Collision and Packet Loss Analysis in a LoRaWAN Network

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1 7 5th Europea Sigal Processig Cferece EUSIPCO Collisi ad Packet Loss Aalysis i a LoRaWAN Network Guillaume Ferré IMS Laboratory - Uiversity of Bordeaux - Bordeaux INP 35 cours de la libérati 334 Talece FRANCE guillaume.ferre@ims-bordeaux.fr Abstract Iteret of thigs IoT is csidered as the ext techological revoluti. Therefore, may solutis are developed either i free, i.e. ISM bads or i free bads with the ultimate aim of affordig cectivity over several kilometers. Based this feature, i urba evirmet the desity of IoT devices will be extremely high. I this paper we propose to aalyze the collisi ad packet loss whe LoRaWAN is csidered. Based the LoRaWAN features, we develop closed-form expressis of collisi ad packet loss probabilities. Simulati results cfirm our theoretical developmets. We also show that our theoretical expressis are more accurate tha the Poiss distributed process to describe the collisis. I. INTRODUCTION The Iteret of thigs IoT deploymet i free bads is maily based two commuicati techologies which are Sigfox [ ad LoRa [. Such techologies are called low power wide area etwork LPWAN ad they share the same objectives, i.e the establishmet of lg rage, low power ad low data rate commuicatis. Despite these similarities, they are techically ad ecomically opposites. Ideed the Sigfox physical layer is based ultra arrow baommuicati, as LoRa uses a spreadig spectrum techique to exchage iformati. I this paper we focus our atteti LoRa i order to aalyze the collisi ad packet loss whe LoRaWAN is csidered. Ideed, if we believe to the IoT success i our every day life, it is predictable to have several millis of IoT devices usig such modulati techiques. Therefore, the coexistece of all these devices, whe the commuicati is performed usig the same rage, will threat IoT commuicati with iterfereces. The chael access i LoRaWAN class A is based a ALOHA priciple. For etworks usig a ALOHA type chael access, may packet collisi studies have bee de before. The csidered system i this paper is differet tha the e used for classical aalysis of packet collisi i a u-slotted ALOHA based protocol [3. Ideed, packet time air ad time betwee two successive trasmissis deped the IoT applicati. Thus, the model allowig to describe the collisi effect based Poiss distributed process PDP is ot accurate eough. However, i our case this modelig is useful to defie lower ad upper bouds of the probability of success for a give spreadig factor. I this paper we propose a more accurate ad specific approach to predict the collisi ad packet loss. Thus, based the LoRaWAN MAC mechaisms we develop closed-form expressis of the probability of collisi ad packet loss ad our theoretical aalyses are cfirmed by the simulatis. The paper is orgaized as follows. I secti II the LoRa modulati ad the LoRaWAN MAC layer are itroduced. I secti III we develop the received sigal model that we csider. Based this model, secti IV is devoted to the collisi ad packet loss theoritical aalyse. Simulati results are give i secti V. II. LORA AND LORAWAN The followig sectis are dedicated to itroduce the LoRa physical ad MAC layers which is based the patet [4. For more details the reader is referred to [5. It should be oted that the LoRa physical Layer is ot published. A. LoRa physical layer priciple LoRa is based Chirp Spread Spectrum CSS modulati. CSS was proposed for the first time for commuicati systems by Wikler [6 ad applicati to digital commuicati by Beri [7. CSS is csidered as a subcategory of Direct-Sequece Spread Spectrum. CSS is compliat with IoT etwork eeds because it permits to come over the receiver s sesitivity issue ad icrease the commuicati rage at the cost of a reduced spectral efficiecy. The spectrum spreadig i LoRa is achieved usig a chirp sigal that ca be described by its istataeous phase φt or a specific time fucti f c t. f c t is called the raw chirp that: icreases liearly, for a up raw chirp, from a iitial value B to a fial value B, decreases liearly, for a dow raw chirp, from a iitial value B to a fial value B, B stads for the ISM sigal badwidth used for the commuicati. The raw chirp time durati is equal to the symbol period T s. f c t is defied as follows: f ct = ± B T s t The relatiship betwee the badwidth ad the symbol period is give by T s = SF B SF stads for the spreadig factor expet SF [7,...,. Let D s be the symbol rate of the B depeds the used ISM bad aa be chose equal to 5, 5 or 5 khz ISBN EURASIP 7 655

