AMACHINE-to-machine (M2M) communication system

Transcription

1 Optial Access Class Baing fo Stationay Machine Type Counication Devices with Tiing Advance Infoation Zehua Wang Student Mebe IEEE and Vincent W.S. Wong Senio Mebe IEEE Abstact The cuent wieless cellula netwoks can be used to povide achine-to-achine MM) counication sevices. Howeve the Long Te Evolution LTE) netwoks which ae designed fo huan uses ay not be able to handle a lage nube of busty ando access equests fo achine-type counication MTC) devices. In this pape we popose a schee that uses both access class baing ACB) and tiing advance infoation to pevent ando access oveload in MM systes. We foulate an optiization poble to deteine the optial ACB paaete which axiizes the expected nube of MTC devices successfully seved in each ando access slot. Hence the nube of ando access slots equied to seve all MTC devices can be iniized. To educe the coputational coplexity and ipove the pacticability of the poposed schee we popose a closed-fo appoxiate solution to the optiization poble and pesent an algoith to estiate the nube of active MTC devices equiing access in each ando access slot. The coectness of the analytical odel and the accuacy of the estiation algoith ae validated via siulations. esults show that both nueical and appoxiate solutions povide the sae pefoance. Ou poposed schee can educe nealy half of the ando access slots equied to seve all MTC devices copaed to the existing schees which use tiing advance infoation only ACB only o coopeative ACB. Index Tes Machine type counications LTE ando access ando access oveload. I. INTODUCTION AMACHINE-to-achine MM) counication syste consists of a lage nube of achine-type counication MTC) devices which can counicate with the eote seve o othe MTC devices in a pee-to-pee anne. MM is leading us to the Intenet of Things. Its applications include sat eteing eote secuity sensing health cae onitoing and fleet tacking. It is expected that oe than 3. billion MTC devices will be deployed by 9 []. Since potential MM applications usually equie sealess coveage ove a lage aea one appoach to povide MM Manuscipt eceived July 4; evised Decebe 6 4 and Mach 6 5; and accepted May 3 5. This wok was suppoted by the Natual Sciences and Engineeing eseach Council NSEC) of Canada. Pat of this wok was pesented at the IEEE Intenational Confeence on Counications ICC) Sydney Austalia June 4. The associate edito coodinating the eview of this pape and appoving it fo publication was Ping Wang. The authos ae with the Depatent of Electical and Copute Engineeing The Univesity of Bitish Colubia Vancouve BC V6T Z4 Canada e-ail: zwang@ece.ubc.ca; vincentw@ece.ubc.ca). Colo vesions of one o oe of the figues in this pape ae available online at Digital Object Identifie.9/TWC.5.XXXXXXX sevices is via the existing wieless cellula netwoks. Meanwhile the 3d Geneation Patneship Poject 3GPP) Long Te Evolution LTE) netwoks allow MTC devices to connect to eote seves o devices in othe netwok doains []. Howeve the LTE netwoks which ae designed fo huan-tohuan HH) counications ay not be optial fo MM taffic. MM counications diffe fo HH counications in seveal aspects [3]. Fo MM taffic the data payload can be only seveal bytes which is uch salle than the payload in HH taffic. Busty ando access equests fo any MTC devices ay be sent to the sae base station o evolved node B enb) siultaneously. Since the nube of MTC devices can be uch lage than the nube of huan uses contention aong MTC devices fo ando access which seldo happens in HH counications can occu in the MM context. This type of contention is called ando access oveload [4]. To undestand how ando access oveload ay degade the pefoance of LTE netwoks we now suaize the pocedues of ando access fo a use equipent UE) o an MTC device. A UE fist synchonizes its downlink by listening to the synchonization signals sent by enb. Then the aste infoation block is eceived which guides the UE to eceive the syste infoation blocks SIBs). The SIBs help UEs to locate the efeence signal obtain valid ando access peables and locate ando access slots [5]. ando access peables in LTE netwoks use Zadoff-Chu ZC) sequences [6]. The afoeentioned SIBs specify the 64 ZC sequences used fo ando access in the cell. ZC sequences ae used as ando access peables since an enb can distinguish each ZC sequence fo the ovelapping eceived signal and can deteine its popagation delay. This is because the discete autocoelation of a ZC sequence ceates an ipulse while two diffeent ZC sequences ae consideed to be othogonal [7]. Conside thee UEs n n and n 3 in Fig. a) as an exaple. Without loss of geneality two popagation paths ae assued fo each UE due to ultipath effect. UEs n n and n 3 tansit ando access peables A A and A 3 espectively. The cyclic pefix CP) is included fo tansission of each ando access peable [8]. Due to diffeent popagation distance fo the UEs to the enb and the ultipath effect oe than one copy of each peable with diffeent factions of CP ae captued by the enb in its obsevation inteval in Fig. b). The captued signal at enb is the suation of all signals eceived as shown in Fig. c). The enb

2 UE:n 3 Peable: A 3 Discete coelation UE: n Peable: A enb UE:n Peable: A Multiple copies of the sae peable sent by a UE ay be eceived by enb via ultiple popagation paths. a) t i: the popagation delay of n i is evaluated by the location of ipulse shifted fo i 3). t Fo peable A Fo peable A t 3 t Tie d) Fo peable A 3 CP A {}}{ CP CP esult signal captued + CP CP A CP A A A 3 A 3 obsevation inteval b) A A A A A 3 A 3 Fig.. Peable detection and popagation delay evaluation in ando access of LTE netwoks. UEs n n and n 3 ae awae of the 64 ando access peables used in the cell. They select peables A A and A 3 and tansit the with cyclic pefix CP) to the enb. By calculating the coelation between the ovelapping cyclic shifted peable sequences and each of the 64 peables the peables A A and A 3 and thei popagation delay can be deteined. deteines whethe the captued signal contains a specific peable by calculating its discete coelation with each of the 64 peables. The signal contains a specific peable as long as its discete coelation contains an ipulse in tie doain. Moe than one ipulse with diffeent enegy fo a peable ay exist since copies of the sae peable with diffeent factions of CP ae contained in the captued signal. The ipulse with the lagest aplitude deteines the popagation delay as shown in Fig. d). Fig. pesents the fist thee steps of ando access in LTE netwoks. Afte eceiving the ando access A) peable tansitted by a UE in Step the peable index and its associated popagation delay ae deteined. Then the enb sends a ando access esponse A) to acknowledge the UE. An A contains the following fields: a) a nube to identify a ando access slot b) the index of the peable eceived c) the tiing advance coand [8] [9] and d) the esouce allocation infoation. With the afoeentioned fields in an A a) and b) ae used togethe to addess an A to the UEs. The tiing advance coand in c) takes an index value called tiing advance to convey the popagation delay by a ultiple of 6 T s whee T s denotes the basic tie unit and is equal to / 3.7 7) second [6]. In othe wods the popagation delay deteined in Fig. d) is quantized to an index value with the ganulaity of 6 T s. Tiing advance coand synchonizes the uplink by infoing the UE the aount of tie that its data should be tansitted in advance so that the data will aive at enb at the anticipated tie. The esouce allocation infoation in d) is used to schedule the tansission of L/L3 essage in Step 3 fo the eceive of the A. Soe UEs ay send the sae peable via the sae ando access slot. Thus these UEs will eceive the sae A and send thei L/L3 essages ove the sae wieless channel. This ay cause packet collisions at the enb as shown in Fig.. Copaed to HH the pobability of this kind of packet collisions inceases in the MM ando access c) Fig.. The fist thee steps of ando access in LTE netwoks. Multiple UEs o MTC devices ay eceive the sae A if they send the sae peable in the sae ando access slot so thei L/L3 essages ay be tansitted on the sae wieless channel and packet collisions ay occu at the enb. oveload scenaio since the nube of MTC devices equiing access to an enb is uch lage than the nube of UEs. Thus ando access oveload of MTC devices ay degade the pefoance of LTE netwoks. Vaious woks have been poposed to ipove the pefoance of LTE netwoks seving MTC devices []. Lee et al. in [] study the thoughput issue and popose to split the ando access peables into two sets to seve conventional data applications of UEs and shot data applications of the MTC devices sepaately. With the goup paging appoach Wei et al. in [] popose a odel to estiate the nube of successful and collided MTC devices in each ando access slot. Liu et al. popose a hybid ediu access contol potocol fo MTC devices in [3]. The MTC devices contend fo the tansission oppotunities in the fist peiod. Only successful MTC devices ae assigned a tie slot fo thei data tansissions in the second peiod. MTC devices can be gouped with soe MTC gateway devices [4]. Tu et al. in [5] and Fu et al. in [6] notice that those non-echageable MTC devices have liited enegy and popose echaniss to aggegate seveal shot data packets at the gateway MTC device and send the togethe in an enegy-efficient anne. Zhou et al. in [7] use a sei-makov chain to deteine the optial nube of shot packets in an aggegated packet with a given packet collision ate. Access class baing ACB) can be used to educe ando access oveload in LTE netwoks by boadcasting an ACB paaete b whee b to all MTC devices via SIBs [8]. When an MTC device wants to connect to an enb it fist geneates a ando nube between [ ] unifoly. It joins the ando access contention only if the geneated value is less than the ACB paaete b boadcasted by the enb. Lien et al. in [9] popose a coopeative ACB schee to contol the ACB paaetes on ultiple enbs to seve MTC devices efficiently. Chou et al. in [] popose to estiate the ACB paaetes by pedicting the nube of MTC devices equiing ando access. Duan et al. in [] popose to dynaically update the ACB paaete based on the nube of packet collisions occued in the past. Fo a stationay MTC device since its popagation delay to the enb is a constant the tiing advance infoation in As sent in ultiple ando access slots is identical. A ando access potocol fo stationay MTC devices is poposed in []. Each MTC device stoes the tiing advance eceived in a successful ando access and copaes the stoed value to the tiing advance in subsequent ando access. It sends

