Analysis of PDCCH performance for M2M traffic in LTE

Transcription

1 1 Anlysis of PDCCH performnce for M2M trffic in LTE Prjwl Osti, Psi Lssil, Smuli Alto, Ann Lrmo, Tuoms Tirronen Abstrct As LTE is strting to get widely deployed, the volume of M2M trffic is incresing very rpidly. From the M2M trffic point of view, one of the issues to be ddressed is the overlod of the rndom ccess chnnel. The limittion in the PDCCH resources my severely constrin the number of devices tht n LTE enb cn serve. We develop Mrkov model tht describes the evolution of the Messge 4 queue in the enb formed by severl users performing the rndom ccess procedure simultneously nd then study its stbility nd performnce. Our model explicitly tkes into ccount the 4 initil steps in the rndom ccess procedure. By utilizing the model, we re ble to determine the stbility limit of the system, which defines the mximum throughput, s well s the probbility of filure of the rndom ccess procedure due to different cuses. We observe tht the shring of the PDCCH resources between Messge 2 nd Messge 4 with different priorities mkes the performnce of the whole rndom ccess procedure deteriorte very rpidly ner the stbility limit. However, we cn extend the mximum throughput nd improve the overll performnce by incresing the PDCCH resource size. Furthermore, we estimte the upper limit of the number of devices tht cn be served by n LTE enb nd determine the minimum PDCCH resource size needed to stisfy given trffic demnd. Index Terms LTE, M2M, Mrkov processes, MTC, PDCCH, Stbility I. INTRODUCTION Mchine-to-mchine (M2M) communiction or mchine type communiction (MTC) is the technology tht enbles severl devices to communicte with ech other without the need of constnt humn intervention. Severl billions of such devices tht use MTC re predicted to exist over the next few yers nd mjority of them re expected to be wireless sensors. This leds to the possibility of developing wide rnge of pplictions over M2M tht cn potentilly generte huge mount of revenue [1]. Even the existing networks, which re primrily designed for more trditionl humn-tohumn (H2H) trffic, re hndling some M2M trffic [2]. However, s the volume of M2M trffic grows more M2M type communiction specific provisions should be included in the Mnuscript received Februry 14, 213; revised... This work ws supported by TEKES s prt of the Internet of Things progrm of DIGILE (Finnish Strtegic Centre for Science, Technology nd Innovtion in the field of ICT nd digitl business). Copyright (c) 213 IEEE. Personl use of this mteril is permitted. However, permission to use this mteril for ny other purposes must be obtined from the IEEE by sending request to pubs-permissions@ieee.org P. Osti, P. Lssil, S. Alto re with the Deprtment of Communictions nd Networking, Alto University School of Electricl Engineering, Finlnd. (Emil: firstnme.lstnme@lto.fi) A. Lrmo, T. Tirronen re with NomdicLb, Ericsson Reserch, Finlnd. (Emil: firstnme.lstnme@ericsson.com) /$. c 213 IEEE design of the stndrds like LTE, which is likely to be the most widely ccepted stndrd for the 4G cellulr network. Indeed 3GPP hs conducted different studies [3], [4] tht ttempt to ddress the issues relted to M2M communiction in the present systems s well s in the future releses of LTE. The M2M trffic is different from the trditionl voice nd dt trffic for which most of the existing networks including LTE re optimized. The sensors re typiclly (but not lwys) sttic nd need not be optimized for mobile use. A huge number of mchines my exist in cell, which my ccess the network periodiclly or in rndom bursts. They lso hve limited power budget which should be used s efficiently s possible. A mchine my need very smll portion of the network s resources t time but s the number of mchines grows, their collective resource demnd cn overwhelm the network very esily, e.g., the rdio ccess network my not be ble to hndle the momentry surge in rndom ccess requests if thousnds of devices mke rndom ccess ttempts t the sme time. Indeed, one of the most discussed issues is the problem of rndom ccess overlod [], [6] t the edge of the network. Slotted Aloh is the bsis of the whole rndom ccess procedure which is n inherently unstble protocol [7] nd hs to be stbilized in some wy to mke it work [8], [9]. In the H2H cse, congestion in the rndom ccess chnnel is rrely problem becuse the number of users requesting rndom ccess t the sme time is lmost never so lrge. However, the deluge of rndom ccess requests from the huge number of MTC devices my overwhelm the network s signling resources. In contention bsed rndom ccess such s Slotted Aloh, when rndom ccess ttempt fils, the device my opt for retrnsmission possibly fter certin bckoff period cusing further increse in the trffic nd deteriortion of performnce. Severl such Slotted Aloh chnnels re operting in prllel for rndom ccess in LTE nd even such multi-chnnel rndom ccess systems re not immune to the inherent instbility issues of Slotted Aloh []. Recent works (e.g., [11], [12]) hve nlyzed the performnce of bckoff lgorithms in LTE for stbilizing nd optimizing the rndom ccess procedure. In fct, the rndom ccess will fil in LTE if ny one of the four steps of the procedure is unsuccessful, leding to wste of resources, retril nd ultimte increse in the trffic which my hve been prohibitively high to begin with. Moreover, different signling nd dt chnnels re involved in the LTE rndom ccess procedure nd congestion in ny one of them will ultimtely ffect the performnce of the whole procedure.

2 2 Since the problem is well known there hve been vrious ttempts to ddress this issue. The uthors in [] present nice overview of the problem when mssive number of devices re to be given ccess. In the context of ssigning seprte rndom ccess resources to vrious trffic types, [13] considers the pproch of dynmiclly shring the rndom ccess resource (prembles) between H2H nd M2M trffic. In [14] highly tilored pproch to overcome RAN overlod is presented by dividing the M2M trffic into severl priority clsses nd providing them different number of rndom ccess opportunities. A more dynmic pproch for RAN overlod control is provided by [1]. It is rgued tht with the incresing trffic rrivl rte the number of subfrmes tht should be llocted for the RACH procedure should lso be incresed to prevent rndom ccess filure. However, we will show here tht doing so my not necessrily solve the problem s other bottlenecks come into effect tht even reduce the rndom ccess throughput. Fundmentlly, the rndom ccess procedure is bsed on prllel Slotted Aloh chnnels. The Slotted Aloh chnnel itself hs stbility limit tht determines the mximum mount of trffic the system cn sustin. Almost ll the existing works focus in just this first step of the rndom ccess procedure (Slotted Aloh), either by employing good bckoff lgorithms (see e.g., [11], [12]) or by devising efficient wys to shre the rndom ccess opportunities mong different types of trffic in the cell (see [13], [14]). As it turns out in our nlysis, the steps following the Slotted Aloh step pose dditionl limittions on the stbility nd the performnce of PDCCH which is shred between Messge 2 s nd Messge 4 s with Messge 2 s receiving the priority, s stted in [16]. These messges re exchnged between the device nd enb in the steps following the initil Slotted Aloh step. In this pper we nlyze the performnce of the whole initil rndom ccess procedure ssuming tht the PDCCH hs limited resources in the form of CCEs (control chnnel elements). The initil rndom ccess is chrcterized by 4 steps, s will be discussed in detil lter on. A study by 3GPP [17] presents simultion results on the overll success probbilities of the ccess ttempts fter ll 4 steps. However, our objective is to derive model menble to mthemticl nlysis to obtin fundmentl insights. More specificlly, we develop Mrkov model tht describes the evolution of the Messge 4 queue nd then study its stbility nd performnce (mesured by the probbility of rndom ccess filure). This pper is the first to nlyze jointly the impct of ll the 4 steps of the initil rndom ccess, s fr s we know. In summry, the min contributions of our pper re the following. We provide trctble model for the nlyzing jointly the impct of the steps 1 4 in the LTE rndom ccess procedure. From the model we re ble to explicitly determine the mximum throughput of the system which gives the upper limit of the rrivl rte of the rndom ccess requests, i.e., the stbility limit. We dditionlly provide numericl method to estimte the filure probbility of the rndom ccess process nd its different components t vrious stges of the rndom ccess process. In our extensive numericl exmples, we illustrte how the different prmeters ffect the performnce, including the possibility to optimize the mximum throughput for certin prmeters. We observe tht the filure probbility is lmost zero when the rrivl rte is below the stbility limit. However, due to the nture of the priority system, this probbility increses very quickly ner the stbility limit. We dditionlly provide two exmples determining the mximum number of devices in cell nd dimensioning the PDCCH resource tht highlight the ppliction of our results. The pper is orgnized s follows: In the next section we explin the bckground nd role of PDCCH in LTE followed by the steps involved in the rndom ccess procedure. This is followed by Section III where we describe our trffic model nd identify the points where the rndom ccess cn fil in our model. Then in Section IV, we present our stochstic model, which describes the evolution of the Messge 4 buffer together with n uxiliry vrible s two-dimensionl Mrkov chin. The Mrkov model is utilized in Section V, where we consider the stbility of the Messge 4 buffer nd derive its mximum throughput. In Section VI, we present the methodology of determining the performnce of the system. The methodology is then pplied in Section VII, where we give numericl results. Finlly, conclusions of the study re presented in Section VIII. II. LTE BACKGROUND A. Downlink control informtion In LTE, downlink control informtion is sent over the Physicl Downlink Control Chnnel (PDCCH). The control informtion includes downlink scheduling ssignments, which re used to crry the informtion needed to receive dt on the Physicl Downlink Shred Chnnel (PDSCH), nd uplink scheduling grnts, which re used to indicte the shred uplink resources (Physicl Uplink Shred Chnnel, PUSCH) the terminl uses to send dt to the bse sttion (enodeb). The smllest physicl resource in LTE time-frequency structure is clled resource element, which consists of one subcrrier during one OFDM symbol. The trnsmissions re divided into frmes of length of ms which re further divided into subfrmes of 1 ms. In time domin, subfrme typiclly consists of 14 OFDM symbols. In every subfrme, 1 3 OFDM symbols re reserved for the control region which crries the PDCCHs. Three OFDM symbols is typicl size for the control region nd this size is ssumed in this work. The totl mount of vilble physicl resources during one subfrme depends on the number of subcrriers, i.e., the totl bndwidth llocted for the LTE crrier. For exmple, MHz cell bndwidth would correspond to 3 subcrriers. The totl PDCCH resource is mesured in control chnnel elements (CCEs), where ech CCE is set of 36 resource elements. One downlink control messge, such s downlink ssignment or uplink grnt, is crried over PDCCH which uses either one, two, four or eight CCEs. The number required depends on the size of the pylod of the messge nd the coding rte. For the numericl exmples, we typiclly mke the ssumption tht the totl resources vilble for the downlink control informtion in subfrme is N = 16 CCEs, the sme number used in 3GPP RAN overlod study [17]. Thus, the resources used to schedule dt in downlink nd used to hnd out uplink grnts re shred. The cpcity