2 7 5th Europea Sigal Processig Cferece EUSIPCO trasmitted sigal ad D b the bit rate, the D s = D b /SF. Lger rage is achieved by varyig the spreadig factor, however to meet highly robust commuicati it is possible to vary the codig rate. With LoRa, symbols are obtaied from a biary combiati of SF bits. Each symbol is associated to a uique chirp. The differet chirps are orthogal to each other i order to retrieve at the receiver the symbols without iter-symbol iterferece IES. If we ote M the set of symbols, the chirp associated to the symbol m, m [, M, is obtaied by delayig the raw chirp f c t by τ m = m B. The chirp outside [ Ts, Ts Ts is cyclically shifted i the iterval [, Ts +τ m. Thus, the chirp associated to the trasmissi of the m th symbol is decomposed of parts: from t [ Ts, Ts + τ m[, raw chirp up ou dow advaced of T s τ m, from t [ Ts + τ m, Ts, raw chirp up ou dow delayed of τ m. For a up chirp, we obtai: f m c t = B T s t τ m + B f m c t = B T s t τ m for t [ T s, T s + m B [ for t [ T s + m B, T s Thus, the expressi of the basebad trasmitted sigal by the ode is give as follows: r t = k S l e jπf c,kt kt st kt s+jφ f c,k t represets the trasmittehirp at time kt s, S l the set of trasmitted symbols iside the packet p ad φ a iitial phase. If we ote K the S l size, thus the silet time durati of the LoRa ode will be at least equal to KTs KT s. Thus, from t + KT s to t + KTs the ode will be silet, is the duty cycle. B. LoRaWAN: a LoRa Mac layer LoRaWAN is a ope stadard developed by the LoRa Alliace. It s e of the possible MAC layer for the LoRa modulati ad obviously the well kow. The LoRaWAN specificati defies 3 categories of odes: Class A: a basic class of LoRa that is implemeted i all LoRa chips. It allows bi-directial commuicatis which is usually origially started by the ode i a asychrous way. The uplik trasmissi triggers two short dowlik receive widows. The trasmissi slot is scheduled whe eeded by the ode i a radom time basis. Accordig to LoRaWAN specificatis, class A is a ALOHA based-protocol. Class B: this class is cceived to guaratee uplik ad dowlik separati. Nodes are sychrized usig a beac trasmitted by the gateway. Thus, they ca receive iformati from Iteret without sedig requests. Class C: the ode has ctiuously ope receive widows that are closed ly while trasmittig. Compared to A ad B classes, C class csumes more eergy to operate but it offers the lowest latecy. The packet time durati is called the time air. This value depeds several parameters, such SF, B, the size of the payload, the codig rate, etc. After each uplik packet trasmissi, the ode waits for a gateway ackowledgmet ACK of the correct packet recepti. LoRaWAN allows two possible ackowledgmets two differet chaels. The first is trasmitted by the gateway with a cstat delay of T after the ed of the uplik packet recepti. The gateway uses the same chael tha the precedig uplik for this ACK. The secd ACK is trasmitted a differet chael tha the uplik after a time T > T + T g, T g is the time air of the packet trasmitted by the gateway. For more details the LoRaWAN MAC structure see [6. III. RECEIVED SIGNAL MODEL Based the complex evelope defiiti of, we express i this secti the model of the received sigal that we use. We csider a system which is composed of e LoRa gateway ad N LoRa odes. The th ode trasmit P packets, ad p represets the packet umber p trasmitted by the ode with a time air T. The badwidth is arrow eough to make the assumpti of flat fadig propagati chael. We ote h the chael coefficiet associated to the ode. If xt is the received sigal at the LoRa gateway, we have: N P xt = h r t Tp 3 = p= Let Tp be the begiig of the LoRa packet p. Based this, we defie: p Tp = p T + Toff + T r u = p T + u= p T r u 4 u= T r u is a statiary iid radom process with etries uiformly distributed. T r u is used for modelig the differet ode cstraits i terms of commuicatis with the gateways. I the followig sectis we csider that T r u is uiformly distributed i the set [T mi, T. This assumpti is justified due to the huge umber of IoT applicatis. T ad Toff are the time air ad time off air, respectively. IV. COLLISIONS AND PACKET LOSS AT THE LORAWAN GATEWAY A. Probability of Collisi at the LoRaWAN gateway I the followig we express the probability of collisi betwee p ad, give that p represets the ode of iterest. If C p deotes this collisi, based 4, the collisi evet is defied as follows: C p = {Tp Ω C = [Tp T, Tp + T} Based this we focus our atteti ly T. ISBN EURASIP 7 656

