2D Time-frequency interference modelling using stochastic geometry for performance evaluation in Low-Power Wide-Area Networks

2D Time-frequency interference modelling using stochastic geometry for performance evaluation in Low-Power Wide-Area Networks


In wireless networks, interferences between transmissions are modelled either in time or frequency domain. In this article, we jointly analyze interferences in the time-frequency domain using a stochastic geometry model assuming the total time-frequency resources to be a two-dimensional plane and transmissions from Internet of Things (IoT) devices time-frequency patterns on this plane. To evaluate the interference, we quantify the overlap between the information packets: provided that the overlap is not too strong, the packets are not necessarily lost due to capture effect. This flexible model can be used for multiple medium access scenarios and is especially adapted to the random time-frequency access schemes used in Low-Power Wide-Area Networks (LPWANs). By characterizing the outage probability and throughput, our approach permits to evaluate the performance of two representative LPWA technologies Sigfox® and LoRaWAN®.


2D time-frequency interference; time-frequency random access; capture effect; stochastic geometry; IoT; LPWANs.

1 Introduction

Trading bit rates for better link budgets, LPWANs provide long range wireless connectivity to IoT devices [1, 2, 3]. Such networks provide a promising alternative to traditional cellular or multi-hop networks and are indeed envisioned to provide nationwide connectivity over industrial, scientific and medical (ISM) bands to battery-powered IoT devices that transmit little amount of data over long periods of time, e.g., water & gas meters. Thanks to the long range, the IoT devices can communicate directly with the base stations in a star topology.

Random access schemes such as Aloha are commonly used in LPWANs in which multiple devices access frequency resources with neither carrier sensing nor contention mechanisms [4, 5]. This reduces the communication overhead and the packet air time, but it increases the risk of collisions between packets when they overlap in time domain. Multiple works have been dedicated to the interferences modelling in time domain [6, 7, 8, 9]. In [8], the product of power and overlapping time duration between a transmission of interest and an interfering transmission is used to represent the quantity of interfering, then the sum is taken over multiple interfering transmissions to give the total interference.

Note that interference modelling of transmission overlapping in frequency domain is also well studied in the partially overlapped channels (POC) scenarios [10, 11, 12], which are commonly used for networks such as IEEE 802.11. The interference factor in the case of POC is evaluated as the accumulated energy in overlapped frequency domain  [10]. It has been proven that the use of POC can indeed improve the network throughput in comparison to common orthogonal channelization schemes [12, 10].

Our work differs from the aforementioned interference models in that it’s the first, to the best of our knowledge, to consider the joint overlapping in both time and frequency domains. Our model based on stochastic geometry is a high-level flexible one which can be adapted to multiple scenarios.

In section 2, existing works on LPWANs performance evaluation are introduced. In section 3, our interference modelling approach is described and expressions of , outage probability and network throughput are given. Then in section 4, we give the results on probabilistic evaluation of overlapping. Finally in section 5, the developed model is used to study performances of two different LPWA technologies, Sigfox® and LoRaWAN®. Section 6 concludes the article and introduces some research perspectives.

2 Related Work

Multiple works exist for the performance evaluation of LPWANs [13, 14, 15, 16]. Most of these works use Poisson processes to model the packet arrival, which we believe is not the most adapted for periodic packet sending scenarios in LPWANs. For example, a device reporting on a daily basis would not send more than one message per day. However, as Poisson models the intensity of packet arrival, an intensity of one message per day represents in fact the mean value, i.e., one message on average per day, which is not quite the case described. In [13], multiple annuli LoRaWAN® cell structure is well modelled and illustrated with a few applicative scenarios. This structure is considered in our article but channel effects and capture effect are added to our model thus making it more complete and realistic. In [14], the performances of a random Frequency Division Multiple Access (random FDMA) scenario are studied in the pure Aloha case, but the capture effect with little overlap between packets is not considered. In [16], the performances in terms of packet delivery ratio and throughput of LoRaWAN® and Sigfox® are simulated. However, the simulation process and the numerous network parameters are not exposed enough thus lacking of transparency and possibility of reuse. In [15], some interesting insights on the limits of LoRaWAN® are given, but again the model is based on Poisson process and there is no possible extension to account for the capture effect. In this article, in order to give the limit of the performance, we study the outage probability and throughput of LoRaWAN® and Sigfox® when every node is transmitting as frequently as possible, according to either the ISM band duty cycle constraints or technology-related constraints, which result in message sending periods in the order of 1 to 10 minutes. This scenario of the saturation throughput could be that of packet and object tracking systems [17]. Other less frequent IoT scenarios such as water & gas metering can be evaluated using our model thanks to its flexibility.

