A Tractable Model of the LTE Access Reservation Procedure for Machine-Type Communications
Jimmy J. Nielsen, Dong Min Kim, Germán C. Madueño, Nuno K. Pratas, Petar Popovski
AA Tractable Model of the LTE Access ReservationProcedure for Machine-Type Communications
Jimmy J. Nielsen, Dong Min Kim, Germ´an C. Madue˜no, Nuno K. Pratas, Petar PopovskiAPNET section, Department of Electronic Systems, Aalborg University, Denmark { jjn,dmk,gco,nup,petarp } @es.aau.dk Abstract —A canonical scenario in Machine-Type Communica-tions (MTC) is the one featuring a large number of devices, eachof them with sporadic traffic. Hence, the number of served devicesin a single LTE cell is not determined by the available aggregaterate, but rather by the limitations of the LTE access reservationprotocol. Specifically, the limited number of contention preamblesand the limited amount of uplink grants per random access re-sponse are crucial to consider when dimensioning LTE networksfor MTC. We propose a low-complexity model of LTE’s accessreservation protocol that encompasses these two limitations andallows us to evaluate the outage probability at click-speed. Themodel is based chiefly on closed-form expressions, except forthe part with the feedback impact of retransmissions, whichis determined by solving a fixed point equation. Our modelovercomes the incompleteness of the existing models that arefocusing solely on the preamble collisions. A comparison withthe simulated LTE access reservation procedure that followsthe 3GPP specifications, confirms that our model provides anaccurate estimation of the system outage event and the numberof supported MTC devices.
I. I
NTRODUCTION
Machine-Type Communication (MTC) is commonly char-acterized by a large number of cellular devices that are activesporadically, where a large number of devices may activate ina correlated way due to a sensed physical phenomenon (e.g.,a power outage in the smart grid). In more traditional human-centric traffic where the associated payloads are relativelylarge, a small number of active devices can cause the networkto become in outage mainly due to the lack of availableresources for data transmission. In contrast, the associatedpayloads are relatively small in MTC such that the divisionof the aggregate available data rate with the small data raterequired by each Machine-Type Device (MTD) leads to theconclusion that the system can support a vast number ofMTDs. Recent studies have shown that such a conclusion ismisleading: the network still becomes in outage, not being ableto provide access to the MTDs, despite plenty of availableresources to support a massive number of MTDs. Here theculprit in the limited number of supported devices, is not theavailable resources as in human-centric traffic, but instead thebottlenecks in the access reservation protocol [1]. Specificallyin LTE, the Access Reservation protocol that is outlined inFig. 1 has two limitations that unveil with MTC. The first isin
MSG 1 , where only a limited number of preambles can beused to signal a sporadic request for uplink resources to theeNodeB, in the RACH phase. The second is in
MSG 2 , where MSG4 - Contention Resolution MSG1 - Preamble MSG3 - Connection Request MSG2 - Random Access Response
Device eNodeB
Fig. 1. Message exchange between a device and the eNodeB during the LTErandom access procedure. a bottleneck may be caused by the limited amount of feedbackresources in the access granting (AG) phase.In the literature, analytical models of the preamble collisionprobability have already been considered in standardizationdocuments [2]–[4] and scientific papers [5], [6]. In [7] thepreamble collision probability is used to estimate the successprobability of transmission attempts. However, we have foundthat existing models are incomplete and inaccurate and in thispaper we introduce a superior model that closely matches thesystem outage breaking point of the detailed simulation.The second limitation in the AG phase has been consideredseparate from collisions in [8] for bursty arrivals followingthe Beta distribution, which is a valuable result for situationswhere many alarm messages are sent simultaneously. In [6] theauthors present an approach to cell planning and adaptation ofPRACH resources that only takes into account the preamblecollisions. As we show in this paper, the AG phase is a limitingfactor before the amount of preamble collisions becomes anissue, since the impact of occasional collisions is