3 3 for downlink ssignments nd uplink grnts sent during one subfrme depends on the sizes of the PDCCHs used to crry these messges (denoted by N Msg2 nd N Msg4 ) nd the totl number of CCEs (denoted by N). In relity, the number of CCEs one PDCCH (nd one downlink control messge) uses depends on the chnnel conditions nd is selected by the enodeb. Using our model, we will lter study the effect of different (sttic) CCE lloction sizes (N Msg2 nd N Msg4 ) for different messges s well s the consequence of vrying the PDCCH resource size (N). B. Rndom ccess procedure Below we briefly describe the Contention bsed Rndom Access Procedure [18] utilized, e.g., by the M2M trffic. Step 1: M2M device (UE) initites the rndom ccess procedure by rndomly choosing one of the vilble RACH prembles, nd sending the premble in Messge 1 over the Physicl Rndom Access Chnnel (PRACH). A collision hppens if two or more UEs choose the sme premble in the sme subfrme. However, this collision is relized only in Step 3, i.e., even if two or more UEs use the sme premble for Messge 1 nd collision occurs, the bse sttion does not detect this event t this stge. 1 The trnsmission of rndom ccess premble is restricted to certin subfrmes. Let b denote their periodicity, i.e., rndom ccess is possible in every bth subfrme. In ddition, let K denote the totl number of vilble prembles. Step 2: enodeb replies with Messge 2,.k.. Rndom Access Response (RAR), which includes n uplink grnt for Step 3. Messge 2 is sent over the PDSCH. For this, we need to schedule the user, i.e., send downlink ssignment control messge over PDCCH. There my be t most one RAR messge in ech subfrme, but ech my hve multiple uplink grnts (ech corresponding to seprte premble). Let c denote the mximum number of uplink grnts per RAR per subfrme. Note tht in our model, n uplink grnt is given for every used premble, whenever not limited by c, s the bse sttion does not detect collision t this stge nd thus mkes no distinction between collided nd n uncollided premble. Step 3: Next the UE sends Messge 3 over PUSCH. The collisions in Step 1 will be relized in Step 3: The two or more UEs tht chose the sme premble in Step 1 will ll try to utilize the sme uplink grnt in Step 3 to send their Messge 3 s. As result, the Messge 3 s interfere with ech other rendering the signl received t the enb undecodble nd none of the UEs involved will be sent the subsequent Messge 4. 2 Step 4: After receiving Messge 3 s relted to uncolliding prembles generted in Step 1, enodeb replies with Messge 4 s using gin the PDSCH, which needs to be scheduled on PDCCH. Let N be the size of PDCCH resource (in CCEs), N Msg2 nd N Msg4 be the number of CCEs used to send Messge 2 nd Messge 4, respectively. Then mximum 1 Note tht this is more conservtive pproch (nd even relistic from the ctul system point of view) thn detecting the collision in Step 1 itself. 2 This is gin conservtive ssumption s UE with much stronger signl thn others my be selected even in the event of collision due to the cpture effect. TABLE I: Model prmeters with typicl vlues in [17] Symbol Prmeter Typicl vlue [17] K Number of prembles 4 b RACH periodicity c Mx number of UL grnts 3 per subfrme N PDCCH resource size in CCEs 16 N Msg2 Number of CCEs used for Messge 2 4 N Msg4 Number of CCEs used for Messge 4 4 M Mx number of Messge 4 s per subfrme 4 (without Messge 2) m Mx number of Messge 4 s per subfrme (with Messge 2) 3 N of M = Messge 4 s cn be sent in one subfrme if N Msg4 Messge 2 is not present in tht subfrme. On the other hnd, when Messge 2 is lso sent in subfrme then t most Msg2 N N m = Messge 4 s cn be sent in tht subfrme. N Msg4 Although the prmeters N, N Msg2 nd N Msg4 re more relevnt from the system deployment point of view, our model is gretly simplified when we use the derived prmeters M nd m. This messging scheme is demonstrted in Figure 1. In ddition, the (model) prmeters introduced in this section re summrized in Tble I including set of typicl vlues for the FDD LTE [17, Tble ], where N = 16, N Msg2 = 4, N Msg4 = 4 which leds to M = = 4 nd m = 4 = 3. UE Messge 2 Messge 1 Messge 3 Messge 4 enodeb Fig. 1: Messge sequence in LTE rndom ccess. III. PERFORMANCE OF THE RANDOM ACCESS PROCEDURE We explore the performnce of the rndom ccess procedure described in the previous section by modeling nd nlyzing its success (or filure) probbility. To simplify the nlysis, we ssume tht no other trffic except this rndom ccess trffic is present in the network. We consider stedy-stte trffic scenrio where new (i.e., fresh) rndom ccess requests rrive ccording to Poisson process with constnt rte λ (new requests per subfrme). This cn be interpreted s trffic model where the requests re generted independently by lrge popultion of synchronous M2M devices. Similr scenrios with thousnds of devices ccessing the system re considered relistic in 3GPP studies, see [4], [17]. If rndom ccess request fils, it needs to be retrnsmitted gin lter