3 its L/L3 essage only if the sae tiing advance is eceived. By copaing the tiing advance infoation the pobability of packet collisions in Step 3 of ando access is educed because not all MTC devices tansit thei L/L3 essages afte they eceive the sae A. Howeve tiing advance is an index value obtained by quantizing the popagation delay in a ganulaity of 6 T s. It ay be identical fo two MTC devices if the diffeence between thei popagation distance to the enb is less than 6 ct s i.e. 56 whee c is the speed of light. Thus only copaing the tiing advance infoation ay not be sufficient to educe the ando access oveload since oe MTC devices ay have the sae tiing advance infoation when the density of MTC devices inceases. In this pape we popose a schee that jointly uses ACB and tiing advance infoation to educe ando access oveload. Ou contibutions ae as follows: We foulate an optiization poble to find the optial ACB paaete which axiizes the expected nube of MTC devices successfully seved in each ando access slot. Diffeent fo ou pevious wok in [3] the inteval analysis is used in this pape to deteine the nueical solution of the optiization poble. To educe the coputational coplexity and ipove the pacticability of ou poposed schee we popose a closed-fo appoxiate solution fo the optiization poble. We futhe pesent an algoith to estiate the nube of MTC devices that equie access to an enb in each ando access slot. Ou syste odel is validated via siulations. Siulation esults show that the appoxiate solution obtains the sae pefoance as the nueical solution with the poposed schee in educing ando access oveload. Futheoe alost 5% ando access slots can be educed by the poposed schee when copaed to the existing schees that use tiing advance infoation only [] ACB only [] o coopeative ACB [9]. The est of the pape is oganized as follows. In Section II we intoduce ou syste odel and poble foulation. The nueical and closed-fo appoxiate solutions fo the foulated poble and the algoith to estiate the nube of MTC devices ae pesented in Section III. Section IV pesents the siulation esults. Conclusions ae given in Section V. II. SYSTEM MODEL AND POBLEM FOMULATION Conside a set of stationay MTC devices N within the coveage of an enb in LTE netwoks whee N {... N}. MTC devices that equie access to an enb send thei ando access peables in peiodic ando access slots. Since the MTC devices ae stationay the popagation delay fo each MTC device is a constant. The tiing advance is an index value i.e....) afte quantizing the popagation delay with the ganulaity of 6 T s [6]. Thus popagation delays that ae close can still be quantized to two diffeent consecutive index values when thee is a ultiple of 6 T s between the. Diffeent popagation delays ay be quantized to the sae tiing advance when they ae within the sae quantization ganulaity. To siplify ou syste odel we assue that popagation delays ae quantized to the sae tiing advance if thei diffeence is less than o equal to half of the quantization ganulaity. This diffeence is denoted by τ. Thus τ is equal to 8 T s. Let t i and TA i denote the popagation delay and tiing advance of MTC device i N espectively. Fo an MTC device j N \ {i} we have TA i T j A if t i t j τ and TA i T j A if t i t j > τ. We efe to the axiu popagation distance of the MTC devices in set N as the deployent ange which is deteined by enb as ax i N c t i. ) Each MTC device i N has stoed its tiing advance infoation TA i in the pevious successful ando access. When an MTC device tansits a ando access peable in a ando access slot and eceives an A it sends its L/L3 essage only if the tiing advance in the eceived A atches TA i. Since the oot ean squae delay spead in icocells of uban aea is.5 µs [4 pp. 443] which is less than τ 8 T s it is feasible to copae TA i fo MTC device i in diffeent ando access slots. Conside MTC devices n n and n 3 in Fig. a) as an exaple and assue thei popagation delays satisfy t t > τ t t 3 > τ and < t t 3 < τ. In this exaple we have N {n n n 3 } TA > T A T A 3 and ct. Assue they tansit the sae ando access peable in a ando access slot. At least thee copies of the peable ae eceived by enb with siila eceived powe [8]. Since only one A is tansitted fo the sae peable n n and n 3 will eceive the sae A. In this A TA T A and T A 3 have the sae pobability to be used as the tiing advance. If TA is included in A since TA and T A 3 ae the sae n and n 3 will send thei L/L3 essages in the sae wieless channel and packet collision will occu. If TA is used in A since TA diffes fo both T A and T A 3 n and n 3 will not send thei L/L3 essages. The L/L3 essage sent by n will be successfully eceived by enb. Let ct denote the popagation distance of an MTC device with popagation delay t. Let d 8 ct s denote the iniu diffeence between the popagation distance of two MTC devices with diffeent tiing advance infoation. We conside that the MTC devices in set N ae unifoly distibuted. The pobability that a andoly selected MTC device has the sae tiing advance infoation as the MTC device that has popagation distance to the enb is +d γ dγ ) +d if < d +d 4d p ) γ dγ d if d d d γ dγ ) d if d <. ) Conside an MTC device u N we denote I u as the event that u passed the ACB check and I u othewise. If the ACB paaete fo the cuent ando access slot is b we have the pobability P I u ) b. Let Y u denote a ando vaiable which epesents the nube of additional MTC devices that also passed the ACB check in the cuent ando access slot. Then Y u follows a binoial distibution BN b). We use Γ u to epesent the