effectivelydiminished with retransmissions. In [9] the authors present ananalysis accounting for preamble collision and the AG phase,which however does not consider retransmissions.In this paper we propose an analytical model of the trans-mission failure probability in an LTE cell for sporadic uplinktransmissions carried over the LTE random access channel.The proposed model captures the features of the existingaccess reservation protocol in LTE, meaning that we are notproposing a new access protocol rather introducing a tool foranalysis of the existing LTE access reservation protocol. Thepurpose of the proposed model is to be able to estimate thecapacity in terms of the number of terminals or RACH arrivaldensity that can be supported by LTE in a given configurationwhile accounting for retransmissions as well as modelingthe bottlenecks that appear in the contention phase and theAG phase. This is a major contribution of the paper, as the a r X i v : . [ c s . I T ] N ov dlePopulation Backlogged RACH Data Phase ✂ I Access Granting(a)(b) ✂ T ✂ R N Tx < T Yes No ✂ A N F N S Fig. 2. Flow diagram of LTE access reservation protocol: (a) one-shot transmission model; (b) m -Retransmissions model (dashed lines). existing models do not capture these bottlenecks. Three othercontributions are: 1) an iterative procedure to determine theimpact of retransmissions using a Markov chain model of theretransmission and backoff procedure; 2) analytical derivationsof the metrics based on a Markov chain, thereby achievingan analytical model that can be evaluated at click-speed. 3)analysis of the protocol breaking point using ever increasingaccess loads to the network.Initially we present the system model and assumptions inSection II, whereafter in Section III we present the proposedanalytical model of the access reservation bottlenecks in LTE.The proposed model is compared numerically to simulationresults and other models from the literature in Section IV, andfinally the conclusions are given in Section V.II. S YSTEM M ODEL
We focus our analysis on a single LTE cell, with N MTDs,also called machine-type User Equipment (UE). We assumethat the MTC applications associated with these MTDs, gen-erate new uplink transmissions with an aggregate rate λ I , asdepicted in Fig. 2. That is, λ I = (cid:80) Ki =1 λ app i , where λ app i isthe transmission generation rate of the i th out of K MTCapplications running on the UEs. We assume this aggregaterate follows a Poisson distribution with rate λ I . For eachnew data transmission, up to m retransmissions are allowed,resulting in a maximum of m +1 allowed transmissions. Whenthese transmissions fail and retransmissions occur, then anadditional stress is put on the access reservation protocol, sincethe rate of retransmissions λ R adds to the total rate λ T .As shown on Fig. 2, we split the access reservation modelinto two parts: (i) the one-shot transmission part in Fig. 2(a)that models the bottlenecks at each stage of the access reser-vation protocol; (ii) the m -retransmission part in Fig. 2(b),where a finite number of retransmissions and backoffs aremodeled. We focus our analysis on MTC, for which the trafficis characterized by having a very small payload. Therefore, inthe one-shot transmission, depicted in Fig. 2(a), we assumethat the RACH and access granting phases are the systembottlenecks. In other words, we assume that the network hasenough data resources to deliver the serviced MTC traffic. A. LTE Access Reservation Protocol
The uplink resources in LTE for frequency division du-plexing (FDD) are divided into time and frequency units denoted resource blocks (RBs) [10]. The time is divided inframes, where every frame has ten subframes, each subframeof duration t s = 1 ms. The system bandwidth determines thenumber of RBs per subframe that ranges between 6 RBs and100 RBs. The number of subframes between two consecutiverandom access opportunities (RAOs) denoted δ RAO variesbetween 1 and 20. Every RAO occupies 6 RBs and up to1 RAO per subframe is allowed.The LTE random access follows the access reservationprinciple meaning that devices must contend for uplink trans-mission resources in a slotted ALOHA fashion within aRAO [11], [12]. As shown in Fig. 1, the access procedureconsists of the exchange of four different messages between aUE and the eNodeB. The first message (MSG 1) is a randomlyselected preamble sent in the first coming RAO. In Fig. 2(a)the intensity of UE requests leading to preamble activations