4 4 on. Whenever the sttion mkes n ccess request (fresh or retrnsmission), the premble is selected rndomly mong the K vilble ones [19]. Our purpose is to determine the mximum throughput θ (successful requests per time unit) of the system. In ddition, we study the behvior of the filure probbility (nd its components described below) s function of the rrivl rte λ of fresh requests. A rndom ccess request my fil due to collision in Step 1 when two or more UEs choose the sme premble. It my lso fil due to loss in Step 2 when the number of chosen prembles exceeds the mximum number of UL grnts. Here we ssume tht the excess requests re not buffered but lost. This is resonble s the UE is expecting response typiclly in ms [17] which is of the sme order s the intervl between trnsmission opportunities (recll our prmeter b). Thus, there is no time for the bse sttion to strt buffering the requests from Step 1 to be sent in subsequent time slots. If the Messge 2 timer expires, the UE will nywy perform bckoff nd eventully mkes retry, see [19]. While Step 3 does not generte ny new ccess filure cuses for requests successful in steps 1 nd 2, we still hve to tke into ccount the finl step. Messge 2 s nd Messge 4 s shre the sme resources in the PDCCH. But due to more strict constrint on the return time of Messge 2 s ( ms in [17]), we ssume tht they get the bsolute priority over the resource, see lso [16]. Only wht remins of the resource is then llocted to Messge 4 s, which hve more lenient time constrint (48 ms in [17]). Thus, in order to void dditionl losses in the finl step, there must be buffer for Messge 4 s. A filure tkes plce in Step 4 if the Messge 4 corresponding to the originl request is delyed in the buffer beyond the threshold tht triggers the retrnsmission timer. Thus, the filure probbility consists of the following components: Pr{filure} = Pr{collision in Step 1}+ Pr{no collision in Step 1, loss in Step 2}+ (1) Pr{no filure in Steps 1 nd 2, dely in Step 4}. It should be noted here tht our im is to nlyze the rndom ccess procedure itself nd therefore we ignore the effect of physicl lyer impirments on different rndom ccess messges for the ske of simplicity. IV. DELAY MODEL FOR MESSAGE 4 In this section we develop the Mrkov chin model used to describe the evolution of Messge 4 buffer. The vrious prmeters nd the vribles of the model re summrized in Tble II for quick reference nd described more elbortely in the text. We consider discrete time model where time slots re indexed by n. The length of one time slot in our model corresponds to the periodicity of RACH opportunities denoted by b, with typicl vlue of b = subfrmes (cf. Tble I), which corresponds to ms in bsolute time units. The model does not tke into ccount processing delys but ssumes tht messge received in time slot genertes response TABLE II: Summry of the symbols. Symbol Prmeter λ Arrivl rte of fresh rndom ccess requests Aggregte rrivl rte (fresh nd retrnsmitted) of the rndom ccess requests K Number of prembles b RACH periodicity c Mximum number of UL grnts per subfrme in Messge 2 M Mximum number of Messge 4 s per subfrme (without Messge 2 s) m Mximum number of Messge 4 s per subfrme (with Messge 2 s) N PDCCH resource size in CCEs N Msg2 Number of CCEs used for Messge 2 N Msg4 Number of CCEs used for Messge 4 n Index of the time slot of the Mrkov model A nk Number of rndom ccess requests using premble k p i Pr{A nk = i} (see (2)) Y n (1) Number of successful Messge 1 s (see (3)) Ỹ n (1) Totl number of prembles chosen (see (4)) q (1) ij Pr{Y n (1) = i, Ỹ n (1) = j} (see ()) Y (2) n Totl number of non-colliding UL grnts included in Messge 2 s Ỹ n (2) Totl number of UL grnts included in Messge 2 s (see (11)) q (2) ij Pr{Y n (2) = i, Ỹ n (2) = j} (see (7)) q (2) i Pr{Y n (2) = i} (see (8)) q (2) j Pr{Ỹ n (2) = j} (see (9) nd ()) X n Queue length of Messge 4 buffer t the beginning of the time slot n (see (12)) Y n (3) Number of successful (non-colliding) Messge 3 s Y n (4) Number of trnsmitted Messge 4 s (see (13)) θ (1) () Throughput of successful Messge 1 s per subfrme (see (14)) θ (2) () Throughput of successful UL grnts per subfrme (see (1)) θ (2) () Throughput of ll UL grnts per subfrme (see (19)) σ (4) () Averge left-over cpcity of Messge 4 s per subfrme (see (2)) θ Mximum throughput of Messge 4 s (see (23)) 2 Aggregte rrivl rte for which the throughput of Messge 2 s is mximum (see (18)) 4 Aggregte rrivl rte for which the throughput of Messge 4 s is mximum (see (22)) immeditely in the following time slot. Thus, for exmple, if enb receives Messge 1 in time slot n, it will reply with Messge 2 in the following time slot n + 1. Consider first the rndom ccess chnnel used in Step 1. For modeling its dynmics, we pply the well-known Slotted Aloh model [7], which is bsed on the pproximtive ssumption tht ll rndom ccess requests together (not only the fresh ones but lso the retrnsmissions) constitute Poisson process, the rte of which is denoted by (ttempts per subfrme). Thus, while the effect of retrnsmissions on the totl trffic is tken into ccount, the ctul retrnsmission mechnism itself is not explicitly modeled. The Poisson pproximtion is justified when the fresh requests rrive ccording to Poisson process, they do not fil too frequently, nd in the event of filure, the retrnsmissions re sufficiently rndomized, i.e., the intervls between successive retrnsmissions re sufficiently long reltive to the time slot durtion. As lredy mentioned,

5 our scenrio of lrge popultion of synchronous M2M devices llows us to model the rrivl process of fresh requests s Poisson process. Furthermore, s we will see, our 4-step model llows lower trffic rtes thn the corresponding Slotted Aloh system, nd the filure probbility remins very smll unless the system is operted in close proximity of its stbility limit. According to LTE specifictions, the Bckoff Prmeter, which gives n upper limit for the retrnsmission intervls, cn be s high s 96 ms [2]. Moreover, the two retrnsmission timers in LTE even seem to help (rther thn hinder) to mke the retrnsmissions sufficiently rndom. This logicl resoning for the justifiction of the Poisson pproximtion is complemented by simultion study presented in Appendix B. Also, s discussed erlier, in LTE ctully K prllel Aloh chnnels re used nd ech time device mkes retrnsmission the premble is selected rndomly. This rndomiztion further helps in mixing the fresh rndom ccess requests together with the retrnsmission ttempts. In ddition, it is good to notice tht the input prmeter of the model is not λ, the rte of fresh requests per subfrme, but, the ggregte rte of ll requests. In the following section, we will first explin how the throughput θ of successful requests cn be determined from the model s function of whenever the system is stble. In ddition, we give necessry nd sufficient condition for stbility. Therefter we utilize the fct tht, in ny stble system, the verge input rte must be the sme s the verge output rte, which implies tht the rrivl rte λ of fresh requests is equl to the throughput θ of successful requests whenever the system is stble. This is how we get the functionl reltionship between the ggregte request rte nd the rrivl rte λ of fresh requests. Now let A nk denote the totl number of rndom ccess requests with premble k (including both the new ones nd the retrnsmissions) in time slot n. Since the ggregte strem of requests (including the fresh ones nd the retrnsmissions) is ssumed to follow Poisson process nd the prembles re chosen independently from the uniform distribution, the A nk re IID rndom vribles obeying Poisson distribution with men b/k nd point probbilities p i () := Pr{A nk = i} = (b/k)i e b/k, i. (2) i! This is n immedite consequence of the so clled splitting property of the Poisson process, see, e.g., Proposition 6.7 in [21] or Proposition in [22]. Now define Y n (1) := #{k : A nk = 1, k = 1,..., K}, (3) Ỹ n (1) := #{k : A nk 1, k = 1,..., K}, (4) where Ỹ n (1) is referring to the totl number of prembles chosen in time slot n, nd Y n (1) is the number of successful (uncolliding) Messge 1 s. We observe tht the joint distribution of the rndom vribles Y n (1) nd Ỹ n (1) is s follows: q (1) ij (1) () := Pr{Y n = i, Ỹ n (1) = j} = ( ) K i j i p K j p i 1(1 p p 1 ) j i, i j K, () where we hve used the multinomil coefficient defined by ( ) i + j + k (i + j + k)! := i j i! j! k! nd shorthnd nottion p i = p i (). Consider now the dynmics of Step 2. Recll (from Tble I) tht c denotes the mximum number of UL grnts included in single Messge 2. There re t most b Messge 2 s nd, thus, t most bc UL grnts per time slot. Messge 2 s in time slot n re generted by Messge 1 s of the previous time slot. Let Ỹ n (2) denote the totl number of UL grnts included in Messge 2 in time slot n, nd Y n (2) the number of successful (uncolliding) UL grnts. No losses pper in this step, Ỹ n (2) = Ỹ (1) n 1 nd Y n (2) = Y (1) n 1, if the totl number of prembles chosen in the previous time slot is sufficiently smll, Ỹ (1) n 1 bc, which is trivilly true if K bc. But if Ỹ (1) n 1 > bc, then losses hppen so tht Ỹ n (2) = bc. We ssume tht the prembles tht re given UL grnt in the ltter cse re chosen rndomly by enb. Thus, we hve (for the non-trivil cse K > bc) q (2) ij (2) () := Pr{Y n = i, Ỹ n (2) = j} = (6) q (1) ij (), i j < bc, K k k=bc l=i q (1) lk () ( l i )( k l bc i with the following mrginl distributions q (2) i () := Pr{Y (2) n = i} = q (2) j () := Pr{Ỹ (2) n = j} = ) ( k bc), i j = bc, bc j=i j i= (7) q (2) ij (), i bc, (8) q (2) ij (), j bc. (9) By utilizing the definition of q (2) ij (), we esily find tht ( ) K p () K j (1 p ()) j, j < bc, q (2) j j () = K ( ) K p () K l (1 p ()) l, j = bc. l l=bc Thus, we hve the following representtion: () Ỹ (2) n = min{b(), bc}, (11) where B() is binomilly distributed rndom vrible with prmeters K nd 1 p (). In Step 3, the collisions (originlly due to Step 1) re relized. Those Messge 3 s in time slot n tht re generted by unsuccessful UL grnts conveyed in the Messge 2 s of the previous time slot re colliding in this step. Clerly we hve Y n (3) = Y (2) (3) n 1, where Y n denotes the number of successful Messge 3 s in time slot n. Consider finlly the dynmics of Step 4. New Messge 4 s in time slot n re generted by successful Messge 3 s of the previous time slot. Recll (from Tble I) tht M denotes the mximum number of Messge 4 s per subfrme. Thus, t most bm Messge 4 s cn be crried in single time slot. Recll

