I Introduction
A wide range of applications will be enabled by the advent of Internet of Things (IoT) technology, among which smart cities, intelligent transportation systems and environmental monitoring. Given the expected proliferation of such IoT devices in the near future, providing tailored wireless communication protocols with high spectral efficiency and low power consumption is becoming more and more urgent. Indeed, many of these services will depend on the future IoT Wireless Sensor Networks (WSNs), supported by the newly developed LowPower WideArea Network (LPWAN) technologies such as LoRa, SigFox or Ingenu [2, 3, 4, 5]. The LoRa physical layer uses the Chirp Spread Spectrum (CSS) modulation technique, where each chirp encodes values, for Spreading Factor (SF) to [6], and which allows multiple enddevices to use the same channel simultaneously. Based on the LoRa physical layer, LoRaWAN defines the MAC layer protocol standardized by LoRa Alliance [7]. It is an increasingly used LPWAN technology, as it operates in the ISM unlicensed bands and enables a flexible adaptation of transmission rates and coverages under low energy consumption [6]. The LoRaWAN architecture is a star topology, where enddevices communicate with the network server through gateways over several channels based on ALOHA mechanism, with duty cycle limitations [4]. In LoRaWAN, smaller SFs provide higher data rates but reduced ranges, while larger SFs allow longer ranges but lower rates [5].
The main issue of LoRabased networks such as LoRaWAN is the throughput limitation: the physical bitrate varies between 300 and 50000 bps [7]. In addition, collisions are very harmful to the system performance as the LoRa gateway is unable to correctly decode simultaneous signals sent by devices using the same SF on the same channel. Such interferences will be referred to as coSF interferences. Although SFs were widely considered to be orthogonal among themselves, some recent studies have shown that this is not the case by experimentally evaluating the effects of interSF interferences [8, 9, 10]. Thus, authors in [11] have analyzed the effect of imperfect SF orthogonality, through the comparison of two scenarios, perfect and imperfect SF orthogonality. Authors in [12] also analyzed the achievable uplink LoRa throughput under imperfect SF orthogonality, and have demonstrated the harmful impact of both coSF and interSF interferences on the overall throughput. More recently, [10] also unveiled a significant drop in performance when taking into account the interSF interferences in highdensity deployments. In [13], the authors proposed a model for analyzing the performance of a multicell LoRa system considering coSF interference, interSF interference, and the aggregated intra and intercell interferences. They also highlighted the necessity for an SF allocation scheme accounting for these interferences.
In order to improve the LoRa system performance, a number of works have proposed resource optimization methods [14, 15, 16]. However, most papers, so far, have assumed perfect orthogonality among SFs. In particular, the authors in [14] designed a channel and power allocation algorithm that maximizes the minimal rate. However, no SF allocation nor SFdependent rates were considered, despite the strong dependency of the rate to SFs. In addition, the solution of [14] requires instantaneous Channel State Information (CSI) feedback, which is not adapted to LoRa networks due to their energy consumption limitations [7]. In [15]
, a heuristic SFallocation is proposed in addition to a transmit power control algorithm, where enddevices with similar path losses are simply assigned to the same channel with different SFs, according to their distance to the gateway. Although the issue of interSF interferences was highlighted, it was ignored in their proposed solution. The authors of
[16, 17] proposed a method for decoding superposed LoRa signals using the same SF, as well as a full MAC protocol enabling collision resolution, the combination of which was shown to drastically outperform LoRaWAN jointly in terms of network throughput, delay, and energy efficiency. Finally, reference [18] extended the channel allocation method of [14]by investigating power allocation, and proposed an algorithm based on Markov decision process modeling.
Therefore, in this work, we jointly investigate the issues of SF and transmit power allocation optimization under both coSF and interSF interferences. Unlike our preliminary work [1] which only considered SF allocation under fixed transmit power, and treated the cases of coSF and interSF interferences separately, we now tackle the joint SF and power allocation under a generalized coSF and interSF interference modeling. We focus on the problem of maximizing the minimum achievable shortterm average rate in the uplink, whereby shortterm average rate is defined as the average rate over random channel fading, but given a fixed position of enddevices.