isrepresented by λ T . LTE has 64 orthogonal preambles, whereonly d = 54 are typically available for contention amongdevices, since the rest are reserved for timing alignment.Commonly, the eNodeB can only detect which preambleshave been activated but not if multiple activations (collisions)have occured. This assumption holds in small cells [13, Sec.17.5.2.3], and refers to the worst-case scenario where thedetected preamble does not reveal anything about how manyusers are simultaneously sending that preamble . In otherwords, the preamble collision is not detected at MSG 1.Thereafter, in MSG 2, the eNodeB returns a random accessresponse (RAR) to all detected preambles. The intensity ofactivated preambles is in Fig. 2(a) represented by λ A , where λ A ≤ λ T since in a preamble collision only 1 preambleis activated. The contending devices listen to the downlinkchannel, expecting MSG 2 within t RAR . It should be notedthat typically a maximum of 3 RAR messages per subframecan be sent by the eNodeB [8]. If no MSG 2 is receivedand the maximum of T MSG 1 transmissions has not beenreached, the device backs off and restarts the random accessprocedure after a randomly selected backoff time within theinterval t r ∈ [0 , W c ] ∩ Z + , where W c is the maximum backofftime. If received, MSG 2 includes uplink grant information,that indicates the RB in which the connection request (MSG 3) When the cell size is more than twice the distance corresponding to themaximum delay spread, the eNodeB may be able to differentiate the preamblehas been activated by two or more users, but only if the users are separablein terms of the Power Delay Profile [13], [14]. hould be sent. The connection request specifies the requestedservice type, e.g., voice call, data transmission, measurementreport, etc. In case of collision the devices receive the sameMSG 2, resulting in their MSG 3s colliding in the RB.In contrast to the collisions of MSG 1, the eNodeB is ableto detect collisions of MSG 3. The eNodeB only replies tothe MSG 3s that did not experience collision, by sendingmessage MSG 4, with which the required RBs are allocatedor the request is denied in case of insufficient resources.The latter is however unlikely in the case of MTC, due tothe small payloads. If the MSG 4 is not received within t CRT since MSG 1 was sent, the random access procedureis restarted. Finally, if a device does not successfully finish allthe steps of the random access procedure within m +1 MSG 1transmissions, an outage is declared.III. M
ODELING THE A CCESS R ESERVATION P ROTOCOL
We now go to the analysis of the access reservation pro-cedure. First, we model the
One-shot transmission and thenextend it to the m -Retransmissions model. The numericalresults cover the complete model, as depicted in Fig. 2. A. One-Shot Transmission Model
We are interested in characterizing how often a transmissionfrom a UE fails. This happens when the transmission is notsuccessful in both the preamble contention and AG phases,i.e., a request from the UE must not experience a preamblecollision and the uplink grant must not become stale anddropped. We model this as two independent events: p f ( λ T ) = 1 − (cid:16) − p c ( λ T ) (cid:17)(cid:16) − p e ( λ A ) (cid:17) , (1)where p c ( λ T ) is the collision probability in the preamblecontention phase given UE request rate λ T , while p e ( λ A ) isthe probability of the uplink grant being dropped from the AGqueue given preamble activation rate λ A .
1) Preamble Contention Phase:
We start by computing p c ( λ T ) . Let d denote the number of available preambles( d = 54 ). Let the probability of not selecting the samepreamble as one other UE be − d . Then the probability ofa UE selecting a preamble that has been selected by at leastone other UE, given N T contending UEs, is: P ( Collision | N T ) = 1 − (cid:18) − d (cid:19) N T − . (2)Assuming Poisson arrivals with rate λ T , then: p c ( λ T ) = + ∞ (cid:88) i =1 (cid:34) − (cid:18) − d (cid:19) i − · P ( N T = i, λ T · δ RAO ) (cid:35) (3) ≤ − (cid:18) − d (cid:19) λ T · δ RAO − , where P ( N T = i, λ T · δ RAO ) is the probability mass functionof the Poisson distribution with arrival rate λ T · δ RAO . Theinequality comes from applying Jensen’s inequality to the con-cave function − (1 − /d ) x , where λ T is the total arrival rate(including retransmissions), and δ RAO is the average number of subframes between RAOs. The computed p c ( λ T ) is thusan upper bound on the collision probability.