4 event that an abitaily chosen MTC device u has popagation distance fo the enb. Given Γ u the conditional pobability that thee ae i additional MTC devices which passed the ACB check and contend with u in the cuent ando access slot is P Y u i I u Γ u ) P Y u i I u ) P Y u i) P I u ) ) N b) N i b i+ i... N. 3) i Note that the ando vaiables I u and Y u ae independent of the position of MTC device u. Conside thee ae peables in total. Let J u j denote the event that u selected peable j fo peables in a unifo anne. Let L u N \ {u} denote the set of othe MTC devices except u that have passed the ACB check and have chosen peable j. The cadinality of L u denoted as L u L u is a ando vaiable. We have P L u k J u j Y u i I u Γ u ) ) ) k i i k 4) k ) k... i j.... With L u k J u j Y u i I u and Γ u i.e. given the event that MTC device u whose popagation distance is has passed ACB check and i othe MTC devices have passed ACB check as well and eanwhile aong these i MTC devices k MTC devices tansitted the sae peable j as u) MTC device u succeeds in the ando access if the following two conditions ae satisfied: a) u s popagation delay is quantized as the tiing advance infoation and included in A; b) the othe k MTC devices that eceive the sae A do not have the sae tiing advance of u. Let S u o S u ) denote the event that MTC device u succeeds o fails) in the cuent ando access. The conditional pobability of S u is P S u L u k J u j Y u i I u Γ u ) k ) p )) p )) k k+ ) p ))k. 5) k + Fo 4) and 5) we have P S u L u k J u j Y u i I u Γ u ) P S u L u k J u j Y u i I u Γ u ) P L u k J u j Y u i I u Γ u ) ) ) k p ))k i ) i k k + ) k ) i ) ) k i p ) 6) k + ) k k... i j.... Thus fo i... N and we obtain P S u I u Y u i Γ u ) i P S u L u k J u j Y u i I u Γ u ) j k ) i i k ) i k + i k ) p ) ) i ) ) k+ p ) i + p ) i + k + k ) i ) p ) + p ) ) i+ ) i + ) i + φ )) i+ 7) φ ) i + ) whee φ ) p) and i+ ) i+) k+ i k+) k) is used in the thid step. Fo 3) and 7) we have P S u I u Γ u ) N i N ) k P S u I u Y u i Γ u ) ) N b) N i b i+ i i ) i + φ )) i+ φ ) i + ) b)n φ ) ) N i + φ )) i+ i + N i ) ) i+ b ). b) 8) Since N ) N i i + ) N i+) equation 8) becoes P S u I u Γ u ) b)n φ ) N ) N ) ) ) i+ + φ )) i+ N b ). 9) i + b) i We futhe have and N i N i + N i + + φ )) i+ N i + + φ )) b ) b) ) b ) b) ) i+ + ) ) i+ b ) b) ) N ) b ) b) ) N. )

5 Equation 9) becoes P S u I u Γ u ) b)n φ ) N ) + + φ )) b ) b) By substituting φ ) p) ) N + into ) we obtain ) ) N b ). b) ) P S u I u Γ u ) b ) N N p )) p ) b ) ) N. 3) An MTC device has to pass ACB check befoe being seved in a ando access slot. The pobability that MTC device u with popagation distance does not pass ACB check but succeeds in the ando access contention is zeo i.e. P S u I u Γ u ). Thus we have P S u Γ u ) P S u I u l Γ u ) l P S u I u Γ u ). 4) Since MTC devices in set N ae unifoly distibuted we have P S u ) N p ) b ) N p ) b ) ) N d. 5) Fo 5) we can obtain the pobability that an abitay MTC device succeeds in the cuent ando access slot with an ACB paaete b. Let ando vaiable Z denote the nube of MTC devices that succeed in ando access. ando vaiable Z follows a binoial distibution. That is Z BN P S u )). The expectation of Z is given by E [Z] NP S u ) p ) b ) N p ) b ) ) N d. 6) Note that equation 6) is fo one ando access slot. In a busty equest scenaio enb needs to seve ultiple MTC devices in a nube of consecutive ando access slots. To educe the total nube of ando access slots equied to seve all MTC devices we need to find the optial ACB paaete b that axiizes E [Z] in each ando access slot. Theefoe the optiization poble can be foulated as axiize b E [Z] subject to b. 7) Fo 6) we notice that the objective function in poble 7) does not have a closed-fo expession and it is difficult to solve. Deteining the nueical solution fo the optiization poble with inequality constaints has been studied in [5 pp. 343] which povides an algoith to find the best solution afte unning the inteval Newton s ethod [6] ove all subintevals of the paaetes being optiized. Howeve with the given algoith we have to evaluate the objective function of poble 7) any ties by nueical integals. Since the nube of MTC devices N vaies in diffeent ando access slots the given algoith in [5] ay not be pactical to be used due to its high coputational coplexity. In Section III we will use inteval analysis [7] and pove that the solution to poble 7) exists on an inteval whee the objective function is stictly concave. With the poposed appoach not only the nueical solution can be deteined efficiently but also a closed-fo appoxiate solution can be obtained. An algoith to estiate the nube of MTC devices equiing access to enb in each ando access slot will be given. Siulation esults to be pesented in Section IV show that the appoxiate solution obtains the sae pefoance as the nueical solution and the poposed estiation algoith woks well with diffeent MTC taffic odels. III. SOLUTIONS AND POPOSED ALGOITHMS We fist pesent ou nueical and appoxiate solutions fo poble 7). We then popose an algoith to estiate the nube of MTC devices equiing access to enb in each ando access slot which is efeed to as the backlog of each ando access slot. A. Nueical Solution Since the nube of peables is up to 64 in LTE netwoks and b p ) < b < the objective function in poble 7) i.e. equation 6) can be appoxiated as E [Z] H N b). ) e Nb p) e Nb d p ) 8) We conside the following poble in ou futhe discussion axiize b H N b) subject to b. 9) We will show that when N is lage the solution to poble 9) always exists on an inteval whee the objective function is stictly concave. Such a naowe inteval which contains the solution of the optiization poble is efeed as the shape inteval in the context of inteval analysis [7 pp. ]. To deteine the shape inteval of b fo poble 9) we study how the value of b affects the objective function H N b). We define ρ 4 d)d to siplify the notation and pesent two popositions as follows: Poposition : Given N and the function H [ N ] b) is stictly inceasing with b on the inteval ln ρ. Poof: Please efe to Appendix A.