3 The “cards tossing” model

3.1 Assumptions

devices share limited time-frequency resources denoted by to send packets to a single base station, with . is the bandwidth and the message sending period. is the user of interest. the packet sent by user . We make the following assumptions:

  1. Messages from all the senders have the same rectangle-shaped time-frequency support, since they are limited in time duration, denoted by and bandwidth occupancy, denoted by . Then takes the form where stands for the indicator function of set and denotes the time-frequency support of node (for the sake of simplicity, we use the same notation for the sender itself and the time-frequency support of its message); is the initial time of transmission and the lowest frequency of the packet. , and (see figure 1).

  2. There is no cooperation between the devices, i.e., random access considered. The couples are thus independent. They are also assumed to be uniformly distributed, as it is probably optimal in terms of dispersing the packets and avoiding collisions.

  3. Given support , the time-frequency energy of the information packet is uniformly distributed over , i.e., , where is the energy density.

  4. The channel is affected by an additive time-frequency white noise of energy density , i.e., .

Assumption 3 is the ideal and most efficient way of using time-frequency resources [10]. In practice, transmit spectrum mask is usually applied to specify the upper limit of power permissible and attenuate the signal outside the mask. An attenuation of to is observed in real-world scenarios [10], so this assumption can be considered as realistic. Notice that the common Aloha scenario is encompassed in this formalism by fixing . In this case, but remains random.

3.2 expression

As mentioned in the introduction, interference can be modelled as the sum of accumulated energy in time-frequency domain, which is calculated as the sum of energy coming from different interfering transmissions. The is defined as the ratio between the energy of the message of interest, and the interference plus noise, i.e.,


where is the surface between the transmission of interest and an interfering one ( is the surface measure). We can normalize with respect to the surface of the time-frequency support of transmissions, i.e., . Thus, the can be recast as


The overlapping phenomenon between packets is similar to the game of players tossing cards onto a table and trying to recognize their own cards afterwards (see figure 1). When there are too many players, the probability of overlapping will increase to the extent that it’s highly probable to be unable to recognize a card. In the next subsection, we illustrate in two different scenarios, the interest of our “cards tossing” model in the derivation of the outage probability and throughput of wireless systems in function of the number of devices .

Figure 1: Illustration of the “cards tossing” game, where each rectangle represents the information packet . The rectangle in bold represents the transmission of interest, i.e., the packet , while the gray areas depict the sub regions in collision (the darker the area, the larger the number of “cards” covering the sub region). The light blue area is defined as the non border area , denoted by . The light green defined as the border area . These two areas constitute .

3.3 Outage and throughput model

With multipath fading and path loss

The first scenario is when the packets from devices suffer from path loss and fading. can be thus expressed as .

  • models the transmission energy density, which is supposed to be identical for all devices.

  • models the distance-dependent attenuation, e.g., path loss, where is the Euclidean distance between device and the base station. We choose the following non singular model [18] expressed as , where is a critical distance to avoid taking infinity when tends to 0. Here we fix it to . is a constant modelling system-level losses and gains which is fixed to 1 in our study. is the path loss exponent assumed to be greater than 2.

  • is a random variable modelling small scale, large-scale or composite distance non dependent fading. We suppose that results from a rayleigh multipath fading which gives exponential cumulative distribution function (cdf), i.e., (with denoting the cdf and ). The mean value is fixed to 1 in our study. Note that other forms of fading can be considered with our paradigm.

The can be thus recast as follows,


In order to study the distance distribution of devices in the cell, we define as the distance to the base station of the most distant devices, in the sense that their transmissions barely satisfy the target , denoted by in the presence of only path loss, i.e., . This gives us .