6 6 lso tht the mximum number (per subfrme) is reduced from M to m if there is Messge 2 in the corresponding subfrme. On the other hnd, there my be t most c successful Messges 3 s per subfrme. Now we hve to consider two different cses seprtely. 1) If c m, then ll new Messge 4 s in time slot n re trnsmitted immeditely in the sme time slot, implying tht the retrnsmission timer is never triggered due to dely, Pr{no filure in Steps 1 nd 2, dely in Step 4} =. 2) On the other hnd, if c > m, then it is not gurnteed tht ll new Messge 4 s cn be delivered in time slot. Thus, in this cse, buffer is needed for Messge 4 s in order to void dditionl losses. Assume now tht c > m, nd let X n denote the number of buffered Messge 4 s in the beginning of time slot n. The evolution of X n is s follows: X n+1 = X n Y (4) n + Y (3) n = X n Y (4) n + Y (2) n 1. (12) Here Y n (4) time slot n, Y (4) n denotes the number of trnsmitted Messge 4 s in = min { } X n, bm Ỹ (2) n /c (M m). (13) Note tht the expression bm Ỹ (2) n /c (M m) on the right hnd side refers to the leftover PDCCH service cpcity for Messge 4 s in time slot n. We observe tht (X n, Y n (3) ) is n irreducible nd periodic two-dimensionl Mrkov chin with stte spce E = {, 1,...} {, 1,..., bc}. Eqution (12) describes the evolution of the first component, while the second one is independent of the previous stte of the process, Y (3) n+1 (X n, Y n (3) ), from which the Mrkov property cn be verified. Irreducibility (under ssumption c > m) nd periodicity follow esily from the construction. To summrize, when c > m, Messge 4 s form queue, nd unlike in the previous cse we get nonzero probbility of filure due to queuing dely in Step 4. This probbility cn be clculted numericlly s described lter in Section VI. V. STABILITY AND THROUGHPUT ANALYSIS In this section we consider the stbility of our buffer model. If the system is stble, the buffer for Messge 4 s does not explode nd, s lredy explined in the previous section, the throughput θ of successful requests must be equl to the rrivl rte λ of fresh requests. Thus, our purpose is first to find conditions for stbility in terms of the totl trffic, nd then to determine the throughput of successful requests, θ(), s function of, s well s the mximum throughput θ = mx θ(). In order to simplify nottion, we ssume throughout this section tht K > bc. The generliztion to the cse K bc is strightforwrd. Recll tht we pply the well-known Slotted Aloh model [7] for the rndom ccess chnnel used in Step 1. Thus, the throughput (per subfrme) of successful Messge 1 s s function of, the rrivl rte of ll rndom ccess requests per subfrme, is given by θ (1) () = E[Y (1) n ]/b = e b/k. (14) The mximum throughput in Step 1 is chieved when b equls the number of vilble prembles, mx θ (1) () = θ (1) (K/b) = (K/b) e 1 (K/b).368, which puts n upper limit for the throughput θ of successful requests, s well s for the rrivl rte λ of fresh requests. For greter vlues of λ, the performnce of the rndom ccess chnnel collpses. With the prmeter vlues given in Tble I, mx θ (1) () = requests/ms. Let us then consider the throughput in Step 2. Since K > bc, the throughput is further reduced by the limited number of UL grnts in Messge 2. The throughput (per subfrme) of successful UL grnts s function of is clerly θ (2) () = E[Y n (2) ]/b = 1 b bc i=1 iq (2) i (), (1) where q (2) i () is defined in (8). With the prmeter vlues given in Tble I (including c = 3), we hve, fter numericl optimiztion of (1) with respect to, the mximum throughput in Step 2 s follows: mx θ (2) () = requests/ms. Note from (1) tht, for ll, implying tht θ (2) () < c, (16) mx θ (2) () c. (17) On the other hnd, since θ (2) () is continuous stisfying θ (2) () θ (1) () for ll, nd θ (1) () is bounded with limits θ (1) () for nd, there is 2 < such tht θ (2) ( 2) = mx θ (2) () mx θ (1) () = (K/b) e 1. (18) Prt of the resources in Step 2 re, however, wsted by colliding UL grnts. Let θ (2) () denote the throughput (per subfrme) of ll UL grnts s function of, θ (2) () = E[Ỹ n (2) ]/b = 1 b bc j=1 j q (2) j (). (19) The remining prt of the downlink control chnnel resources re vilble for Messge 4 s. The verge leftover service

7 7 cpcity (per subfrme) for Messge 4 s in time slot n is given by [ ] σ (4) () = E bm Ỹ (2) n /c (M m) /b = b 1 (b i)c m + (M m) q (2) () + (). i=1 i b j=(b i 1)c+1 q (2) j where q (2) j () is defined in (9). Note tht we clerly hve (2) σ (4) () m. (21) Below we give necessry nd sufficient condition for the stbility of the Messge 4 buffer, which is the min theoreticl result of the pper. For clrity, we hve plced the proof in Appendix. Proposition 1: The buffer for Messge 4 s is stble if nd only if θ (2) () < σ (4) (). If the buffer is stble, the throughput of successful requests, θ(), is clerly equl to θ (2) (). Define now 4 = sup{ 2 : θ (2) () < σ (4) ()}. (22) As direct corollry of Proposition 1, we get the following result. Corollry 1: The throughput of successful requests is given by θ() = θ (2) () for ll < 4, nd the mximum throughput by θ = θ (2) ( 4). (23) In ddition, we recll from the previous section tht the rrivl rte λ of fresh requests is equl to the throughput θ of successful requests whenever the system is stble. Thus, for ll < 4, λ() = θ (2) (). VI. PERFORMANCE ANALYSIS METHODOLOGY In this section, we describe how we use the dely model to determine the performnce of the system. The evlution cn only be done numericlly nd it depends on the following functions θ (1) (), see (14), θ (2) (), see (1), θ(2) (), see (19) nd σ (4) (), see (2). These in turn depend on further definitions given in Section IV. Note tht our tble of symbols nd definitions, Tble II, lso provides references to the equtions chrcterizing our derived quntities. We strt by describing the evlution of the mximum throughput θ nd then continue with determining the filure probbility nd its vrious components. A. Determining the mximum throughput θ Consider first the mximum throughput θ, which cn be thought s the cpcity of the system. Functions θ (2) () nd σ (4) () re clerly continuous stisfying lim θ(2) () = < M = lim σ (4) (). While not t ll esy to prove, it is, however, intuitively cler tht θ (2) () is n incresing function for ll < 2 nd it intersects with the decresing function σ (4) () t most once in the intervl [, 4]. If there is no intersection, then there is no stbility issue (of the Messge 4 buffer) nd the mximum throughput is determined by Step 2, θ = θ (2) ( 2) = mx θ (2) (). The sitution is illustrted in Figure 2, where we hve plotted functions θ (1) (), θ (2) (), θ (2) (), nd σ (4) () with the (bsic) prmeter vlues given in Tble I. Note tht for these vlues c m so tht there will be no queue t ll in the Messge 4 buffer. As lredy mentioned in the previous section, the mximum throughput is θ = requests/ms, which is lso visible from Figure Σ 4 Θ Θ 2 2 Θ Θ 2 Θ 1 Fig. 2: Illustrtion of the throughput curves (in different steps) nd the mximum throughput θ in the cse where θ (2) () nd σ (4) () do not intersect. The prmeter vlues re tken from Tble I. On the other hnd, if curves θ (2) () nd σ (4) () intersect, then the mximum throughput is equl to the stbility limit of the Messge 4 buffer, θ = θ (2) ( 4) = σ (4) ( 4). (24) Figure 3 gives n exmple of this sitution. The prmeters used in this cse re otherwise the sme s in the previous figure but now c = 6 (insted of c = 3) so tht there re more UL grnts in Step 2, which mkes the stbility of the Messge 4 buffer n issue. With these prmeter vlues, the mximum throughput is θ = 3.22 requests/ms, which is clculted by numericlly solving (24), where θ (2) () is defined in (1) nd σ (4) () in (2). The numericl result cn lso be verified by the figure.