6 Poposition : Given N and the function [ H N ] b) is stictly concave with b on the inteval. ln ρ Poof: Please efe to Appendix B. Now we have the following theoe: Theoe : Given N and let b denote the solution of poble 9). We have b if N ] ln ρ ρ b ln ρ if ln ρ ln ρ ρ < N < ) ρ ) b ln ρ ln ρ ln ρ if N ρ. Poof: Please efe to Appendix C. Fo Theoe we have b fo N ln ρ ρ. By futhe consideing the concavity of the[ function H ] N b) with espect to b on the inteval Poposition ) ln ρ the value of b fo N > ln ρ ρ can be deteined as follows. We fist) apply the bisection seach on the inteval to deteine the value of b that axiizes ln ρ ln ρ the function H N b). We denote this value by ˆb. Then the solution of poble 9) is obtained by b in { ˆb }. The pocedues fo an enb to seve N initial backlog is given in Algoith. In Line N c is the cuent backlog i.e. the nube of eaining MTC devices that have not been seved) which is initialized by N. ɛ is the teination theshold to seach ˆb. Lines 4 5 ae the steps within one ando access slot. Accoding to above analysis we have the optial ACB paaete b if N c ln ρ ρ Lines 4 ] 5). Othewise b is eithe on the inteval ln ρ N cρ ) when ) ln ρ N cρ ) > o on the inteval ln ρ N ln ρ cρ ) N cρ ) when ln ρ. Thus the enb fist deteines the value of N cρ ) ˆb that axiizes the objective function in poble 9) by nueical seach. Then the optial ACB paaete fo the cuent ando access slot is b in { ˆb } Lines 7 8). Afte boadcasting the ACB paaete b in the cuent ando access slot enb acknowledges each ando access peable eceived Lines ). The MTC devices being acknowledged send thei L/L3 essages to the enb. Let z denote the nube of L/L3 essages that the enb eceived successfully in Line 3. Then the enb seves those z MTC devices by allocating wieless channel to each of the. Afte seving those z MTC devices successfully the nube of MTC devices that still need to be seved is updated. The enb continues to seve the eaining MTC devices in the subsequent ando access slots until all of the ae successfully seved. B. Closed-fo Appoxiate Solution Note that ando access oveload usually happens when a lage nube of MTC devices equie access in the LTE netwoks. When N c > ln ρ ρ enb still needs to evaluate the nueical integal any ties to deteine ˆb accoding to Line 7 in Algoith. Thus Algoith ay not be Algoith Pocedues fo an enb to seve MTC devices with ACB paaete b in ando access slots. : Initialize d N. : Set N c : N ρ : 4 d)d ɛ : 3. 3: epeat 4: if N c ln ρ ρ then 5: Set b :. 6: else 7: Deteine ˆb by using bisection seach ) on the concave inteval ln ρ ln ρ N cρ ) N cρ ) until b bˆb < ɛ. 8: Set b : in { ˆb }. 9: end if : Boadcast b fo the next available ando access slot via syste infoation blocks. : Listen and eceive the ando access peables. : Send As fo the eceived peables. 3: Set z : nube of L/L3 essages which ae successfully eceived. 4: Seve these z MTC devices by allocating a wieless channel to each of the. 5: Set N c : N c z. 6: Until N c. pactical due to its high coputational coplexity. We popose a closed-fo solution b to appoxiate ˆb. Then the appoxiate solution to poble 9) is given by { b app in b }. ) We now descibe how to deteine b. We fist define a set of netwok scenaios S { } ζ... ζ S whee the theetuple ζ i i N i i ) fo i... S denotes a netwok scenaio. Fo each netwok scenaio ζ i we deteine ˆb i i ln ρi ag ax b βiβ i) H i N i i b) whee β i N and iρ i ) ρ i 4i d)d. Note that β i i β i ) is a local concave inteval of H i N i i b) with espect to b by Poposition ) and the length of the inteval is invesely popotional to N i. When i ln ρi N i is lage i.e. N i > ρ i the length of the inteval β i β i ) is sall. Theefoe we intoduce a vaiable γ S < γ S < ) fo the set S. Fo each netwok scenaio ζ i S we use the inteio point b i γ S ) γ S β i + γ S ) β i on the inteval β i β i ) to appoach to ˆb i. Fo netwok scenaio ζ i let δ i γ S ) denote the elative eo of b i γ S ) fo the optial value ˆb i which is deteined by the value of γ S. We have δ i γ S ) b i γ S ) ˆb i ˆb i γ S β i ˆb i ˆb i + γ S ) β i ˆb i ˆb i ) < γ S < i... S. We now deteine the optial value of γ S i.e. the value of γ S that iniizes the oveall elative eos of all netwok scenaios in set S. The sign of elative eo δ i γ S ) given by ) can be positive o negative depending on the netwok scenaio ζ i S. To evaluate the oveall elative eos fo all netwok scenaios in S we deteine the squae oot of the su of squaes of the elative eos fo all netwok scenaios

7 in set S. Thus with the given set S we need to solve the following poble to deteine the optial value of γ S iniize γ S S i ) δi γ S ) subject to < γ S <. 3) Poble 3) can be solved by tansfoing it to a geoety poble as follows. ) We denote two vectos a ) β ˆb... β S ˆb S β and b ˆb... β S ˆb S. ˆb ˆb ˆb S ˆb S Note that a and b can be taken as two points in a hypespace with S diensions. Fo any γ S on ) we have the thid point in the S -diensional space given by c γ S a + γ S ) b. Since βi ˆb i ˆb i < and βi ˆb i ˆb i > hold fo any netwok scenaio ζ i S the point a is in the S ) th quadant of the S -diensional space and b lies in the fist quadant. That is poble 3) ais to deteine a point on the open line segent fo a to b in the S diensional space which obtains the iniu distance to the oigin. Such a geoety poble is eady to be solved and the solution to poble 3) is given by γs a )T b a) b a 4) whee ) T denotes the tanspose. To bette evaluate γs fo typical LTE netwoks we copose the set S by enueating netwok scenaios with the paaetes given as follows: i fo to k with inceent of 5 N i fo 4 devices to 34 devices with inceent of 5 devices and i fo peables to 64 peables with inceent of peables. Theefoe netwok scenaios ae included in the set S. We evaluate γs with the set S by equation 4) and we obtain γs.83. Note that b in ) is used to ) appoxiate ˆb in Theoe on the inteval ln ρ ln ρ. When ando access oveload occus the nube of MTC devices ) N is lage and the length of inteval ln ρ ln ρ is sall. Thus we apply γs.83 given by the set of netwok scenaios S to obtain a value of b ) on the inteval ln ρ ln ρ. We have b γs ln ρ N ρ ) + ln ρ.7 ln ρ γ S) N ρ ) N ρ ). 5) Accoding to ) the closed-fo appoxiate solution of poble 9) is given by { } b.7 ln ρ app in. 6) N ρ ) We study the elative eo between the nueical solution b and the closed-fo appoxiate solution b app of poble 9). With all netwok scenaios in set S the cuulative distibution function is shown in Fig. 3. We find that the elative eo fo b app to b is bounded by ±.6%. The pocedues fo an enb to seve all N MTC devices ae updated by Algoith whee the enb does not need to seach fo ˆb nueically as Line 7 in Algoith. Instead Cuulative distibution function elative eo of b app fo b Fig. 3. Cuulative distibution function of the elative eo between b app and b. Algoith Pocedues fo an enb to seve MTC devices with ACB paaete b app in ando access slots. : Initialize d N. : Set N c : N ρ : 4 d)d 3: epeat { 4: Set b app : in..7 ln ρ N cρ ) 5: Boadcast b app fo the next available ando access slot via syste infoation blocks. 6: Listen and eceive the ando access peables. 7: Send As fo the eceived peables. 8: Set z : nube of L/L3 essages which ae successfully eceived. 9: Seve these z MTC devices by allocating a wieless channel to each of the. : Set N c : N c z. : Until N c. the ACB paaete b app is given by a closed-fo expession as Line 4 in Algoith. Siulation esults to be pesented in the next section show that b app obtains the sae pefoance as b with ou poposed schee. C. Backlog Estiation Algoith fo Poposed Schee The initial backlog N in Algoiths and ay not be available to enb. Thus the actual backlog N c in each ando access slot ay not be used to deteine b o b app. Meanwhile MTC devices ay not equie access to enb at the sae tie since they ay not be activated siultaneously. Futheoe it is difficult to deteine the duation equied to activate all MTC devices and deteine the pobability that an MTC device is activated in a specific ando access slot. We popose to estiate the actual backlog N c in the cuent ando access slot based on the peable collision atio in the pevious ando access slot. The peable collision atio is defined as the atio of the nube of diffeent peables that ae tansitted by MTC devices in a ando access slot but those fail to seve any MTC device ove the total nube of peables available. Let denote an estiation of the N c backlog in the cuent ando access slot. When N c MTC devices pefo ando access the sub-optial ACB.7 ln ρ paaete is b in{ } which is deteined ˆ by 6) based on the backlog estiation. We conside }.