2) Access Granting Phase:
The mean number of activatedpreambles in the contention phase per RAO, is given by λ A . Asdiscussed in Section II, we assume that the eNodeB is unableto discern between preambles that have been activated by asingle user and multiple users, respectively. This will lead to ahigher λ A , than in the case where the eNodeB is able to detectthe preamble collisions. The main impact of this assumptionis that there will be an increased rate of AG requests, eventhough part of these correspond to collided preambles, whicheven if accepted will lead to retransmissions.The λ A can be well approximated, while assuming thatthe selection of each preamble by the contending users isindependent, by, λ A = [1 − P ( X = 0)] · d, (4)where P ( X = k ) is the probability of k successes, which canbe well approximated with a Poisson distribution with arrivalrate λ T /d , i.e.: P ( X = k ) ≈ ( λ T /d ) k e − λ T /d k ! . (5)To compute the outage probability due to the limitation inthe AG phase, i.e., due to the maximum number of uplinkgrants per subframe and a maximum waiting time of t RAR subframes, we consider that this subsystem can be modeled asa queuing system. We assume that the loss probability p e ( λ A ) can be seen as the long-run fraction of costumers that are lost in a queuing system with impatient costumers [15].In LTE, pending uplink grants are served with a determin-istic time interval (1 subframe) between each serving slot [8].A straightforward approach would be to use an M/D/1 modelstructure, as presented in [15], in order to compute the dropprobability. Unfortunately, the expression to compute p e ( λ A ) for the M/D/1 queue does not have a closed-form solution.However the equivalent expression for the M/M/1 queue in[15] has a closed-form solution, see eq. (6). We have comparedthe results of the two model types and found no noticeabledifference in the computed outage numbers in practice. Thus,in the following we use the M/M/1 model to compute p e ( λ A ) : p e ( λ A ) = (1 − ρ ) · ρ · Ω1 − ρ · Ω , with Ω = e − µ · (1 − ρ ) · τ q . (6)where ρ = λ A µ is the queue load, µ is the number of uplinkgrants per RAR ( µ = 3 ), with τ q = T d − µ and T d is the maxwaiting time (in terms of requests) in the uplink grant queue,i.e., T d = µ · t RAR .The fact that we are using an M/M/1 model instead of anM/D/1 model, may cause a discrepancy between the simulationand model results when the queue becomes congested ( ρ > ).However, we are interested in the switching point ( ρ = 1 ) fromwhich we then estimate accurately the outage breaking point,as shown in the results in section IV. E.g., δ RAO =1 if 10 RAOs per frame and δ RAO =5 if 2 RAOs per frame. (cid:2) (cid:2) (cid:2) f / c p W (cid:2) f / c p W (cid:2) f / c p W (cid:2) f / c p W (cid:2) (cid:2) (cid:2) f / c p W f p − Connect off on p − on p drop f p Fig. 3. Markov Chain backoff model to estimate number of requiredtransmissions. The states in the red dashed box are used to calculate N TX . B. m -Retransmissions Model When UEs are allowed to make retransmissions the proba-bility of an UE becoming in outage is the probability that noneof the allowed m +1 transmissions attempts are successful.When retransmissions are allowed ( m > ), the total arrivalrate λ T must include the extra arrivals caused by the UEs’retransmissions. The number of retransmissions λ R is howevera result of the limit m and transmission error probability p f , which in turn depends on the number of retransmissions λ R . This chicken and egg problem can be solved iterativelyusing a derivative of the Bianchi model [16] applied to oursystem model. Specifically, we are using a model adapted toLTE, with a structure similar to the one presented in [17].The following derivations of the number of transmissions andoutage probabilities have, to the best of our knowledge, notbeen presented previously.The mean number of required transmissions N TX and outageprobability P outage , are computed with help of the Markovchain model depicted in Fig. 3. In the Markov chain model, thestate index { i, k } denotes the i th transmission attempt stageand k th backoff counter. The number of allowed retransmis-sions is then given by m .Whenever the one-shot transmission is successful, depictedin Fig. 2(a), the UE enters the connect state: P ( connect | i,
