I Introduction
Smart Utility Networks (SUN) have gained increased attention in recent years as key enablers of Smart Cities [1]. In such smart environments, the InternetofThings (IoT) will play a paramount role in connecting a massive number of devices such as smart meters, smart light bulbs, and smart appliances. Besides the existence of several potential network technologies, e.g., LoRaWAN, SigFox, and WiSUN, the efficient connection of massive numbers of devices is still a challenge if the reliability requirements are high [2].
The industry has backed two significant initiatives: LoRa Alliance and WiSUN Alliance. The LoRa Alliance – supported by Semtech, IBM, Cisco, Orange, among others – maintains the LoRaWAN specification [3]. LoRaWAN is a LowPower Wide Area Network (LPWAN) technology operating in the subGHz ISM band, using chirp spreadspectrum modulation, allowing increased signal robustness and range at low power consumption and low data rates [1]. LoRaWAN uses the LoRa physical (PHY) layer designed by Semtech and specifies the upper layer protocols to enable IoT deployments. The WiSUN Alliance – supported by Cisco, Analog Devices, Toshiba, and others – maintains the WiSUN specification [4, 5]. WiSUN is a Field Area Network (FAN) technology built upon the physical and link layers defined by the IEEE 802.15.4g standard. IEEE 802.15.4g operates in different bands of the ISM, including the same subGHz bands used by LoRaWAN, where it uses a Gaussian Frequency Shift Keying (GFSK) modulation over narrowband channels. Besides those bottom layers, WiSUN also defines network and applicationlevel services, including profiles for specific utility applications (e.g., energy, gas, water).
While utility service providers modernize their systems to use smart meters, deployments can use different communication technologies in the same geographical region, thus raising the question of how intertechnology interference affects network scalability. Coordination among transmissions in different technologies is unfeasible at the network or lower layers, so it is essential to understand the impact of these sources of interference. To achieve a realistic model, one should take into account that LPWAN devices in the ISM radio band are subject to interference generated by other networks sharing the same part of the spectrum. For instance, different authors report the analysis of the interaction of LoRa with other technologies considering IEEE 802.15.4g [6], SigFox [7, 8] and IEEE 802.11ah [8]. The results in those papers suggest that LoRa susceptibility to interference arriving from other technologies depends not only on the activity on those interfering networks but also on the configuration of the LoRa signal, mainly the spreading factor (SF). In this paper, we consider LoRaWAN as our target technology and model its performance in the presence of IEEE 802.15.4g interference sources in the subGHz ISM band (e.g., around MHz in Europe and MHz in USA/Brazil). Please note that the restriction of the model to IEEE 802.15.4g interference comes without loss of generality since one can extend it to other network technologies provided that appropriate isolation thresholds between the technologies are available.
While we focus on modeling how IEEE 802.15.4g affects LoRaWAN, we evolve from previous developments in [9] and [10] to approach the problem from an analytic perspective. We use the proposed models to derive two optimization algorithms that allow for the exploration of the configuration space of LoRaWAN in the presence of internal and external IEEE 802.15.4g interference. The proposed algorithms are tools for network planning, guiding the tradingoff between the number of nodes and coverage area. We validate our analytic finds using simulations configured according to experimental results on the interplay between these networks published in [6].
The contributions of this work include a closedform expression for the interSF LoRaWAN interference analytic model of [10]; the extension of the analytic models of [9] and [10] to consider external interference; the performance analysis of LoRaWAN considering the experimental results on intertechnology interference from [6]; and two novel algorithms to optimize LoRaWAN configuration, either in terms of network load or communication range, under reliability constraints.
The the remaining of this paper is organized as follows. Section II summarizes related work, and Section III briefly introduces the characteristics of LoRaWAN and IEEE 802.15.4g. Section IV introduces the proposed system model and the outage models. Section V presents the proposed algorithms. Section VI evaluates the models and algorithms. Section VII concludes the paper.