This metric is especially suited for LoRa networks, since the enddevices will likely be fixed for a certain period of time (at least for a few seconds) in many applications, and their positions known at the gateway, as in conventional signalstrengthbased SF allocation methods [7]. Firstly, we formulate a joint SF assignment and power allocation problem by modeling the achievable uplink shortterm average rate under coSF and interSF interferences, and power constraints. Next, given the mathematical intractability of this mixedinteger optimization problem, we split it into two subproblems: SF assignment under fixed transmit power, then transmit power allocation given the previous SF assignment solution. To solve the first subproblem, we propose an SFallocation algorithm based on matching theory. We show its stability and convergence properties, and analyze its computational complexity. Next, we transform the second subproblem into an equivalent feasibility problem with nonlinear constraints. To make it tractable, we propose to approximate the constraints in two different ways: linear and quadratic. The numerical results demonstrate that, compared to baseline schemes, our proposed method not only provides larger minimum rates, but also jointly improves the network throughput and fairness level. Moreover, the proposed power control further improves the system’s performance in terms of minimum achievable rates and user fairness, while realizing massive power savings.
The remainder of this paper is organized as follows. Section II describes the system model. Section III presents our joint SF and transmit power allocation problem and its contraints. Section IV details a lowcomplexity manytoone matching algorithm for the first subproblem. Section V discusses our transmit power allocation scheme for the second subproblem. Section VI studies the performance of the proposed algorithms. Finally, Section VII presents our conclusions.
Ii System Model
We consider a gateway located at the center of a circular cell or radius km and enddevices randomly distributed within it and simultaneously active, as depicted in Figure 1. We denote by the set of enddevices and by the set of SFs. We assume that all enddevices transmit on the same channel of bandwidth , with a duty cycle of 100% without loss of generality ^{2}^{2}2LoRaWAN imposes a duty cycle of 1% in some channels [7], in which case the theoretically achievable throughput would be 100fold, see Section VI.. The data bitrate of , is given by [6],
(1) 
where is the coding rate, with .
Let be the channel gain between the enddevice and the gateway, the carrier frequency and the deterministic pathloss [12]. Then, the uplink instantaneous ChanneltoNoise Ratio (CNR), , for enddevice at is given by [12],
(2) 
where is the distance from enddevice to the gateway, is the path loss exponent and dBm is the Additive White Gaussian Noise (AWGN) and is the receiver noise figure. Assuming Rayleigh fading channels, the CNR
is modeled as an exponential random variable with mean
.The area covered by each SF is given by the distance ranges in Table I [12],
(3) 
where is the link budget of the defined as , given the receiver sensitivity of each in Table I and the maximal transmit power. Hence, larger SFs result in larger communication ranges, with .
SF  Bitrate [kb/s]  Receiver sensitivity [6] [dBm]  Reception thresh. [dB]  InterSF thresh. [19] [dB]  Distance ranges 
7  5.47  123  6  7.5  [0,] 
8  3.13  126  9  9  (,] 
9  1.76  129  12  13.5  (,] 
10  0.98  132  15  15  (,] 
11  0.54  134.5  17.5  18  (,] 
12  0.29  137  20  22.5  (,] 
Next, we denote the SF assignment by and define it as,
If there is only one enddevice assigned to , this enddevice is only subject to interSF interferences caused by enddevices using a different SF. Hence the interSF SignaltoInterferenceplusNoiseRatio (SINR) of enddevice can be expressed as
(4) 
where is the transmission power of the enddevice at , and .
When there is more than one enddevice assigned to a SF, these devices are subject to both interSF and coSF interferences. Therefore, the coSF SINR of device on is written as,
(5) 
Note that this is a more general model as compared to that of [1], which assumed the dominance of coSF interferences over interSF interferences. In conformity to LoRaWAN standards, instantaneous CSI feedback is not assumed, unlike [14]. Hence, the SF allocation is performed every period of time, during which the longterm fading instance, i.e., path loss, can be assumed to be fixed. This is well suited to a wide range of applications envisioned for IoT systems based on LoRa, expected to be static, or with low mobility [20]. Therefore, the achievable uplink shortterm average rate for enddevice at is given similarly to [12] by,
(6) 
where
is the probability of successful reception analyzed in the following section.