8 Algoith 3 Pocedues fo an enb to seve MTC devices with ACB paaete b based on backlog estiation. : Initialize d. : Set ρ : 4 d)d. 3: Initialize k : ) : 4: epeat 5: Set k : k + b k) 6: Boadcast b k).7 ln ρ ρ..7 ln ρ :. k) ρ ) fo the next available ando access slot via syste infoation blocks. 7: Listen and eceive the ando access peables. 8: Set x k) : nube of peables which ae not selected by any MTC device. 9: Send As fo the eceived peables. : Set z k) : nube of L/L3 essages which ae successfully eceived. : Seve these z k) MTC devices by allocating a wieless channel to each of the. : Set { k+) : ax.7 ln ρ k) f ρ x k) } +zk) ). 3: Until x k) and bk)..7 ln ρ ρ ) since we have b fo any backlog estiation.7 ln ρ [ ρ ) ]. Let ando vaiable Ψ denote the peable collision atio when the sub-optial ACB paaete.7 ln ρ b is utilized. We have the following theoe: ˆ Theoe : Given and the atio of actual backlog and its estiation Nc the expected peable collision atio E[Ψ ] is appoxiated by E [ ] Ψ e Nc.7 ln ρ ρ ) e Nc.7 ln ρ ρ p)) e Nc.7 ln ρ ρ ) ) d p ) ) Nc f 7) which is a stictly inceasing function of Nc. Poof: Please efe to Appendix D. Thus the invese function of f in 7) exists. Since the deployent ange of the MTC devices is known by enb the invese function f can be stoed on the enb by a lookup table to educe coputational coplexity. N c is equal to f E[Ψ ]). Howeve only one instance ψ of ando vaiable Ψ can be obtained fo the actual backlog N c. This is because N c ay change ove ando access slots since soe MTC devices ay have been seved and an unknown nube of MTC devices ay be activated. Fo two consecutive ando access slots N c does not change significantly. We popose to estiate N c in a ando access slot based on the value of ψ in the pevious ando access slot with the backlog estiation. k) We denote by the backlog estiation in the k th ando access slot. The backlog estiation fo the k + ) th ando access slot is obtained by k+) k) f ψk) ). The pocedues ae given in Algoith 3. The backlog estiation fo the fist ando access slot is initialized by ).7 ln ρ ρ in Line 3. With the ACB paaete b k) deteined fo the kth ando access slot in Line 5 the nube of peables not used and the nube of MTC devices successfully seved ae denoted by x k) and zk) which ae deteined in Line 8 and Line espectively. The backlog estiation fo the next ando access slot is obtained with the cuent backlog estiation k) and the peable collision atio xk) +zk) Line ). The loop is teinated by Line 3 when no peable is selected i.e. x k) ) while no MTC device is blocked i.e. ) which eans all MTC devices have been seved. b k) IV. PEFOMANCE EVALUATION In this section we fist validate ou syste odel by copaing the analytical and siulation esults of the nube of successfully seved MTC devices in a ando access slot. Then we pesent that using the optial ACB paaete b takes the least nube of ando access slots to seve all MTC devices copaing with using sub-optial ACB paaetes. We show that the closed-fo appoxiate solution b app in Algoith achieves the sae pefoance as the nueical solution b in Algoith. With the sae netwok settings we also pesent the pefoance of Algoith 3 which uses ACB paaete b deteined by the backlog estiation in each ando access slot. By applying the MTC taffic odels fo [8] we futhe copae b with b in each ando access slot of a siulation un to show its accuacy. A. Model Validation To pesent the coectness of ou syste odel we copae the aveage nube of MTC devices seved in a ando access slot in siulations to its expectation calculated analytically. We conside N MTC devices equie access to enb in a ando access slot togethe. They ae deployed within the deployent ange of.5 k. Fo the ganulaity of quantizing the popagation delay to tiing advance we have τ 8T s.6 µs and d cτ 78 [8] whee T s / 3.7 7) sec is the basic tie unit [6] and c 3 8 /sec. We vay the nube of MTC devices N fo 5 to 5 in siulations. In each siulation un we fist conside the ACB check on each MTC device with paaete b and then let those MTC devices which passed the ACB check contend fo 64 peables [6]. We check each peable and incease the nube of successfully seved MTC devices by when one of the following two cases happens: ) the peable is selected by exactly one MTC device; ) the peable is chosen by ultiple MTC devices but the tiing advance of the selected one is diffeent fo the othes. The nube of MTC devices successfully seved in the ando access slot with a given ACB paaete b is deteined afte each siulation un. We plot the aveage esult of 5 3 siulation un and the coesponding analytical esult given by 6) in Fig. 4. We find that the analytical esult given by 6) closely atches with the aveage of the siulation esults. Fo a sall ACB paaete oe MTC devices can be successfully seved in a ando access slot if a elatively lage ACB paaete is used because MTC devices ae excessively blocked. When the value of ACB paaete is lage than the

9 E[Z] Ana-N 5 Si-N 5 5 Ana-N 45 Si-N 45 Ana-N 75 Si-N 75 Ana-N 5 Si-N b Fig. 4. Analytical and siulation esults of the expected nube of MTC devices seved E[Z] in a ando access slot with diffeent ACB paaetes. optial value the nube of successfully seved MTC devices educes since letting oe MTC devices paticipate in ando access will incease the packet collision ate at the enb. B. Effect of Optial ACB Paaetes With the poposed schee that uses both tiing advance infoation and ACB to educe the ando access oveload we conside thee ae N MTC devices at the beginning of a siulation un. Each MTC device within the initial backlog N needs to be seved exactly once. An MTC device which has not been seved will keep on tying to pass the ACB check and equest access to enb until it is seved in a ando access slot. We un siulations ove consecutive ando access slots and count the nube of ando access slots equied to seve all MTC devices. We show that using the optial ACB paaete b can educe the nube of ando access slots equied to seve all N backlog MTC devices when copaed with using sub-optial ACB paaetes. We intoduce a positive ultiplie α and use the ACB paaete b in { αb } in siulations. That is the optial ACB paaete b is utilized when α. We change the value of α fo.5 to.48 with step size.6 fo each siulation un. Thus ando access slots equied to seve N MTC devices with the optial o sub-optial ACB paaetes ae copaed. We un siulations ties with vaious and the initial backlog N is equal to. The aveage esults ae pesented in Fig. 5. We obseve that using the optial ACB paaete b i.e. α) in ou poposed schee equies the iniu nube of ando access slots to seve all MTC devices. C. Pefoance Copaison with Othe Schees We copae ou poposed schee that uses both ACB and tiing advance infoation with the following schees in tes of total ando access slots equied to seve all MTC devices: a) the schee that uses only tiing advance infoation in [] b) the schee that uses only ACB in [] c) the coopeative ACB schee that coodinates ultiple enbs in [9]. We fist conside the taffic odel that all N MTC devices ae activated siultaneously. Fo ou poposed Total ando access slots equied Siulation esults with 75 Siulation esults with k Siulation esults with.5 k α Fig. 5. Total ando access slots equied vesus α and diffeent deployent ange. The optial ACB paaete b i.e. when α ) equies the iniu nube of ando access slots to seve all MTC devices. N ) schee we un siulations with Algoiths 3 espectively. That is we not only copae the pefoance of the poposed schee with the nueical solution b and its closedfo appoxiation b app when the actual backlog N c in each ando access slot is available to the enb Algoiths and ) but also study the pefoance of the poposed schee.7 ln ρ ˆ by using the ACB paaete b deteined by the backlog estiation in each ando access slot Algoith 3). To obtain the best pefoance of the schee that uses ACB only we efe to the wok in [] and use its optial ACB paaete /N c in each ando access slot with the actual backlog N c. To siulate the coopeative ACB schee we use fou enbs to seve backlog MTC devices and allocate a nube of peables to each of the andoly in each siulation un. To copae coopeative ACB with othe schees in a fai anne the total nube of peables used by the enbs in coopeative ACB is the sae as the nube of peables used by the enb in othe schees. To achieve the best pefoance fo coopeative ACB we conside that each of the fou enbs is awae of the actual nube of MTC devices equiing access to it. We fist copae the total ando access slots equied by the afoeentioned schees when N 8 MTC devices ae activated togethe. The aveage esults of 5 siulation un with vaying deployent ange ae shown in Fig. 6. Ou poposed schee consues the least nube of ando access slots to seve all MTC devices. Both schees that use tiing advance infoation have bette pefoance when the deployent ange inceases because fewe MTC devices have the sae tiing advance in spase netwoks and the collision pobability deceases accodingly. esults also show that the coopeative ACB schee that uses fou enbs with total 64 peables obtains the sae pefoance as the schee that uses only ACB with one enb of 64 peables. This is because each enb in the coopeative ACB schee deteines its optial ACB paaete based on the actual nube of MTC devices equiing access to it. esults show that fo schees only using ACB the axiu nube of MTC devices that can be seved in a ando access slot is