Ii Related Work
Georgiou and Raza [11] propose an analytic model of LoRaWAN
which considers both disconnection and collision probabilities in Rayleigh fading channels. They show that
LoRaWAN is sensitive to node density because it affects collision probability. In [9], we extend the work of [11] to exploit message replications and multiple receive antennas at the gateway. We show that message replication is an interesting option for lowdensity networks, while the performance gains from spatial diversity are significant in all cases. Mahmood et al. [10], as well as we [9] and Georgiou and Raza [11], use stochastic geometry and Poisson Point Processes (PPP) to derive analytic models of the coverage probability of LoRaWAN. Contrasting with [11] and [9], the work in [10] takes into account the effect of interference from the imperfect orthogonality of LoRa signals using different SFs. In this sense, Croce et al. [12] show that LoRa SF separation is not perfect, so LoRa nodes are subject to some interference generated by nodes using other SFs.Sørensen et al. [13] propose analytic models to investigate the performance of LoRaWAN uplink regarding latency, collision, and throughput. They also explore the network resource allocation for given QoS requirements and traffic models. Pop et al. [14] evaluate how LoRaWAN downlink impacts uplink. They consider the medium access control (MAC) layer and, through simulations, verify that if too many enddevices request delivery confirmation, the downlink becomes unstable and unable to deliver several acknowledgment packets, thus forcing nodes to retry, ultimately flooding the network.
Orfanidis et al. [6] report an experimental evaluation of the interference between IEEE 802.15.4g and LoRa by measuring the packet error ratio in different SINR scenarios inside an anechoic chamber. The measurements consider a single IEEE 802.15.4g interferer over one LoRa link. Their results show that lower SFs are more susceptible to interference. Although these measurements show interesting results, it is important to note that they consider a limited number of nodes, thus making it hard to extrapolate the conclusions. To the best of our knowledge, no other work has studied the susceptibility of LoRa to external IEEE 802.15.4g interference, and there is none published work that investigates this relationship in a networkscale scenario with several active links in both IEEE 802.15.4g and LoRaWAN.
Iii LPWAN Networks
LPWANs employ lowpower communication technologies that enable the connection of thousands of IoT devices. Most of these technologies work in frequencies below GHz and employ modulation techniques that enable link budgets of dB, resulting in robust communication channels with low energy consumption reaching distances in the order of kilometers [1]. Additionally, for reducing complexity and energy consumption, LPWAN technologies use MAC protocols that may decrease the efficiency of channel use. For instance, both LoRaWAN and SigFox transmit data using unslotted ALOHA, which is known to present high collision probability when a large number of stations are connected [15].
Iiia LoRaWAN
LoRa is a proprietary subGHz chirp spread spectrum modulation technique optimized for longrange applications and low power consumption at slow transmission rates [3]. LoRa modulation depends, basically, on three parameters [16]: bandwidth (), usually set to kHz or kHz for uplink and kHz for downlink; SF, which assumes values from 7 up to 12; and the forward error correction (FEC) rate, varying from to . From these parameters, one can extract the packet TimeonAir (ToA), the receiver sensitivity, and the required signal to noise ratio (SNR) for successful detection in the absence of interference. Table I features the relation among LoRa parameters, given a packet configuration, showing that ToA grows exponentially with SF, reducing the bit rate while increasing the receiver sensitivity, thus allowing higher coverage.







7  41.22  5.47  123  6  
8  72.19  3.12  126  9  
9  144.38  1.76  129  12  
10  247.81  0.98  132  15  
11  495.62  0.54  134.5  17.5  
12  991.23  0.29  137  20 
The implementation of LoRa PHY is agnostic of higher layers. LoRaWAN is the most widely used protocol stack for LoRa networks. It implements a star topology where enddevices (nodes) connect through a singlehop to one or more gateways, which in turn connect to a network server via an IP network. LoRaWAN MAC uses the unslotted ALOHA [15]. Moreover, a LoRa gateway can process up to nine channels in parallel, combining different subbands and SFs [1]. Besides, LoRa features the capture effect, making it possible to recover a LoRa signal when two or more signals are received simultaneously, in the same frequency and SF, provided that the desired signal is at least dB above interference [12].