Let us denote the probability measure. Suppose that every packet is repeated times, and the repetitions are independent. The outage probability can be defined as , which is further expressed as follows,


Note that repetition mechanism is a commonly used scheme in LPWANs to trade efficiency for robustness of transmission.

Naturally, depends on the position of the device of interest. Distant devices with larger suffer from greater . Suppose that the devices are uniformly distributed in the cell of the base station, which is defined as in the shape of an annulus formed with smaller radius and larger radius . The probability density function (pdf) of can be expressed as ( is used to denote the pdf). The notation is omitted for the sake of simplicity.

Global outage probability is defined as the outage probability averaged over , i.e.,


The effective throughput is defined as the average number of non repetitive packets received per unit time and is denote by , which can be expressed as follows,


Recall that is the number of devices, and the message sending period. In the case of pure Aloha, i.e., packets considered lost when they collide in time or frequency domain, i.e., . Assuming and independent, our outage probability can be recast as,


One can observe that (7) is greater than (4), as (4) includes the capture effect, i.e., certain packets not considered lost even in case of collision. and can be calculated in the similar way. By definition, and .

With perfect power control

We consider another scenario in which the packets of different devices are supposed to arrive at the base station with identical energy density, i.e., , thanks to a certain power control mechanism. The can be recast in this case as,


where . The outage probability can be recast as,


In this case the non fairness between devices in terms of distance to the base station is resolved. does not depend on any more, but only on , the target and , the energy density that results from the power control. The average throughput can be recast as . The quantities to be simulated are listed in table 2.

In both scenarios, we should first study the probabilistic distributions of and . In the next section, we derive the probabilistic evaluations of and . In the case of multipath fading and path loss, the exact distribution of remains difficult to evaluate even with the distribution of derived and , and assumed to be independent random variables. We use Monte Carlo method to evaluate it.

4 Probabilistic Evaluations

The results of and are different in 1D case and 2D case. We first give the results of in the easier 1D case in section 4.1, i.e., so interference happens only when there is overlap in time domain. Physical layer technologies such as spreading spectrum fall into this case. Then the results of in the more complicated 2D case are given in section 4.2, where overlapping can happen in both time and frequency domains, random FDMA approach belongs to this case. The results on are given in section 4.3.

We denote by (resp. ) the pdf of (resp.), by (resp. ) the cumulative distribution function (cdf) of (resp. ).

The overlapped surface between two packets is determined by their relative position in . Recall that and are defined over and . We can thus define and as the normalized absolute time and frequency difference between emission and , see figure 2. and are defined over and respectively. From the assumption 2, also have identical distributions. For the sake of brevity, we will omit index in the expressions, i.e., pdf of (resp. and ) is denoted by (resp. and ) , and the (cdf) denoted by (resp. and ).

Figure 2: When an emission collides with , the overlapped surface, represented in gray, is .

4.1 1D “cards tossing” game

In the 1D game, so that . and become deterministic and independent of . We have


One can easily deduce that


which gives and the probability of collision is given by .

In the case where are assumed uniformly distributed over , the cdf of , can be derived as follows,


where . can be obtained by deriving for , and .

4.2 2D “cards tossing” game

Recall that and are assumed independent and uniformly distributed over and . Denote by and by .

A simple look at the geometrical configuration plotted in figure 2 allows to express the normalized surface as


Thus, for , we have , we immediately get,


where the bound of the first integral is due to the fact that when , , the inner integral is zero. The probability of collision is . When and are independent and uniformly distributed over and , some long algebra leads to,


Similar procedures as in 1D game should be taken to find .

4.3 Probabilistic evaluation of

Let’s first consider the probabilistic evaluation of in the 2D case. Notice that can be divided into two areas i.e., the non border area denoted by and the border area denoted by . We have , and . See figure 1. For the sake of brevity, we define the event that falls into the non border area also as . The event that falls into the border area as .

In fact because a packet in has greater chance to be corrupted by an interfering one as it can come from all directions. A packet in cannot be interfered from certain positions of out of border, resulting in a smaller probability of being corrupted. In section 4.1 and 4.2, we could have separated the derivation in and and obtained the same results. For , instead of evaluating it separately in and , we give the approximation as follows,


where stands for convolution, and the times convolution. When (This hypothesis is realistic in the case where and , which is verified in most LPWANs scenarios [1, 2, 3, 13, 19, 15, 16]), the first approximation is obviously valid. The second approximation comes from , and it’s the same with . We use and obtained in the case of independent and uniform distribution in section 4.1 and 4.2 to evaluate (16).

