Estimation and Adaptation for Bursty LTE RandomAccess

Guan-Yu Lin, Shi-Rong Chang, and Hung-Yu Wei∗

Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan

Abstract—With the potential to generate numerous connectionrequests, an explosive growth in the volume of data trafficand the number of mobile and machine-to-machine (M2M)devices has drawn new attention on the radio access network(RAN). Surging random access attempts cause not only severepreamble collisions but also downlink resource shortage, andthus degrade the performance of random access procedure.However, the effect of downlink resource shortage on systemperformance is not yet comprehensively studied. In addition,most existing random access contention resolution mechanismssacrifice RACH (random access channel) throughput for a highsuccess probability, and thus the price is that low-throughputmechanisms need long time to deal with access attempts. In thiswork, we evaluate the MAC-level performance for the 4-steprandom access procedure in LTE systems, for both with andwithout constrained downlink resources. Further, we propose anovel RACH contention resolution scheme, the dynamic backoff(DB) scheme. DB can achieve high RACH throughput yieldinga high random access success probability under various RACHoverloaded scenarios.

Index Terms—LTE, Machine Type Communications, Randomaccess, RACH procedure

I. I NTRODUCTION

By integrating innovative applications with advanced wire-less communication technique, smartphones, tablets, andMachine-to-Machine (M2M) communications have generatedan explosive growth in the volume of generated data traffic andthe number of mobile devices. According to the forecast givenby Cisco [1], mobile data traffic will increase 18-fold, andthe number of mobile-ready devices and M2M connectionswill exceed 8 billion and nearly 2 billion respectively by2016. In view of this, great attention has been drawn to theissue that whether current 4G communication systems such asLong Term Evolution (LTE) can support accesses from a largenumber of devices, particularly at the radio access network(RAN) level.

Both smartphones and M2M traffic impose significant loadat the RAN level: a smartphone user equipment (UE) willperform a random access procedure (known as the RACHprocedure in LTE systems) to request resources for uplink data

Copyright (c) 2015 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

Guan-Yu Lin is with the Department of Electrical Engineering, NationalTaiwan University, Taipei 10617, Taiwan (email: b93901086@ntu.edu.tw)

Shi-Rong Chang is with the Department of Electrical Engineering, NationalTaiwan University, Taipei 10617, Taiwan (email: b97901131@ntu.edu.tw)

Hung-Yu Wei* is the corresponding author with the Department of Elec-trical Engineering, National Taiwan University, Taipei 10617, Taiwan (email:hywei@ntu.edu.tw)

transmission if not assigned with a dedicated scheduling re-quest (SR) resource. In contrast, an M2M device may attemptto build a network connection by RACH procedure. To managean enormous quantity of RACH attempts in the near future,RACH resource efficiency requires improvement to ensuresmooth wireless access without severe congestion or block-ing. For RAN improvement, 3GPP evaluates performancemetrics, including the system signaling overhead and userQoS metrics for different traffic scenarios on various devicessuch as smartphones and tablets in [2]. In addition, RACHperformance in an M2M-involved RAN overload scenario hasbeen evaluated in [3]. Since M2M is also known as MachineType Communications (MTC) in 3GPP [4], the two terms areinterchangeably used in the following context.

To perform RACH procedure in LTE, a UE competes withother UEs for resources in the random access channel (RACH)and in the physical downlink control channel (PDCCH)1.When competing simultaneously, numerous devices generatesevere collisions in RACH resources and exhaust PDCCH,resulting in a high RACH failure probability as well asperformance degradation of both M2M and H2H (human-to-human) communications [4, 5]. MTC, characterized bynumerous devices [6], has the greatest potential to cause suchRACH degradation.

Previous research has extensively studied the RACH colli-sion probability, throughput, and capacity in terms of differentcommunication systems [7–14]. Furthermore, some adaptiveschemes have been proposed to dynamically adjust RACHresources or tune the RACH preamble transmission probabilityaccording to the observed RACH traffic loading [11, 15, 16].

However, at present only a preliminary study [17] evaluatesthe performance of a RACH system with PDCCH constraints.In addition, existing adaptive schemes such as those in [11,15, 16] have two main drawbacks. First, these schemes donot consider the performance of RACH throughput, i.e., theysacrifice RACH throughput in pursuit of a high RACH successprobability. Secondly, they apply some artificial configurationparameters or thresholds for overload control, and thus inducea significant degradation in RACH success probability whenartificial configurations cannot catch up with the property orchange speed of bursty RACH arrival traffic.

Our contribution is summarized as follows:1) Evaluation of the PDCCH-limited 4-step RACH pro-

1Both PDCCH and PDSCH (physical downlink shared channel) arecon-sumed to complete RACH procedure, but the amount of PDCCH is muchmore deficient than that of the PDSCH. Therefore, in this workonly PDCCHshortage is considered.

cedure: We provide a comprehensive MAC-level analysisfor the 4-step RACH procedure, whose model followsthe evaluation methodology in [3], e.g., thepreambledetection probability is applied to model the effect ofpath loss and power ramping, and the effect of limitedPDCCH resources on RACH performance is evaluated.

2) Proposal of a RACH overload resolution protocol: Wepropose a novel RACH contention resolution method toestimate the number of potential RACH access attemptsand then disperse them over time. The theoretical andsimulation results show that the scheme can reach a highRACH throughput and a high RACH success probabilitywithout suffering from performance degradation due tothe unpredictable RACH arrival traffic.

The rest of this paper is organized as follows. SectionII reviews previous work for RACH performance evaluationand RACH contention resolution. Section III introduces thesystem model. RACH performance for different DL resourceconstraints is evaluated in Section IV. Section V proposes ourmechanism for RACH overload resolution. In Section VI, wesimulate the performance of the baseline RACH procedure andthe proposed dynamic backoff mechanism. We conclude ourwork in Section VII.

II. RELATED WORK

Random access channel (RACH) is an uplink channel incommunication systems for UE to perform random accesssuch as transmitting RACH preambles or uploading small data.Since RACH resources are shared by all UEs, the capacityof RACH resources is limited by collisions. RACH collisionprobability, RACH capacity, and throughput for different com-munication systems have been evaluated further in [7–14].However, early research evaluates only the performance of theRACH preamble transmission rather than the performance of4-step RACH procedure, and thus cannot be directly appliedto a RACH system with a downlink resource constraint.

With the fast development in M2M applications, the RACHperformance affected by numerous M2M devices has raised anextensive attention. To understand the M2M-involved RACHperformance, 3GPP specifies [3] to provide evaluation method-ology and the corresponding simulation results. In addition,ACB (access class barring) and EAB (enhanced access bar-ring) mechanisms have been standardized in LTE to preventMTC devices from overloading RACH.

Group paging is a good approach to resolve RACH con-tention. The authors in [18] propose a group paging method,in which only those paged M2M devices are allowed to sendtheir preambles. In other words, since eNB (a base station inLTE) knows the number of paged M2M devices, it can avoidsevere RACH collisions and PDCCH shortage by adjustingthe group size and related timer for random access response(RAR). The authors then provide comprehensive analysis onperformance metrics like collision probability, access successprobability, and average access delay. Despite its satisfactoryRACH success probability, group paging has some drawbacks.First, if the number of M2M devices is large, a M2M devicemay waste a long time waiting to be paged. Secondly, the

eNB needs to manage group information of M2M devices.And thirdly, since the eNB might not know whether an M2Mdevice wants to connect to the network or not, low efficiency inpaging resource might occur when only part of M2M deviceswish to connect. Therefore, group paging is more suitablefor M2M applications in terms of synchronized transmissionbehavior or somewhat static group divisions. Notice thatalthough the authors in [18] have modelled the limited PDCCHand 4-step RACH procedure, the group paging scheme avoidsPDCCH shortage by adjusting RAR timer, and thus the effectof PDCCH shortage on RACH performance is not evaluated.

To handle dynamic and bursty RACH traffic, some adaptivemechanisms have been proposed to tune RACH parametersfor a better performance. The authors in [16] propose toadaptively determine the value of persistent factor according tothe history of the throughput and backlog value. Sunet al. [11]propose two dynamic schemes, namely the dynamic RACHAssignment Algorithm and the Adaptive Access ProbabilityScheme. The former dynamically adjusts the volume of RACHresources based on the loading of RACH resources, and thelatter focuses on the adaptive adjustment of the access proba-bility for various traffic types. The work in [19] provides ananalysis for both uniform backoff (UB) and binary exponentialbackoff (BEB) algorithms, and a new algorithm is proposed toadaptively control the window size of the UB algorithm underunsaturated traffic conditions. To assist the EAB mechanismin determining the timing of activation and the parameters touse, a prioritized random access with dynamic access barring(PRADA) framework is proposed in [15]. PRADA consistsof two components: pre-assignation of different amount ofRACH slots to different classes of devices, and the dynamicaccess barring (DAB) approach. DAB estimates the RACHload by observing the average number of successful preamblesin each two neighboring RACH slots. If RACH loading isheavy, EAB is triggered to have those UEs that are readyto send their first preamble defer their RACH attempts fora long time. However, we have two concerns about DAB.First, since for bursty RACH arrival traffic the number ofsuccessful preambles drastically varies with time, EAB maybe frequently activated and then deactivated, and thus causesunreliable performance on RACH success probability. Andsecondly, DAB applies up to seven parameters for overloadcontrol, but no guide to set parameters is provided.

In [17], the authors investigate the effect of PDCCHshortage on the RACH performance. Considering limitedPDCCH resources, the authors proposes a two-dimensionalMarkov chain model to analyze the status of Msg4 queueand the expected RACH throughput by assuming that boththe preamble arrival rate and the preamble transmission rateare Poisson-distributed and are independent to each other.Infact, the preamble transmission rate is correlated with thepreamble arrival rate and RACH related parameters (such asthe maximum number of preamble transmission attempts andthe power ramping effect). Without considering these RACHparameters, the proposed model in [17] cannot reflect therealistic situation of RACH contention.

In summary, the effect of the limited PDCCH resourceson RACH performance is not yet comprehensively studied.

In addition, existing adaptive mechanisms cannot guaranteeRACH success probability for diverse RACH arrival traffic, letalone RACH throughput. The reason is that existing researchapplies artificial thresholds or parameters to determine thecollision status according to real-time observation on RACHthroughput. Therefore, the RACH success probability degradessignificantly when the RACH throughput drastically changes,or when these artificial thresholds or parameters cannot catchup with the diversity or change speed of the RACH arrivaltraffic. The two scenarios may happen all the time in that eNBtypically has no full information about the upcoming RACHtraffic in advance, especially for bursty M2M traffic.