IiiB IEEE 802.15.4g
IEEE 802.15.4g is an amendment to the IEEE 802.15.4 standard focusing on Smart Utility Networks (SUN) that has been playing an important role in smart grid deployments [18]. The standard specifies different modes able to operate in different frequency bands, including the SubGHz ISM bands used by LoRaWAN. Data rates range from 40 to 1000 kbps, depending on the modulation and bandwidth in use [19].
MultiRate FSK (MRFSK), with 2FSK or 4FSK, is the predominant modulation version in SUN applications due to its communication range [20]. In this configuration, the transceiver combines FSK modulation with Frequency Hopping Spread Spectrum (FHSS) to increase robustness [19]. The data rate may vary from 2.4 to 200 kbps, depending on region and frequency band. The mandatory configuration for all regions is 2FSK operating at kbps, which implies a channel spacing of 200 kHz. The receiver sensitivity for a PER of must be better than dBm or dBm at, respectively, 200 and 50 kbps, without FEC. If FEC is enabled, the sensitivity must drop to dBm and dBm for the same data rates. Transmit power depends on regional regulations, but must be at least dBm [5]. Most configurations use dBm transmit power and duty cycle [21].
IEEE 802.15.4g proposes extensions to the MAC mechanisms defined by IEEE 802.15.4e amendment [22] and makes extensive use of lowenergy modes. The IEEE 802.15.4g networks are expected to form multihop, mesh networks. Before sending data, the MAC layer performs either carrier sense or a simplified version of channel monitoring named Coordinated Sampled Listening (CSL) [23].
Iv System Model
Following the developments in [9] and [10], we use a set of Poisson Point Processes (PPP) [24]. Our model considers nodes deployed uniformly in a circular region around a gateway. Figure 1
illustrates a possible setup where SF increases according to the distance from the gateway. The vector
defines the limits of each SF ring. Note that is the maximum network communication range, i.e., the coverage radius. LoRaWAN devices transmit in the uplink at random using the ALOHA protocol and transmit once in a given period . Considering that all nodes run the same application, the network usage is different for each SF because of different data rates (see ToA in Table I). Figure 1 also shows the ToA difference graphically. Hence, we model the transmission probability of LoRaWAN devices as a vector , and , where is the ToA for SF of ring . Note that, for the sake of simplicity, we define the set to denote the SF rings and that each ring uses a respective SF in .Each LoRaWAN SF ring constitutes a separated PPP, denoted , making it possible to attribute different densities to each SF. has density in its area , where and are, respectively, the inner and outer radii of SF ring (from ), and is the spatial density of nodes in . Similarly, we model the IEEE 802.15.4g network as an additional PPP where nodes transmit with probability . The PPP has density in the area , where is the spatial density of nodes in . The average number of nodes in is , for LoRaWAN, and , for IEEE 802.15.4g. The average number of nodes in the LoRaWAN network is .
For instance, take the ring in Figure 1, defined by two circles of radii km and km. Nodes in this ring use . The ring area is km. If there are, on average, nodes in this ring, then its spatial density is nodes/km. Finally, if nodes transmit probability is (), the density of is .
In our analysis, is the Euclidean distance between the th node and the gateway, and denotes the distance of the node of interest. All nodes use the same transmit power to send signal , while both path loss and Rayleigh fading affect the received signals of LoRaWAN and IEEE 802.15.4g. Path loss follows , with wavelength , path loss exponent , while represents a device in either network. Finally, denotes the Rayleigh fading. Therefore, a LoRaWAN signal received at the gateway is the sum of the attenuated transmitted signal , interference, and noise,
(1) 
where
(2) 
accounts for intranetwork interference, considering both coSF and interSF interference by summing all other received signals from all SFs, and
(3) 
models external interference arising, in our case, from all active nodes in the IEEE 802.15.4g network. Finally,
is the additive white Gaussian noise (AWGN) with zero mean and variance
dBm, where dB is the receiver noise figure and kHz is the LoRa channel bandwidth. The remainder of this section uses this model to derive the outage probability of LoRaWAN links.Iva Outage Condition 1: Disconnection
Following [9], we consider the disconnection probability, which depends on the communication distance.