The reasoning and the evaluation of in the 1D case is similar and thus omitted.

5 Application

Let us now illustrate how our model can be used to evaluate the performance of two LPWA technologies.

5.1 Sigfox®

2D “cards tossing” parameters

First, we consider Sigfox®, an LPWA technology based on Ultra Narrow Band (UNB) [14]. The packet takes only a bandwidth around 100\siHz. In doing so, the noise power is greatly reduced and the transmission range is thus increased. In the physical layer, binary phase-shift keying (BPSK) is used. In the medium access control (MAC) layer, random FDMA scheme is adopted [19], i.e., due to transmitter oscillator’s jitter, it’s not possible to channelize, so a packet is transmitted at a randomly chosen frequency in the available frequency band of 40\sikHz.

Sigfox® also limits the number of messages per node to 140 messages per day, which equals to a message around every 617\sis [16]. is fixed to , i.e., devices transmit as frequently as possible. The maximal allowed payload size per packet is 12 bytes. With the preamble and cyclic redundancy check (CRC) fields, the transmission duration of a packet is around 1.76\si[20]. Note that this and satisfy the European Telecommunications Standards Institute (ETSI) requirement of 1% duty cycle constraints in the 868\siMHz band [21]. The “cards tossing” game of Sigfox® falls into the 2D case described in 4.2 and its parameters, i.e., , , and are given in table 1.

Outage and throughput model for Sigfox®

For now, there is no report of any power control mechanism in Sigfox®, so the model introduced in 3.3.1 is chosen. Parameters , , , , are also listed in table 1. The derivation of these parameters is as follows.

The maximum transmission power in 868\siMHz is fixed to 14\sidBm i.e.,  [21]. Thanks to UNB, Sigfox® benefits from a reduced noise floor around -154\sidBm, i.e., . This gives us a link budget around 168\sidB. Let’s consider a reception threshold of 8\sidB, a shadow fading margin of 10\sidB as well as a penetration loss of around 15\sidB for urban environment. This gives us a target around 33\sidB, i.e., . Finally let’s consider a path loss exponent of 3.6 for urban environment. All of these parameters give us a of 5.2\sikm for urban scenario.

Sigfox® LoRa®
Outage and throughput model With path loss Perfect power
and fading control
“cards tossing” model 2D 1D
Transmission power 14 See section 5.2.3
Noise floor (\sidBm) -154 -117
Target (\sidB) 33 See table 3
Path loss exponent 3.6 3.6
Range (\sikm) 5.2 See table 3
Cell form Single annulus Multiple annuli
Payload size (bytes) 12 See table 3
Application bit rate (\sibits/s) 54.5 See table 3
F(\sikHz) 40 125
100\siHz 125\sikHz
T(\sis) 617 See table 3
(\sis) 1.76 See table 3
Number of channels 1 3
Table 1: Table of Notations


Taking all the parameters of Sigfox®, formulae  (4)–(5)–(6)–(7) expressing the first 6 quantities listed in 2 (For the definition of , see section 5.2) are simulated and the results are given in figures 3 and 4.

Outage probability in function of and
Global outage probability
averaged over or
Average effective throughput in function of
Outage probability in pure Aloha scenario
in function of and
Global outage probability in pure
Aloha scenario averaged over or
Average effective throughput in function of
in pure Aloha scenario
Outage probability in function of and
Average effective throughput of a certain
in function of
Table 2: Table of simulated quantities

Figure 3: Outage probability in function of and . The solid line represents the pure Aloha case, the dashed line the case with capture effect. The five series of curves in each sub figure represent from top to bottom the case where .

Figure 4: Global outage probability and average effective throughput in one hour for Sigfox® scenario. The five colors in each sub figure from top to bottom represent . Solid line represents the pure Aloha case, dashed line the case with capture effect.

Figure 3 shows that capture effect represented by the difference between the solid and dashed lines decreases with the distance , because naturally the devices nearer to the base station have more chances to benefit from the capture effect. Repetitions do reduce the outage probability.