8 Θ Σ Θ 2 Θ 2 Θ 1 Fig. 3: Illustrtion of the throughput curves (in different steps) nd the mximum throughput θ in the cse where θ (2) () nd σ (4) () intersect. The prmeter vlues re otherwise the sme s in Figure 2 but now c = 6 (insted of 3). B. Determining the filure probbility In ddition to the mximum throughput, we re interested in determining the filure probbility s function of the rrivl rte λ of fresh requests. Recll from (1) tht Pr{filure} = Pr{collision in Step 1}+ Pr{no collision in Step 1, loss in Step 2}+ Pr{no filure in Steps 1 nd 2, dely in Step 4}, The first two probbilities on the right hnd side re clerly given by the following equtions: Pr{collision in Step 1} = 1 θ(1) (), Pr{no collision in Step 1, loss in Step 2} = θ (1) ( ) () 1 θ(2) () = θ(1) () θ (2) (). θ (1) () The third one stisfies with Pr{no filure in Steps 1 nd 2, dely in Step 4} = Pr{no filure in Steps 1 nd 2} Pr{dely in Step 4 no filure in Steps 1 nd 2} Pr{no filure in Steps 1 nd 2} = θ(2) (). If c m, then we know (from Section IV) tht Pr{dely in Step 4 no filure in Steps 1 nd 2} =. However, if c > m, we do not hve ny explicit expression for the conditionl probbility Pr{dely in Step 4 no filure in Steps 1 nd 2}, but we resort to simultions of the two-dimensionl Mrkov chin (X n, Y n (3) ). In this wy, we get n estimte of the conditionl probbility for ny fixed totl rte. TABLE III: Different scenrios used in the numericl studies. Note tht P is the bsic scenrio tht is tken from [17] nd lso mentioned in Tble I. Scenrio b c M m N Msg2 N Msg4 N P P P P P P P P In the finl step, we determine the corresponding rrivl rte λ of fresh requests from the eqution below, λ() = θ (2) (), (2) which is vlid whenever the system is stble s explined in the previous section. By utilizing its inverse function (λ), we re finlly ble to express the filure probbility nd its components s function of the rrivl rte λ of fresh requests. VII. NUMERICAL RESULTS In this section we illustrte the properties of our model through numericl exmples. First we observe how the mximum throughput behves when the prmeter c, which is the mximum number of uplink grnts in Messge 2, is vried. Then we study the queuing behvior of the Messge 4 buffer. In ddition, we nlyze vrious components of the rndom ccess filure probbility with vrying trffic lod. We then provide method to estimte the mximum number of users tht cn exist in cell bsed on trffic models presented in [17] nd finlly dimension the PDCCH resource to sustin such trffic. For the numericl study, we mke use of vrious scenrios formed by different combintions of the prmeters b, c, M nd m provided in Tble III. In every scenrio we hve totl of K = 4 prembles vilble for the contention bsed rndom ccess, nd in every subfrme N = 16 CCEs re used in the PDCCH for purposes of rndom ccess (i.e., sending Messge 2 s nd Messge 4 s). Scenrio P is the bsic scenrio mentioned in the Typicl vlue column of Tble I. An equivlent prmeter set is considered in the RAN overlod study [17] by 3GPP. In Scenrio P1 we ssume c = 6 UL grnts re given per Messge 2 while the rest of the prmeters re the sme s P1. In Scenrios P2 nd P3, N Msg2 = 8 CCEs re used for Messge 2 nd N Msg4 = 4 CCEs for Messge 4, while Scenrios P4 nd P ssume tht ech Messge 2 nd Messge 4 use N Msg2 = N Msg4 = 8 CCEs. The finl two scenrios P6 nd P7 re lmost equivlent to Scenrios P nd P1, respectively, except tht they consider the rndom ccess opportunity to be vilble in every subfrme (rther thn every subfrmes considered in P nd P1), i.e., b = 1 is ssumed in these two finl scenrios. A. Mximum throughput θ From Figures 2 nd 3 we observe tht the mximum throughput, θ, clerly increses with c, the number of UL

9 9 M m 1 (m + (b 1)M ) = M. b b 4 P2 P3 q* P6 P7 2 P4 P 1 1 c c = 1 3. c = 6 2. c = c = b Fig. : The mximum throughput s function of periodicity prmeter, b, for different vlues of uplink grnts, c, sent in Messge 2. We see tht for ny given c, there is n optiml vlue of b for which we get the lrgest mximl throughput. B. Behvior of the Messge 4 queue P P1 3 we see tht the optiml b is 1 for smller vlues of c. This optiml periodicity becomes lrger s we increse c. q* grnts in Messge 2. Mximum throughput is clculted ccording to Corollry 1 s explined in detil in Section VI-A. Clerly, s c increses, more nd more UL grnts cn be sent in Messge 2 of the first subfrme (out of b), freeing up the resources to send the Messge 4 s in the lter subfrmes. Since the number of PDCCH resources is limited, this growth cnnot continue indefinitely, nd in the limit c, the mximum throughput pproches vlue which is pproximtely equl to M (M m)/b, s cn be observed from Figure 4. Intuition behind this pproximte limit is s follows. Assume tht b is sufficiently smll so tht the bottleneck of the whole system is due to competition of common PDCCH resources between Messge 2 s nd Messge 4 s. Now, when c, ll the UL grnts (if ny) relted to single time slot cn be sent in Messge 2 of the first subfrme (out of b). Thus, there re typiclly m Messge 4 s in the first subfrme nd M Messge 4 s in the remining b 1 subfrmes of the time slot in considertion so tht the totl number of Messge 4 s per subfrme is pproximted by 3 Fig. 4: The mximum throughput s function of prmeter c for different scenrios. We see tht the throughput increses with c until it sturtes pproximtely to level M (M m)/b (represented by dshed line). The vlues of prmeters b, M, nd m used here re tken from Tble III, while prmeter c is llowed to vry. Moreover, from Figure 4, we observe tht when lrge mount of UL grnts re sent in Messge 2 (lrge c), the prmeter combintion (b, M nd m) of P/P1 performs best in terms of mximum throughput mong ll the scenrios considered here. Moreover there is lso n optiml trdeoff between prmeters b nd c. For given vlue of c, incresing b will llow us to send more uplink grnts in time slot (less loss in Step 2) while incresing the collision probbility in Step 1. For smller vlues of c (typiclly ), different scenrios give optiml performnce. For exmple, when M = 4 nd m = 3 (scenrios P nd P1), the optiml mximl throughput is function of the periodicity of the rndom ccess opportunities, b, when the number of uplink grnts in Messge 2, c, is fixed. This cn be observed in Figure where we plot the mximl throughput ginst the periodicity prmeter, b for different vlues of c. In such cses To study the queuing behvior of the Messge 4 buffer, the Mrkov chin (12) ws simulted for 6 time slots for different vlues of λ nd for Scenrios P1 P7 (except P nd P6, which do not form queue) in Tble III. Note tht in the simultion, the totl rrivl rte is the given prmeter which then corresponds to certin rte of new requests λ() given by (2). The simultion results re presented in Figure 6, which shows the men queue length of the Messge 4 buffer s function of λ for different scenrios. As cn be seen, it is chrcteristic for the system tht the Messge 4 queue remins nicely under control until λ is quite close to the stbility limit, θ, given by (24). Intuitively, this lso mens tht the likelihood of user experiencing timeout event due to buffering dely is very low until the lod is close to the stbility limit. In the next section, we will observe the contribution of this queuing dely in the timeout event of the rndom ccess procedure nd compre it with the other cuses of filure. C. Contributions of vrious components in rndom ccess filure To gin further insight to wht re the most likely cuses for rndom ccess filures, we look t the probbilities given by (1). The three components of the filure probbility re depicted, on logrithmic scle, in Figure 7 s function of λ for Scenrio P1 (with lbels Collision in 1, Loss in 2, nd Dely in Step 4 ). The filure probbilities for collision in Step 1 nd loss in Step 2 re obtined numericlly s described in Section VI-B, while the conditionl probbility Pr{dely in Step 4 no filure in Steps 1 nd 2} needed for the third component in (1) hs been estimted from simultions of the Mrkov chin (12) for 6 time slots. We cn observe tht, even with rrivl rtes close to the stbility limit, the collision probbility in Step 1 remins reltively low (of