Iii Problem Formulation
In this section, we formulate the joint SF and power allocation optimization problem in our considered LoRabased system, under imperfect SF orthogonality. In particular, the goal will be to improve the overall fairness of the system by maximizing the minimal uplink average rate over enddevices and SFs, under coSF and interSF interferences. We first derive the expression of the probability of successful reception, . Assuming , there are two cases:
Iii1 One enddevice at
enddevice is only subject to interSF interferences. The transmission can be successfully decoded if the node satisfies the interSF as well as the signal reception conditions. In this case, interSF interferences are more critical than the signal reception condition since there are always interSF interferences for . Hence the probability of successful transmission can be written as,
(7) 
where is given in (4) and is the interSF interference capture threshold for , defined in Table I. Using the random instantaneous CNR variables for all and marginalizing over them, it has been shown in [1] with similar calculations as in [12] that (7) can be written as,
(8) 
Iii2 More than one enddevice at
in this case, the coSF interferences as well as the interSF interferences largely dominate the signal reception condition [12]. Therefore, the success probability is expressed as in [21],
(9) 
where is given in (5) and is the coSF capture threshold which is equal to 6dB for all [6, 21]. With similar calculations as in [1], we obtain
(10) 
Given the above analysis, the joint SF and transmit power allocation optimization underlaying LoRaWAN network is formulated as follows (for ),
(11) 
where the minimization is over the that are nonzero, and
(12) 
where is the indicator function, i.e., it equals 1 if the condition is verified and 0 otherwise.
Finally, the overall optimization problem becomes
(13)  
(13a)  
(13b)  
(13c)  
(13d)  
(13e) 
Our objective function (13) expresses the maximization of the minimum datarate over all served enddevices (i.e., for which ) and SFs. Constraint (13a) is the power budget, where the maximum transmit power per enddevice is fixed to . Constraint (13b) defines the binary SF allocation variables . Constraints (13c) and (13d)^{3}^{3}3Setting enables to control the harmful effects of coSF interferences, and reduces the computational complexity of the proposed method, as shown in Sections IVD and VIC. ensure that an enddevice is assigned to at most one , and that the maximal number of enddevices sharing is . Finally, (13e) ensures that if there are enough enddevices (), no SFs should remain unused, i.e., at least one enddevice should be allocated to each SF. Clearly, is a mixedinteger problem with a nonconvex objective function, as it includes both binary allocation variables and continuous power allocation variables . Such problems are known to be generally NPhard [22]
, making them difficult to solve. We therefore propose to solve this problem by decomposing it into the following two optimization phases: (1) the discrete optimization phase of the allocation of binary variables
while keeping the power allocation variables fixed to , (2) the continuous optimization phase of the power allocation variables , where the allocation variables have been fixed to their previous solution. These two phases may be iterated until convergence, or until the maximum number of iterations has been reached.Denoting by and
, the SF assignment vector and transmit power vector for all enddevices, respectively, Algorithm
1 provides the overview of the general proposal.In the next sections, we describe each of the optimization phases.
Iv Proposed Spreading Factor Allocation
Iva Formulation of the proposed SF allocation optimization
In this section, the problem of SF allocation is addressed. We assume that all enddevices transmit with the maximum transmission power, i.e., . This problem can be formulated as follows,
(14)  
(14a)  
(14b)  
(14c)  
(14d) 
is an integer programming problem, given the binary variables , with a nonlinear objective function, hence it is difficult to obtain its optimal solution. Therefore, we
propose an optimized SF allocation method, using tools from matching theory.