Figure 4 shows that is not greatly reduced in the capture case than in the pure Aloha case; the improvement in increases with , as more collisions happen with higher , thus amplifying the capture effect, but the high collision regime is not the optimal zone for the low-power devices to function. Device density increases with distance and further devices barely benefit from the capture effect, so devices at the cell edge are probably the bottleneck of the network performance, i.e., it’s them that stops the network performance from getting better. One can also observe that repetitions reduce the global outage probability but also result in lower effective throughput because of the introduced redundancy. Our abacuses permits to find the optimal in function of the target outage probability, , throughput and energy cost.

5.2 LoRaWAN®

1D “cards tossing” parameters

LoRaWAN® is another LPWA technology based on spectrum spreading [22]. The spreading factor is denoted by and can vary from 6 to 12. Every packets are spread in the available bandwidth , i.e., . In Europe 3 default channels are used, each with a bandwidth of  [22]. Different result in different bit rate. The smaller the , the higher the bit rate. Different payload sizes are also specified for different . can thus be calculated, the detail can be found in [13, 23]. is fixed to according to the ETSI duty cycle constraint of  [21] in the 868\siMHz band. Transmissions from different are considered orthogonal and do not interfere with each other [13, 16, 23], so the “cards tossing” game of LoRaWAN® can be seen as seven orthogonal and parallel 1D games described in section 4.1, each having three orthogonal channels of bandwidth , different and . The corresponding game parameters are listed in table 1 and 3  [13, 23].

Multiple annuli cell structure

The greater the , the lower the sensitivity of transmission associated  [23] . This results in smaller required for greater . Transmission with greater can thus reach further. is defined as the maximum distance that a transmission with a certain can barely reach, with only path loss considered, i.e., , where takes the maxmimum allowed power  [21]. The noise power is around . Let’s add a shadow margin of 10 \sidB as well as a penetration loss of around 15 \sidB, to give us the for different , see table 3. The path loss exponent is fixed to the same 3.6 as in Sigfox® scenario. With the maximal allowed transmission power of 14 \sidBm, the communication ranges in terms of path loss for different are thus calculated and listed in table 3. These are the ranges with which the reception barely satisfies , in presence of only path loss and without considering fading and interferences, i.e., .

Further devices should use greater to simply reach out to the base station, while nearer devices can benefit from higher bit rate of smaller . Let’s consider a ideally pre-configured network where all the devices located in the annulus defined by and take spreading factor (For , it’s the annulus between and ) so that they make use of the smallest possible and thus the highest possible bit rate while guaranteeing the communication range at the same time [13, 15]. The probability of a node falling into a certain annulus denoted by is proportional to its surface [15]. The number of devices taking a certain is thus just . , and are listed in table 3.

LoRaWAN® network support over-the-air activation of the node which requires the node to open 2 successive downlink windows after a uplink transmission, in order to receive the MAC layer commands from the network [22]. Also, LoRaWAN® network infrastructure can manage the and date rate by means of an ADR (Adaptive Data Rate) scheme, which also necessitates the node to listen to gateway downlink transmissions. Note that the downlink and uplink in LoRaWAN® share the same channels, so collision phenomenon may be aggravated by the use of downlink. The pre-configured network that we consider is the scenario without nodes listening to gateway, i.e., there is only uplink transmissions. Each node is set to an appropriate according to its distance from the gateway.

SF Sensitivity Range Payload Bit rate
(\sidBm) (\sidB) (\sikm) (bytes) (\sis) (\siKb/s) (\sis)
6 -121 21 1.13 0.13 242 0.233 8.309 23.3
7 -124 18 1.37 0.06 242 0.400 4.840 40.0
8 -127 15 1.67 0.09 242 0.707 2.738 70.7
9 -130 12 2.02 0.13 115 0.677 1.359 67.7
10 -133 9 2.45 0.19 51 0.698 0.585 69.8
11 -135 7 2.78 0.17 51 1.561 0.261 156.1
12 -137 5 3.16 0.23 51 2.793 0.146 279.3
Table 3: Parameters of LoRaWAN®

Outage and throughput model with power control scheme for LoRaWAN®

To improve the fairness of devices located in the same annulus and having the same , we consider the following ideal power allocation strategy : Allocate to the devices with distance to make sure they get covered; Make sure that the reception power density attenuated by path loss of all devices in the same annulus, denoted by , is identical and equal to that of devices with distance , i.e., . Fading effect is neglected in this case. This setting falls into the perfect power control paradigm introduced in sub section 3.3.2. according to the section 5.2.2. By shrinking all the annuli, our power allocation scheme can in fact result in non zero . With cdf of already given in section 4, there is no problem in evaluating this adaptation.