In this work, we tackle the two issues as follows. First, weevaluate the effect of limited PDCCH resources on RACHperformance by considering all RACH parameters in [3].And secondly, we resolve the problem of artificial settingsby directly estimating the number of RACH attempts, andthereby calculating the backoff window size to maximizeRACH throughput while satisfying the desired RACH successprobability. Therefore, the backoff window size is not anartificial setting and thus our scheme works well for variousheavy RACH load scenarios.

III. SYSTEM MODEL

A. RACH Procedure Model

In LTE, a UE in radio resource control (RRC) idle modeperforms a RACH procedure to build a network connectionbefore data transmission. Following [20], we model the 4-stepRACH procedure as shown in Fig. 12a. In the first step, theUE transmits a RACH preamble, whose sequence is uniformlyselected from an available preamble pool. LetR denote thenumber of available preamble sequences in each RACH slot,where RACH slots are the specific subframes for RACHpreamble transmissions, and thus each preamble sequence hasan equal probability1

Rto be selected by a UE. After sending

the preamble, the UE sets a random access response (RAR)window and waits for the eNB (a base station in LTE) toreply to it with an uplink grant (Msg2) contained in a RAR.If the UE successfully receives its Msg2 in the RAR window,the UE then sends the RRC connection request (Msg3) to theeNB and starts the Msg4 timer, and waits to receive its ownRRC Connection Setup message (Msg4) from the eNB.

RACH procedure may fail due to the following reasons:

1. Preamble Transmission Failure: A transmitted preamblemay be ignored by the eNB due to either insufficientpreamble transmission power or RACH preamble collision,i.e., two or more UEs transmit preambles with the samepreamble sequenceat the same time. Our model followstwo assumptions for MAC-level evaluation in [3]:

A. Ignore the capture effect, i.e., the eNB cannot decodeany collided preambles.

B. If no collision occurs to a preamble, the preamble hasa preamble detection probability pd,i to be detectedby the eNB, wherei indicates thei-th preamble trans-mission.The modeling of the preamble detection probabilityconsiders the effects of radio channels such as path

loss, power ramping, and fading2. Due to the powerramping mechanism applied in the LTE RACH proce-dure, (i.e., when performing preamble retransmission,a UE increases the preamble transmission power), it isreasonable to havepd,i ≤ pd,j ∀i < j.

2. Msg2 Reception Failure: In our model, we assume thatif the eNB sends a Msg2 to a UE, the UE can alwayssuccessfully receive it. Thus, the reason of failed Msg2reception is that the eNB has no sufficient downlink radioresources (PDCCH) to transmit Msg2 in response to allreceived preambles in UEs’ RAR windows.

3. Msg3 Transmission Failure: A UE applies a HARQ pro-cess to send Msg3 to the eNB. In our modelling, the HARQprocess of Msg3 contains at mostNMsg3 max transmission,whosei-th transmission has a failure probabilityPf,i,Msg3,∀i = 1, · · · , NMsg3 max. If a UE fails to send Msg3 to theeNB by HARQ, it fails in Msg3 transmission.

4. Msg4 Reception Failure: A eNB also applies a HARQprocess to send Msg4 back to the UE. In our modelling,the HARQ process of Msg4 contains at mostNMsg4 max

transmission, whosei-th transmission has a failure prob-ability Pf,i,Msg4, ∀i = 1, · · · , NMsg4 max. A UE fails inMsg4 reception if it does not receive its Msg4 before Msg4timer expiration, which is because of insufficient PDCCHresources or imperfect channel condition.

A RACH trial denotes the action to perform the 4-stepRACH procedure once, and a RACH trial fails if any ofthe four failures mentioned above occurs. If a failed RACHtrial occurs to a UE, the UE performs the next RACH trialinitialized by retransmitting a preamble after a random backoffperiod, whose length is uniformly selected from0, 1, · · · ,W−1, whereW is specified by theBackoff Indicator (known asBI). A UE has at mostK RACH trials, and a UE withKfailed RACH trials fails in its RACH procedure.

UE’s behavior in RACH procedure is illustrated in Fig. 12b.Further details on the RACH procedure can be found in [20].

B. Traffic Model

To evaluate the RACH capacity, we follow the assumptionin [3] that all RACH traffic are from M2M UEs and nobackground noise is from H2H UEs. We consider the uniformtraffic model (traffic type 1) in [3], i.e., the traffic model isexpressed by

N(t) =

{

N ∀ t = 1, 2, · · · , TR

0 ∀ t > TR,(1)

whereN(t) is the number of UEs that are activated to sendtheir first preambles in RACH slott, and RACH slots arethe specific subframes for RACH preamble transmission.N ,

2In LTE, eNB expects to receive the same preamble power, i.e.,PREAMBLE_RECEIVED_TARGET_POWER, from UEs in the same preambletransmission attempts. To guarantee the same received preamble power fromdifferent UEs, a UE boosts the preamble transmission power,PRACH, to bePREAMBLE_RECEIVED_TARGET_POWER added by its path loss so as tocancel the effect of different path loss between UEs and the eNB. Since UEsin the same preamble transmission attempts have the same expected preamblereceiving power, the assumption that they have the same preamble detectionprobability is valid.

UE eNodeB

1. RACH preamble

2. Random access response (RAR, or Msg2)

3. RRC Connection Request (Msg3)

4. RRC Connection Setup (Msg4)

(a) Signaling Flow

No

Yes

Start RACH procedure

Preamble TX �mes ← 0

• Preamble TX �mes ← Preamble TX �mes +1

• Send preamble at a RACH slot

• Wait for Msg2

Receive Msg2

in RAR window ?

• Send Msg3

• Start Msg4 timer

Receive Msg4 before

Msg4 timer expires?

RACH procedure

completes successfullyRACH procedure

failure

Preamble TX times =

Max Preamble TX

times ?

No

No

Yes

Yes

Select a backoff

value and do backoff

(b) Flowchart of UE behavior

Fig. 1. RACH procedure

as a positive constant integer, is thepreamble arrival rate .Note that different from the model in [19], in our trafficmodel each UE performs the RACH procedure only once(includingK preamble transmission opportunities), i.e., oncea UE succeeds or fails in its RACH procedure, no retrial willbe performed throughout the evaluation. This uniform trafficmodel can be regarded as a realistic scenario in which theMTC devices access the network uniformly over a period oftime, i.e. in a non-synchronized manner as suggested in [3].For example, some delay-tolerant applications may drive smartmeters to perform non-synchronized measurement reports, andthese smart meters then randomly or even uniformly select thetiming for RACH preamble transmissions. Different from thepreamble arrival rate N , the preamble transmission rate,m, is the number of transmitted preambles in a RACH slot.Since in a RACH slott ∀t ≤ TR, N UEs send their firstpreambles and some UEs send their preamble retransmission,m is clearly the sum ofN and the preamble retransmissionrater, as illustrated by Fig. 2.

C. PDCCH constraint

In a PDCCH-constrained RACH system, a UE may fail inRACH procedure due to Msg2 or Msg4 timer expiration. Asmentioned in [5], in the simulation scenario of the LTE FDDmode in [3], the eNB has 16 physical downlink control channel(PDCCH) dedicated for RACH procedure in each subframe(1 ms). In each subframe, the eNB can send a random accessresponse (RAR) message, which always consumes 8 PDCCHand can carry 1 to 3 Msg2 (UL grants). In addition, it takes4 PDCCH to transmit a Msg4. Note that since the evaluationin [3] assumes that up to three Msg2 (i.e., 1 RAR) can besent to UEs in each subframe, in a subframe the eNB maytransmit at most three Msg2 and two Msg4 (8+4×2=16)or four Msg4 (4×4=16), rather than six Msg2 (8×2=16).

Since the RAR window size3 is 5 ms in [3], in a RARwindow the possible transmission quantity of (Msg2, Msg4)could be(3i, 20 − 2i) ∀i ∈ {0, 1, · · · , 5} if all the 80 (by16×5) PDCCH are used. For ease of indication, letNRAR,NMsg2, andNMsg4 be the number of RAR, Msg2, and Msg4that can be transmitted in five subframes respectively, andwe have8NRAR + 4NMsg4 ≤ 80, 0 ≤ NRAR ≤ 5, and0 ≤ NMsg2 ≤ 3NRAR.

The simulation in [3] applies theMsg2-first policy, whichassumes that the Msg2 has absolute priority over Msg4, i.e.,the eNB always sends a RAR (and at most 1 RAR) in asubframe if in that subframe at least one Msg2 is waiting to betransmitted. Prioritizing Msg2 in Msg2-first policy can avoidunnecessary Msg2 expiration since RAR window size is muchsmaller than the Msg4 timer length. However, when PDCCHshortage occurs, Msg2-first policy may not maximize the over-all RACH throughput since the probability of Msg4 expirationincreases. Consider that when PDCCH shortage occurs, Msg2and Msg4 compete for limited PDCCH, and thereby the overallRACH throughput is determined by the minimal throughputof Msg2 and Msg4, i.e.,min(NMsg2, NMsg4). The RACHthroughput maximization problem for each five subframes(i.e., RAR window size) can then be modelled as (2).

max(min(NMsg2, NMsg4))

s.t. 8NRAR + 4NMsg4 ≤ 16× 5 = 80

0 ≤ NRAR ≤ 5 , 0 ≤ NMsg2 ≤ 3NRAR,

NMsg2 ∈ Z∗ , NMsg4 ∈ Z∗ , NRAR ∈ Z∗ (2)

whereZ∗ is the set of non-negative integers, the objectivefunctionmax(min(NMsg2, NMsg4)) means to maximize theoverall throughput (min(NMsg2, NMsg4)), and the constraintinequalities consider the PDCCH constraints as mentioned

3In this work, we do not tune the length of Msg2 or Msg4 timer. Thereason is that increasing timer values helps little to improve PS when thebusy period is much larger than the two timers. Instead, applying larger timervalues extends the time for a UE to wait for Msg2 and Msg4, and thus causeslarger delay for successful RACH procedure.

0 500 1000 1500 20000

50

100

150

200

250

300

Time (RACH Slot)

Num

ber

of T

rans

mitt

ed P

ream

bles

Preamble Transmission Rate mPreamble Arrival Rate N

Preamble Transmission Rate m

Preamble Arrival Rate N

Preamble Retransmission Rate r

m = N +r

Expected Preamble TransmissionRate E(m) in the steady state

Fig. 2. Preamble transmission ratem is the sum of preamble arrival rateN and preamble retransmission rater. For uniform traffic model,N is assumed tobe a constant integer during the preamble arrival period (TR is 2000 in this figure); in contrast,r andm vary with time.

before. We let the proposedthroughput-first policy denotesthe resource management approach adopting the solution of(2). By solving (2)4, we can find that the optimal RACHthroughput is 12 whenNRAR is 4 andNMsg4 is 12. To reachthis optimum, in each RACH slot the eNB grants at most 12Msg2 even if it has the capacity to grant up to 15 Msg2, sothat it retains eight more PDCCH to send two more Msg4.