3 7 5th Europea Sigal Processig Cferece EUSIPCO A collisi betwee betwee p ad will happe if Tp Ω C. P C p represets the probability of this collisi ad we have: PC p = Tp Ω C 5 The radom variable Tp is the summati of multiple uiform radom variables see 4. For p >> we suppose, usig the cetral limit theorem, that T = p u= T ru is Gaussia distributed with a mea ad a variace µ = p T + T mi 6 = p T T mi 7 Uder this assumpti, Tp is ow a summati betwee two radom variables T ad T ad T T N µ, 8 UT mi, T 9 Ideed, due to the varity of IoT applicatis, the time air T of the ode ca be csidered uiformly distributed betwee the miimal time air T mi ad the imal time air T. Based this, PC p ca be rewritte give T as PC p = T T = T T mi Tp Ω C T dt T mi. With 8 we have T Tp Ω C T p T = Q µ T p + T Q µ µ = µ + p T give Qx = Q x, ca be rewritte as follows: PC p = T T T mi T T mi Q αt + β dt Q γt + δ dt α = p 3 β = [ µ T T + p 4 γ = p + = α + 5 δ = [ µ T T p = β + T 6 with the two chage of variables y = αt + β ad z = γt + δ, is rewritte as PC p = y Q y dy α T y y = αt mi z = γt mi γ T + β, y = αt + δ, z = γt + β + δ z z Q z dz usig: [ x Qxdx = x Qx x Qx + e x e x x π we get the probability of collisi: 7 PC p = fy, y fz, z 8 α T γ T PC p represets the probability of collisi betwee p ad. As we csider p as the ode of iterest, a fiite set Ω p of packets p of the ode have a ull probability to be i collisi with p. We ote Np the umber of packet of that ca be i collisi with p. Thus, the more we icrease p, the higher Np will be. For p, the evets C p with p Ω p are disjoit. If we ote C p the evet associated to the collisi of p with, we have C p = p Ω p C p 9 Thus the probability of collisi of p with is: PC p = PC p p Ω p Numerically, we observed that is idepedet of p. This result is iterestig because it demstrates that is ot depedet the packet umber p, csequetly it does t deped the time. Logically PC p is a fucti of T, ad the T spreadig i time i.e the support of the distributi. Based we express the probability of at least e collisi i a evirmet composed of N idepedet odes. Thus the probability of havig at least e collisi betwee ad the rest of the odes usig the same SF is give by: N [ P c N = PC p = B. Packet loss at the LoRaWAN gateway If o collisi occurs p we csider that the packet will be ackowledge by the gateway. I this case, after a fixed delay T the LoRaWAN gateway will use the same badwidth to trasmit the ACK. We ote T g the time air used by the gateway to aswer ad we suppose that it ca be csidered as a cstat value. Durig T g, the gateway is uable to receive a packet. Thus, a packet loss will occurs if a ode sed a ISBN EURASIP 7 657

4 7 5th Europea Sigal Processig Cferece EUSIPCO iformati durig the gateway ACK of p. This meas that the packet loss of e ode depeds the success of aother ode. Our goal is to express the probability of packet loss of based the success of p. This will occur whe: T p Ω L = [T p + T, τ T, τ + T g τ = Tp + T + T, ad x, y refers to x if x > y ad y otherwise. Based 3 cases must be csidered: case whe T < T mi, case whe T > T ad case 3 whe T [T mi, T. We defie the probability of packet loss of as P L p, L p is the evet associated to the loss give the success of p. Usig the same approach as the e developed i the previous secti, we have : I case : P L p p = T p T Ω L T dt I case : P L p p = T I case 3: P L p T mi T mi =.5 T +.5 T T mi T Tp Ω L T dt Tp Ω L T Tp Ω L 3 4 dt 5 T dt 6 Ω L = [T p + T, τ + T, g Ω L = [τ T, τ + T, g T = T T mi ad T = T T. Based 8 we have: Tp Ω L T = Q αt + µ τ T g Q αt + β 7 ad Tp Ω L T = Q αt T g Q γt 8 Followig the same maipulatis as the es from to 8, we ca express P for the 3 previous cases. L p Regardless the value of T, we ote L p the evet associated to the packet loss of give the success of p. Thus we have: P L p = P L p 9 p Ω l p Ω l p is the set of p that ca be lost give the success of p. Thus the probability of packet loss i.e a collisi with the ACK trasmitted by the gateway to p give the success of p is: P l N = P s N N = [ P L p 3 P s N = P c N is the probability of success of p i.e o collisi. V. SIMULATION RESULTS We csider a system which is composed of e gateway that ca establish cectis with N odes usig the same spreadig factor. We also csider that the sigals trasmitted by odes usig differet SF are orthogal. The odes access the chael radomly i time. The badwidth is fixed to B = 5kHz ad = %. T is uiformly distributed. The support of his distributi begis from the miimum time air to the imum time air. These values are reported i table I ad obtaied usig the lie LoRaWAN calculator. The miimum ad imum umber of packet per day are fixed by the miimum ad imum chael time durati used, respectively. For our simulatis we csider a miimum chael time use of 3s ad a imum of 864s. Time air Payload size bytes SF mi ms ms mi Table I: LoRaWAN imum ad miimum time air ad associated umber of bytes i the payload. Time air values obtaied from the LoRaWAN calculator. First of