Matching theory is a promising tool for resource allocation in wireless networks [23]. According to this theory, our considered allocation problem
can be classified as a manytoone matching problem with conventional externalities and peer effects. There are two sets of players, the set of SFs and the set of enddevices, where each player of the one set seeks to be matched with players of the opposing set. An enddevice prefers to be matched to the SF offering the highest utility, while each SF prefers to be matched with the group of enddevices with the highest utility. The difficulty of our problem is that there is an interdependency between nodes’ preferences, i.e., whenever an enddevice is matched to an SF, the preferences of the other enddevices may change due to coSF and interSF interferences. In addition to these conventional externalities (preference interdependency) and unlike the problem in
[14] where only orthogonal channels (not SFs) were considered, our problem exhibits peer effects that are caused by interSF interferences. That is, the preferences of an enddevice depend not only on the identity of the SF and the number of enddevices assigned to it, but also on the assignment of enddevices to other SFs (since they cause interSF interferences). Therefore, to solve , we propose a manytoone matching algorithm between the set of SFs and the set of enddevices. Next, we define the basic concepts of matching theory.IvB Fundamentals of Matching Theory
In order to describe our proposed matchingbased algorithm, we describe the basic concepts of matching theory that have been used in our algorithm:

Matching pair: a couple (, ) assigned to each other.

Utility of an enddevice: defined for our problem as its shortterm average rate. If it is the only enddevice at ,
(15) If it shares the with other enddevices,
(16) 
Utility of an SF: defined for our problem as the minimum shortterm average rate among the enddevices assigned to it. If is matched to one enddevice only:
(17) otherwise is given as
(18) where is the set of enddevices assigned to .

Preference relation: a player prefers a player over the player , if the utility of is higher when it is matched to than when it is matched to .

Blocking pair: a matching pair is a blocking pair when or is higher when uses , than when they use their current matches, without lowering the utilities of any other enddevice nor SF. In this case, will leave its current match to be matched to .

Twosided exchange stable matching: a matching solution where there is no blocking pair.
IvC Proposed SFAllocation algorithm
In this subsection, we describe the steps of the proposed matchingbased algorithm which exploits matching techniques as in [14, 23], tailored to our specific problem. First, the gateway performs an initial matching between the set of SFs and the set of enddevices by the Initial Matching in Algorithm 2. Next, it swaps the matching pairs obtained in the previous step until reaching a twosided exchange stable matching by the Matching Refinement in Algorithm 3. Details of these steps are given below.
Let denote the set of enddevices that are not allocated to any SF, the requests received by , and the set of enddevices assigned to .
We suppose that the gateway knows its distance with all enddevices.
Initialization: the gateway starts by initializing the preference lists of enddevices and SFs. Each enddevice with a distance to the gateway, can only use SFs if they are included in the coverage area () of the gateway for these SFs, therefore,
(19) 
is sorted according to the increasing order of the distance threshold of the SFs (), i.e., an SF with higher achievable rate is preferred. On the other hand, only considers enddevices having a distance to the gateway lower than ,
(20) 
is ordered such that a user is ranked before another user if is located in the ring of but not , or both are in the ring of but is closer to the gateway than .
Unmatched enddevices are added to .
Initial Matching: for each enddevice in the unmatched list , if , requests its first preferred SF and removes it from , otherwise the enddevice is removed from since all SFs it can use have already reached their quota. Then, each either accepts all current requests if its quota allows it, or it accepts the requests of its most preferred enddevices that fulfill its quota, if not. This process is repeated until becomes empty.
Matching Refinement: for each matching pair (), the algorithm calculates using (17) if it is only assigned to enddevice and (18) in the other case. The utility of enddevice is calculated by (15) if it is the only one at , and with (16) otherwise. Firstly, if there is an that is not assigned to any enddevice that allows to increase , the enddevice leaves to be matched with . Then, the algorithm calculates the utilities of every pair (), and makes a swap between () and () and determines their new utilities. Secondly, if () or () is a blocking pair, the algorithm makes a swap. This swapping step is repeated until reaching a twosided exchange stable matching.
IvD Proposed SFAllocation Algorithm Analysis
We now prove the stability and convergence of the proposed SFAllocation algorithm, and analyze its computational complexity.
Proposition 1.
Stability: When the proposed algorithm terminates, it finds a twosided exchange stable matching.
Proof.
Let us assume that the proposed SFallocation algorithm terminates and the final matching is not twosided exchange stable. Then, the matching contains at least one more blocking pair or where the utility of at least one player among , can be improved without lowering the others’ utility. Accordingly, the proposed algorithm would continue, thereby the matching would not be final, which contradicts the initial assumption. ∎
Proposition 2.
Convergence: After a finite number of swap operations, the algorithm eventually converges to a twosided exchange stable matching.
Proof.