In summary, Msg2-first and throughput-first policy are twomethods to eNB’s resource allocation for Msg2 and Msg4.In the former method, eNB gives Msg2 an absolute priorityover Msg4; in contrast, in the latter method the eNB solves(2), whose solution instructs the eNB to limit the numberof transmitted Msg4 up to 12 per five subframes. In thefollowing section, we evaluate the RACH performance for 3scenarios: (a) no PDCCH constraint, (b) applying Msg2-firstpolicy for limited PDCCH, and (c) applying throughput-firstpolicy for limited PDCCH. Table I summarizes notations forthe following analysis.

IV. PERFORMANCEANALYSIS

A. Capacity of RACH Preamble Transmission

Consider a specific RACH slot in whichNi UEs transmittheir i-th preambles∀i ∈ [1,K] and m =

∑K

i=1 Ni, where[a, b] represents for the set of integerx that {x|a ≤ x ≤ b}throughout this paper. Then the probability that a UE suc-ceeds in itsi-th preamble transmission givenm simultaneouspreamble transmissions,PS1,i|m, is

PS1,i|m = pd,i

(

R− 1

R

)m−1

, (3)

wherepd,i is the preamble detection probability. Eq. (3) statesthat a UE succeeds in the preamble transmission if all the

4To solve (2), we can divide (2) into two linear integer programming sub-problems: to maximizeNMsg2 given NMsg2 ≤ NMsg4, and to maximizeNMsg4 given NMsg2 > NMsg4. Either of the two sub-problems can besolved by existing algorithms such as cutting plane algorithm or branch-and-bound algorithm [21]. The solution of (2) is then equal to that of the sub-problem with a larger maximum.

otherm− 1 UEs select the otherR− 1 preamble sequences,and the non-collided preamble is detected by the eNB.

Let S1 be the number of successfully transmitted preamblesgivenm simultaneous preamble transmissions.E(S1 | m), asderived by (53), is

E(S1 | m) = pdm(R− 1

R)m−1

, (4)

wherepd is the expected preamble detection probability of them UEs, i.e.,

∑Ki=1

Nipd,i∑Ki=1

Ni. Let RACH capacity be the maximal

RACH throughput, i.e.,maxm

E(S1 | m). Sincem representsfor the number of UEs transmitting preambles, we considerm

as an integer and we then have

E(S1 | m)

E(S1 | m− 1)=

m

m− 1

R− 1

R

> 1 if m < R

= 1 if m = R

< 1 if m > R

(5)

0 50 100 150 200 250 3000

10

20

30

the number of transmitted preambles m

E(

S1

m)

0 20 40 60 800

0.05

0.1

0.15

the number of successful preambles S1

P(

S1

m, R

)

m = 30m = 54m = 88

similar E(S1 m) and P(S

1 m, R)

for m = 30 and m = 88

RACH capacity is R(R−1R

)R−1

≃Re−1 at m = R − 1 or m = R

Fig. 3. The expectation and distribution of successful preambles (S1) givenpd,i = 1 ∀i ∈ [1,K] andR = 54.

Thus, RACH capacity is achieved by settingm equal toRor R − 1 as illustrated in Fig. 3, where we can seem = 30

TABLE INOMENCLATURE FORRACH PERFORMANCE ANALYSIS

Notation Description1. Conventional Form

[a, b] The set of integerx that {x|a ≤ x ≤ b}E(A)/P (A) The expectation / probability distribution of random variable A

2. Parameters for MAC-level RACH ModellingK The maximum number of RACH trials (or preamble transmissionattempts) in a RACH procedureR The number of available preamble sequences in each RACH slotpd,i The probability a non-collided preamble in thei-th transmission is detected by the eNB∀i ∈ [1,K]

NMsg3 max/NMsg4 max The maximum number of Msg3/Msg4 transmissionPf,i,Msg3 The probability a UE fails to transmit thei-th Msg3 to the eNB,i ∈ [1, NMsg3 max]Pf,i,Msg4 The probability the eNB fails to transmit thei-th Msg4 to a UE,i ∈ [1, NMsg4 max]

W The backoff window size (in RACH slots)TRAR The period length a UE waits for the RAR containing its Msg2THARQ The period length between two neighboring Msg3 or Msg4 transmissionsTMsg4 The period length for Msg4 timer expirationTretry The period length between the end of a RACH trial to the start of the next RACH trialT23 The period length from the timing UE receives UL grants to thetiming UE sends Msg3

3. System InputN The number of UEs activated to perform RACH procedure in a RACH slot t, ∀t ∈ [1, TR]

4. Performance MetricsS RACH throughput: the number of successful RACH procedure during a RACH slot (equal toS4)PS RACH success probability : the probability a UE succeeds in its RACH procedure duringK trialsD The average delay of a successful RACH procedure

5. Intermediate variables for RACH performance evaluationNi(t)/Ni The number of UEs that transmit theiri-th preambles in RACH slott/ in a steady-state RACH slotm(t)/m The number of UEs that transmit preambles in RACH slott/ in a steady-state RACH slot

S1(t)/S2(t)/S3(t)/S4(t) The number of successful preamble/Msg2/Msg3/Msg4 during RACH slot tS1/S2/S3/S4 The number of successful preamble/Msg2/Msg3/Msg4 during asteady-state RACH slot

NMsg2/NMsg4/NRAR The number of supportable Msg2/Msg4/RAR in a RAR window

N4,max/S4,maxThe number of transmitted/ successful Msg4 in a RAR window when all the 80 PDCCH resourcesin a RAR window are used for Msg2 and Msg4 transmission

E(tRAR) The average waiting time for a UE to receive its own UL grant after sending a preambleE(tMsg3)/E(tMsg4) The expected time a UE takes for unsuccessful Msg3 transmission/ successful Msg4 transmission

Di The average delay of a successful RACH procedure succeedingin the i-th RACH trial ∀i ∈ [1, K]DF12

/DF3/DF4

The average delay of a RACH trial that a UE fails in the step of Msg2/ Msg3/ Msg4DS4

The average delay of a RACH trial that a UE succeeds in Msg4 receptionPS,i (or PS4,i) The probability a UE succeeds in itsi-th RACH trial ∀i ∈ [1, K]

PF12,i/PF3,i/PF4,i The probability a UE fails in the step of Msg2/Msg3/Msg4 in its i-th RACH trial ∀i ∈ [1, K]

andm = 88 have similar distribution and expectation ofS1,derived by (49) and (53) respectively.

B. Expected Preamble Transmission Rate and RACH SuccessProbability

We start from the notations.Ni(t) is the number of UEs thattransmit theiri-th preambles∀i ∈ [1,K] in RACH slot t, andm(t) is the number of transmitted preambles in RACH slott,i.e., m(t) =

∑K

i=1 Ni(t). Moreover,S1(t), S2(t), S3(t), andS4(t) are respectively the number of UEs who successfullyfinish their signaling exchange of preambles, Msg2, Msg3,and Msg4 with the eNB respectively. Further, letNi andSj

represents for the steady-state value5 of Ni(t) andSj(t) ∀i ∈

[1,K] and∀j ∈ [1, 4] respectively. Sincem(t) is∑K

i=1 Ni(t),

5Because of the uniform traffic model and the uniform backoff counterselection, the expected steady state value ofm(t), Ni(t) ∀i ∈ [1,K], andSj(t) ∀j ∈ [1, 4] exist. However, since the random variablesNi ∀i ∈ [2,K]are correlated and the preamble detection probability in the i-th and j-thpreamble transmission∀i 6= j are different, the derivation of the probabilitymass function ofm involves a complicatedK-dimensional Markov chain.To make the analysis tractable and reduce calculation complexity, in thispaper we deriveE(m), E(Ni) ∀i ∈ [1,K], andE(Sj) ∀j ∈ [1, 4] ratherthan distribution ofm, the joint distribution ofN1, · · · , NK and the jointdistribution ofS1, · · · , S4.

we have

E(m) = E(

K∑

i=1

Ni) =

K∑

i=1

E(Ni) (6)

By (4), E(S1) is obtained by

E(S1) ≃ E(S1 | m = E(m)) = pdE(m)(R− 1

R)E(m)−1

=

K∑

i=1

E(Ni)pd,i(R− 1

R)E(m)−1

, (7)

where the third equality follows the approximation thatpd ≃∑Ki=1

E(Ni)pd,i∑Ki=1

E(Ni)=

∑Ki=1

E(Ni)pd,i

E(m) .

Let NMsg3 max and NMsg4 max be the maximal HARQtransmission attempts of Msg3 and Msg4 respectively.Pf,i,Msg3 andPf,j,Msg4 are respectively the probability thata UE fails to transmit thei-th Msg3 and fails to receive thej-th Msg4,∀i ∈ [1, NMsg3 max] and∀j ∈ [1, NMsg4 max]. Inthis way,E(S3) can be expressed as

E(S3) = E(S2)(1−

NMsg3 max∏

i=1

Pf,i,Msg3), (8)

which follows the fact that a UE who has received its Msg2has a probability1 −

∏NMsg3 max

i=1 Pf,i,Msg3 to succeed intransmitting Msg3 to the eNB through HARQ transmission.

We now approximateE(S2) andE(S4) for different down-link resource constraints.