10 Total ando access slots equied Use tiing advance info only in [6]) Use ACB only with actual backlog in []) Use coopeative ACB with actual backlog fo each enb in wok [9]) Use tiing advance info and ACB by b with backlog estiation Alg. 3) Use tiing advance info and ACB by b app with actual backlog Nc Alg. ) Use tiing advance info and ACB by b with actual backlog Nc Alg. ) Total ando access slots equied 4 3 Use tiing advance info only in [6]) Use ACB only with actual backlog in []) Use coopeative ACB with actual backlog fo each enb in [9]) Use tiing advance info and ACB by b with backlog estiation Alg. 3) Use tiing advance info and ACB by b app with actual backlog Nc Alg. ) Use tiing advance info and ACB by b with actual backlog Nc Alg. ) Deployent ange of MTC devices Fig. 6. Total ando access slots equied vesus the deployent ange of MTC devices. N 8 64) Initial backlog N Fig. 7. Total ando access slots equied vesus initial backlog N. k 64) deteined by the total nube of peables in the netwok. This also explains the eason why using both tiing advance infoation and ACB obtains uch bette pefoance than using eithe tiing advance infoation o ACB only. We futhe notice that in spase netwoks using only tiing advance infoation ay equie fewe ando access slots than using ACB only. The eason is that the nube of MTC devices N 8 is not vey lage and the optial ACB paaete is equal to one in spase netwoks. The effect of educing ando access oveload with ACB vanishes. Howeve copaing tiing advance in the eceived A befoe tansitting the L/L3 essage is still helpful to avoid packet collisions. We also find that the poposed schee equies oe ando access slots to seve all MTC devices if these MTC devices ae deployed in a salle aea. Eventually when the deployent ange is equal to the poposed schee equies the sae nube of ando access slots to seve all MTC devices as the schee that uses ACB only. This phenoenon can be explained as follows. When N MTC devices ae located in a salle aea the density of MTC devices inceases and oe MTC devices have the sae tiing advance infoation. Thus the schee that uses ACB only with one enb is a special case of ou poposed schee when is sall enough such that all MTC devices have identical tiing advance infoation. We copae Algoiths 3 fo ou poposed schee. esults in Fig. 6 show that using b app Algoith ) equies the sae nube of ando access slots to seve N 8 initial backlog as using b Algoith ) when the actual backlog N c is available to the enb. Without the actual backlog N c alost the sae pefoance is obtained by using the ACB paaete b deteined by the backlog estiation fo each ando access slot Algoith 3). We now pesent how the nube of initial backlog N affects the nube of ando access slots equied. Siulation esults ae pesented in Fig. 7. Fo the schees using ACB the nube of ando access slots equied to seve N backlog MTC devices inceases with N linealy. The coopeative ACB schee obtains the sae pefoance as the schee that uses only ACB. This follows the sae easons that we have explained fo Fig. 6. Fo the schee which uses tiing advance infoation only the equied nube of ando Total ando access slots equied Use tiing advance info only in [6]) Use ACB only with actual backlog in []) Use coopeative ACB with actual backlog fo each enb in [9]) Use tiing advance info and ACB by b with backlog estiation Alg. 3) Use tiing advance info and ACB by b app with actual backlog Nc Alg. ) Use tiing advance info and ACB by b with actual backlog Nc Alg. ) Total nube of available peables Fig. 8. Total ando access slots equied vesus nube of peables. N k) access slots inceases exponentially. This is because when N inceases oe packet collisions occu at the beginning of each siulation un. Thus using only tiing advance infoation equies oe slots to seve all MTC devices. Ou poposed schee that uses both tiing advance infoation and ACB equies the least ando access slots in all scenaios which educes half of the nube of ando access slots copaed to the othe two schees. Fig. 7 also copaes the pefoance of the poposed schee by using ACB paaetes b in Algoith and b app in Algoith when actual backlog N c is available. The siulation esults obtained by Algoiths and coincide with each othe. Copaed with Algoiths and Algoith 3 obtains nealy the sae pefoance by using the ACB paaete b deteined by backlog estiation in each ando access slot. We pesent siulation esults fo diffeent nube of peables in Fig. 8. Fo all schees the nube of ando access slots equied inceases exponentially when fewe peables ae available. Fo the schee that uses tiing advance infoation only the nube of slots equied to seve N backlog MTC devices is an ode of agnitude highe than those equied by othe schees. Copaed with schees that use tiing advance infoation only ACB only o coopeative ACB ou poposed schee only equies half of ando ac-

11 Total ando access slots equied Poposed schee by b fo backlog estiation in taffic odel λ 3 µ 4 Poposed schee by b app fo actual backlog Nc in taffic odel λ 3 µ 4 Poposed schee by b fo actual backlog Nc in taffic odel λ 3 µ 4 Poposed schee by b fo backlog estiation in taffic odel λ µ Poposed schee by b app fo actual backlog Nc in taffic odel λ µ Poposed schee by b fo actual backlog Nc in taffic odel λ µ %.75% %.75% %.38% %.5% Total ando access slots used to activate N MTC devices Fig. 9. Total ando access slots equied to seve N MTC devices vesus the nube of ando access slots used to activate N MTC devices. N 3 k 64) cess slots to seve all MTC devices because copaing tiing advance infoation educes the packet collision pobability at enb. The pefoance of the poposed schee by using ACB paaete b and its appoxiation b app coincide with each othe when the actual backlog N c is available to the enb Algoiths and ). Moeove using the ACB paaete b deteined by backlog estiation in each ando access slot Algoith 3) obtains alost the sae pefoance as using b o b app fo the actual backlog N c. D. Pefoance with Diffeent Taffic Models The MTC devices ay not be activated siultaneously but ay be activated within a peiod of tie. Let V denote the length of tie that all N MTC devices ae activated. Accoding to the wok in [8] the pobability density function that a given MTC device is activated at tie v v V ) is given by qv; λ µ V ) vλ V v) µ whee Betaλ µ) V λ+µ Betaλ µ) is the beta function. Two taffic odels ae suggested in [8] by changing λ and µ. We have λ and µ when MTC devices ae unifoly activated within the activation peiod. Othewise we have λ 3 and µ 4. We incease V fo 5 to 5 ando access slots to activate N 3 MTC devices. Fo each pai of λ and µ we copae the pefoance of ou poposed schee by using the ACB paaetes b and b app when the actual backlog N c in each ando access slot is available to the enb. We also siulate ou poposed schee with ACB paaete b fo the backlog estiation in each ando access slot. The aveage esults of siulations ae given in Fig. 9. We find that the sae pefoance is obtained with ACB paaetes b and b app when the actual backlog N c is available. Without the actual backlog N c ou poposed Algoith 3 which applies ACB paaete b deteined by the backlog estiation takes only.38%.75% oe ando access slots to seved N MTC devices copaed with using ACB paaetes b and b app. With the sae values of N and given above Fig. copaes the ACB paaetes b deteined by the actual backlog N c and b deteined by the backlog estiation in each ando access slot of a siulation un. We conduct two siulation un with two taffic odels λ 3 µ 4 and Values of ACB paaetes b and b 3 b deteined by actual backlog Nc in taffic odel λ 3 µ 4 b deteined by backlog estiation in taffic odel λ 3 µ 4 b deteined by actual backlog Nc in taffic odel λ µ b deteined by backlog estiation in taffic odel λ µ Index of each ando access slot in a siulation un Fig.. Copaison of b and b deteined by the actual backlog N c and its estiation in each ando access slot espectively. N 3 k 64) λ µ espectively. We find that the ACB paaete.7 ln ρ b deteined by the backlog estiation in ˆ each ando access slot is close to the b deteined by the actual backlog N c in the coesponding ando access slot. Moeove both b and b ae equal to in the fist 33 ando access slots with the taffic odel λ 3 µ 4. This is because the backlog inceases slowly at the beginning of the siulation. This also explains the eason why oe ando access slots ae equied to seve all MTC devices when they ae activated within a longe activation duation V with the taffic odel λ 3 µ 4 in Fig. 9. V. CONCLUSIONS In this pape we poposed to use both ACB and the tiing advance infoation to elieve the ando access oveload in MM systes. We deteined the optial ACB paaete b which axiizes the expected nube of MTC devices successfully seved in each ando access slot. To educe the coputational coplexity in deteining the optial ACB paaete b we poposed a closed-fo solution b app to appoxiate b. We also pesented an algoith to estiate the nube of MTC devices that equie access to enb in each ando access slot. Though siulations we validated ou analytical esults and showed that the closed-fo appoxiate solution b app obtains the sae pefoance as the nueical solution b. We futhe showed that ou schee woks well with the poposed backlog estiation algoith in vaious taffic odels. We found that alost 5% of the ando access slots can be saved to seve all MTC devices when copaed with othe schees that use eithe tiing advance infoation o ACB only. Fo futue wok ou poposed schee can be extended by using diffeent ACB paaetes fo MTC devices with diffeent pioities o QoS equieents. Moeove fo unifoly distibuted MTC devices the MTC devices that ae close to the enb ay have a highe chance to be seved. This is because the nube of MTC devices with the sae tiing advance infoation deceases when thei distance to the enb is educed. This fainess issue in ou poposed schee is expected to be addessed in the futue wok. We expect that

