it is considered that only one user can leave the network in a given time slot.

This system can be modeled as a one-dimensional discrete-time Markov chain (DTMC). Figure 1 shows the DTMC for modeling the cellular network considering that the basic time unit is the system’s time slot, $T_{slot}$ , and the state variable represents the number of users being served by the network. Hence, the chain goes from state $n$ to state $k$ with probability $Q (n, k)$ . Let us define $S$ as the number of servers and $P_{s} (n)$ as the probability that one user finishes its session in the $n th$ time slot. For any state $n$ , $1 \leq n \leq S - 1$ , the system can remain in the same state, move to state $n + 1$ , or go to state $n - 1$ . In state $n = 0$ , the system can only move to state $n = 1$ or remain in the same state. Accordingly, in state $n = S$ , the system can either go to state $n - 1$ or remain in the same state. The possible transitions from state $n$ to state $k$ are as follows

$Q (n, k) {\begin{matrix} P_{s} (n) & k = n - 1 (n \geq 1) \\ P_{a} & k = n + 1 (n \leq S - 1) \\ 1 - P_{a} - P_{s} (n) & k = n (1 \leq n \leq S - 1) \\ 1 - P_{s} (n) & k = n (n = S) \\ 1 - P_{a} & k = n (n = 0) \end{matrix}$ (5)

where $P_{s} (n) = P (U_{n} = 1) = n P_{s} (1 - P_{s})^{(n - 1)}$ . This chain is solved considering the next set of linear equations

$Π P = Π$ (6)

where $Π$ is the SS probabilities given by $[π_{0}, π_{1}, \dots, π_{S}]$ and $P$ is the transition probabilities matrix composed by the valid transitions in the system described above. As such, the SS probabilities can be calculated as follows

$π_{i} = \frac{P_{a}^{i}}{i! P_{s}^{i} {(1 - P_{s})}^{\frac{i (i + 1)}{2}}} π_{0}$ (7)

$π_{0} = \frac{1}{\sum_{j = 0}^{S} \frac{P_{a}^{j}}{j! P_{s}^{j} {(1 - P_{s})}^{\frac{j (j + 1)}{2}}}}$ (8)

Figure 1.

BCC system in discrete time.

Then, the probability that users are blocked in a time slot is found as $P_{b} = P_{a} π_{S}$ . It is important to notice that in this model, since only one user can be blocked in a time slot (due to the assumption that only one arrival can occur in a time slot), the time slot–blocking probability corresponds to the user’s blocking probability. The value of the time slot is crucial in order to have a good approximation to the continuous model described by the Erlang-B formula which is analyzed in the “Numerical Results” section.

Poisson-geometric model

In this model, the consideration that only one arrival or one departure can occur in a time slot is no longer employed. Therefore, arrivals follow a Poisson process, while service time is modeled by a geometric random variable as previously described. This system is also modeled by a DTMC similar to the approximated Bernoulli-geometric (B-G) model but with the difference that there are transitions from one state to any other state of the chain. Suppose that the chain goes from state $i$ to state $j$ . For the case when $j < i < S (S > j > i)$ , this transition takes place when during a time slot there are $i - j + k (j - i + k)$ departures (arrivals) and there are $k$ arrivals (departures) for $k = 0, 1, \dots$ . The system remains in state $i (j)$ in case that there are same number of departures and arrivals during a time slot. In case that $i = S (j = S)$ , additional cases have to be considered. For instance, the case where there is one service termination and there is one or more arrivals. Hence, the valid transition probabilities $Q (i, j)$ are described as follows

$Q (i, j) {\begin{matrix} \sum_{k = 0}^{i} P_{A} (j - i + k) P_{S} (i, k); j > i; j \leq S \\ \sum_{k = 0}^{i} [1 - \sum_{n = 0}^{j - i + k - 1} P_{A} (n)] P_{S} (i, k); j > i; j = S \\ \sum_{k = 0}^{i} P_{A} (k) P_{S} (i, i - j + k); j < i \\ \sum_{k = 0}^{i} P_{A} (k) P_{S} (i, k); j = i; j, i \leq S \\ {(1 - P_{s})}^{i} + \sum_{k = 1}^{i} [1 - \sum_{n = 0}^{k - 1} P_{A} (n)] P_{S} (i, k); j = i; j, i = S \end{matrix}$ (9)

where $P_{S} (i, k) = (\begin{matrix} i \\ k \end{matrix}) P_{s}^{k} (1 - P_{s})^{i - k}$ . This chain is numerically solved using the Gauss–Seidel method considering the set of linear equations in equation (6). For this model, the time slot–blocking probability is calculated as follows: in state $S - j$ , there are blocked calls if during the time slot interval occur $k + j$ arrivals and $k + j - 1$ or less departures, for $j = 0, 1, \dots, S$ , and $k = 1, 2, \dots$ . As such, blocking probability can be mathematically expressed as follows