In our scenario, the non fairness between devices with the same but different distance is removed, i.e., doesn’t depend on any more, but non fairness exists between different as with greater have greater device number and thus greater expressed as . should be recast as follows,


where the factor three comes from the three available channels. The outage probability averaged over is expressed as . Quantities to be simulated are listed in table 2. Note that in the scenario considered, coincides with , and with as there is no tolerance of overlapping.


The simulation results in the LoRaWAN® case are given in figures 5 and 6. Figure 5 shows the non fairness in terms of outage probability between different , which is directly related to , which again is dictated by non uniformity of device density in function with .

Figure 5: Outage probability of LoRaWAN® in function of and . From top to bottom, the five curves in each sub figure represent . Sub figure on the left is the case where , on the right .

Figure 6: The figure in the top represents average effective throughput in one hour. The five different colors from top to bottom represent the case . The figure in the middle represents average effective throughput in one hour of different , with . The numbers on the curves stand for the . The figure in the bottom represents the global outage probability. The colors have the same meaning as described earlier.

Several observations can be made from figure 6. First, repetition mechanism reduces overall outage probability but also the average effective throughput as redundancy is introduced. It increases energy cost as well. Second, the differences between are multiple, dictated by and . determines , while determines the speed of message sending. For example, is the fastest of all , but as , the curve of reaches its maximum much earlier than , which limits its performance. has the worst performance as it has the longest and biggest , which again confirms that the devices at the cell edge are probably the bottleneck of the network performance.

At last, if we try to compare Sigfox® and LoRaWAN®, we can see that even though Sigfox® can support more devices but in terms of throughput, it’s of the same order as LoRaWAN®. Note that the payload sizes in LoRaWAN® are much more important than that of Sigfox® (see table 1 and 3). It seems that Sigfox® is more suitable for applications with a lot of devices having smaller traffic, while LoRaWAN® can support applications with more important traffic but less devices.

6 Conclusion and perspectives

In this article, we provide a high-level flexible model whose interest is illustrated by the performance evaluation of two LPWA technologies. To the best of our knowledge, this is the first model which considers joint time-frequency interference. Note that our paradigm can be adapted to other systems to evaluate the relationship between number of devices, repetition times, outage probability, throughput and energy cost. Capture effect is also taken into account in our model so further questions such as how to amplify it intelligently can be investigated in the future. We believe that our model provides a useful dimensioning tool for the future IoT scenarios.

Our model can be completed in the mathematical level. First, when the hypotheses and are not satisfied, algorithmic approach seems to be more adapted due to the difficulty in the probability evaluation on the border area. Second, the proportion between and has an influence on the probability distributions of and , thus comes the question of the best strategy of proportioning the time duration and frequency occupancy of the information packet, in order to minimize the overlapping. At last, it’s possible to formulate a more general problem of finding the best strategy to use a 2D time-frequency resource, always in terms of minimization of overlapping phenomenon. We have the options between orthogonal division of the frequency band, division into multiple partially overlapping bands (POC), and random FDMA if we go to the extreme.

In both the LPWANs scenarios considered, cell edge devices seem to be the bottleneck of the global network performance. A possible solution is the densification of the infrastructure, knowing that IoT devices can communicate with multiple base stations i.e., multiple reception or macro-diversity. A study on the -coverage of devices is given in [24]. By combining the -coverage model for LPWANs and our “cards tossing” model, we can jointly design the infrastructure deployment and MAC layer of devices, in order to improve the network performance while limiting the cost.


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  24. Zhuocheng Li, Tuong-Bach Nguyen, Quentin Lampin, Isabelle Sivignon, and Steeve Zozor. Ensuring k-coverage in low-power wide area networks for internet of things. https://hal.archives-ouvertes.fr/hal-01353801.
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