10 EN P4 P Fig. 6: Men queue length s function of λ for vrious scenrios. The verticl dotted lines represent the stbility limit, θ, for ech scenrio. Scenrios P nd P6 do not produce ny queue s c = m in those two cses. the order 2 ). Also, the probbility of loss in Step 2 is considerbly lower nd mkes no difference whtsoever in the system. Thus, the number of prembles K is not bottleneck nd the limittion of c is even less of bottleneck. Ultimtely, s λ grows, the probbility of filure becomes dominted by the event tht the queuing dely of Messge 4 grows too lrge. However, this hppens in very shrp mnner close to the stbility limit. For the bsic scenrio, P, we get no filures due to dely s discussed in Section IV but we still hve two other components s demonstrted in Figure 8. LogFilure Component Probbilities Collision in Step 1 P2 P3 P7 P1 Dely in Step 4 8 Loss in Step Fig. 7: A brekdown of the filure probbility (logrithmic scle) s function of λ for Scenrio P1. The verticl dotted line represents the mximum throughput θ for the prmeter set used. On the other hnd, in Scenrio P6, where rndom ccess opportunity is vilble in every subfrme, different picture emerges (see Figure 9). Now the probbility of loss in Step 2 increses more rpidly thn the collision probbility in Step 1. Since no queue is formed s c = m, there is no loss due to queuing dely. Even t moderte rrivl rte, the limittion of the control chnnel resource begins to contribute to the filure of the rndom ccess procedure more thn the collisions in Step 1. This is becuse very few UL grnts cn be sent in one Messge 2 (c = 3) nd some users who were successful in Step 1 hve to be dropped in Step 2. For Scenrio P7, the loss probbility in Step 2 remins below the collision probbility LogFilure Component Probbilities Collision in Step 1 Loss in Step Fig. 8: A brekdown of the filure probbility (logrithmic scle) s function of λ for Scenrio P. The verticl dotted line represents the mximum throughput θ for the prmeter set used. We see tht the probbility of filure due to Loss in Step 2 is considerbly higher compred to tht of Scenrio P1. in Step 1 while pproching it t higher rrivl rtes. Ner the stbility limit, the dely in Step 4 gin domintes the probbility of rndom ccess filure (see Figure ). LogFilure Component Probbilities Collision in Step 1 Loss in Step Fig. 9: A brekdown of the filure probbility (logrithmic scle) s function of λ for Scenrio P6. The verticl dotted line represents the mximum throughput θ for the prmeter set used. Note how the limittion of c cuses the loss probbility in Step 2 to increse beyond the collision probbility in Step 1 for higher rrivl rtes. No queue is formed in this cse s c = m so tht there is no dely component in the rndom ccess filure. D. Estimtion of mximum number of devices in cell From our model it is possible to estimte the mximum number of MTC devices tht cn exist in cell if the trffic chrcteristic of the mchines is similr to Trffic Model 1 in [17, Tble 6.1.1], which ssumes tht there re fixed number, D, of devices in cell ech generting request for rndom ccess uniformly over period of 6 seconds. In our model, this corresponds to rrivls t rte (or mximum throughput) λ = θ = D/6 per millisecond, where θ is the mximum throughput the respective scenrios support given by (23). Therefore the mximum number of devices cn

11 11 LogFilure Component Probbilities Collision in Step 1 Loss in Step 2 Dely in Step Fig. : A brekdown of the filure probbility (logrithmic scle) s function of λ for Scenrio P7. The verticl dotted line represents the mximum throughput θ for the prmeter set used. TABLE IV: An estimte of the mximum number of devices tht cn exist in cell ccording to our model when the rrivl rte of the request follows Trffic Model 1 [17, Tble 6.1.1]. Scenrio D mx P 143 P1 194 P2 132 P3 164 P4 8 P 97 P6 166 P7 182 be estimted by the quntity 6 θ. For different scenrios, the mximum number D mx of such devices is tbulted in Tble IV. We cn now understnd the reson for prcticlly no losses when there re 3 devices in cell when Trffic Model 1 [17, Tble 6.1.1] is used ccording to our model s mny s 143 devices cn be served nd 3 is too smll number to produce ny kind of discernible filure. On the other hnd, when Trffic Model 2 [17, Tble 6.1.1] is used, which hs n intensity six times tht of Trffic Model 1 nd is more bursty in comprison, fr fewer thn 3 devices my relibly be offered services. Clerly, we will hve low probbility of rndom ccess success with this trffic if 3 devices re used, s is evident from the pcket-level simultion in [17, Tble ]. Thus, we cn conclude tht the model cn be used to mke predictions bout the cpcity of cell s well. E. Dimensioning the PDCCH resource In this section we will demonstrte method to dimension the PDCCH resource to be used in cell under the two trffic models mentioned in [17], i.e., determine the minimum number of CCEs, N min, tht is necessry for the system to operte properly under these two trffic models for scenrios P P7. More specificlly, we fix the prmeters b, c, N Msg2 nd N Msg4 nd determine the vlues of N (nd consequently those of M nd m s well) for different scenrios tht will TABLE V: PDCCH resource size in CCEs needed to support Trffic Model 1 [17, Tble 6.1.1]. Scenrio M m N Msg2 N Msg4 N min P P P P P P P P TABLE VI: PDCCH resource size in CCEs needed to support Trffic Model 2 [17, Tble 6.1.1]. Scenrio M m N Msg2 N Msg4 N min P P P2 P P4 P P6 P llow them support the trffic models described in [17]. In Trffic Model 1 3 devices mke rndom ccess ttempts uniformly over 6 second period. This mens tht mximum throughput of θ 3 = 6 sec =. requests per subfrme should be supported. In Tble V, we show the minimum number of CCEs necessry to sustin t most 3 devices. In summry, in scenrios P nd P1 we need just N min = 4 CCEs in the PDCCH to send Messge 2 s nd Messge 4 s. In P2, P3, P4, nd P minimum of N min = 8 CCEs is sufficient in the PDCCH for sending Messge 2 s nd Messge 4 s. In the finl two scenrios P6 nd P7, where we hve rndom ccess opportunity in every subfrme (b = 1), we gin need minimum of N min = 4 CCEs to sustin the rrivl rte described in Trffic Model 1. It should be noted tht in ll our scenrios t lest 4 or 8 CCEs re necessry to send ech Messge 2 or Messge 4, which constrins the choice of these minimum number of necessry CCEs to be the multiples of 4 or 8. The rrivl rte is six times higher in Trffic Model 2 compred to the first. Under these conditions, mximum 3 sec throughput of θ = = 3. rrivls per subfrme should be supported by the system. From (1), we see tht with c = 3, this throughput is never chieved mking it impossible for scenrios P, P2, P4 nd P6 to sustin the rrivl rte described in Trffic Model 2 with ny number of CCEs. In P1, P3, P nd P7, where c = 6 uplink grnts re provided in Messge 2, we require N to be t lest 16, 24, 32 nd 16, respectively to sustin the trffic. The results for Trffic Model 2 re summrized Tble VI. VIII. CONCLUSIONS The PDCCH of LTE my become bottleneck when very lrge number of devices wnt ccess to the network. We hve presented Mrkov chin model to describe the shring of the PDCCH resources between Messge 2 s nd Messge 4 s. Using the model we hve clculted the contribution of