$\begin{matrix} P_{b} = π_{S} (\sum_{j = 0}^{S} (\begin{matrix} S \\ j \end{matrix}) P_{s}^{j} {(1 - P_{s})}^{S - j} \sum_{i = j + 1}^{\infty} P_{A} (i)) \\ + π_{S - 1} (\sum j = 0^{S - 1} (\begin{matrix} S - 1 \\ j \end{matrix}) P_{s}^{j} {(1 - P_{s})}^{S - 1 - j} \sum_{i = j + 2}^{\infty} P_{A} (i) \\ + \dots + π_{0} (\sum_{S + 1}^{\infty} P_{A} (i))) \end{matrix}$ (10)

Hence, it can be written as follows

$P_{b} = \sum_{k = 0}^{S} π_{k} \sum_{j = 0}^{k} (\begin{matrix} k \\ j \end{matrix}) P_{s}^{j} {(1 - P_{s})}^{k - j} (1 - \sum_{i = 0}^{j + S - k} P_{A} (i))$ (11)

Teletraffic analysis for the CR network

In this section, we develop the Markovian model for the CR network. Specifically, we consider the discretized birth and death process described in the previous section in order to identify the empty slots in the primary network on a time slot basis. Then, we calculate the average CF delay, average energy consumption at the CF phase and the SS phase, and the average reporting delay in the SS phase.

In the following analysis, we consider that idle resources of the cellular network are assigned in a random manner, that is, empty slots are not assigned sequentially. This assumption is based on the fact that for low traffic conditions, the slots at the beginning of the frame would be more likely to be occupied than the slots at the end of the frame. Following this assumption, any slot can be occupied with the same probability $P_{oc} (i) = i / S$ when there are $i$ users in the primary network.

In addition, in the CF phase, where collisions can happen since multiple secondary nodes can attempt a transmission when an empty slot is detected, a successful transmission occurs when only one node transmits its control packet. The success probability when there are $k$ contending nodes remaining in the CF phase is calculated as $P_{suc} (k) = (\begin{matrix} k \\ 1 \end{matrix}) τ (1 - τ)^{k - 1} = k τ (1 - τ)^{k - 1}$

CF phase

As described earlier, the CF occurs whenever a vibration outside the normal range of values is detected in the building either because a seismic activity occurred or due to strong winds, among other reasons. At this point, all the nodes in the system become active and transmit a control packet to the sink in order to be part of a cluster in the network. Nodes transmit in each time slot with probability $τ$ only if they detect that the slot is empty, that is, not being used by a PU. The CF phase finishes when all the nodes in the system have successfully transmitted their control packet. In this work, we consider that the first $N_{CH}$ nodes that successfully transmit their control packet become CHs, while the rest of the nodes are assigned as CM of their nearest CH.

In Figure 2, we show the DTMC for the WSN in CF phase, where $i$ represents the number of active calls in the cellular system and $k$ represents the number of nodes that have to transmit their control packet in order to be aggregated to a particular cluster and be then classified as CM or CH.

Figure 2.

Discrete-time Markov chain for the CF phase.

The transition probabilities for this Markov chain are now described in detail:

$P_{1}$ corresponds to the probability that one PU arrives to the system, no PU terminates its service and only one secondary node transmits in the current slot which is empty. Then

$P_{1} = P_{a} (1 - P_{S}^{(i)}) P_{suc} (k) (1 - P_{OC} (i))$ (12)

$P_{2}$ corresponds to the probability that no PU arrives to the system, no PU terminates its service and only one secondary node transmits in the current slot which is empty. Then

$P_{2} = (1 - P_{a}) (1 - P_{S}^{(i)}) (1 - P_{OC} (i)) P_{S} (k)$ (13)

$P_{3}$ corresponds to the probability that no PU arrives to the system, one PU terminates its service, and only one secondary node transmits in the current slot which is empty. Then