12 ou wok will otivate oe efficient echaniss that can bette suppot MM applications in LTE netwoks. APPENDIX A POOF OF POPOSITION The fist-ode patial deivative of H N b) with espect to w..t.) b is given by b N H N b) e Nb p ) e Nb p)) d. 8) p ) Let function g b ) e Nb p ) e Nb p). The fist-ode patial deivative of g b ) w..t. is given by g b ) p ) g b ) p ) p ) Nb p ) ) e Nb p). 9) Note that e Nb p) >. The sign of 9) is deteined by p) Nb p ) ). It can be shown that p ) inceases with on [ d) and deceases with on d ]. Let p u denote the axiu value that p ) can obtain. We have p u 4 d)d. When Nb p ) < we have b < Np). To ake the inequality hold fo [ ] we have b < in [] Np). That is b < Np u. 3) [ Fo b Np u ) we have g b ) < fo [ d) g b ) > fo d ]. Thus g b ) obtains and the iniu value at d on the inteval [ ]. When g b d) i.e. e Nb p u e Nb pu we obtain anothe inteval of b as follows: b ln p u N p u ). 3) Note that < p u < we intoduce anothe vaiable ϑ p u whee < ϑ <. We have ln p u ln ϑ) k ϑk k ϑ k p u ϑ ϑ k + 3) and k k p u ϑ ϑ k. 33) Since k ϑk k+ < k ϑk the inteval of b given by 3) [ contains the ] inteval of b given by 3). Thus fo b and [ ] we have g b ) and ln p u Np u ) ln pu Np u ) the equality holds when b and d. [ ] Conside 8) and note that N ln p >. Fo b u Np u ) the integand of the integal is positive fo d) and d ] and is nonnegative when o ] d. Thus b [ H N b) > fo b ln p u Np u ). That [ is H N ] b) is stictly inceasing with b on the inteval by noting that ρ p u which copletes the poof. ln ρ APPENDIX B POOF OF POPOSITION The second-ode patial deivative of H N b) w..t. b is given by b N H N b) p ) ) p ) e Nb p) e Nb d. 34) Let function fb ) p ) e Nb p) e Nb. The fistode patial deivative of f b ) w..t. is given by f b ) p ) f b ) p ) p ) Nb p ) ) p ) e Nb p). 35) Since p ) e Nb p) < the sign of 35) is deteined by p) Nb p ) ). Denote p u as the axiu value that p ) can obtain i.e. p u 4 d)d ). When Nb p ) < we have b < Np). To ake the inequality hold fo [ ] we have b < in []. That is Np) b < Np u. 36) Since p ) inceases with [ on [ d) and deceases with on d ]. Fo b Np u ) we have f b ) > fo [ d) and f b ) < fo d ]. Thus f b ) obtains its axiu value at d on the inteval [ ]. By aking f b d) i.e. p ue Nb pu e Nb we obtain anothe inteval of b which is given by ln pu p u < b ln p u N p u ). 37) Since p u the inteval of b in 37) is contained by the [ ] inteval of b in 36). Thus fo b and [ ] ln pu Np u ) ln pu Np u ) we have f b ) and the equality holds when b and d. N Conside equation 34) and note that [ ] >. Fo ln pu b Np u ) the integand of the integal is negative fo d) and d ] and is nonpositive when o d. Thus b H N b) <. That is [ H N ] b) is stictly concave with b on the inteval by noting that ρ p u which copletes the poof. ln ρ APPENDIX C POOF OF THEOEM Fist of all we intoduce the following lea: Lea : Thee exists exactly one value of b that akes b H N b). Poof: We assue both b b and b b + σ can ake 8) equal to. We have p ) Nb e d p ) Nb e p) d 38) p )

13 and p ) e Nb +σ) d p ) p ) e Nb +σ) p) d. 39) We ultiply e Nσ on both sides of 38) and note that the left hand side of the esult is identical with the left hand side of 39). Thus the diffeence of thei ight hand sides is i.e. Note that p ) p ) p) p) Nb e. Moeove e Nσ p) e Nσ ) p) e Nσ p) e Nσ d. 4) Nb e p) with the equality holds at > when σ >. In addition e Nσ p) e Nσ < when σ <. Thus σ is the only passibility which copletes the poof. ln ρ By substituting b into 8) which is the ight [ ] bounday of inteval in Poposition we have ln ρ b H N b) b N p ) ln ρ ρ ρ p ) ρ ρ p)) d. 4) Note that the sign of 4) depends on only and it is always negative fo > d. Based on Lea and Popositions and the value of b that akes b H N ) b) equal to ust be on the inteval ln ρ ln ρ. By taking the constaint in poble 9) i.e. b ) into account we have b when ln ρ ln ρ i.e. N ρ )). By ) ln ρ letting we have b ln ρ ln ρ fo ln ρ ln ρ ln ρ N ρ ). Othewise fo ρ ) < N < ρ ) we have b ln ρ ] which copletes the poof. APPENDIX D POOF OF THEOEM We conside N c MTC devices ae equiing access to the enb in the cuent ando access slot. With the backlog estiation.7 ln ρ the ACB paaete b is used ˆ in the ando access slot. By using ACB paaete b let ando vaiables X and Z denote the nube of peables not used by any MTC device and the nube of MTC devices successfully seved in the cuent ando access slot espectively. Thus Z is equal to the nube of selected peables that succeed to seve MTC devices. Since the nube of available peables is the expected peable collision atio is given by E [ [ ] X Ψ ] E Z ] E [ X E [ Z ]. 4) ecall that I u o I u ) denotes the event that an abitay MTC device u passes the ACB check o not) and J u j denotes the event that MTC device u selects peable j unifoly fo available peables. The pobability that u selects peable j is given by P I u J u j) P I u ) P J u j I u ).7 ln ρ ρ ). 43) Fo an abitay peable j let K j denote the event that peable j is not selected by any MTC device. The pobability of K j is given by ) ) ) N N.7 ln ρ.7 ln ρ P K j ) ρ ) ρ ).7 ln ρ ρ ) ) N. 44) Fo lage and N c we have the following appoxiation E [ ] X P K j ) e Nc.7 ln ρ ρ ). 45) By substituting ACB paaete b eaanging the esult we have E [ ] Z e Nc.7 ln ρ ρ p ).7 ln ρ ˆ p)) e Nc into 8) and.7 ln ρ ρ ) ) d. 46) Equation 7) is obtained by substituting 45) and 46) into 4). We now conside anothe estiation of the N c MTC devices we have Nc > Nc if and only if <. Thus anothe sub-optial ACB paaete b.7 ln ρ ρ ) fo ust be geate than the sub-optial ACB paaete.7 ln ρ b fo. ˆ That is inceasing N c actually inceases the ACB paaete used in a ando access slot. Thus the expected peable collision atio in 7) stictly inceases with the value of Nc which copletes the poof. EFEENCES [] Cisco Cisco visual netwoking index: Global obile data taffic foecast update 4-9 Feb. 5. [Online]. Available: pape c-586.pdf [] 3GPP TS.368 V3.. Sevice equieents fo achine-type counications MTC) stage Jun. 4. [3] F. Boccadi. W. Heath J. A. Lozano T. L. Mazetta and P. Popovski Five disuptive technology diections fo 5G IEEE Counications Magazine vol. 5 no. pp Feb. 4. [4] J.-P. Cheng C.-H. Lee and T.-M. Lin Pioitized ando access with dynaic access baing fo AN oveload in 3GPP LTE-A netwoks in Poc. IEEE Globeco Wokshop on Machine-to-Machine Counications Houston TX Dec.. [5] 3GPP TS V.5. E-UTA adio esouce contol C) potocol specification Ma. 5. [6] 3GPP TS 36. V.5. E-UTA physical channels and odulation Ma. 5. [7] D. C. Chu Polyphase codes with good peiodic coelation popeties IEEE Tans. on Infoation Theoy vol. 8 no. 4 pp Jul. 97. [8] 3GPP TS 36.3 V.5. E-UTA physical laye pocedues Ma. 5.