1) No PDCCH constraint: With sufficient PDCCH re-sources, the eNB can grant all requested Msg2 and Msg4.Thus, we have

E(S2) = E(S1) (9)

E(S4) = E(S3)(1 −

NMsg4 max∏

i=1

Pf,i,Msg4) (10)

2) Msg2-first policy: In Msg2-first policy, Msg2 has abso-lute priority over Msg4, and thus at most 15 Msg2 can besupported in a RAR window. By total probability theoremand Bayes’ rule [22], the distribution ofS2 is

P (S2 = s2) =

∑

{Ni}Ki=1

P ({Ni}Ki=1)P (S1 = s2 | {Ni}

Ki=1, R) ∀s2 ∈ [0, 14]

min(∑K

i=1 Ni,R)∑

s=s2

∑

{Ni}Ki=1

P ({Ni}Ki=1)P (S1 = s | {Ni}

Ki=1, R)

, if s2 = 15,(11)

where P (S1 | {Ni}Ki=1, R) is the distribution of

S1 derived in (54), andP ({Ni}Ki=1) is the joint dis-

tribution of N1, · · · , NK . As mentioned, the derivationof P ({Ni}

Ki=1) involves complicated calculation, so we

approximateP ({Ni}Ki=1) by INi=E(Ni), ∀i∈[1,K], where

IA is the indicator function, i.e., ifA is true, IA is 1;otherwise,IA is 0. We then have

P (S2 = s2) ≈

P (S1 = s2 | {E(Ni)}Ki=1, R) ∀s2 ∈ [0, 14]

min(∑K

i=1 Ni,R)∑

s=s2

P (S1 = s | {E(Ni)}Ki=1, R) , if s2 = 15

(12)

As mentioned before, in a RAR window the eNB has 80PDCCH available, support up to 5 RAR, and each RARalways consumes 8 PDCCH and can carry 1 to 3 Msg2.Thus, givenS2, the eNB would take8min(5, ⌈S2

3 ⌉)PDCCH for Msg2 transmission. Since each Msg4 takes4 PDCCH, the number of transmittable Msg4 in a RARwindow,N4,max, is

N4,max =1

4

(

80− 8min(5, ⌈S2

3⌉)

)

= 20− 2min(5, ⌈S2

3⌉) (13)

By (12) and (13), we can derive the distribution ofN4,max, i.e., P (N4,max). Next, when PDCCH is insuf-ficient for all UEs’ request, the eNB will use all the 80PDCCH resources to transmit Msg2 and Msg4. In thisway, N4,max Msg4 are transmitted, and on average theeNB sends at most one Msg4 to each UE waiting for

Msg4. Therefore, the RACH throughput in the PDCCHshortage scenario,S4,max, is given by

P (S4,max = s) =

20∑

n=0

P (N4,max = n)∗

((

n

s

)

(1− Pf,1,Msg4)sPf,1,Msg4

n−s

)

∀s ∈ [0, 20], (14)

E(S4,max) =20∑

s=0

sP (S4,max = s), (15)

where we haveS4,max ≤ 20 since in a RAR window80 PDCCH can support up to 20 Msg4, i.e.,1

4 × 80. Wethen have

E(S4) =

E(S4,max) , if E(S2) > E(S4,max)

E(S3)(1 −

NMsg4 max∏

i=1

Pf,i,Msg4) , otherwise

(16)

which states that if PDCCH resource is the bottleneck(E(S2) > E(S4,max)), the eNB uses all remainingPDCCH for Msg4 transmission, and thus RACH through-put is determined byE(S4,max); otherwise, the RACHthroughput is determined byE(S3) and the HARQ re-transmission probability.

3) Throughput-first policy : In the throughput-first policy,the eNB grants at most 12 Msg4 in a RAR window. Sim-ilar to (12) and (13),P (S2) andN4,max are respectivelygiven by

P (S2 = s2) ≈

P (S1 = s2 | {E(Ni)}Ki=1, R) ∀s2 ∈ [0, 11]

min(∑K

i=1Ni,R)

∑

s=s2

P (S1 = s | {E(Ni)}Ki=1, R), if s2 = 12

(17)

N4,max = 20− 2min(4, ⌈S2

3⌉) (18)

With P (S2) andP (N4,max), E(S4,max) andE(S4) canthen be derived by (15) and (16).

Let PS,i be the probability that a UE succeeds in thei-thRACH trial ∀i ∈ [1,K]. Since on average all UEs havethe same successful probability in the transmission of Msg2,Msg3, and Msg4,PS,i is proportional topd,i ∀i ∈ [1,K]. So,PS,i can be obtained fromE(S4) by

PS,i =pd,i

∑Kr=1 E(Nr)pd,r

E(S4) ∀i ∈ [1,K], (19)

which means that each UE in thei-th trial has a probabilitypd,i∑

Kr=1

E(Nr)pd,rto be one of theE(S4) UEs that successfully

receive Msg4. To justify (19), we can see when in a scenariowith a perfect Msg2, Msg3, and Msg4 transmission (i.e.,sufficient PDCCH,Pf,i,Msg3 = 0 ∀i ∈ [1, NMsg3 max], andPf,j,Msg4 = 0 ∀j ∈ [1, NMsg4 max]), (19) can be reduced to(3), i.e.,PS1,i|m=E(m).

With PS,i, we can approximateE(Ni), ∀i ∈ [2,K] by

E(Ni) = E(Ni−1)(1 − PS,i−1), (20)

which means that all UEs failing in theiri − 1 RACH trialwill proceed to theiri-th RACH trial after the backoff time∀i ∈ [2,K].

In summary, we can deriveE(m), PS,i andE(Ni) ∀i ∈[1,K] by the iterative procedure6 in Fig. 4. After derivingPS,i ∀i ∈ [1,K], the RACH success probability,PS , is thengiven by

PS = 1−

K∏

i=1

(1− PS,i). (21)

Note that the derivation ofE(m) andPS in the steady statedoes not involveW . This is because in the steady stateW isfar smaller than the RACH traffic arrival period and thus doesnot contribute to RACH contention resolution.

C. Delay analysis

Let PF12,i, PF3,i, andPF4,i be the probability that a UEfails in its i-th RACH trial due to failed Msg2 reception, failedMsg3 transmission, and failed Msg4 reception respectively. Inaddition, PS4,i, equal toPS,i, is the probability that a UEsucceeds in thei-th RACH trial. Given the definition, wehavePF12,i + PF3,i + PF4,i + PS4,i = 1 ∀i ∈ [1,K]. Morespecifically,

PF12,i = 1−pd,i

∑K

r=1E(Nr)pd,rE(S2), (22)

PF3,i =pd,i

∑K

r=1E(Nr)pd,r[E(S2)− E(S3)], (23)

PF4,i =pd,i

∑K

r=1E(Nr)pd,r[E(S3)− E(S4)], (24)

PS4,i = PS,i =pd,i

∑K

r=1 E(Nr)pd,rE(S4), (25)

where the derivation ofPF12,i, PF3,i, andPF4,i is similar tothat ofPS,i in (19).

Let TRAR be the window size of Msg2,TMsg4 be thewindow size of Msg4,T23 be the time interval length betweena UE receives a Msg2 and the UE sends a Msg3, andTretry

be the time for a UE to prepare for the next RACH preambleretransmission. In each RACH trial, the average delay whena UE fails to receive a Msg2 (DF12

), fails to send theMsg3 (DF3

), fails to receive a Msg4 (DF4), and succeeds in

receiving a Msg4 (DS4) are respectively given by

DF12= 1 + TRAR + Tretry +

W + 1

2, (26)

DF3= 1 + E(tRAR) + T23 + (NMsg3 max − 0.5)THARQ

+ Tretry +W + 1

2, (27)

DF4= 1 + E(tRAR) + T23 + E(tMsg3) + TMsg4

+ Tretry +W + 1

2, (28)

6Fig. 4 summarizes the relation amongE(m), PS,i ∀i ∈ [1, K],E(Ni) ∀i ∈ [1,K], and Sj ∀j ∈ [1, 4]. For example, we can deriveE(S1) by E(Ni), E(m) and (7) as shown in Fig. 4. To solve (6) to (20)simultaneously for these metrics, we can initializePS,i ∀i and then iterativelyupdate their values according to the closed loop in Fig. 4. The answer isobtained when convergence is achieved.

DS4= 1 + E(tRAR) + T23 + E(tMsg3) + E(tMsg4)

+ Tretry +W + 1

2, (29)

whereE(tRAR) is the average waiting time for a UE to receiveits own UL grant after sending a preamble;THARQ is the timerinterval length between two neighboring Msg3 transmissions;(NMsg3 max − 0.5)THARQ is the total time for a UE to findits NMsg3 max-th failure in sending Msg3, where0.5THARQ

is the time interval length between a Msg3 transmission andthe UE’s awareness of Msg3 transmission failure (i.e., noACK is received);E(tMsg3) is the expected time a UE takesfor unsuccessful7 Msg3 transmission; andE(tMsg4) is theexpected time for a UE to successfully receive Msg4.

We now introduce the derivation ofE(tRAR), E(tMsg3) andE(tMsg4). E(tRAR) can be derived by consideringE(S2) andthe starting time of the RAR window. For example, if a UEsends its preamble in subframen, the RAR window starts inn+3, andE(S2) is 12,E(tRAR) is about3− 1+ ⌈E(S2)

3 ⌉ ifthere is no PDCCH constraint;E(tMsg3) can be derived by

E(tMsg3) =

NMsg3 max∑

i=1

(i− 1)THARQ∗

(1− Pf,i,Msg3)∏i−1

r=1 Pf,r,Msg3∑NMsg3 max

i=1 (1− Pf,i,Msg3)∏i−1

r=1 Pf,r,Msg3

, (30)

which is the average time for unsuccessful Msg3 transmissionsgiven that the last Msg3 transmission is successful.E(tMsg4),the time a UE waits until it receives the Msg4 successfully,depends onE(S4) and can only be derived through someapproximation approaches such as techniques in queueingtheory.

The average delay that a UE succeeds in thei-th RACHtrial, Di, is then given by

Di =

i−1∑

j=1

PF12,jDF12+ PF3,jDF3

+ PF4,jDF4

PF12,j + PF3,j + PF4,j

+DS4

(31)

Finally, the average delay for a successful RACH procedure,D, is given by:

D =

K∑

i=1

Di

PS,i

∏i−1j=1 (1 − PS,j)

∑K

i=1 PS,i

∏i−1j=1 (1− PS,j)

(32)

V. DYNAMIC BACKOFF : A NOVEL RACH CONTENTION

RESOLUTION SCHEME

A. Motivation and Illustration of the dynamic backoff scheme

RACH procedure is an efficient protocol, but it is notadaptive to bursty traffic load, and thus preamble collisionswill be severe if many UEs begin their preamble transmissionsat a similar time. In view of this, we propose the dynamic

7Please note that because a UE (re)starts the Msg4 timer upon(re)transmitting Msg3, the period for the last Msg3 transmission, which issuccessful transmitted, is always overlapped with the beginning period ofthe Msg4 timer. Thus, to avoid calculating twice the time forthe last Msg3transmission,E(tMsg3) is defined as the expected time for unsuccessfulMsg3 transmission rather than for successful Msg3 transmission.

• Initialization

• An arbitrary scalar �, 0 < � < 1

• ��,� = �, ��, = ��,���,

��,� ∀ � ∈ [1, � − 1]

��, ∀ � ∈ [1, �]

�(�) ∀ � ∈ [1, �]

�(�)

�(��) �(��)

• No PDCCH constraint

• Eq. (8)-(10)

• Msg2-first policy

• Eq. (8), (12)-(16)

• throughput-first policy

• Eq. (8), (14)-(18)

• � �� = N

• Eq. (20)

Eq. (6)

Eq. (7)

Eq. (19)

Fig. 4. Summary for the iterative derivation ofE(m), PS,i andE(Ni) ∀i ∈ [1,K]

backoff (DB) scheme, which dynamically configure the back-off window size according to the RACH traffic load. For easeof indication, in the following context we letbaseline RACHand baseline schemedenote the RACH procedure specifiedin the LTE specification, as we introduced in Section III.A.