$P_{3} = (1 - P_{a}) P_{S}^{(i)} P_{S} (k) (1 - P_{OC} (i))$ (14)

$P_{4}$ corresponds to the probability that no PU arrives to the system, no PU terminates its service, and no secondary node successfully transmits its control packet. This can occur either because the current slot was occupied by a PU or because no or more than two secondary nodes transmitted in the slot. Then

$P_{4} = [P_{OC} (i) + (1 - P_{S} (k))] (1 - P_{a}) (1 - P_{S}^{(i)})$ (15)

$P_{5}$ corresponds to the probability that no PU arrives to the system, one PU terminates its service, and no secondary node successfully transmits its control packet. This can occur either because the current slot was occupied by a PU or because no or more than two secondary nodes transmitted in the slot. Then

$P_{5} = [P_{OC} (i) + (1 - P_{S} (k))] P_{S}^{(i)} (1 - P_{a})$ (16)

$P_{6}$ corresponds to the probability that one PU arrives to the system, no PU terminates its service, and no secondary node successfully transmits its control packet. This can occur either because the current slot was occupied by a PU or because no or more than two secondary nodes transmitted in the slot. Then

$P_{6} = [P_{OC} (i) + (1 - P_{S} (k))] P_{a} (P_{S}^{(i)})$ (17)

The average CF time is given by the absorption time from state $(i, N)$ , for $0 \leq i \leq S$ , to absorbant state $(j, 0)$ , for $0 \leq j \leq S$ , which is numerically evaluated.

The average power consumption in the CF phase is calculated by considering the following: in this phase, all remaining nodes to transmit their control packet must first determine the channel status to detect transmissions of the cellular system. This procedure requires only to detect the channel for a small amount of time at the beginning of the time slot; if during this time the slot is sensed as occupied, it is not necessary for the nodes to continue to listen to the channel for the remaining time of the slot. The energy consumed by all nodes in the secondary network when a PU uses a time slot is $E_{Rx}^{Det}$ , which is a fraction of the power consumption needed for receiving a packet $(E_{Rx}^{CF})$ . We consider the case where there are $k$ remaining nodes to transmit their packet in the CF phase and there are $i$ active calls in the cellular system. Then, the energy consumption depends on the following:

Slot not available. When the time slot is being used by a PU, the secondary network nodes detect this slot as busy and the power consumption is

$E C_{OC} (k) = {kE}_{Rx}^{Det}$ (18)

Slot available. When the channel is idle, that is, there is no PU occupying the slot, the power consumption is as follows:

(a) When the available time slot is unsuccessful, either because there are no transmissions or there are two or more simultaneous transmissions, the energy consumption is given as follows

$E C_{fail} (k) = \sum_{j = 0, j \neq 1}^{k} [{jE}_{tx}^{CF} + (k - j) E_{rx}^{CF}]$

which entails the energy consumption of the j transmissions and the energy consumption due to the reception of the rest of the nodes.

(b) When a successful transmission occurs, the energy consumption in that time slot is

$E C_{succ} (k) = [E_{tx}^{CF} + (k - 1) E_{rx}^{CF}]$

which corresponds to the consumption of only one transmission and $k - 1$ packet receptions.

As such, the total average energy consumption is calculated by considering a Markov chain with rewards, where $r_{OC} = {kE}_{Rx}^{Det}$ is the reward for an occupied slot, $r_{succ} = E C_{succ} (k)$ is the reward for a successful transmission, and $r_{fail} = E C_{fail} (k)$ is the reward of a collision when there are $k$ remaining to transmit in the CF phase. The reward Markov chain is numerically solved.

SS phase

While the seismic activity occurs, all nodes in the system register their sensed data and store them in their respective internal memory in form of data packets. When the SS begins, all nodes must transmit all of the gathered packets to the sink node. In this phase, clusters are already formed and the nodes already belong to a particular CH. Then, each CH can broadcast to their members a schedule for the collision-free transmission whenever an empty slot is detected. This is achieved by means of a TDMA-based protocol. In this phase, each node knows its transmission time and has to wait until the previous node in the schedule has achieved its transmission. The SS ends when all nodes have transmitted all of their packets to the sink. Building on this, it can be seen that for a specific earthquake or vibration epoch, the system generates $N_{eq}$ packets which corresponds to the sum of all packets sensed by each node.