all, we compare our theoretical results with simulatis. Fig. shows the evoluti of PC p whe p =, SF = for T = s ad s. As it ca be see, our theoretical results are csistet with the simulatis. Ideed, we verify the accuracy of the theoretical aalysis, both theoretical expressi ad simulati curves coicide. I fig. we show the evoluti of P c N. O the e had we compare our theoretical results with the simulati whe T = s ad s. I both cases, simulati ad theory are superimposed. O the other had, we compare our results with a Poiss distributed process PDP geerally use to describe collisis i such a etwork ad defied as follows: D p = T P P DP N = e T + T+Tmi. Dp N As we ca see whe T is closed to the mea value of T, our approach ad the e based PDP give the same results. However, whe it s ot the case the PDP based approach is ot accurate to predict the collisi. Ideed, whe T = s we ca see a differece with our results. For example, for P c N =.3 our theoretical results ad also the simulati show that 58 odes ca be maaged by the gateway, as the PDP predict 8 odes. I fig. 3 we compare the probability of at least e collisi for a payload size betwee ad 59 bytes, whe odes are csidered. These sizes are available for each SF i LoRaWAN. The results ca be grouped i three parts providig the same performace. The first is composed of SF = 7 ad ISBN EURASIP 7 658

5 7 5th Europea Sigal Processig Cferece EUSIPCO 6 5 to a importat umber of collisi at the gateway. C p Probability of collisi of p - P T = s, SF = - Theory T = s, SF = - Simulati T = s, SF = - Theory T = s, SF = - Simulati Probability of at least e packet loss T = T mi, T = s T = T mi, T = T mi T = T mi, T T T = T, T = s T = T, T = T mi T = T, T T Packet umber p Figure : Probability of collisi P C p whe p =, SF = ad for T = s ad s 4 6 8,,,4,6,8,,,4,6,8 3, Number of odes Figure 4: Probability of at least e packet loss, SF = Probability of at least e collisi T = s, SF = - Theory T = s, SF = - Simulati T = s, Poiss distributed process T = s, SF = - Theory T = s, SF = - Simulati T = s, Poiss distributed process 5,,5,,5 3, Number of odes Figure : Probability of at least e collisi for SF =, whe T = s ad s 8, the secd of SF = 9 ad the third of SF =,,. From the probability of collisi poit of view it s better to use SF = 8, SF = 9 or SF =. Ideed, SF = ad give the same probability of collisi tha SF = but they offer a lower sesitivity. Probability of at least e collisi SF=7 SF=8 SF=9 SF= SF= SF= Payload size i Bytes Figure 3: Probability of at least e collisi for the differet SF whe odes are csidered I fig. 4 ad 5 we show the evoluti of P l N whe SF = ad SF = 7, respectively. This allows to show the worst ad the best case. For each SF, T g = T mi which correspds to a ACK composed of byte. We ca see that the P l N is bouded ad pass through a imum value. From the etwork poit of view ly the icreasig part of the curves are iterestig. Ideed the decreasig part correspds Probability of at least e packet loss ,,,4,6,8,,,4,6,8 3, Number of odes T = T mi, T = s T = T mi, T = T mi T = T mi, T T T = T, T = s T = T, T = T mi T = T, T T Figure 5: Probability of at least e packet loss, SF = 7 VI. CONCLUSION I this paper we proposed a aalysis of packet collisi ad loss whe LoRaWAN is csidered. Based the LoRaWAN features, we developed theoretical expressis for both the collisi ad the packet loss. These developmets have bee cfirmed by simulatis results. We have also showed that our approach allows to more accurately describe the collisi tha the classical PDP approach. The perspectives of this work are umerous. We are curretly workig the theoretical demstrati of the idepedece of PC p i p ad the developmet of a software that ca predict the iterferece level of each odes based their locatis, duty-cycle, time air, etc. Csider that the odes with differet SF ca be orthogal is also a iterestig perspective of this work. REFERENCES [ SigFox, 6, [accessed [Olie. Available: [ Lora modem desig guide : Sx7/3/6/7/8. [Olie. Available: [3 R. Rom ad M. Sidi, Multiple Access Protocols: Performace ad Aalysis. New York, NY, USA: Spriger-Verlag New York, Ic., 99. [4 O. Seller ad N. Sori, Low power lg rage trasmitter, Aug. 7 4, us Patet App. 4/7,7. [Olie. Available: [5 A. Augusti, J. Yi, T. Clause, ad W. M. Towsley, A study of lora: Lg rage ad low power etworks for the iteret of thigs, Sesors, vol. 6, o. 9, p. 466, 6. [Olie. Available: [6 M. Wikler, Chirp sigals for commuicatis, IEEE WESCON Cveti Record, p. 7, 96. [7 A. Beri ad W. Gregg, O the utility of chirp modulati for digital sigalig, IEEE Trasactis Commuicatis, vol., o. 6, pp , Jue 973. ISBN EURASIP 7 659

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