12 12 vrious events in the rndom ccess procedure s filure. In ddition, we hve derived method to determine the mximum throughput of the system, which revels the upper limit for the rrivl rte of fresh rndom ccess requests. This method is then used to dimension the PDCCH resource size to support the trffic models studied by 3GPP in [17]. We hve observed tht ner the stbility limit, the probbility of filure increses very shrply. Indeed, it is esy to see tht by dmitting more users to the system in Step 2, the cpcity (mesured by the mximum throughput) of the rndom ccess chnnel cn be modestly incresed. However, this lso increses the probbility of the rndom ccess filure due to lrge queuing dely in the Messge 4 buffer. This is becuse the size of the PDCCH resource is fixed nd Messge 2 s lwys hve priority over Messge 4 s. Moreover, using our model we re lso ble to predict the mximum number of devices tht cn exist in cell under uniform trffic conditions. Anlysis lso shows tht reducing the number of CCEs in ech Messge 2 or Messge 4 increses the cpcity of the rndom ccess chnnel but the behvior of the queue ner the stbility limit remins more or less the sme. So limiting the rrivl rtes by some kind of dmission control is necessry to mnge this overlod. Our model cn be extended to study the reliztion of collision in Step 1 of the rndom ccess procedure including the impct of the cpture effect. These extensions will, no doubt, give better performnce bounds but it remins to be seen how much improvement cn they relly offer. Additionlly, the impct of physicl lyer impirments on the rndom ccess procedure cn be studied by modifying the model, e.g., by explicitly tking into ccount the retrnsmission mechnism nd fding effects. As prt of future work, we cn lso study the impct of sending further messges in the PDCCH fter the rndom ccess procedure is finished. Moreover, techniques like enhnced PDCCH hve been proposed by 3GPP to overcome the overlod issues of PDCCH which cn lso be subject of further study. APPENDIX A PROOF OF PROPOSITION 1 Here we give the proof for the necessry nd sufficient condition for the stbility of the Messge 4 buffer. Proposition 1: The buffer for Messge 4 s is stble if nd only if θ (2) () < σ (4) (). Proof: ) Assume first tht c m. Then we know from the previous section tht there will be no queue t ll in the Messge 4 buffer (which, of course, is one form of stbility). On the other hnd, we hve in this cse θ (2) () < c m σ (4) () by (16) nd (21). So the clim is true whenever c m. b) Assume now tht c > m, nd consider the Mrkov chin (X n, Y n (3) ) defined on E = {, 1,...} {, 1,..., bc}. The buffer for Messge 4 s is stble if nd only if this irreducible nd periodic Mrkov chin is positive recurrent, i.e., there is unique stedy-stte distribution π ij = lim Pr{X n = i, Y n (3) = j}. n Now, depending on, we hve two cses to consider: 1 θ (2) () < σ (4) () nd 2 θ (2) () σ (4) (). 1 Assume first tht is such tht θ (2) () < σ (4) (), (26) nd define δ = b(σ (4) () θ (2) ()) >. For ny (i, j) E, we hve E[X n+1 + Y (3) n+1 X n = i, Y (3) n = j] = E[X n Y n (4) + Y n (3) X n = i, Y n (3) = j] + E[Y (3) n+1 ] E[X n + Y n (3) X n = i, Y n (3) = j] + E[Y n (2) ] = i + j + bθ (2) () = i + j δ(1 ɛ), where ɛ = bσ (4) ()/δ. In ddition, for ny (i, j) E such tht i > bm, we hve E[X n+1 + Y (3) n+1 X n = i, Y (3) n = j] = E[X n Y n (4) + Y n (3) X n = i, Y n (3) = j] + E[Y (3) n+1 [ ] ] = i E bm Ỹ (2) n /c (M m) + j + E[Y n (2) ] = i bσ (4) () + j + bθ (2) () = i + j δ. Thus the non-negtive function V defined on E by stisfies Foster s criterion: V (i, j) = (i + j)/δ E[V (X n+1, Y (3) n+1 ) V (i, j) X n = i, Y (3) n = j] 1 + ɛ 1 F (x), where F is the finite set F = {, 1,..., bm} {, 1,..., bc}, nd 1 F (x) = 1 if x F, otherwise. It follows from Foster s Theorem (see e.g. [23]) tht the Mrkov chin (X n, Y n (3) ) is positive recurrent under condition (26). 2 Assume now tht is such tht For ny (i, j) E, we hve Thus, θ (2) () σ (4) (). (27) E[ X n+1 + Y (3) n+1 i j X n = i, Y (3) n = j] = E[ Y (4) n + Y (3) n+1 X n = i, Y (3) n = j] E[bM + Y (3) n+1 X n = i, Y n (3) = j] = bm + E[Y n (2) ] = b(m + θ (2) ()). sup E[ V (X n+1, Y (3) n+1 ) V (i, j) X n = i, Y n (3) = j] <, (i,j) E

13 13 where V is now defined on E by V (i, j) = i + j. In ddition, s shown in 1, we hve, for ny (i, j) E such tht i > bm, E[X n+1 + Y (3) n+1 i j X n = i, Y (3) n = j] = b(θ (2) () σ (4) ()), implying, by (27), tht E[V (X n+1, Y (3) n+1 ) V (i, j) X n = i, Y (3) n = j] for ll x E \ F, where the finite set F is defined s in 1. These conditions together re sufficient (see e.g. [23]) to show tht the Mrkov chin (X n, Y n (3) ) is not positive recurrent under condition (27). APPENDIX B SIMULATION STUDY ON THE POISSON APPROXIMATION Recll tht due to lrge number of mchines mking independent rndom ccess ttempts, we ssume tht the fresh requests rrive ccording to Poisson process with n verge of λ rrivls per ms. Here we exmine, through simultions, the ccurcy of our Poisson pproximtion, which sttes tht the ggregte rndom ccess requests (the fresh ones s well s the retrnsmissions) constitute, pproximtely, Poisson process. The simultor mimics the four-step rndom ccess procedure in n LTE system nd explicitly tkes into ccount bckoff lgorithm, which ws not directly considered in the model presented in Section IV nd the subsequent nlyses. This mechnism hndles the bckoffs which my be cused by one of three events if there is collision in Step 1 (relized only in Step 3), or if there is loss in Step 2, or if the dely is beyond the cceptble limit in Step 4 s explined in Section III. More specificlly, if fresh request fils (due to ny one of the three events described erlier), our simultor models the bckoff mechnism by ssigning some probbility of retrnsmission, P ReTx, to tht request. This mens tht such bcklogged users will mke n ttempt for retrnsmission in the subsequent RACH opportunities with probbility P ReTx. After the first filure, mximum of N mx 1 more retril ttempts cn be mde to send Messge 1 until the ttempt is successful. If the request is not successful even fter N mx ttempts, the request is dropped. We work t the grnulrity of time slots of length b = ms. A rndom ccess request will bckoff immeditely if it does not receive Messge 2 in the following time slot (loss in Step 2). A bckoff due to collision in Step 1 occurs time slots fter the corresponding Messge 3 is sent (collision in Step 1, relized in Step 3). A request queued up in the Messge 4 buffer (nd not yet sent) enters the bckoff stte if the corresponding Messge 3 ws sent time slots erlier (dely in Step 4). Note tht time slots correspond to time of ms, which is close to the timer vlue of 48 ms used in [17]. The bse sttion hs wide rnge of choices for Bckoff Prmeter vlues [2, Tble 7.2-1]. After filed trnsmission ttempt, user will uniformly select time between nd the Bckoff Prmeter, T B, to remin in the bckoff stte. This mens tht if T B = 2 ms (s done in [17, Tble ]), retrnsmission ttempt is mde by such user, on the verge, fter T B = ms (i.e, fter two time slots in our simultion model) nd probbility of retrnsmission P ReTx = b/ T B = / =. cn be ssigned to ll bckoff users. Similrly, if the T B = 16 ms (which is lso possible ccording to [2, Tble 7.2-1]), retrnsmission tkes plce fter of T B = 8 ms in verge, mking P ReTx = b/ T B = /8 =.62. As the upper bound for the number of trils, we use N mx = (s in [17, Tble ]) but lso N mx = to see how vrying the mximum number of trils ffects the distribution of the ggregte number of rndom ccess requests. Numericl results We hve run simultions for Scenrio P1 described in Tble III, where the different prmeter vlues used re b = ms, c = 6, M = 4, nd m = 3. The dditionl prmeters for the simultion re N mx nd P ReTx. Recll tht the highest rte of fresh rrivls (λ) tht this scenrio cn support is θ = 3.22 requests per ms s observed from Figure 3 nd the discussion preceding it. To be s exhustive s possible, we run the simultion for three trffic conditions low trffic (λ = 1.), medium trffic (λ = 2.), nd high trffic (λ = 3.), where ll the rrivl rtes re expressed per ms. Ech simultion run consists of time slots. We first present the results for the cse where P ReTx = 1/2 =. nd N mx =. In Figure 11 (nd ll the subsequent ones), we hve plotted the empiricl distribution of the ggregte number of ll rndom ccess requests rriving in time slot s the br chrt nd overlyed it with the probbility mss function of the Poisson distribution with the men b predicted from the theoreticl model presented in the pper. Recll tht refers to the ggregte rrivl rte of ll rndom ccess requests (per ms). We see tht for ll the three trffic conditions, the empiricl distribution is very close to the corresponding theoreticl Poisson distribution. The empiricl ggregte request rte âb is lso very close to the predicted vlue b s shown in Tble VII. As expected, for low nd medium rrivl rtes the empiricl distribution is very close to Poisson distribution. However, the empiricl distribution bers striking resemblnce to Poisson distribution even t hevy trffic (close to the stbility limit θ ). Similrly, if we reduce the vlue of N mx to, hrdly ny chnge is noticed in the empiricl distribution. This cn be seen in Figure 12. In ddition, if we use longer Bckoff Prmeter of 16 ms, corresponding to P ReTx = 1/16 =.62, we notice tht the distributions re even closer to Poisson distribution s observed from Figure 13. Moreover, the Bckoff Prmeter cn be s high s 96 ms, in which cse P ReTx = 1/96., which helps to mke the empiricl distribution even closer to the Poisson distribution. In summry, we cn sy tht the distribution for the ggregte number of ll rndom ccess requests my be pproximted by Poisson distribution.