In Figure 3, we show the DTMC for the SS phase, where $i$ is the number of active calls in cellular system and $x$ is the number of packets that remain to be transmitted $(0 \leq x \leq N_{eq})$ .

Figure 3.

Discrete-time Markov chain for the SS phase.

We now describe in detail the possible transitions and their corresponding probabilities:

$P_{1}$ corresponds to the probability that no PU arrives to the system, one PU terminates its service, and the secondary node finds the time slot idle (this occurs with probability $(1 - P_{OC} (i))$ ). Then

$P_{1} = (1 - P_{a}) P_{S}^{(i)} (1 - P_{OC} (i))$ (19)

$P_{2}$ corresponds to the probability that no PU arrives to the system, no PU terminates its service, and the secondary node finds the time slot idle. Then

$P_{2} = (1 - P_{a}) (1 - P_{S}^{(i)}) (1 - P_{OC} (i))$ (20)

$P_{3}$ corresponds to the probability that one PU arrives to the system, no PU terminates its service, and the secondary node finds the time slot idle (this occurs with probability $(1 - P_{OC} (i))$ ). Then

$P_{3} = P_{a} (1 - P_{S}^{(i)}) (1 - P_{OC} (i))$ (21)

$P_{4}$ corresponds to the probability that no PU arrives to the system, no PU terminates its service, and the secondary node does not find the time slot idle (this occurs with probability $P_{OC} (i)$ ). Then

$P_{4} = (1 - P_{a}) (1 - P_{S}^{(i)}) P_{OC} (i)$ (22)

$P_{5}$ corresponds to the probability that no PU arrives to the system, one PU terminates its service, and the secondary node does not find the time slot idle. Then

$P_{5} = (1 - P_{a}) P_{S}^{(i)} P_{OC} (i)$ (23)

$P_{6}$ corresponds to the probability that one PU arrives to the system, no PU terminates its service, and the secondary node does not find the time slot idle (this occurs with probability $P_{OC} (i)$ ). Then

$P_{6} = P_{a} (1 - P_{S}^{(i)}) P_{OC} (i)$ (24)

Consider that the system is composed of $N$ nodes, of which $N_{CH}$ are $CH s$ and $N - N_{CH}$ are $CM s$ . Hence, there are $N_{CH}$ clusters in the system.

To determine the adequate number of clusters, it is proposed that $N_{CH} = CN$ , where $C$ is the percentage of nodes that will be classified as $CH$ . For the development of this work, it is proposed that $C = 10 %$ . The case where a different perecentage of nodes is considered is not studied and remains open for future research since the performance of the network can be affected by a different value of $C$ .

The average number of $CM s$ per cluster (including CH) is $N_{CM} = N / N_{CH}$ , which is for an ideal case, where $CH s$ are well distributed and all have the same amount of CMs. Recall that we consider that the first $N_{CH}$ nodes that successfully transmit in the CF phase become CHs. In practice, this usually does not happen, since CH nodes can be clustered in a certain region of the surveilled area. However, classical clustering algorithms, such as LEACH,²⁹ do not guarantee such uniform distribution either. In fact, for $N \leq 100$ , this CH selection achieves a similar performance as LEACH as shown in the work by Romo Montiel et al.³⁷ Furthermore, different techniques to assure a good node distribution can be considered at the expense of having a more computational complex CF algorithm or CF delay. We consider this issue to lay outside the scope of the work, and we leave this research area open for a future work.

From this, the average frame size in the SS is given as follows

$T_{frame} = N_{CM} T_{slot}$ (25)

where $N_{CM}$ is composed of 1 CH and $N_{CM} - 1$ CMs. If the earthquake or vibration affecting the building is considered to be of a duration of $T_{eq}$ seconds, then the average number of frames needed to transmit the data acquired by the sensors is given as follows

$n_{f} = \frac{T_{eq}}{T_{frame}}$ (26)

and the number of total packets to report the event are $N_{eq} = N * n_{f}$ . The average number of time slots required to transmit all the packets, $D_{SS}$ , is then the absorption time from state $(i, N_{eq})$ , for $0 \leq i \leq S$ , to absorbent states $(i, 0)$ which is numerically evaluated form the Markov chain described above.