A node is not connected to the gateway if the SNR of the received signal is below the threshold that allows successful detection in the absence of interference.
Receiver sensitivity is different for each SF, what results in different SNR reception thresholds defined in
],
where denotes the SNR threshold vector, and denotes the th element of , i.e., the SNR threshold for SF ring .
Then, we model the connection probability as
(4) 
Since we assume Rayleigh fading, the instantaneous SNR is exponentially distributed
[25], and therefore(5) 
IvB Outage Condition 2: IntraNetwork Interference
Intranetwork interference arises from the activity of other devices in the same network. We follow [10] to model both coSF and interSF interference. To recover a packet, the signaltointerference ratio (SIR) at the gateway must be above a given threshold. The transceiver manufacturer informs that SFs are orthogonal and that the coSF SIR threshold is dB [17]. Goursaud et al. [26] propose theoretical SIR thresholds that match Semtech coSF value but show that different SFs are not completely orthogonal. However, Croce et al. [12] showed, experimentally, that the SIR thresholds for Semtech SX1272 LoRa transceiver are lower with regards to coSF interference (dB) but significantly higher with respect to (w.r.t.) interSF interference. In this paper, we assume the experimental SIR thresholds of [12]
where is the SIR threshold matrix, and denotes the element of at the th line and th column, i.e., the SIR threshold for the desired signal using SF and interference using SF. Note that relates to the coSF SIR while relates to interSF SIR. If one takes the SF column as an example, it shows how SF interference affects the LoRa signals. Desired signals using higher SF are more robust to interSF interference, allowing for the decoding of LoRa packets even if the interference power is much higher than the signal (e.g., dB higher if the signal uses SF).
Following this rationale, we first use the formulations in [10] to analyze the success probability considering the interference from only one different SF. Let
(6) 
where the interference received from nodes using SF is
(7) 
Since the desired node at uses SF, the success probability is
Since ,
(8) 
Note that (8) has the form of the Laplace Transform w.r.t. , where , . Thus, using (7) and applying the property of the sum of exponents,
Solving the expectation over yields
Finally, we solve the expectation over the PPP using the probability generating functional of the product over PPPs where , with as the density of , converting to polar coordinates, and replacing , obtaining
(9) 
As a contribution over [10], we solve the integral in (9), called here. Assuming a generic SIR threshold , if we rearrange and evaluate the indefinite integral to apply the binomial theorem we have
Since is continuous in , we interchange the summation and the integration and solve the integral, yielding
Finally, we put the summation in the form of the Gauss Hypergeometric function [27] and resort to the Pocchhamer function, , and to , which yields the final form of as
(10) 
Now, as in [10], we consider interference from all SFs, i.e.,
(11) 
Following an analysis similar to the case for one SF, we arrive in a combination of the received signals from all SFs weighted by the isolation thresholds , where denotes the SF ring of the desired node. Thus, the capture probability is
(12) 
IvC Outage Condition 3: External Interference
Orfanidis et al. [6] report the selectivity of LoRa receivers in the presence of IEEE 802.15.4g signals. We use Orfanidis et al. experimentally obtained isolation thresholds to analyze the SIR in the presence of external interference generated by an IEEE 802.15.4g network.
Here, we model the IEEE 802.15.4g network as PPP and consider [6]
],
where denotes the LoRa vs. IEEE 802.15.4g SIR threshold vector, and denotes the th element of , i.e., the SIR threshold for the desired signal in SF ring and interference from the IEEE 802.15.4g network.