14 14 b obtined from simultions for different vlues TABLE VII: The predicted vlue of b nd the corresponding empiricl vlue b of prmeters λ, PReTx, nd Nmx. λ b PReTx =., Nmx = b b PReTx =., Nmx = PReTx =.62, Nmx = b =.4 () λ = 1., b b = (b) λ = 2., b b = 23.7 (c) λ = 3., b Fig. 11: Empiricl distribution of Messge 1 when PReTx =. nd Nmx = b =.6 () λ = 1., b b = 12.6 (b) λ = 2., b b = (c) λ = 3., b Fig. 12: Empiricl distribution of Messge 1 when PReTx =. nd Nmx = b =.3 () λ = 1., b b = (b) λ = 2., b b = 23. (c) λ = 3., b Fig. 13: Empiricl distribution of Messge 1 when PReTx =.62 nd Nmx = R EFERENCES [1] J. Conti, The internet of things, Communictions Engineer, vol. 4, no. 6, pp. 2 2, Dec.-Jn. 26. [2] M. Z. Shfiq, L. Ji, A. X. Liu, J. Png, nd J. Wn, A first look t cellulr mchine-to-mchine trffic lrge scle mesurement nd chrcteriztion, in Proceedings of the 212 ACM SIGMETRICS interntionl conference on Mesurement nd modeling of computer systems. ACM, 212. [3] 3GPP, Bckoff enhncements for RAN overlod control, 3rd Genertion Prtnership Project (3GPP), R , 211. [4], System improvements for Mchine-Type Communictions (MTC), 3rd Genertion Prtnership Project (3GPP), TR , 211, V1.6. (211-11). [] S.-Y. Lien, K.-C. Chen, nd Y. Lin, Towrd ubiquitous mssive ccesses in 3GPP mchine-to-mchine communictions, IEEE Communictions Mgzine, vol. 49, no. 4, pp , April 211. [6] A. Lrmo nd R. Susitivl, RAN overlod control for Mchine Type Communictions in LTE, in GLOBECOM Workshops (GC Wkshps), 212 IEEE, Dec [7] D. Bertseks nd R. Gllger, Dt Networks, 2nd ed. Prentice-Hll, [8] M. Rivero-Angeles, D. Lr-Rodriguez, nd F. Cruz-Perez, A new EDGE medium ccess control mechnism using dptive trffic lod slotted ALOHA, in Technology Conference. IEEE VTS 4th, vol. 3, 21, pp [9] G. Wng, X. Zhong, S. Mei, nd J. Wng, An dptive medium ccess control mechnism for cellulr bsed Mchine to Mchine (M2M) communiction, in Wireless Informtion Technology nd Systems (ICWITS) Interntionl Conference on, September 2. [] I. E. Pountourkis nd E. D. Syks, Anlysis, stbility nd optimiztion of Aloh-type protocols for multichnnel networks, Computer Communictions, vol. 1, no., pp , [11] J.-B. Seo nd V. Leung, Design nd nlysis of bckoff lgorithms for

15 1 [12] [13] [14] [1] [16] [17] [18] [19] [2] [21] [22] [23] rndom ccess chnnels in UMTS-LTE nd IEEE systems, IEEE Trnsctions on Vehiculr Technology, vol. 6, no. 8, pp , 211., Performnce modeling nd stbility of semi-persistent scheduling with initil rndom ccess in LTE, IEEE Trnsctions on Wireless Communictions, vol. 11, no. 12, pp , 212. K.-D. Lee, S. Kim, nd B. Yi, Throughput comprison of rndom ccess methods for M2M service over LTE networks, in GLOBECOM Workshops (GC Wkshps), 211 IEEE, Dec. 211, pp J.-P. Cheng, C.-h. Lee, nd T.-M. Lin, Prioritized Rndom Access with dynmic ccess brring for RAN overlod in 3GPP LTE-A networks, in GLOBECOM Workshops (GC Wkshps), Dec. 211, pp S. Choi, W. Lee, D. Kim, K.-J. Prk, S. Choi, nd K.-Y. Hn, Automtic configurtion of rndom ccess chnnel prmeters in LTE systems, in Wireless Dys (WD), Oct. 211, pp M.-Y. Cheng, G.-Y. Lin, H.-Y. Wei, nd A.-C. Hsu, Overlod control for mchine-type-communictions in lte-dvnced system, IEEE Communictions Mgzine, vol., no. 6, pp. 38 4, June GPP, Study on RAN improvements for mchine-type communictions, 3rd Genertion Prtnership Project (3GPP), TR , 211, V1.. (211-8)., Evolved Universl Terrestril Rdio Access (E-UTRA) nd evolved universl terrestril rdio ccess network (E-UTRAN); Overll Description, 3rd Genertion Prtnership Project (3GPP), TS 36.3, 211, V.6. (211-12). E. Dhlmn, S. Prkvll, nd J. Sko ld, 4G LTE/LTE-Advnced for Mobile Brodbnd. Acdemic Press, GPP, Evolved Universl Terrestril Rdio Access (E-UTRA); Medium Access Control (MAC) protocol specifiction, 3rd Genertion Prtnership Project (3GPP), TS , 212, V11.. (212-9). R. Nelson, Probbility, Stochstic Processes nd Queueing Theory The Mthemtics of Computer Performnce Modeling. Springer, 199. S. Ross, Stochstic Processes, 2nd ed. Wiley, S. Meyn nd R. Tweedie, Mrkov Chins nd Stochstic Stbility. Springer, Smuli Alto received his M.Sc. nd Ph.D. degrees in Mthemtics from the University of Helsinki in 1984 nd 1998, respectively. From 1984 to 1997, Dr. Alto worked s Reserch Scientist t VTT Technicl Reserch Center of Finlnd. Since 1997, he hs been with TKK Helsinki University of Technology, which is now prt of Alto University. Currently he cts s Chief Reserch Scientist leding the Teletrffic nd Performnce Anlysis Group in the Deprtment of Communictions nd Networking. Dr. Alto s reserch interests include queueing theory, teletrffic theory, nd performnce nlysis of modern communictions systems nd networks. Ann Lrmo is senior resercher t Ericsson Reserch in Finlnd. She received her MSc in communiction technology in 2 from Helsinki University of Technology. She hs been with Ericsson since 24, working with 3G nd 4G technologies. Her current reserch interests include the Internet of Things technologies, rdio protocols, performnce evlution, rdio resource mngement, nd simultor development. She hs lso been ctive in the res of innovtion nd ptenting. Prjwl Osti received his bchelor s degree in electronics nd communictions engineering from from Tribhuvn University Institute of Engineering, Nepl in 27 nd MSc in communiction engineering from Alto university, Finlnd in 211. Currently he is pursuing PhD degree t the Alto University School of Electricl Engineering. His current reserch interests include scheduling in wireless networks nd Internet of Things communictions. Psi Lssil is senior reserch scientist t the COMNET Deprtment in the Alto University School of Electricl Engineering. Dr. Lssil received his Ph.D. degree in 21 nd since then hs published widely on the mthemticl modeling nd performnce evlution of networking technologies. His current reserch interests include flow-level performnce of scheduling nd resource mngement methods in cellulr networks, cpcity limits of multihop wireless networks, mobility modeling nd its impct on wireless networks. Tuoms Tirronen received his D.Sc. in Communictions Engineering in 2 from Alto University. Since 212 he hs been working in Ericsson Reserch s wireless ccess networks resercher. His reserch interests include 4G nd G, Internet of Things, performnce evlution, rdio protocols nd resources. He is lso ctive in 3GPP stndrdiztion work nd innovtion nd ptenting.