The average power consumption per idle frame is obtained by considering that for each transmission the nodes consume $E_{Tx}^{SS}$ energy units and each node receiving a packet in the SS phase consumes $E_{Rx}^{SS}$ units of energy. In addition, at the end of each frame, the CH receives all the data transmitted by the CMs and sends all the information collected to the sink node consuming $E_{Tx}^{Sink}$ units of energy. In case that the slot is being used by a PU, the energy consumed is $E_{Rx}^{Det}$ . As such, for each transition of the Markov chain, the energy consumption is quantified rendering a reward Markov chain where $r_{OC} = E_{Rx}^{Det}$ is the reward for an occupied slot and $r_{tx} = E_{Tx}^{SS} + E_{Rx}^{SS}$ is the reward for an idle slot. The reward Markov chain is numerically solved.

Numerical results

In this section, numerical results are presented to quantitatively evaluate the performance of the CR network for different traffic loads and seism duration. We study both the CF and SS phases in terms of average delay and average energy consumption. The results obtained from the mathematical model were verified using discrete simulations. Good match between simulation and analytical results was found.

First, we study the accuracy of the proposed discretization of the Erlang-B formula by observing the impact of the time slot duration on our proposed approaches. Also, the conditions under which our discrete-time Erlang-B approaches are suitable are provided. Numerical and simulation results presented in this section consider that $1 / λ = 10 s$ , and we vary the value of $1 / μ$ . The effect of the time slot duration on the precision of the approximation for the B-G model is first studied. This effect can be seen for the time slot–blocking probability for different values of the time slot. It can be seen that the mismatch between the proposed B-G model and the Erlang-B formula becomes noticeable large as the value of $T_{slot}$ approaches $1 / μ = 180$ as it is shown in Figure 4. For a value of the time slot interval 2 orders of magnitude smaller than $(1 / μ)$ , there is a good match between both models.

Figure 4.

Analytical and simulation results for time slot–blocking probability versus time slot size: $λ = 0.1$ , $μ = 0.006$ , and $S = 20$ .

This observation is further confirmed by observing Figure 5. The blue surface presents the blocking probability for the continuous time model. Hence, it can be seen that it is unaffected by the different values of the time slot. Conversely, the red surface corresponds to the first discrete-time model based on the approximation where only one or no arrivals can occur in a time slot period and one or no departures can occur in a time slot. As such, the accuracy of this B-G model is highly dependent on the value of the time slot. Figure 5 clearly shows that when the time slot is in the same order of magnitude or 1 order of magnitude smaller than the average service time, this B-G model does not render accurate results. On the other hand, for values of the average service time 2 or more orders of magnitude higher than to the value of the time slot, this approximation, that greatly simplifies the mathematical analysis, provides accurate results.

Figure 5.

Analytical and simulation results for time slot–blocking probability versus number of servers and average service holding time $1 / μ$ seconds for the Bernoulli-geometric model.

For the Poisson-geometric model, Figure 6 shows the time slot–blocking probability for different values of the time slot period for both analytical and simulation results. Note that the value of the time slot has no visible impact on the analytical model compared to the simulation results and there is a good match between these numerical results.

Figure 6.

Analytical and simulation results for time slot–blocking probability versus number of servers for the Poisson-geometric model and different values of average service holding time $1 / μ$ seconds.

For the CR system, since the time slot has a duration of $T_{slot} = 0.015 s$ , we considered the approximated model as basis for the teletraffic analysis, we considered values of $1 / μ$ 2 orders of magnitude higher than $T_{slot}$ and vary $λ$ in order to consider different traffic loads and assure a good approximation of the Erlang-B formula.

We now study the CF time. Figure 7 shows the average time slots required for the CF procedure for different number of nodes in the network for offered traffic loads of 3, 30, and 60 E (erlangs). In this figure, we compare the simulation results obtained through the developed simulator detailed in Algorithm 1 with analytical results obtained by numerically solving the Markov chain described in section “Cluster formation phase.” We can see a good match between these models, which validates the mathematical analysis. As expected, when traffic load is high in the primary system, average CF time is also high due to the lack of empty spaces. However, when $τ$ is in the range of 0.1 and 0.2, low CF times can be found. Based on this observation, we conclude that the selection of $τ$ is of major importance for the performance of the SHM system. Based on this, these results can be used as guidelines for selecting the value of $τ$ according to the number of nodes in the network.

Figure 7.