A swap operation occurs if it improves the utility of at least one player without decreasing the others’, hence the utilities can only rise. Additionally, the maximal throughput that can be achieved on an is upperbounded by the data bitrate , meaning that each and the enddevices assigned to it have utilities upper bounded by .
The number of potential swap operations is finite: enddevice assigned to can make at most swap operations. The total number of swap operations is thus upperbounded by .
∎
Proposition 3.
Complexity: The running time of our proposed algorithm is upperbounded by , where .
Proof.
Initial matching complexity:
in the worst case, all the enddevices have the same preference list, and they are located in the area covered by all the SFs. At round the gateway receives requests, at round it receives requests, at round it receives requests. Therefore, the total number of requests equals
.
The complexity of the initial matching is upper bounded by:
.
Matching refinement complexity:
in each iteration, for each , the algorithm considers at most enddevices and examines swap operations for each of these enddevices. Therefore, the number of swap operations that are examined in one iteration is upper bounded by .
Let , thus the computational complexity of the matching refinement is upper bounded by .
In summary, the computational complexity of our algorithm is upper bounded by .
∎
Note that this complexity is not excessive as our algorithm is run at the gateway which is not computationallylimited.
V Proposed Power Allocation Optimization
Once the enddevices are assigned to SFs, we next optimize the power allocation variables in order to maximize the minimal throughput achieved on each SF. Given the fixed assignment variables from the previous step, the power allocation problem can be written as follows,
(21)  
(21a) 
It can be observed that the objective function of problem (14), unlike in previous works such as [14], is nonlinear nonconvex, for which a global optimum is difficult to obtain. This greatly increases the difficulty of this optimization problem. Instead, we seek for a nearoptimal solution by transforming the initial problem as follows. Let be the set of transmit power vectors such that the minimum throughput over enddevices and SFs is above a certain parameter , namely
(22) 
Since the minimal throughput value is above , all throughput values should be above as well. Hence, defining
(23) 
we can write . Introducing a new variable , problem (21) is equivalent to the following optimization problem,
(24)  
s.t.  (24a)  
(24b) 
Therefore, we take the following approach: for a given , we solve the feasibility problem
Find  (25)  
s.t.  (25a) 
then is increased until no feasible can be found. In practice, parameter can be updated using the bisection method [14] as detailed in Algorithm 4, as follows. Initially, is lowerbounded by , upperbounded by which is equal to the minimal bitrate over allocated SFs and enddevices. First, setting as the midpoint of the interval , problem (25) is solved and if a feasible solution is found, it is denoted as and we update the lower bound as . Otherwise, if no feasible power vector is found, is set as . This procedure is iterated until the interval length is smaller than the desired accuracy .
However, contains nonlinear inequalities, making it difficult to solve the feasibility problem (25). Hence, we devise two methods for making this problem tractable: linear approximation (A) and quadratic approximation (B) of these nonlinear inequalities.
Va Feasibility problem with linear approximation
In this subsection, in order to make problem (25) tractable, we first approximate the nonlinear inequalities in the set by linear ones. We distinguish two cases, one where only a single enddevice is assigned to and the second, where more than one enddevices are assigned to .
VA1 Case 1
a single enddevice is assigned to , hence is only subject to interSF interferences. Therefore, given (8), is given by,
(26) 
Rearranging and taking the logarithm of both sides, the inequalities in (26) are equivalent to
(27) 
The term is dominated by the interSF interference capture threshold , which takes very small values as can be observed from Table I. Thus, the term will be generally close to zero, as confirmed by the numerical evaluations in Section VI. Therefore, we can approximate the logarithmic term using the TaylorMaclaurin series,
(28) 
where denotes the remainder of the Taylor series.
VA2 Case 2
if is shared by more than one enddevice, from (10), the set is given by
(30) 
Similarly to Case 1, we perform the following linearization in order to make problem (25) tractable. By rearranging the inequalities, we obtain for all ,
(31) 
However, in this case, the coSF interference capture threshold no longer induces small values of , since in practice, dB [21]. Therefore, we now make use of a different approximation based on Taylor’s theorem.
Let . Clearly, is a twice continuously differentiable function.
From Taylor’s theorem, we have
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