Given the same RACH arrival traffic, we compare thebaseline RACH and the DB scheme in Fig. 5. In Fig. 5(a), weobserve that for the baseline RACH a high preamble arrivalrate causes a high preamble transmission rate, and thus ahigh preamble collision probability. As a result, successfulpreamble transmissions occur only at the beginning and theend of the RACH arrival period. Applying a large backoffwindow sizeW may help resolve preamble contention, yetthe eNB does not know the value ofW that can maximizethe RACH throughput while guaranteeing an acceptablePS

since typically the eNB has incomplete information aboutRACH arrival traffic. In contrast, the DB scheme estimatesthe number of UEs that attempts to send preambles, and thenaccordingly calculates the backoff window size to reshape thepreamble transmission traffic, as shown in Fig. 5(b) and Fig.5(c) respectively. In this way, both RACH throughput andPS

can be significant improved without the need to guess theoptimalW value.

B. Protocol design

The proposed scheme modifies only the first step of thebaseline RACH and keeps the remaining steps unchanged. Inthe baseline scheme, a UE sends its preamble upon activatedto access the network, so the preamble transmission rateis determined by the preamble arrival rate. Instead, in theproposed scheme the eNB directly reshapes the preambletransmission traffic according to the RACH attempt estimationresult, and thus RACH performance does not suffer from theunpredictable preamble arrival rate.

Time structure: We divide the time axis into disjointRACH configuration periods, as shown in Fig. 6.Ti meansthe i-th RACH configuration period, andTi(j) denotes thej-th RACH slot in Ti. Ti hasti RACH slots, containingNG,i

(equal toti−1) normal RACH slots and 1RACH estimation

slot, i.e., thee-th RACH slot,Ti(e), is dedicated for the eNBto estimate the number of RACH attempts. In our designe isequal toti − TB, whereTB is the period length (in RACHslots) for RACH configuration broadcast and is decided bysystem implementation.

In the DB scheme, each UE runs at mostK rounds oftransmission processuntil it successfully sends a preamble. Ineach transmission process, a UE first joins theRACH attemptestimation processin a RACH estimation slot, and then trans-mit a preamble in a normal RACH slot. In a RACH attemptestimation process, a UE first retrieves the probabilitype,i andthe timing ofTi(e) for Ti, and then performs a Bernoulli trialwith a probabilitype,i. If the trial is successful, the UE sendsa preamble inTi(e) with the maximal preamble transmissionpower level, i.e., with detection probabilitypd,K ; otherwise,the UE does not send a preamble. If a UE successfully sendsa preamble inTi(e), it proceeds with its RACH procedure. Ifa UE successfully sends a preamble inTi(e) and then finishesRACH procedure successfully, it leaves the RACH system;otherwise, the UE prepares to send a preamble after receivingthe updated RACH configuration.

In Ti(e), the eNB observes the number of successful (S)preamble sequences andempty (E) preamble sequences (i.e.,not selected by any UE to send preambles). LetMi be thetotal number of UEs that wants to (re)transmit a preamble inTi, andMi is the approximation ofMi. The eNB usesS andE to get Mi, and then decides and broadcasts 3 parameters(NG,i+1, pe,i+1, andti+1(e)) for the next configuration period(Ti+1).

After receiving the 3 parameters forTi+1, a UE who hasfinished itsj-th estimation process inTi but not yet transmitteda preamble successfully finishes itsj-th transmission processby uniformly selects one of theNG,i+1 normal RACH slotsfor preamble transmission.

C. Parameter Determination

We now show the way to deriveMi, NG,i+1, pe,i+1, andTi+1(e) as follows.

0 500 1000 1500 2000 2500 3000 3500 40000

500

Time (RACH slot)

UE

num

ber

(a)

RACH transmission rateRACH arrival rateRACH success rate

0 500 1000 1500 2000 2500 3000 3500 40000

2

4x 10

4

Time (RACH slot)RA

CH

acc

ess

atte

mpt

s

(b)

realistic estimated

0 500 1000 1500 2000 2500 3000 3500 40000

100

200

Time (RACH slot)

UE

num

ber

(c)

RACH transmission rateRACH arrival rateRACH success rate

Baseline RACH

Dynamic Backoff Scheme

Dynamic Backoff Scheme Preamble transmission rate is reshaped to ensurehigh RACH success rate (throughput)

Fig. 5. Comparison of the baseline RACH and the dynamic backoff scheme

Ti-1 Ti Ti+1

• Ti : the i-th RACH configuration period

• Ti (j) : the j-th RACH slots in Ti

• Ti (e) : the RACH estimation slot in Ti

• ti : the number of RACH slots in Ti

• R : the number of preambles

• : the RACH estimation slot

• ��� : the number of RACH attempts estimated in Ti (e)

• �: the desired preamble transmission rate

• NG,i : the number of normal RACH slots in Ti

• pe,i : the probability for a UE to send a preamble at Ti (e)

• TB : broadcast period to update RACH configuration

TB TB TBNG,i-TB NG,i+1-TB

(1) NG,i-1 = TB

(2) pe,i-1 = �

���

(3) Ti-1(e) = Ti-1(1)

(1) NG,i = ����

�

(2) pe,i = �

����

(3) Ti(e) = Ti(NG,i-TB+1)

(1) NG,i+1 = ���

�

(2) pe,i+1 = �

���

(3) Ti+1(e) = Ti+1(NG,i+1-TB+1)

time

1 1 1

ti ti+1ti-1

Fig. 6. The time structure of the proposed dynamic backoff scheme

• Derivation of Mi: We can approximateMi by Mi asfollows

Mi = argmaxM′

P (S,E | M ′, R)

= argmaxMi

∑

M1,U

P (S | U, pd,K)P (U,E | M1, R)∗

P (M1 | Mi, pe,i)

≈

argmaxM1

P (U = ⌈ Spd,K

⌉, E | M1, R)

pe,i, (33)

whereR is the number of preambles per RACH slot,and M1 is the number of UEs sending preambles inTi(e), and M1 is the estimation ofM1. Note thatboth M1 and S are binomial random variables, i.e.,P (M1 | Mi, pe,i) =

(

Mi

M1

)

pe,iM1(1− pe,i)

Mi−M1 and

P (S | U, pd,K) =(

US

)

pd,KS(1− pd,K)

U−S respectively.And P (U,E | M1, R), as expressed in (34), has a similarderivation to that ofP (U | m,R) in (49). The readerscan refer to [23] for a recursive derivation of (34).

P (U,E | M1, R) =1

RM1

(

M1

U

)(

R

U

)

U !

(

R− U

E

)

∗

R−U−E∑

i=0

(−1)i(

R − U − E

i

)

i∑

j=0

(

M1 − U

j

)(

i

j

)

j!∗

(R− U − E − i)M1−U−j (34)

• Determination of NG,i+1: To guarantee that the trans-mission rate in each normal slot is about or less thanG, we setNG,i+1 as max(TB,⌈ Mi

G⌉). The selection ofG

determines the RACH performance and will be discussedin the following subsection.

• Determination of Ti+1(e): To guaranteeTB RACH slotsfor eNB to broadcast RACH configuration,Ti+1(e) is setas the lastTB + 1 RACH slots ofTi+1.

• Determination of pe,i+1: We set pe,i+1 as ⌈ R

Mi⌉ for

a good estimation result. Note that ifpe,i+1 is toolarge, numerous preamble transmissions may cause thesituation that each RACH preamble sequence suffers fromcollision, i.e.,S = 0 andE = 0. In this way, the eNBonly knows that the number of transmitted preambles islarge, but has insufficient information to estimate howlarge it is. In contrast, ifpe is too small, a bad estimationresult happens when no UEs send their preambles in

Ti+1(e). The selection ofpe,i+1 affects the estimation

error, i.e.,ˆMi+1−Mi+1

Mi+1, and will be discussed in the

following subsection.

Without loss of generality, in the following analysis wereplaceMi, pe,i, andNG,i with M , pe, andNG to simplifynotations. Table II summarizes notations for analysis.

D. Estimation Error of the RACH access attempts

In this subsection we assumepd,K is very close to 1, asspecified in [3]. In this case, the number of non-collidedpreambleU is approximately equal to the number of successfulpreamblesS, and thusP (S,E | M1, R) ≃ P (U,E | M1, R).

To evaluate| M1−M1

M1|, we first checkP (U,E | M1, R) in

(34) and Fig. 7. We can seeP (U = 0, E = 0 | M1, R) inFig. 7(a) is a monotonically increasing function ofM1 since(U = 0, E = 0) means that collision occurs to each preamble,which is more likely to happen asM1 increases. Therefore,(U = 0, E = 0) cannot be used to estimateM1. In contrast,eitherU > 0 or E > 0 means that the collision situation isless severe and thusP (U,E | M,R) has a unique maximum∀ U + E ≤ R, as shown in Fig. 7(b).

GivenM1, the distribution ofM1 is given by

P (M1 | M1)

=∑

0≤U+E≤R,U∈Z∗, E∈Z∗

IM1=argmax

m′

P (U,E|m′,R)P (U,E | M1, R),

(35)

whereZ∗ is the set of non-negative integers, andIA is theindicator function, i.e., ifA is true, thenIA is 1; otherwise,IA is zero. SinceM1 cannot be easily expressed as a closed-form function ofM1, R, U , andE, we numerically examineE(| M1−M1

M1|) in Fig. 8, which shows that smallerM1

Rcauses

smaller estimation error. Although we can apply smallerpe toobtain smallerM1

Rand thus smaller estimation error, too small

pe reduces the RACH throughput in the RACH estimation slot.Therefore, in this paper we controlM1 close toR by lettingpe =

R

M≈ R

M.

GivenM , the number of normal RACH slotsNG is given by⌈ MG⌉ (assuming⌈ M

G⌉ > TB), so the probability a UE selects

a normal RACH slot,pn, is given bypn = 1NG

= 1

⌈ MG

⌉≃ G

M.