Average CF time vs number of nodes and different values of $τ$ for an offered traffic of 3, 30, and 60 E.

In order to study the data reporting time and energy consumption in the SS phase, that is, when the clusters are already formed and the nodes transmit in an ordered fashion to their CH using a TDMA-based protocol, three historical earthquakes are considered, which are as follows:

Mexico City earthquake, on 19 September 1985, with a magnitude of 8.1 on the Richter scale and a duration of 2 min (120 s).

Japan earthquake, on 11 March 2011, with a magnitude of 9.0 on the Richter scale and a duration of 6 min (360 s).

Chile earthquake, 22 May 1960, with a magnitude of 9.5 on the Richter scale and duration of 10 min (600 s).

For the average reporting delay, that is, the time for all nodes in the system to report the information detected by the nodes to the base station, eight slots in the primary network are considered to be assigned to the users. Table 1 shows the time required to transmit the complete information of the data, while the offered traffic load is 0 E. This represents the case where the WSN has dedicated channels and we use it as a reference value since the WSN can find all time slots unoccupied. From these results, we can compare the performance of the system when the CR capabilities are enabled to the case of a conventional WSN for SHM applications.

Table 1.

Average time of data reporting on SS phase and different loads of offered traffic on cellular system.

	Mexico City	Japan	Chile
0 E	8000 slots	24,000 slots	40,000 slots
	2 min	6 min	10 min
3 E	17,051.89 slots	63,670.5 slots	81,255.8 slots
	5 min	15 min	20 min
30 E	209,851.3 slots	484,827 slots	854,057 slots
	53 min	121 min	213 min
60 E	439,370.25 slots	945,221 slots	1,703,310 slots
	110 min	236 min	425 min

It can be seen that the use of CR capabilities greatly increases the average reporting time specially in highly congested environments, as it is expected after a seismic activity. Indeed, the traffic load in a normal operation environment is of 3 E with eight servers, since this traffic load entails a blocking probability of 0.02. We consider high-traffic scenarios since after an event we expect the traffic load to drastically increase in the cellular system. However, for the proposed application of the studied system, which is to report all the gathered information about the seismic activity, we believe that a time as high as 425 min, like the one in the worst-case scenario considered in this work, is acceptable. Indeed, the proposed system is not intended to work as an early alarm but rather to make available all the important information about the structural health of the building in question. As such, if the technical team in charge of the SHM has the relevant data 7 h after the event, this means that the structure of the building could be revised the same day the event occurred.

Based on the presented results, it is possible to calculate the optimal number of nodes. To this end, the network administrator has to establish the average time to begin event reporting reception. This corresponds to the average CF delay, which is the time required for the nodes to organize among them in order to initiate event packet transmissions; prior to the CF, nodes can sense the event data but are unable to transmit their information to the sink. Note that this time is related to the specific requirements of the seism monitoring center. In addition, the traffic load of the primary system (cellular network) has to be known or estimated, as in a conventional cellular system where blocking probability is achieved by selecting the appropriate number of traffic channels. Building on this, the number of nodes that guarantee a lower average CF delay can be selected. Finally, the appropriate transmission probability can be selected, that is, the transmission probability that entails the lower average delay. It is important to note that the reporting delay depends on the event duration (duration of the seism or vibration) and traffic load in the primary system. As such, we consider that the optimal number of nodes is mainly related to CF delay and not on reporting delay. In Tables 2 –4, we show the optimal number of nodes to achieve an average target CF delay of 10, 6, and 3 s for different traffic loads in the primary network. For instance, in order to achieve an average time to start reporting the seism of 10 s with a traffic load in the primary network of 60 E, more than 10 nodes should compose the WSN with a transmission probability of 0.1 or 0.2. A higher number of nodes or a different transmission probability would not achieve less than 10 s. Conversely, if this average CF delay is desired to be of 3 s, and the offer load in the primary network is 3 E, the WSN can be composed by 20 nodes or less with a transmission probability of 0.1, or 10 nodes or less with τ = 0.2 or 0.3. As such, the results derived in this work provide sound guidelines for the selection of the system’s parameters for different conditions.

Table 2.

Optimal number of nodes for average cluster formation delay lower than 10 s.

$τ$	10 nodes	20 nodes	30 nodes
0.001	–	–	–
0.002	–	–	–
0.01	★	–	–
0.1	$★, \circ, \oplus$	$★$	$★$
0.2	$★, \circ, \oplus$	$★$	–
0.3	$★, \circ$	–	–

★—3 E, ∘—30 E, ⊕—60 E.