The analysis of the LoRa capture probability in the presence of IEEE 802.15.4g interference is similar to the case for one SF (), but taking the vector and the IEEE 802.15.4g network parameters into account. For
(13) 
the capture probability with respect to external interference is
(14) 
IvD Coverage Probability
The coverage probability is the probability that the selected node is in coverage (not in outage), i.e., it can successfully communicate considering all the outage conditions defined above. The coverage probability of the desired node is thus
(15) 
V Optimum LoRaWAN Configuration
The expressions in Section IV determine the expected reliability of a single node located at a given distance from the gateway. However, what if one wants to plan the network deployment? In this section, we consider the use of the previous model to this end. We first consider the inversion of the expressions in the model to obtain network configurations for a targeted minimum average reliability. Afterward, we propose two algorithms that derive optimum network configurations supporting the desired minimum reliability requirement.
Va Guaranteeing the Reliability Target
To start our search for optimal LoRaWAN configurations we invert the previously described outage expressions defined in (5) and (12), so the network parameters can be extracted from them to achieve a minimum desired reliability level. Note that (14) does not depend on the LoRaWAN configuration. It is, however, taken into account in the optimization algorithm proposed in Sections VB and VC to consider external interference. One can assume that, in our network model, the nodes presenting the worst average reliability in each SF ring are those on the ring outer limit. It happens because the signals emitted by those nodes suffer greater pathloss and are, therefore, more susceptible to interference.
VA1 SF Ring Limits
As a first step, we find the maximum distance that ensures the required minimum average reliability level w.r.t. the connection probability . We denote this threshold by . We rewrite (5) to perform operations over the SNR threshold in and the outer SF ring limit ,
(16) 
and then it is straightforward to obtain
(17) 
Note that the radius of the overall coverage area is .
VA2 Ring Densities
Since (17) defines the network geometry, it is now possible to obtain the outage due to external interference observed by the nodes at each ring edge from . After that, we compute the maximum densities of the PPPs in that satisfy the given final reliability target , the previously assumed connection reliability target , and the external interference of each SF . Thus, following (15), making and , we have for each SF ring that and thus
(18) 
In (18), the function is independent of if the SF limits are predefined. Then, let and , yielding, for each SF ring ,
(19) 
If we name the vectors and the matrix , from (19) we derive a system of linear equations and solve it for the PPPs densities by making
(20) 
Note that is a square matrix, both and are row vectors of length , and all values in are positive real numbers. Considering the diagonal method to compute the determinant of , it is possible to observe that, due to , the values at are significantly higher than when , thus making the positive diagonal greater than the negative diagonal, incurring in a determinant that is virtually never zero.
VB Maximization of Communication Range
The expressions presented above allow us to obtain twelve network parameters: six communication range limits from (17), and six PPP densities from (20). Note that (20) depends on (17) because of . Combining both equations generates an incomplete linear system of six equations and twelve variables. In order to search for optimized feasible network configurations, we propose an algorithm that uses (17) and (20) in an iterative method, trying to extend the SF ring outer limits as much as possible, while preserving the targeted final reliability level and ensuring service to a minimum quantity of nodes (). The algorithm extends the SF ring outer limits by reducing . Similarly, increasing shortens these limits. The algorithm, iteratively guesses values for and then, after obtaining through (17), analyzes the maximum possible densities . As gets closer to , the capture probability increases and, with fixed and , higher is possible only with lower densities. It may lead to configurations breaking the restriction. Conversely, if is too close to , the outer limits of the SF rings will be shorter, leading to small coverage areas that are useless in practice. However, the proposed algorithm identifies feasible ranges for the network parameters, thus dealing with this parameter tradeoff.
Algorithm 1 employs a branchandbound technique to explore the network design space (i.e., possible values of ), seeking to maximize the width of each SF ring and, as a consequence, the network communication range (disk radius), while preserving the targeted minimum reliability and ensuring service to, at least, a given number of nodes (). The branchandbound technique fits well to our problem because it reduces the design space in half in each iteration, accelerating convergence. The inputs of the algorithm are the targeted reliability , the dutycycle vector , , and the density of IEEE 802.15.4g interfering nodes . The algorithm outputs a variable stating if the algorithm converged () or not (), the achieved number of nodes , and the vectors and containing, respectively, the SF limits and ring densities that ensure the desired reliability.