0) = 1 − p f , ≤ i ≤ m. Where, p f is short for p f ( λ T ) . Whenever the one-shot accessfails, the UE increases the backoff counter: P ( i, k | i − ,
0) = p f W c , where ≤ k ≤ W c − and ≤ i ≤ m .At the last stage of the Markov chain, the UE enters the drop state if the transmission fails: P ( drop | m,
0) = p f ( λ T ) . The UE enters the off state after the connect or the drop states,with probability: P ( off | drop ) = P ( off | connect ) = 1 . From the off state, the node enters the first transmissionstate { , } with probability p on : P ( 0 , | off ) = p on . where the probability p on is defined as p on = 1 − e − λ I .Let b i,k be the steady state probability that a UE is at state { i, k } . Then b i,k can be derived as: b i,k = W c − kW c p f b i − , = W c − kW c p i f b , = W c − kW c p i f p on b off , (7)for ≤ i ≤ m and ≤ k ≤ W c − .Let b connect be the steady state probability that a node is at connect state: b connect = m (cid:88) i =0 (1 − p f ) b i, = m (cid:88) i =0 (1 − p f ) p i f p on b off = (cid:0) − p m +1 f (cid:1) p on b off . By imposing the probability normalization condition, asdetailed in Appendix A, we find b off as: b off = 2 (1 − p f )2 (1 − p f ) (1 + 2 p on ) + p on ( W c + 1) p f (1 − p m f ) . Since a transmission will eventually either finish success-fully in the connect state or unsuccessfully in the drop state,the outage probability can be computed as: P outage = b drop b drop + b connect = p m +1 f , (8)where b drop and b connect , whose derivations are shown in theAppendix A, can be computed as: b connect = 2 (1 − p f ) (cid:0) − p m +1 f (cid:1) p on − p f ) (1+2 p on )+ p on ( W c +1) p f (1 − p m f ) , (9) b drop = 2 (1 − p f ) p m +1 f p on − p f ) (1+2 p on )+ p on ( W c +1) p f (1 − p m f ) . (10)The number of required transmissions can be estimated fromthe steady state probabilities, keeping in mind that b i, /b , represents the probability of using i + 1 or more transmissionattempts to deliver a packet, and b m, /b , is the probabilityof using exactly m +1 transmission attempts: N TX ( λ T ) = (cid:18) m − (cid:80) i =0 ( i + 1) · ( b i, − b i +1 , ) (cid:19) + ( m + 1) · b m, b , = (1 − p f ) m − (cid:88) i =0 ( i +1) p i f +( m +1) p m f = 1 − p m +1 f − p f . (11)From the number of transmissions, the value of λ T can besolved iteratively using the fixed point equation: λ T = N TX ( λ T ) · λ I = λ I − p f ( λ T ) m +1 − p f ( λ T ) . (12) rrival rate (attempts/sec)0 1000 2000 3000 4000 P ou t age Proposed modelSimulationTR 37.868 modelEricsson model (a) Outage probability, m = 0 Arrival rate (attempts/sec)0 1000 2000 3000 4000 N T X Proposed modelSimulation (b) Expected number of transmissions, m = 9 Arrival rate (attempts/sec)0 1000 2000 3000 4000 P ou t age Proposed modelSimulationTR 37.868 modelEricsson model (c) Outage probability, m = 9 Fig. 4. Plots for RACH configuration with 2 RAOs per frame ( δ RAO = 5 ). Ericsson model refers to [6].
Arrival rate (attempts/sec)0 1000 2000 3000 4000 P ou t age Proposed modelSimulationTR 37.868 modelEricsson model (a) Outage probability, m = 0 Arrival rate (attempts/sec)0 1000 2000 3000 4000 N T X Proposed modelSimulation (b) Expected number of transmissions, m = 9 Arrival rate (attempts/sec)0 1000 2000 3000 4000 P ou t age Proposed modelSimulationTR 37.868 modelEricsson model (c) Outage probability, m = 9 Fig. 5. Plots for RACH configuration with 10 RAOs per frame ( δ RAO = 1 ). Ericsson model refers to [6].