Table 3.

Optimal number of nodes for average cluster formation delay lower than 6 s.

$τ$	10 nodes	20 nodes	30 nodes
0.001	–	–	–
0.002	–	–	–
0.01	–	–	–
0.1	$★, \circ$	★	★
0.2	$★, \circ, \oplus$	★	-
0.3	$★, \circ$	-	-

$★$ —3 E, ∘—30 E, ⊕—60 E.

Table 4.

Optimal number of nodes for average cluster formation delay lower than 3 s.

$τ$	10 nodes	20 nodes	30 nodes
0.001	–	–	–
0.002	–	–	–
0.01	–	–	–
0.1	$★$	$★$	–
0.2	$★, \circ$	–	–
0.3	$★$	–	–

$★$ —3 E, ∘—30 E, ⊕—60 E.

We now analyze the energy consumption in the data reporting stage. Figure 8 shows the total energy units consumed in the SS phase for a traffic load of 3, 30, and 60 E in the primary network and 10, 20, and 30 nodes in the WSN.

Figure 8.

Energy units consumed in the SS phase with different traffic loads in the primary network.

As expected, the energy consumption increases as the number of nodes and the traffic load increase. This is because the number of packets are proportional to the number of nodes and the traffic load determines the probability of finding empty slots in the secondary network. Unlike conventional, dedicated bandwidth sensor networks, where nodes wake up in the time slot allocated to transmit their data packet and remain in energy-saving mode for the rest of the time, this is no longer possible in CR-based networks since all nodes have to remain in reception mode to detect the empty slots of the primary network and the successful transmissions of all other members of the cluster. This is the reason for a higher energy consumption in a CR sensor network. We believe that the results presented in Figure 8 are of major importance. Even if nodes can be energized by the building and batteries are not required in normal conditions, whenever an earthquake occurs, it is not uncommon to have power cuts or complete blackouts in the buildings affected. As such, nodes still have to operate on batteries to relay their information to the sink node. As such, the results presented show the initial level of energy required for the complete transmission of the gathered data. The results presented in this work were verified using a home-made discrete simulation. In Table 5, we present the simulation parameters and Algorithm 1 describes in detail the general operation of the simulator.

Table 5.

Simulation specifications.

Parameter	Values	Description
Frame size	125 ms	Using GSM standard
Number of slots in the frame	8 slots	Using GSM standard and TDMA protocol, each channel is sharedup to eight users
Slot size	15 ms	$\frac{Frame size}{No . of slots in the frame} = \frac{125 ms}{8 slots}$
Total nodes	10, 20, and 30 nodes	Total number of nodes that are placed in the area of networkcoverage
Cluster head nodes	10% of total nodes	The 10% of total nodes are cluster head nodes, that is, for 10nodes, we used 1 cluster head
Average number of nodesby cluster	10	The average number of nodes by cluster is calculated dividing $\frac{Total nodes}{CH nodes}$
Transmission probability $(τ)$	0.001, 0.002, 0.01, 0.1, 0.2, 0.3, 0.4, 0.5	We proposed different values of $τ$
Offered traffic (a)	3, 30, and 60 E	We proposed the values of offered traffic to study the systemwhen we have a low, medium, and high traffic in the network
$λ$	1/180	Arrival rate
$μ$	0.016667, 0.166667, and 0.33333	Channel holding time, offered traffic (a) = $\frac{lambda}{μ}$ , $μ = \frac{λ}{offered traffic}$
Simulation limit	1,000,000	We repeated the cluster formation time and SS 1,000,000 times to obtain the average of the CF, energy consumption in the CF phase, energy consumption in the SS phase, and delay of the packets

TDMA: time division multiple access; CF: cluster formation; SS: steady state.

Algorithm 1.

General operation of the simulator.