After initializing the variables (lines 14), the optimization loop starts and runs until the algorithm converges (line 26) or diverges (line 23). The optimization procedure “guesses” values for , trying to reduce it to enlarge the width of each SF ring, thus increasing the coverage area. Note that since depends on from (15), must be greater than ; otherwise, both and would have to be , what is impossible in practice. Thus, Algorithm 1 sets the initial search region for to . The guessed value for in each iteration is at the center of this region, as expressed in line 6. At each iteration, if the selected generates a configuration where the number of nodes is above , it is assumed that can be enhanced by decreasing , what allows for further decreasing in the next iteration. Conversely, if , is increased so that can be lower, allowing for more nodes in the network. This “binding” part of the algorithm is in lines 3034.
Provided the branchandbound technique guesses in line 6, the range limits for all SF are computed using (17) in line 7. In the following, the algorithm uses the newly computed vector to obtain vector and matrix (lines 1116), allowing for the computation of the PPPs densities in (line 17), using (20). Following that, the number of nodes fitting the generated configuration is computed by first obtaining the area of each SF ring in lines 1819 as . The algorithm converges and stops when the difference in the radius of the overall coverage area between two consecutive iterations is less than (line 20) and (line 21), where defines the precision of . If the variation of , i.e., , is too small and the algorithm did not achieve yet, the algorithm keeps trying to converge until the variation in the guessed is below a threshold (line 22), in which case the algorithm stops and announces a divergence. After evaluating the proposed algorithm for a set of test scenarios, we concluded that good values for the stopping thresholds are and .
Algorithm 1 always converges if there are feasible solutions to the problem. If the requirements of minimum network density ( and ), targeted reliability (), or both, are too high, however, the algorithm may take too long to converge. Hence, we stop the algorithm when the changes in get too small (line 22). Note that this guarantees the algorithm to stop because decreases every iteration. It is important to note, however, that achieving higher network density is always possible by reducing the reliability requirement . Moreover, highly demanding scenarios without feasible solutions or with lengthy convergence are not typical in LoRaWAN, since the technology has been conceived to support massive rather than critical IoT applications.
VC Maximization of Number of Nodes
In this section, we consider the case of maximizing the total number of nodes, given restrictions of minimum coverage radius () and average reliability (). In order to do that, we use a more straightforward approach than in Algorithm 1. The problem of maximizing the number of nodes is equivalent to the problem of minimizing . Thus it is straightforward to conclude, from (15), that we should maximize because higher allows for lower . Since we assume that the worst cases are at the edge of the SFs and we have a restriction on the coverage radius, the maximum possible is that yielding . Thus, from (5), we conclude that . Assuming the same for all SFs, we use (17) to compute and obtain the geometry of the network.
Once we have obtained , we get the maximum allowable densities ensuring through (20). Algorithm 2 shows the proposed procedure. Line 1 uses (5) to compute the maximum satisfying . Line 2 uses the computed to obtain the geometry of the network . The loop in lines 510 computes matrix and vector , so the maximum device density vector can be computed in line 11. Finally, after computing the areas of the rings and storing them in vector (line 12), we obtain the maximum number of nodes in line 13.
Note that Algorithm 2 is not iterative since there is no loop searching for the optimum solution. It merely describes how to use the proposed models to determine the optimum LoRaWAN configuration considering the restrictions. The approach produces unfeasible configurations if the restrictions are too strict. Thus, lines 1418 check whether the method generated nonnegative densities for all SFs in order to assess whether the results are feasible or not.