LetUn be the number of non-collided preambles in a normalRACH slot. We have

E(Un) =∑

M

E(Un | M, M)P (M | M), (36)

whereE(Un | M, M) is given by

E(Un | M, M) =M∑

n=0

(

M

n

)

pnn(1− pn)

M−n

(

n(R − 1

R)n−1)

= Mpn(1−pn

R)M−1 R≫1

≃ Mpnexp(−Mpn

R)

≃ GM

Mexp(−

G

R

M

M) ≃ G

R

M1

exp(−G

R

R

M1

), (37)

where the first equality follows (51), the fourth equality fol-lows pn ≃ G

M, and the last equality is obtained byM1 = Mpe

andMpe ≃ R. In addition, we have

P (M | M) =∑

M1

P (M | M1)P (M1 | M,pe)

M≫1,Mpe≃R=

∑

M1

P (M1 = Mpe | M1)e−RRM1

M1!, (38)

where the last equality is obtained by applying Poissondistribution to approximate the binomial distribution. From(37) and (38), we can seeE(Un | M, M) and P (M | M)are not dependent onM value whenM is much larger than1. Thus, by (36)E(Un) is also a constant regardless ofM ,which suggests that degradation from estimation error has aupper bound. Note that if there is no estimation error, wehave M

M= 1 and P (M | M) = δ(M − M), where δ(·)

is the dirac delta function. In this way,E(Un) in (36) isequal to the desired value, i.e.,E(Udesired) = G(R−1

R)G−1

by substitutingG into m in (51). By numerical examination,in the scenario without PDCCH constraint, theE(Un) degra-dation ( E(Un)

E(Udesired)− 1) is quite small as shown in Fig. 9.

Due to negligibleE(Un) degradation from estimation error,in the following performance analysis we assume perfectRACH attempt estimation and useE(Udesired) to approximateE(Un).

E. Performance Analysis

The DB scheme shares the same evaluation methodologywith the baseline scheme because the two schemes are bothanalyzed under a steady preamble transmission ratem: in thebaseline scheme we analyze the RACH performance when thesteady-statem is reached, while in the DB schemem is ad-justed to be steady aboutG. The difference of their evaluationis that in the baseline scheme, we know the preamble arrivalrateN and want to deriveE(m). In contrast, in DB schemewe knowE(m) ≃ G and aim to obtain the time average ofpreamble arrival rate.

We introduce notations as follows.m(t), Ni(t) ∀i ∈ [1,K],Sj(t) ∀j ∈ [1, 4] have been defined in Table I.Tend is thenumber of RACH slots for the DB scheme to manage thearriving RACH traffic, i.e.,Tend RACH slots are taken for allUEs arriving during[1, TR] to finish their RACH preambletransmissions.Ni,DB and Sj,DB are respectively the timeaverage ofNi(t) andSj(t) over t ∈ [1, Tend], i.e., Ni,DB =∑Tend

t=1Ni(t)

Tend∀i ∈ [1,K] and Sj,DB =

∑Tendt=1

Sj(t)Tend

∀j ∈[1, 4]. Note that since the total number of transmitted RACHattempts duringt ∈ [1, Tend] is equal to TendE(m) or∑K

i=1 Ni,DBTend, we haveE(m) =∑K

i=1 Ni,DB = G. Witha steady transmission rateE(m), equations for the schemewithout PDCCH constraint are similar to (7)-(10), i.e.,

S1,DB ≃K∑

i=1

Ni,DBpd,i(R− 1

R)E(m)−1

(39)

S2,DB = S1,DB (40)

S3,DB = S2,DB(1−

NMsg3 max∏

i=1

Pf,i,Msg3) (41)

S4,DB = S3,DB(1−

NMsg4 max∏

i=1

Pf,i,Msg4) (42)

TABLE IINOMENCLATURE FOR THE DYNAMIC BACKOFF SCHEME

Notation DescriptionTi the i-th RACH configuration periodti the number of RACH slots inTi

Ti(j) The j-th RACH slot inTi

Ti(e) The RACH estimation slot inTi

TB A constant period to broadcast RACH configurationpe,i The probability for a UE with a RACH attempt to send a preamblein Ti(e)NG,i The number of normal RACH slots inTi

pn The probability for a UE to select a normal RACH slot for preamble transmissionMi The number of UEs that join the RACH attempt estimation process inTi(e)M1 The number of UEs sending preambles in a RACH estimation slotMi The estimation ofMi

M1 The estimation ofM1

R The number of available preambles in a RACH slotS The number of success preambles in a RACH slotE The number of unused (empty) preambles in a RACH slot

U/Un The number of non-collided preambles in a RACH / normal RACH slotG The desired number of transmitted preambles in a RACH slot

Tend The number of RACH slots for all UEs to finish sending their preamblesPS,i,DB The probability that a UE succeeds in itsi-th RACH trials during[1, Tend]

Ni,DB The time average ofNi(t) over [1, Tend], i.e.,Ni,DB =∑Tend

t=1Ni(t)

Tend∀i ∈ [1,K]

Si,DB The time average ofSi(t) over [1, Tend], i.e.,Si,DB =∑Tend

t=1Si(t)

Tend∀i ∈ [1, 4]

0 100 200 300 400 500 600 7000

0.5

1

the number of transmitted preambles M1

P(

U, E

M

1, R)

(a)

U = 0, E = 0

0 100 200 300 400 500 600 7000

0.1

0.2

0.3

the number of transmitted preambles M1

P(

U, E

M

1, R)

(b)

U = 1, E = 1U = 2, E = 0

Fig. 7. Illustration ofP (U,E | M1, R)

Note that if we consider limited PDCCH,S2,DB andS4,DB

can be derived by just substitutingS2,DB and S4,DB intoE(S2) and E(S4) in (12)-(16) for Msg2-first policy or in(14)-(18) for throughput-first policy respectively. Moreover,let PS,i,DB be the probability that a UE succeeds in itsi-thRACH trial during t ∈ [1, Tend]. Analogous to (19)-(21), wehave

PS,i,DB =pd,i

∑K

r=1 Nr,DBpd,rS4,DB ∀i ∈ [1,K] (43)

Ni,DB = Ni−1,DB(1− PS,i−1,DB) ∀i ∈ [2,K] (44)

PS,DB = 1−K∏

i=1

(1− PS,i,DB) (45)

By simultaneously solvingE(m) =∑K

i=1 Ni,DB, and (39) to(45), we can obtain the RACH success probabilityPS,DB andthe RACH throughputS4,DB.

Finally, note thatNi(t), Si(t), andPS,i(t) have steady-statevalues (i.e.,Ni, Si, andPS,i respectively) and time-averagedvalues (i.e.,Ni,DB, Si,DB, andPS,i,DB respectively). It is thetime-averaged values that reflects the RACH performances. Inthe baseline scheme, the steady-state values approach the time-averaged values, so we derive the steady-state values for easeof calculation. In contrast, in the DB scheme the steady-statevalues are quite different from the time-averaged values, andwe directly derive the time-averaged values.

VI. SIMULATION RESULTS

Our simulation parameters follow the evaluation methodol-ogy of LTE FDD mode in [3], as shown in Table III. In the“No PDCCH constraint” scenario, we assume the quantity ofPDCCH is always sufficient to send all requested Msg2 andMsg4 in a RACH slot.

TABLE IIISIMULATION PARAMETERS FORLTE FDD

Parameter Setting Parameter SettingCell bandwidth 5 MHz PRACH Configuration Index 6

Total number of preambles 54 Maximum number of preamble transmission 10Number of UL grants per RAR 3 Number of CCEs allocated for PDCCH 16Number of CCEs per PDCCH 4 Ra-ResponseWindowSize 5 subframes

mac-ContentionResolutionTimer 48 subframes Backoff Indicator 20 msHARQ retransmission probabilityfor Msg3 and Msg4 (non-adaptive HARQ)

10%Maximum number of HARQ TXfor Msg3 and Msg4 (non-adaptive HARQ)

5

preamble detection probabilityfor the i-th preamble transmission 1−

1

ei

0 1 2 30

0.02

0.04

0.06

0.08

M1

R

E

(

|M1−

M1

M1

|)

Fig. 8. E(| M1−M1

M1|) for different M1

Rwith R = 54

0 50 100 150 200−0.03

−0.02

−0.01

0

0.01

desired preamble transmission rate G

E(U

)−E

(Ud

es

ir

ed)

E(U

de

sir

ed)

Fig. 9. Degradation in the number of non-collided preamblesU due to RACHattempt estimation error for differentG given R = 54 andpe ≃ R

M

A. Evaluation for DL Resource Shortage

In Fig. 10 we show the RACH performance for RACHsystems with different PDCCH constraints. We can see that thethree schemes have different performance only when preamblearrival rateN is between 10 and 17. This is because whenN isless than 10, the Msg2 throughput (S2) is low due to lowN . Inthis case, 16 PDCCH are sufficient and the RACH throughputis dominated byN . WhenN ranges between 10 and 17, Msg2throughput exceeds the capacity of 16 PDCCH, and thus for

a PDCCH-limited RACH system the RACH throughput isdominated by the available PDCCH. ForN > 17, high N

causes severe preamble collisions and low Msg2 throughput.In this case 16 PDCCH are sufficient to support all Msg2and Msg4 again, and thus RACH throughput is dominated byRACH collisions. As we expected, for10 ≤ N ≤ 17 theproposed throughput-first policy has higher RACH throughputthan the original Msg2-policy. Lastly, the theoretical resultsefficiently approximate the simulation results.

B. Performance evaluation of the proposed dynamic backoffmechanism

1) Uniform Traffic: In Fig. 11 the RACH throughput fora RACH system without a PDCCH constraint is shown. Wesee that whenN is less than 15, the RACH throughput ofboth the baseline and the dynamic backoff (DB) schemesare close to the RACH arrival rate, i.e., very high RACHsuccess probability. AsN increases above the RACH capacity(about 17), the RACH throughput of DB remains close to theRACH capacity regardless ofK andN . In contrast, the RACHthroughput of the baseline scheme drops for largerN or largerK since either largerN or K results in a larger preambletransmission rate, a larger RACH collision probability, andthus a lower RACH success probability and a lower RACHthroughput.

For DB scheme, RACH throughput for differentG values isshown in Fig. 12. First, we compare the maximal throughputfor different preamble detection probability. In Fig. 12(a),when we configureG close toR (54), the achievable maximalthroughput without PDCCH constraint is about19.5. Remem-ber that the maximum throughput of a slotted-ALOHA systemis about 19.87 (by54 ∗ e−1 [24]), and thus we reach98.14%(by 19.5

19.87 ) capacity. In contrast, in Fig. 12(b) we considerlower preamble detection probability. When we configureG

as 60 and there is no PDCCH constraint, the theoretical andsimulated RACH throughput are respectively 16.96 and 16.66,so the resource efficiency is98.23% (by 16.66

16.96 ). Thus, RACHcapacity is related to the preamble detection probability,andthe proposed scheme can approach the theoretical capacity ofslotted Aloha even if affected by the estimation error.

