Limitations and Sidelink-based Extensions of 3GPP Cellular AccessProtocols for Very Crowded Environments

Paolo Castagnoa, Vincenzo Mancusoc, Matteo Serenoa, Marco Ajmone Marsanb,c

aUniversita degli Studi di Torino, Turin, ItalybPolitecnico di Torino, Turin, Italy

cIMDEA Networks Institute, Madrid, Spain

Abstract

An experience common to smartphone users is the difficulty in accessing services in crowded scenarios, such asa rock concert or a football match. In these cases, to (partially) mitigate frustration, users generically claimthat network congestion is occurring, and try again and again to access the network with their smartphones:the result is that user frustration and network congestion reinforce each other! This paper investigatesthe root causes of poor performance of cellular networks in crowded environments and shows that thecommonly adopted random access procedure can prevent full utilization of wireless resources. We developa simple yet accurate analytical model to analyze why attempting random access to wireless resources canbecome a problem even when access congestion avoidance is enforced, e.g., with the Access Class Barringtechnique. The model we propose suggests that cluster-based network access, leveraging device-to-devicecommunications, significantly alleviates access problems. Moreover, it sheds light on scalability laws thatgovern network utilization and quality of experience, in terms of cell capacity, number of access channels,and cluster size.

Keywords: Crowded radio access network, 3GPP RAN, performance analysis and modelling

1. Introduction

Our common experience is that wireless access net-works perform poorly in very crowded environments.When we enjoy a football match or a rock concert inan extremely crowded stadium, and we try to shareour emotions with friends, we discover that placinga phone call or sending a short video, even postinga picture, is not possible, due to network congestion.When large numbers of networking experts gather attop international conferences in their field to discussthe latest research results, reading emails during theoccasionally uninteresting talk is a problem, becausethe wireless access network is not able to sustain thevery large number of email clients. These phenom-ena were quantitatively observed in [1], by collecting

measurements over a tier-1 cellular network in the USduring crowded events, and showing substantial per-formance degradations with respect to normal condi-tions.

The problem can only get worse. The Cisco Vi-sual Networking Index forecast 2017-2022 [2] esti-mates that by 2022 the number of devices connectedto IP networks will be about three and a half times ashigh as the world’s population, generating an overalltraffic of 4.8 ZB (equal to 4.8 ·1021 B) per year. Over70 % of this traffic will come from wireless devices,and 44% of the total IP traffic will be generated bysmartphones. The total mobile data traffic in 2020will reach 77 EB (almost 8 · 1019 B) per month, withthe highest volume in the Asia Pacific region (55.7%of the total), and the highest growth in the Middle

Preprint submitted to Computer Networks February 10, 2020

East Africa region (56%).The 5G Infrastructure Public Private Partnership,

in short 5G PPP, initiated by the European Com-mission, together with companies and research insti-tutions of the field, shares those extreme visions [3].Among the key challenges for 5G, a prominent posi-tion is given to the connection of over 7 trillion wire-less devices serving over 7 billion people, and to theservice of extremely crowded environments, such asa stadium, providing capacities of the order of 0.75Tb/s over the stadium area, or an automated factory,comprising terminal densities up to 100 devices perm2, and requiring sub-ms latency.

The work we present here looks at the performanceof wireless access networks in extremely crowded en-vironments, focusing as an example on the case of agroup of 3GPP cells covering a stadium. The maincontributions of our work are the following:

• We develop a simple analytical model that cap-tures the key aspects of the behaviour of a celland we use it to understand the main sources ofpoor performance.

• We validate the analytical model with detailedsimulations, which prove the validity of the as-sumptions introduced for analytical tractability.

• We show how the model can be instrumental fora correct dimensioning of crowded cellular sys-tems.

• We propose the adoption of device-to-device(D2D) communications [4] as a means to im-prove performance in extremely crowded envi-ronments, and we quantify the benefits that canbe achieved with the D2D approach, showingthat D2D clusters of size k are more beneficialto system performance than a costly increase ofsystem capacity by a factor k (e.g., through thedeployment of k more cells).

This manuscript extends our previous conferencepaper [5]. Besides refreshing the related work and theterminology used for D2D in the context of 3GPP rec-ommendations, and adding new extensive numericalresults, this manuscript provides simple yet effectiveapproximate expressions for the notable operational

points of the system, and analyzes the asymptotic be-havior of the Random Access CHannel (RACH) loadvs. the size of the cell population. Moreover, herewe also discuss the practical validity of the proposedmodel, identify its limitations, and we analyze theconvergence of our proposed iterative approach.

The rest of this paper is structured as follows. Sec-tion 2 discusses the stadium scenario that we considerin this work; Section 3 overviews resource allocationrequest procedures. Section 4 presents the analyt-ical model. Section 5 uses the model to illustratethe system behavior. Section 6 discusses numericalresults. Section 7 discusses the practical relevanceof the model with respect to operating parametersthat are not considered in the analysis. Section 8addresses related work, and Section 9 concludes thepaper.