14 [9] 3GPP TS 36.3 V.5. E-UTA ediu access contol MAC) potocol specification Ma. 5. [] A. Laya L. Alonso and J. Alonso-Zaate Is the ando access channel of LTE and LTE-A suitable fo MM counications? A suvey of altenatives IEEE Counications Suveys & Tutoials vol. 6 no. pp. 4 6 Jan. 4. [] K.-D. Lee M. eisslein K. yu and S. Ki Handling andoness of ulti-class ando access loads in LTE-Advanced netwok suppoting sall data applications in Poc. of IEEE Globeco Wokshop on Machine-to-Machine Counications Anahei CA Dec.. [] C.-H. Wei.-G. Cheng and S.-L. Tsao Modeling and estiation of one-shot ando access fo finite-use ultichannel slotted ALOHA systes IEEE Counications Lettes vol. 6 no. 8 pp Aug.. [3] Y. Liu C. Yuen J. Chen and X. Cao A scalable hybid MAC potocol fo assive MM netwoks in Poc. of IEEE Wieless Counications and Netwoking Conf. WCNC) Shanghai China Ap. 3. [4] 3GPP T.888 V.. Study on enhanceents fo achine-type counications MTC) Ma. 3. [5] C.-Y. Tu C.-Y. Ho and C.-Y. Huang Enegy-efficient algoiths and evaluations fo assive access anageent in cellula based achine to achine counications in Poc. of IEEE Vehicula Technology Conf. VTC-Fall) San Fancisco CA Sep.. [6] H.-L. Fu H.-C. Chen P. Lin and Y. Fang Enegy-efficient epoting echaniss fo ulti-type eal-tie onitoing in achine-to-achine counications netwoks in Poc. of IEEE INFOCOM Olando FL Ma.. [7] K. Zhou and N. Nikaein Packet aggegation fo achine type counications in LTE with ando access channel in Poc. of IEEE Wieless Counications and Netwoking Conf. WCNC) Shanghai China Ap. 3. [8] 3GPP T V.. Study on AN ipoveents fo achinetype counications Oct.. [9] S.-Y. Lien T.-H. Liau C.-Y. Kao and K.-C. Chen Coopeative access class baing fo achine-to-achine counications IEEE Tans. on Wieless Counications vol. no. pp. 7 3 Jan.. [] C. M. Chou C. Y. Huang and C.-Y. Chiu Loading pediction and baing contols fo achine type counication in Poc. of IEEE Int l Conf. on Counications ICC) Budapest Hungay Jun. 3. [] S. Duan V. Shah-Mansoui and V.W.S. Wong Dynaic access class baing fo MM counications in LTE netwoks in Poc. of IEEE Global Counications Conf. GLOBECOM) Atlanta GA Dec. 3. [] K. S. Ko M. J. Ki K. Y. Bae D. K. Sung J. H. Ki and J. Y. Ahn A novel ando access fo fixed-location achine-to-achine counications in OFDMA based systes IEEE Counications Lettes vol. 6 no. 9 pp Sep.. [3] Z. Wang and V.W.S. Wong Joint access class baing and tiing advance odel fo achine-type counications in Poc. of IEEE Int l Conf. on Counications ICC) Sydney Austalia Jun. 4. [4] S. Sesia I. Toufik and M. Bake LTE The UMTS Long Te Evolution: Fo Theoy to Pactice Second Edition. John Wiley & Sons. [5] E. Hansen Global Optiization Using Inteval Analysis: evised and Expanded nd ed. M. Dekke 3. [6] E. Hansen and. Geenbeg An inteval Newton ethod Applied Matheatics and Coputation vol. no. -3 pp May 983. [7]. E. Mooe. B. Keafott and M. J. Cloud Intoduction to Inteval Analysis. SIAM 9. [8] ZTE -4663: [7bis#] LTE: MTC LTE siulations in 3GPP TSG AN WG Meeting 7 Madid Spain Aug.. Zehua Wang S ) eceived the B.Eng. degee in Softwae Engineeing fo Wuhan Univesity Wuhan China in 9 and the M.Eng. degee in Electical and Copute Engineeing fo Meoial Univesity of Newfoundland St Johns NL Canada in. He is cuently a Ph.D. candidate at the Univesity of Bitish Colubia UBC) Vancouve BC Canada. His eseach inteests include achine-type counications device-todevice counications social netwoks and outing and fowading in obile ad hoc netwoks. He has been the ecipient of the Fou Yea Doctoal Fellowship 4YF) at UBC since. He was also awaded the Gaduate Suppot Initiative GSI) Awad fo UBC. M. Wang seved as technical poga coittee TPC) ebes fo seveal confeences including the IEEE Intenational Confeence on Counications ICC) 5 and the IEEE Global Counications Confeence GLOBECOM) 4 5. Vincent W.S. Wong SM 7) eceived the B.Sc. degee fo the Univesity of Manitoba Winnipeg MB Canada in 994 the M.A.Sc. degee fo the Univesity of Wateloo Wateloo ON Canada in 996 and the Ph.D. degee fo the Univesity of Bitish Colubia UBC) Vancouve BC Canada in. Fo to he woked as a systes enginee at PMC-Siea Inc. He joined the Depatent of Electical and Copute Engineeing at UBC in and is cuently a Pofesso. His eseach aeas include potocol design optiization and esouce anageent of counication netwoks with applications to wieless netwoks sat gid and the Intenet. D. Wong is an Edito of the IEEE Tansactions on Counications. He has seved on the editoial boads of IEEE Tansactions on Vehicula Technology and Jounal of Counications and Netwoks. He has seved as a Technical Poga Co-chai of IEEE SatGidCo 4 as well as a Syposiu Co-chai of IEEE SatGidCo 3 and IEEE Globeco 3. He eceived the 4 UBC s Killa Faculty eseach Fellowship. He is the Chai of the IEEE Counications Society Eeging Technical Sub-Coittee on Sat Gid Counications and IEEE Vancouve Joint Counications Chapte.