Secondly, we observe the RACH throughput bounded by thePDCCH resources in Fig. 12(a) and (b). For the two scenarioswith PDCCH constraint, the maximal throughputs are around9.0 and10.8 respectively whenG ranges from30 to 120. The

5 10 15 20 250

0.5

1

Preamble Arrival Rate N

Suc

cess

Pro

babi

lity

PS

5 10 15 20 250

5

10

15

20

Preamble Arrival Rate N

RA

CH

thro

ughp

ut

5 10 15 20 250

100

200

300

400

500

Preamble Arrival Rate N

Tra

nsm

issi

on R

ate

E(m

)

5 10 15 20 250

50

100

150

Preamble Arrival Rate N

succ

ess

dela

y (m

s)

simu, No PDCCH constrainttheo, No PDCCH constraintsimu, Msg2−first policytheo, Msg2−first policysimu, throughput−first policytheo, throughput−first policy

Fig. 10. RACH performance metrics for different arrival rate N and DL resource constraints

10 20 30 40 500

5

10

15

Preamble Arrival Rate N

RA

CH

thro

ughp

ut

Baseline, K = 6Baseline, K = 8Baseline, K = 10Baseline, K = 12DB, K = 6DB, K = 8DB, K = 10DB, K = 12

Fig. 11. RACH throughput without a PDCCH constraint.TR is 2000.

result 9.0 and 10.8 can be easily derived by multiplying thePDCCH capacity (10 and 12 respectively) with1−Pf,1,Msg4.

2) Beta Traffic: A beta distributed RACH traffic model isdescribed in [3]. In this traffic model the preamble arrivalperiod is during[0, T ] and consists ofTR access opportunities(RACH slot). The preamble arrival rate in RACH sloti isgiven byNdevice

∫ ti+1

tip(t)dt , whereNdevice is the number

of M2M devices,ti is the i-th RACH slot, andp(t) is a betadistribution, i.e.,

p(t) =tα−1(1− t)

β−1

Tα+β−1Beta(α, β), α = 3, β = 4, (46)

where Beta(α, β) is the beta function and∫ T

t=0 p(t) = 1.In Fig. 13 we show the performance tradeoff of the baseline

RACH procedure for different downlink resource constraints.The simulation scenario applies the beta traffic model with40,000 UEs arriving during 2,000 RACH slots. It can be seenthat applying a large backoff windowW efficiently increasesthe RACH success probabilityPS yet at the same timeincreases the average delay for successful RACH procedures.

0 50 100 150 2000

5

10

15

20

Configured G

RA

CH

th

rou

gh

pu

t

simu, No PDCCH constraintsimu, Msg2−first policysimu, throughput−first policytheo, No PDCCH constrainttheo, throughput−first policytheo, Msg2−first policy

(a) pd,i = 1 ∀i ∈ [1,K]

0 50 100 150 2000

5

10

15

20

Configured G

RA

CH

th

rou

gh

pu

t

simu, No PDCCH constraintsimu, Msg2−first policysimu, throughput−first policytheo, No PDCCH constrainttheo, throughput−first policytheo, Msg2−first policy

(b) pd,i = 1−1

ei∀i ∈ [1,K]

Fig. 12. RACH throughput of the dynamic backoff scheme for differentconfiguredG

Moreover, given the same backoff windowW , a PDCCH-constrained RACH system has much longer delay and muchlower PS than a PDCCH-unconstrained RACH system does.Furthermore, we can see that the throughput-first policy hasasmaller delay than that of the Msg2-first policy. This is becausethe throughput-first policy guarantees the Msg4 throughputto be at most 12 per subframe, and thus prevents UE fromsuffering from Msg4 timer expiration.

0.2 0.4 0.6 0.8 12000

3000

4000

5000

6000

7000

8000

9000

RACH success probability PS

RA

CH

suc

cess

del

ay (

ms)

No PDCCH constraintMsg2−first policythroughput−first policy

large W

small W

Fig. 13. Tradeoff betweenPS and success delay for different DL resourceconstraints

Finally, we compare the performance of the DB scheme withthat of the baseline scheme, the binary exponential backoff(BEB) scheme, and the dynamic access barring (DAB) schemein Fig. 14. BEB scheme has been broadly used in WiFi forCSMA contention resolution, and in our simulation the backoffwindow size of BEB for thei-th preamble transmission isgiven by Wi = 2CWmin+i−1 ∀i ∈ [1,K]. In addition, wesimulate the DAB scheme proposed in [15]. DAB maintainsa state machine to judge the RACH load (light, medium, andheavy) by observing the change of the number of successfullytransmitted preambles. When RACH is heavily loaded, EABis activated to defer the preamble transmission time of thoseUEs who are ready to send their first preamble byT1 RACHslots per barring. If the preamble transmission of a UE hasbeen deferred for more thanTextra RACH slots, the deferredtime for each of the following EAB barring extends fromT1

to T2 RACH slots. In this DAB simulation, we set thresholdparametersα = 12 and β = 9 for RACH load estimation8.Timer values areTextra = 2500, Treset = 2000, Nreset =1000, T1 = 1000, and backoff window sizeW = 20 in RACHslots.

From Fig. 14, we can observe similar tendency from thebaseline, BEB, and DAB schemes. First, with the increaseof backoff window or delay time,PS grows until reaching1. And secondly, RACH throughput increases and then de-creases with the increase of the backoff window size or delaytime. This is because a small backoff window size or delaytime causes low RACH throughput because of a high levelof RACH collisions, while a large one causes low RACH

8Hereα andβ are not the ones to describe RACH traffic model in (46).

throughput due to low RACH utilization. Fig. 14 also showsthe theoretical performance of the DB scheme. Each point ofthe DB performance curve is obtained by applying differentG

value, and the performance curve holds for different types ofheavy RACH traffic. Each of those points in the boxed area ofthe performance curve has either higher RACH throughput orhigherPS than all the other operating points on this curve. LetPareto operating regiondenote the set of those points in theboxed area. By selectG equal to 40, we can operate RACHin the Pareto region to maximize the RACH throughput whileguaranteeingPS ≥ 99%.

Fig. 14 illustrates two advantages of the DB scheme: first,without any information of RACH arrival traffic, DB is ableto reach the desired operating point in the Pareto operatingregion. In contrast, the other schemes cannot obtain their bestconfigurations, which depend heavily on the RACH arrivaltraffic. And secondly, even when the baseline and BEB schemeare configured to their optimal configurations, the DB schemestill outperforms them in RACH throughput.

VII. C ONCLUSION

In this work, we propose a MAC-level evaluation for the4-step RACH procedure considering imperfect preamble de-tection probability and a limited quantity of PDCCH resources.In addition, we propose a PDCCH allocation method toimprove the RACH performance when the quantity of PDCCHresources is the performance bottleneck. The simulation resultsshow that our theoretical analysis matches the simulationresults for scenarios with different PDCCH constraints andPDCCH management policies.

To address the problem that existing RACH overload reso-lution protocols sacrifice RACH throughput for a high RACHsuccess probability, we propose a novel RACH overloadresolution mechanism, termed dynamic backoff (DB) scheme.DB is capable of progressively estimating the number ofRACH attempts to be transmitted, and accordingly changingthe UE backoff window size. As a result, in RACH overloadedscenarios DB can significantly boost the RACH throughput to98 % of the RACH capacity with a RACH success probabilitymore than 99 % according to the justification of theoretical andsimulation results. Moreover, DB requires little signaling over-head. The only signaling DB needs is for the eNB to broadcastthe updated RACH configuration parameters in each RACHconfiguration period, and there is no additional demands forUE grouping, information about preamble traffic, historicalrecord, or training in advance. Therefore, we believe that DBhas potential to manage bursty random access attempts, andis advantageous to future RACH design in LTE systems.

APPENDIX

DERIVATION OF P (U | m,R), E(S1 | {Ni}Ki=1, R), AND

P (S1 | {Ni}Ki=1, R)

Given each ofm UEs sending a preamble selected fromone of theR available RACH preambles,U among themUEs have non-collided preambles andS1 among theU non-collided preambles are detected (successfully received) by theeNB. Of the m UEs, Ni UEs are in theiri-th preamble

0 0.5 1 1.50

2

4

6

8

10

12

14

16

18

RACH success probability PS

RA

CH

thro

ughp

ut

baseline scheme (W = 3000i, i = 1,2,...,5)BEB scheme (CWmin = 1,2,...,6)

DAB scheme (T1 = 1000, T

2 = 1000*2i, i = 0,1,...,4)

DB, G = 40theoretical performance of DB

Large W / CWmin / T2

Small W / CWmin / T2

Optimal configuration of baseline schemefor current traffic condition( W = 9000 )

Pareto Operating Region(applied to all kinds of traffic conditions)

Optimal configuration of BEB schemefor current traffic condition( CWmin = 3 )

Fig. 14. Tradeoff between RACH success probabilityPS and throughput for different backoff scheme.TR is 10,000 andNdevice is 500,000.

transmission∀ ∈ [1,K], i.e.,m =∑K

i=1 Ni. We are interestedin the mean and distribution ofU andS1, i.e.,P (U | m,R),E(S1 | {Ni}

Ki=1, R), andP (S1 | {Ni}

Ki=1, R).

We definen : A → Z∗ as a function wheren(A) meansthe number of possible combinations of eventA, andZ∗ isthe set of non-negative integers. Moreover,E1(U |m,R) is theevent thatU among them UEs select distinctU preamblesamong a total ofR preambles, andE2(m−U,R− U) is theevent that each of the remainingm − U UEs selects one ofthe remainingR−U preambles and all them−U UEs sufferfrom RACH collisions. In other words, each of theR − U

preambles is not transmitted by exactly one UE.By the principle of permutation and combination [22], we

have n(E1(U |m,R)) =(

mU

)(

RU

)

U !. To derive n(E2(m −U,R − U)), we give each of theR − U preambles a uniqueindexi, ∀i ∈ [1, R−U ]. LetAi denote the event that preamblei is selected by exactly one UE. By the principle of inclusionand exclusion, we have

n(E2(m− U,R− U))

= n(

R−U⋂

i=1

A′

i) = (R − U)m−U − n(

R−U⋃

i=1

Ai)

= (R − U)m−U −

min(R−U,m−U)∑

i=1

(−1)i−1∑

I⊂{1,··· , R−U},|I|=i

n(⋃

k∈I

Ak)

=

min(R−U,m−U)∑

i=0

(−1)i(

m− U

i

)(

R− U

i

)

i!(R − U − i)m−U−i,

(47)

where the last equality holds since we have

∑

I⊂{1,··· , R−U},|I|=i

n(⋃

k∈I

Ak)

=

(

m− U

i

)(

R− U

i

)

i!(R − U − i)m−U−i (48)

P (U | m,R) is then given by

P (U |m,R) =n(E1(U |m,R))n(E2(m− U,R − U))

Rm

=

(

m

U

)(

R

U

)

U !