For the results presented in Sec. IV we found that less than20 iterations were needed to reach convergence (less than 1%change between consecutive iterations).IV. N
UMERICAL RESULTS
In our evaluation, we consider two PRACH configurations,namely the typical configuration with 5 subframes betweenevery RAO [18] and the configuration with one RAO everysubframe. Further, we consider first the case where only asingle transmission is allowed (one-shot, m = 0 ) and then themore realistic configuration of m = 9 allowed retransmissions.The model results are compared with a simulator that im-plements the full LTE access reservation protocol as definedin [11] and [12] given parameters in Table I. Parameter Value
Preambles per RAO (d) 54Subframes between RAOs ( δ RAO ) 1 or 5Max number of retransmissions ( m ) 0 or 9Uplink grants per RAR ( µ ) 3System bandwidth 5 MHzeNodeB processing time 3 msMSG 2 window ( t RAR ) 5 ms or 10 msContention time-out ( t CRT ) 48 msBackoff limit ( W c ) 20 msUE processing time 3 msTABLE ILTE SIMULATION AND MODEL PARAMETERS
A. One-shot Transmission ( m = 0 ) In Fig. 4(a) and 5(a) the outage probabilities are depictedfor m = 0 . There, the proposed model has a much better fitto the simulation results than the 3GPP TR 37.868 model [4,Sec. B.1] and the Ericsson model in [6, eq. (6)]. Specifically,in Fig. 4(a) where the preamble collisions are the main errorcause, the TR 37.868 and Ericsson models are much worsethan the proposed model. From Fig. 5(a) it is clear that thosemodels are not accounting for the limited number of uplinkgrants per RAR that starts to have an impact around λ I = 2700 attempts/sec, causing an upward bend in the outage curve. B. m = 9 Retransmissions
In the typical configuration where retransmissions are al-lowed, a necessary feature of our model is that it is able toaccount for the feedback impact of retransmissions on thearrival rate λ T . An intermediate metric that allows to studythis is the number of transmissions per new data packet N TX .This is shown in Figs. 4(b) and 5(b). In Fig. 4(b) the number oftransmissions is estimated accurately leading to a well-fittingestimation of the outage in 4(c). For the case of 10 RAOsper frame, the Markov chain model slightly overestimates thenumber of transmissions. However, the breaking points in thecurves are the same, meaning that the supported arrival raten the simulation on Fig. 5(c) is closely matched by the onein the model.Finally, the results show that the proposed model is superiorto the existing models from the literature, as they do notcapture the feedback impact of the retransmissions and aretherefore not able to estimate the system outage capacity.The presented results also reveal an interesting insight indimensioning the LTE access reservation parameters. Giventhat there is a times difference in resource usage for RAOs(2 vs 10 RAOs per frame), the gain in supported arrival rate λ I is quite modest, increasing from around λ I = 2250 to around λ I = 2800 , i.e., a 25% increase. In order to further increasethe capacity of the system, it is necessary to simultaneouslyincrease the number of RARs per subframe.V. C ONCLUSIONS AND O UTLOOK
In this paper we have presented a low-complexity, yetaccurate model to estimate the outage capacity of the LTEaccess reservation protocol for machine-type communications,where the small payload sizes mean that system resourcesare typically not the limiting factor. The model accounts forboth contention preamble collisions and the limited number ofuplink grants in the random access response message, as wellas the feedback impact that the resulting retransmissions hason the random access load. For the considered typical LTEconfigurations, the model is able to very accurately estimatethe system outage capacity. This puts it forward as a usefultool in system dimensioning, as it allows to replace time-consuming simulations with click-speed calculations.Future work should look into how diverse channel condi-tions and diverse traffic patterns of users can be efficientlyincluded in the model. While the outage metric is veryimportant from a planning perspective, other metrics suchas access delay or transmission time would be very relevantto be able to estimate accurately when considering real-timemachine-type communications.A
CKNOWLEDGMENT
This work is partially funded by EU, under Grant agreementno. 619437. The SUNSEED project is a joint undertakingof 9 partner institutions and their contributions are fullyacknowledged. This work has been partially supported by theDanish High Technology Foundation via the Virtuoso project.A
PPENDIX
The following steps are taken to derive b off by imposing theprobability normalization condition: b off + b conn + b drop + b , + m (cid:88) i =1 W c − (cid:88) k =0 b i,k = b off + 2 p on b off + m (cid:88) i =1 W c − (cid:88) k =0 W c − kW c p i f p on b off = b off + 2 p on b off + p on b off (cid:18) W c + 12 (cid:19) p f − p m f − p f . The derivation of b connect is as follows: b connect = 1 − b off − p on b off ( p m +1 f − − (cid:18) W c +12 (cid:19) p f − p m f − p f )= 2(1 − p f ) (cid:0) − p m +1 f (cid:1) p on − p f )(1+2 p on )+ p on ( W c +1) p f (1 − p m f ) . (13)The derivation of b drop is as follows: b off = (1 − p on ) b off + b drop + b connect b drop = p on b off − b connect = 2 (1 − p f ) p m +1 f p on − p f )(1+2 p on )+ p on ( W c +1) p f (1 − p m f ) . (14)R EFERENCES[1] G. Corrales Madue˜no, C. Stefanovic, and P. Popovski, “ReengineeringGSM/GPRS Towards a Dedicated Network for Massive Smart Me-tering,” in
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