Vi Numerical Results
This section evaluates the proposed model and algorithms. In the figures, lines represent theoretical probabilities while marks are the average of Monte Carlo simulations with random scenarios. Results consider the expressions for success probabilities , , , and as defined by (5), (12), (14), and (15), respectively. Moreover, we assume dB, , m, m/s (speed of light), MHz for both LoRaWAN and IEEE 802.15.4g. LoRaWAN channel bandwidth is kHz, and IEEE 802.15.4g channel bandwidth is kHz. We also assume that nodes in LoRaWAN and IEEE 802.15.4g transmit with dBm. These parameters configure typical suburban scenarios following European regulations.
Concerning IEEE 802.15.4g interference, we evaluate the algorithms considering three scenarios. In real deployments, the designer of a LoRaWAN network may not know the operational parameters of the interfering IEEE 802.15.4g network. Thus, in a practical situation, the designer should assume worstcase configurations for the external network.
Via Model Validation
Figure 2 aims to validate the presented models by showing the success rates , , , and as a function of the distance from the gateway. The scenario considers an average number of nodes , transmitting with duty cycle % in a circular area around the gateway with radius m. The IEEE 802.15.4g network generating external interference has nodes transmitting with duty cycle %, also in a circular area with radius m. As can be seen, all theoretical expressions (lines) match the simulation results (marks). One can observe in that a relatively light interference from IEEE 802.15.4g () has little impact in lower SFs due to the smaller ToA and reduced probability of concurrent transmissions. Higher SFs, on the other hand, have higher ToA and thus suffer more from this external interference.
ViB Algorithm 1: Maximization of
Now we evaluate Algorithm 1 of Section VB. These results use the same network parameters employed to validate the network model. Figure 3 presents a series of graphs for varying optimization objectives. Plots in the same row consider the same reliability target , while plots in the same column use the same packet generation interval , expressed in minutes. Each graph shows three curves, each one considering a different amount of IEEE 802.15.4g interference, which varies by changing the number of IEEE 802.15.4g nodes (), always with duty cycle . Each optimization point considers different values, evaluated for every m.
The first conclusion when comparing the curves in all plots is that different IEEE 802.15.4g interference leads to shorter communication ranges when following our proposed optimization procedure. That makes sense since shorter distances feature smaller path loss, making signals less susceptible to external interference. It is also possible to observe that less stringent reliability targets lead to larger coverage areas. Again, that makes sense since smaller yields smaller , which in turn enables longer communication range.
Also in Figure 3, plot (a) shows the more rigorous scenario; the configuration allowing the required reliability is only practical for nodes, with a radius varying from to meters, depending on and the external interference. The coverage radius with either converged to unpractical distances of less than meters or diverged, meaning that it was not possible to place such a number of LoRaWAN nodes with packet generation interval of minutes while ensuring minimum reliability of .
For , more feasible and useful scenarios exist if one reduces network usage. Plot 3(b) shows that configurations with up to nodes are possible if the packet transmission interval is minutes. For with minutes, it is possible to find reasonably good network configurations up to . However, shrinks the communication range to unpractical distances.
Figure 4 illustrates the behavior of Algorithm 1 and the network performance when taking the circled case in Figure 3(a) as an example. Figure 4(a) shows the convergence of and for this scenario. We see that in the last iteration converges to , thus “stretching” the network range as much as possible.
Table II shows numerical results of the same scenario in two columns: “All sources”, with the results for our complete model, and “IntraSF only”, disregarding both interSF and external interference sources. We get the results in the “IntraSF only” column using the same models, but setting in , and for and otherwise. When considering all sources of interference, as expected, the fact that ToA impacts the duty cycle induces the optimization procedure to allocate most of the nodes with lower SF. That happens because signal attenuation increases with distance, making more distant nodes more vulnerable to both internal and external interference. Recall that a shorter ToA reduces the collision probability. Moreover, longer ToA generates more internal interference to other SFs. In some cases, higher SFs may not be used to ensure minimum reliability. However, note that (17), (20) and Algorithm 1 can be extended to change the restriction to represent a vector with the minimum number of nodes using each SF. One can achieve that by revisiting the computation of the densities in (20) to consider such a minimum number of nodes when computing the spatial density. Since doing that will possibly result in more nodes using higher SFs, it is expected that fewer nodes use lower SF, resulting in a smaller total number of nodes, as well as a shorter network radius, since the algorithm will converge to a higher to compensate the increased . When disregarding interSF and external interference, we observe that higher SFs are highly affected by interSF and external interference, especially due to their extended ToA. In particular, we observe that interference, rather than path loss, is the main factor for which our method disfavors the use of higher SF. Moreover, it is clear that interference considerably affects coverage.