Rm×

min(m−U,R−U)∑

i=0

(

m− U

i

)(

R − U

i

)

i!(−1)i∗

(R− U − i)m−U−i (49)

From (49) and∑

U P (U | m,R) = 1, we have

Rm =

∑

U

(

m

U

)(

R

U

)

U ! n(E2(m− U,R− U)) (50)

The expected number of non-collided preambles,E(U),comes to

E(U) =∑

U

U P (U | m,R)

=1

Rm

∑

U

U

(

m

U

)(

R

U

)

U ! n(E2(m− U,R − U))

=1

RmmR

∑

U

(

m− 1

U − 1

)(

R − 1

U − 1

)

(U − 1)!

∗ n(E2(m− U,R − U))

=1

RmmR

∑

U

(

m− 1

U − 1

)(

R − 1

U − 1

)

(U − 1)!

∗ n(E2((m− 1)− (U − 1), (R − 1)− (U − 1)))

=1

RmmR

(

(R− 1)m−1)

= m(R− 1

R)m−1

, (51)

where we obtain the fifth equality by (50).E(S1 | U) is thengiven by

E(S1 | U) = E(∑

j∈{u}⊂{m}

pd,j) = UE(pd) = Upd, (52)

where{u} is the set of non-collided preambles,{m} is the setof transmitted preambles, andpd,j is the preamble detectionprobability.E(S1|{Ni}

Ki=1, R) is then given by

E(S1 | {Ni}Ki=1, R) = E(E(S1 | U)) = E(Upd)

= pdE(U) = pdm(R − 1

R)m−1

(53)

To deriveP (S1 | {Ni}Ki=1, R), consider that

P (S1 | {Ni}Ki=1, R) =

min(m,R)∑

U=0

P (S1 | U, {Ni}Ki=1)P (U | m,R),

(54)

whereP (U | m,R) has been derived by (49), andP (S1 |U, {Ni}

Ki=1) is the probability distribution ofS1 given that

Ni among them UEs send theiri-th preambles∀i ∈ [1,K],∑K

i=1 Ni = m, and the number of non-collided preambles isU , ∀U ∈ [0,min(m,R)].

To deriveP (S1 | U, {Ni}Ki=1), we consider the following

two facts. First, given thatU among them UEs have non-collided preambles, each of them UEs has the same prob-ability U

mto have a non-collided preamble. And secondly, a

non-collided preamble for thei-th preamble transmission hasa detection probabilitypd,i ∀i ∈ [1,K]. Thus, the z-transformfor the collision and detection probability of a UE in itsi-thpreamble transmission∀i ∈ [1,K] is

Si(z1, z2) = z1U

m(pd,iz2 + (1− pd,i)) + (1−

U

m), (55)

wherez1 represents that whether the transmitted preamble isnon-collided, andz2 represents that whether the non-collidedpreamble can be detected by the eNB.

The z-transform expression for them UE is then given by

S(z1, z2) =

K∏

i=1

Ni∏

j=1

Si(z1, z2) =

K∏

i=1

Si(z1, z2)Ni

=∑

a1∈[0,min(m,R)], a2∈[0,a1]

ca1,a2z1

a1z2a2 , (56)

whereca1,a2, the coefficient of the termz1a1z2

a2 , is exactlythe probability thata1 UEs have non-collided preambles anda2 among thea1 UEs have detected preambles. Thus, we have

P (S1 = s | U, {Ni}Ki=1) =

cU,s∑U

j=0 cU,j

∀s ∈ [0, U ] (57)

Thus,P (S1|m,R) can be derived by (54) to (57).

ACKNOWLEDGMENT

This work was also supported by Chunghwa Telecom Co.,Ltd., and Ministry of Science and Technology under Grants102-2221-E-002-077-MY2 and 103-2221-E-002-086-MY3.

REFERENCES

[1] “Cisco visual networking index: Global mobile data traf-fic forecast update, 2011-2016.” http://www.cisco.com.

[2] 3GPP TR 36.822 V11.0.0, “LTE Radio Access Network(RAN) enhancements for diverse data applications (Re-lease 11),” Sep. 2012.

[3] 3GPP TR 37.868 V11.0.0, “Study on RAN Improve-ments for Machine-type Communications,” Sep. 2011.

[4] S. Lien, K. Chen, and Y. Lin, “Toward ubiquitous mas-sive accesses in 3GPP machine-to-machine communica-tions,” IEEE Commun. Mag, vol. 49, no. 4, pp. 66–74,2011.

[5] M.-Y. Cheng, G.-Y. Lin, H.-Y. Wei, and A.-C. Hsu,“Overload control for machine-type-communications in

LTE-Advanced system,”IEEE Commun. Mag, vol. 50,no. 6, pp. 38–45, 2012.

[6] 3GPP TS 22.368 V13.1.0, “Technical SpecificationGroup Services and System Aspects; Service require-ments for Machine-Type Communications (MTC),” Dec.2014.

[7] Y. Yang and T. Yum, “Throughput analysis of RACH inUTRA-TDD on AWGN channel,” inIEEE VTC, vol. 2,pp. 572–575, Fall, 2001.

[8] I. Koo, S. Shin, and K. Kim, “Performance analysis ofrandom access channel in OFDMA systems,”SystemsCommunications, Springer, 2005.

[9] R. Soares and J. Brito, “Throughput Analysis of RandomAccess Channel of the UMTS system,”InternationalConference on Wireless and Mobile Communications,pp. 20–20, Jul. 2006.

[10] J. He, Z. Tang, D. Kaleshi, and A. Munro, “Sim-ple analytical model for the random access channel inWCDMA,” IET Electronics Letters, vol. 43, no. 19,pp. 1034–1036, 2007.

[11] L. Sun, Y. Gao, H. Tian, H. Xu, and P. Zhang, “AnAdaptive Random Access Protocol for OFDMA System,”in IEEE VTC, pp. 1827–1831, Fall, 2007.

[12] I. Vukovic and T. Brown, “Performance analysis of therandom access channel (RACH) in WCDMA,” inIEEEVTC, vol. 1, pp. 532–536, Spring, 2001.

[13] P. Zhou, H. Hu, H. Wang, and H. Chen, “An efficient ran-dom access scheme for OFDMA systems with implicitmessage transmission,”IEEE Trans. Wireless Commun,vol. 7, no. 7, pp. 2790–2797, 2008.

[14] C.-H. Wei, R.-G. Cheng, and S.-L. Tsao, “Modeling andEstimation of One-Shot Random Access for Finite-UserMultichannel Slotted ALOHA Systems,”IEEE Commun.Lett., vol. 16, pp. 1196–1199, Aug. 2012.

[15] T.-M. Lin, C.-H. Lee, J.-P. Cheng, and W.-T. Chen,“PRADA: Prioritized Random Access with DynamicAccess Barring for MTC in 3GPP LTE-A Networks,”IEEE Trans. Veh. Technol, vol. 63, pp. 2467–2472, Jun.2014.

[16] P. Choudhary, A. Ghosh, and L. Jalloul, “Simulatedperformance of W-CDMA random access channel,” inIEEE VTC, vol. 4, pp. 2700–2704, Spring, 2001.

[17] P. Osti, P. Lassila, S. Aalto, A. Larmo, and T. Tirronen,“Analysis of PDCCH performance for M2M traffic inLTE,” IEEE Trans. Veh. Technol, vol. 63, pp. 4357–4371,Nov. 2014.

[18] C.-H. Wei, R.-G. Cheng, and S.-L. Tsao, “PerformanceAnalysis of Group Paging for Machine-Type Commu-nications in LTE Networks,”IEEE Trans. Veh. Technol,vol. 62, pp. 3371–3382, Sep. 2013.

[19] J.-B. Seo and V. Leung, “Design and Analysis of BackoffAlgorithms for Random Access Channels in UMTS-LTEand IEEE 802.16 Systems,”IEEE Trans. Veh. Technol,vol. 60, pp. 3975–3989, Oct. 2011.

[20] 3GPP TS 36.321 V12.4.0, “E-UTRA Medium AccessControl (MAC) Protocol Specification,” Dec. 2014.

[21] G. L. Nemhauser and L. A. Wolsey,Integer and combi-natorial optimization, vol. 18. Wiley New York, 1988.

[22] D. P. Bertsekas,Introduction to Probability: Dimitri P.Bertsekas and John N. Tsitsiklis. Athena Scientific, 2002.

[23] Y.-C. Pang, S.-L. Chao, G.-Y. Lin, and H.-Y. Wei, “Net-work access for M2M/H2H hybrid systems: a game the-oretic approach,”IEEE Commun. Lett., vol. 18, pp. 845–848, May 2014.

[24] D. P. Bertsekas, R. G. Gallager, and P. Humblet,Datanetworks, vol. 2. Prentice-hall Englewood Cliffs, NJ,1987.

Guan-Yu Lin is now a PhD student in the depart-ment of Electrical Engineering of National TaiwanUniversity. He received his B.S. degree in the depart-ment of Electrical Engineering at National TaiwanUniversity. He is interested in evaluating systemperformance and designing protocols for wirelesscommunication systems. Also, he has interest instudying resource allocation problems in communi-cation systems with game-theoretical approaches.

Shi-Rong Chang received the B.S. degree in elec-trical engineering from National Taiwan University,Taipei, Taiwan in 2012 and the M.S. degree inelectrical and computer engineering from GeorgiaInstitute of Technology, Atlanta, GA in 2014. Hisresearch interests include network performance anal-ysis and cognitive radio networks. He is currently asoftware engineer working on network testing anddebugging in Oracle America, Inc.

Hung-Yu Wei received the B.S. degree in electri-cal engineering from National Taiwan University in1999. He received the M.S. and the Ph.D. degreein electrical engineering from Columbia Universityin 2001 and 2005 respectively. He was a summerintern at Telcordia Applied Research in 2000 and2001. He was with NEC Labs America from 2003to 2005. He joined Department of Electrical Engi-neering at the National Taiwan University in July2005. He received NTU Excellent Teaching Awardin 2008. He received “Recruiting Outstanding Young

Scholar Award” from the Foundation for the Advancement of OutstandingScholarship in 2006. He is a consulting member of Acts and RegulationCommittee of National Communications Commission from 2008to 2009. Hisresearch interests include wireless networking, and game theoretical modelsfor communications networks. He actively participates in wireless networkstandardization activities and is a voting member of IEEE 802.16 workinggroup.