Require: Total nodes (N), Number of CH nodes (N_CH), Area, Transmission Probability (τ), Offered Traffic Load (λ/µ), Earthquake time (T_earthquake), Number of slots in the frame (N_slot)Ensure: Average Cluster Formation Time, Energy Consumption in the CF phase, Energy Consumption in the SS phase and Delay of the packets to transmit the earthquake1: An arrival is generated in the primary network to start the simulation.2: while times ≤ 1000000 do3: We take the first element of the queue4: if element of the queue == arrival then5: A new arrival is generated6: if there are not slots available then7: blocked ++; //Number of blocked call in the primary network8: else9: We assign a slot to the call.10: A death is generated to ends the calls.11: end if12: else13: A slot is free.14: end if15: In background of the primary network we are sensing the state of the slots in the frame.16: fori = 0 to N_slotdo17: if there are nodes to classify in the CF time then18: ifFrame[i] == 0 then19: //Slot is free20: Decide which nodes can transmit in the slot21: if Just one node can transmit in the slot then22: We assign the node to the CH node with the shortest Euclidian Distance.23: else24: There is a collision and de CF fails in the slot.25: end if26:

N_{slot s_{CF}}

++27: else28: //Slot is taken by an user of the primary network29: end if30: We calculate E_CF31: else32: //There are not nodes to classify in the CF phase. SS phase starts33: ifFrame[i] == 0 then34: First, CM Nodes transmit to CH node and at the end, CH node transmit to Base Station.35:

N_{slot s_{SS}}

36: else37: Slot is taken by a user of the primary network38: end if39: We calculate E_SS40: end if41: end for42: times ++;43: end while

Conclusion

In this work, we study, analyze, mathematically model, and evaluate the performance of an SHM system–based event-driven WSNs. The proposed system is not focused on producing alerts for people in the buildings but rather to store all information regarding the health of the structure during the seismic event in order to study afterward such information by specialists in a disaster control facility. As such, experts can determine the damage of the monitored buildings without having to inspect the structure in person which can take many days or even weeks. Conversely, we guarantee recollecting data times in the order of hours even if traffic load is high in the primary network. Our model is developed based on DTMC that can be easily evaluated. To this end, we developed a new analytical tool to discretize the well-known Erlang-B formula.

The Erlang-B model is proposed, studied, and numerically analyzed in discretized time. Two mathematical models are proposed: the B-G and Poisson-geometric models. The main advantage of the B-G model is that it provides a very good match when the time slot is much smaller than the average inter-arrival time and it provides a closed expression on the blocking probability which is straightforward to evaluate. On the other hand, the main advantage of the Poisson-geometric model is that it provides an exact description of the system irrespective of the slot time value; however, a closed expression for the blocking probability is not provided and thus numerical evaluations need to be performed. As such, this model is suited for systems where the time slot is in the order magnitude of the average inter-arrival time. From this, we use the approximation given by the B-G model for simplicity reasons and we studied the system in which conditions such as good approximations are provided.

From these results, the network administrator can select the appropriate number of nodes, that is, the number of nodes that entail a certain CF delay and event reporting delay. Specifically, a high number of nodes produce high CF and event reporting times but a high resolution of the event is achieved. A lower number of nodes reduces such delays at the cost of having less information regarding the structural health of the building. In addition, the appropriate transmission probability can be selected for that specific number of nodes. Furthermore, the appropriate number of nodes also depends on the specific structure to monitor. For instance, sensor nodes can be placed in strategic junctions of the building as well as particular load-bearing walls where loads of the buildings are of major importance.

In future research work, we intend to use the exact Poisson-geometric model in scenarios where the time slot is in the same order of magnitude than the arrival and departure times. Also, the proposed model can be used in environments where all nodes in a given floor of the structure can communicate among each other and with the base station. In larger monitored areas or denser networks, the developed mathematical model is no longer accurate since a single-hop architecture cannot be used. As such, the clustering algorithm should consider hierarchical clustering and/or different routing protocols to address this issue. This is a current studied topic.

Footnotes

Handling Editor: Jia-Wei Xiang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: We thank the support to the Cuerpo Académico “Área de Comunicaciones” by PRODEP under the “Convocatoria de Fortalecimiento 2016”. Also,we thank the support of the Universidad Autónoma Metropolitana (UAM) through research projects EL001-130 and EL-001-17,and Instituto Politécnico Nacional (IPN).

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Performance analysis of a wireless sensor network with cognitive radio capabilities in structural health monitoring applications: A discrete model

Abstract

Keywords

Introduction

System operation

Secondary network: WSN

Primary network: cellular system

CR network

Erlang-B discretization

Bernoulli-geometric model

Poisson-geometric model

Teletraffic analysis for the CR network

CF phase

SS phase

Numerical results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References