Interference:  All sources  IntraSF only  

Scenario  SF  Range  Range  
(m)  (m)  
3(a)  7  261.6  162.8  300.1  370.0  124.6  300.03 
8  336.4  67.5  475.7  87.1  
9  432.4  32.5  611.6  43.5  
10  555.9  21.8  786.2  25.3  
11  685.4  10.6  969.3  12.9  
12  845.0  4.7  1195.1  6.4 
Finally, Figure 4(b) shows the success probabilities of the example scenario, where the optimized configurations are w.r.t. the minimum average reliability target for all distances from the gateway. As expected, the success probability approaches the desired minimum at the edge of each SF. We can see that collisions () are kept almost constant or increase slightly with SF. That happens because the algorithm reduces the number of nodes using each SF to keep in pace with and , to ensure the minimum .
ViC Algorithm 2: Maximization of
Now we evaluate Algorithm 2 of Section VC. The plots in Figure 5 show the results for different scenarios of required minimum reliability () and message generation period (). For all plots, the xaxis represents the input to the algorithm, while the yaxis shows the achieved maximized number of nodes. In each plot, the xaxis grows up to the value for which the requirements yield practical results.
In Figure 5, if we analyze each row of plots independently, we see that the maximum number of nodes is a linear function of the transmission period . For instance, considering m and , in plots 5(a) and 5(b) are, respectively, and , i.e., doubles when doubles. That is expected since these variations ensure the same network load in all scenarios. We can also observe that, in all plots, increased external interference reduces both the number of supported nodes and the achievable coverage radius.
Interference:  All sources  IntraSF only  

Scenario  SF  Range  Range  
(m)  (m)  
5(a)  7  278.7  149.2  274.9  278.7  211.1  508.2 
8  358.3  61.8  358.3  147.6  
9  460.6  29.9  460.6  73.7  
10  592.1  20.2  592.1  42.9  
11  730.0  9.5  730.0  21.8  
12  900.0  4.0  900.0  10.9 
Table III shows the achieved geometry and number of nodes of the marked scenario of Figure 5(a). Again, the “All sources” column presents the results of our complete model, while the “IntraSF only” column disregards interSF and external interference. Since the method assumes that the maximum number of nodes is achieved with the shortest possible distances, the maximum range of a node using SF has to be (m for this case). As for Algorithm 1, Algorithm 2 also favors lower SFs. Moreover, Table III shows that the maximum number of nodes almost doubles when disregarding interSF and external interference, emphasizing the importance of taking such impairments into account to avoid overestimating the network performance. Finally, Figure 6 shows the success probabilities for the example scenario, which approach at the SF edges, but stay above the required minimum for all distances smaller than .
Vii Conclusion
This paper presents two algorithms to optimize the configuration of LoRaWAN under imperfect SF orthogonality and IEEE 802.15.4g interference. We use models of LoRaWAN networks to derive success probabilities of packet delivery under internal and external (IEEE 802.15.4g) interference. The presented algorithms search for optimum LoRaWAN configurations given restrictions of minimum network density or coverage radius, meeting a target minimum reliability level. The analytic results are validated using simulations.
Regarding IEEE 802.15.4g interference over LoRaWAN, although higher SF should be more robust to this type of interference, they tend to suffer more from that impairment because their increased ToA makes it more likely that transmissions overlap with IEEE 802.15.4g activity. Finally, regarding the proposed algorithms, they can provide a tool for exploring tradeoffs between network load and coverage range by showing the feasible region of LoRaWAN network configurations.
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