2. Scenario

The reference scenario that we use in our analysisis a large stadium, with capacity roughly comprisedbetween 50 and 100 thousand spectators. Many suchstructures exist around the world, including, e.g.: theMaracana in Rio de Janeiro, the San Siro Stadium inMilan, the Bernabeu in Madrid, the Stade de Francein Paris, the Wembley Stadium in London, the CampNou in Barcelona, the Rose Bowl in Pasadena, theAzteca Stadium in Mexico City, and the the Mel-bourne Cricket Ground, just to name a few. Thesestructures regularly host important sport events, andoccasionally also music concerts of famous rock andpop stars, and in the latter case the structure capac-ity grows by up to 50 thousand attendees.

Of course, such extraordinary numbers of people(terminals) imply a wide variety of services: specta-tors may want to send to their friends short videosor pictures of the event, may receive all sort of mes-sages, as well as phone calls, and at the same timeterminals may be involved in content downloads.

We primarily focus on services which imply humanintervention, such as the transmission of a picturewith a messaging application. In this case, the hu-man user is in the service loop, so that the basicsequence of the service operations is made of a re-quest for the radio access network resources, possibly

2

automatically repeated several times, until resourcesare granted, then the use of the network resources,followed by a think time before the next service re-quest.

We will see that in some cases the system bottle-neck is in the request for the radio access networkresources, mostly because 3GPP-compliant cellularsystems use an Aloha-like contention-based schemefor this operation. It may thus happen that, whilethe network resources are available, request colli-sions do not allow their allocation. Under these cir-cumstances, a reduction of the number of requestsis mandatory to restore acceptable network perfor-mance. This can be obtained by reducing the numberof users who are allowed to issue requests, or by forc-ing users to coalesce during the request phase. Thisis where D2D comes into play. If end user terminalsare allowed to form clusters or are instructed to formclusters by the network and use sidelinks for intra-cluster relay—through appropriate commands issuedby the cellular base station (BS), according to 3GPPstandards developed starting with 3GPP release 12and afterwards with 5G specifications [6]—only onerequest is issued whenever multiple terminals of thesame cluster require access to the network resources,as proposed in [7] for opportunistic scenarios.

3. Accessing Resources in 3GPP Networks

In 3GPP standards like LTE, LTE-A and the up-coming 5G, end user terminals (called User Equip-ments – UEs in the LTE jargon) have to proceedthrough the Random Access CHannel (RACH) pro-cedure to access data channels, if not already con-nected to the BS (called evolved NodeB – eNodeB inLTE and generalized NodeB – gNodeB in 5G). Theaccess procedure begins with the UE sending a mes-sage on the Physical RACH (PRACH). Two types ofrandom access procedures are defined: contention-based (implying an inherent risk of collision) andcontention-free [8]. In each 3GPP-compliant cell,a fixed number (64) of orthogonal preamble signa-tures (PSs) are available, and the operation of thetwo types of RACH procedure depends on a partition-ing of these PSs between those for contention-basedaccess and those reserved for allocation to specific

UEs on a contention-free basis. The contention-freeRACH procedure is reserved to delay-sensitive cases,such as incoming traffic and handovers [9]. Other-wise, a contention-based random access PS is chosenat a UE to send a random access signal to the BS.A conflict occurs if more than one UE uses the samePS and time-frequency resources, resulting in unde-codable messages at the BS. The contention-basedprocedure consists of an exchange of four messagesto set up a connection among UE and BS.

Step 1: UE → BS (Random Access Preamble). Afirst message conveys the randomly chosenRACH PS. The UE selects one of the availablePSs and transmits it in a time-frequency slot.Several UEs may choose the same PS and theBS may not be able to decode it. After thePS transmission, UE begins to monitor thedownlink control channel (PDCCH) looking foran answer.

Step 2: UE ← BS (Random Access Response).This message is sent by the BS on the PD-CCH, and addressed with an ID identifyingthe time-frequency slot in which the PS wasdecoded. Whether multiple UEs have collidedor not, if no Random Access Response (RAR)matching message has been received within theRAR window, they must repeat the RACHprocedure, after a backoff delay. The durationof such backoff is randomly chosen in the range(0, B] where B is the maximum number ofsubframes in a backoff period, and varies in(0− 960] ms.

Step 3: UE → BS (Scheduled Transmission). TheUE that receives the RAR message respondswith a scheduled transmission request that in-cludes the ID of the device and a radio resourcecontrol (RRC) connection request message onthe uplink shared channel (UL-SCH).

Step 4: UE ← BS (Content Resolution). Con-tention resolution is released from the BS on thePDSCH. This identifies that no conflict on theaccess procedure exists. The UE can transferdata to BS.

3

Once a UE has successfully performed the RACH pro-cedure, it owns an active duplex connection and is inthe RRC CONNECTED state. Keeping a connection run-ning requires that the BS reserves physical resourcesdevoted to this connection, even if there is no trafficavailable for the intended UE. Therefore the BS canhandle only a limited number of connected devices.Moreover, the connected UE has to continuously ver-ify if there is any incoming traffic, monitoring controlchannels, and therefore incurs high battery consump-tion.

As long as the communication is alive, the UE re-mains in the RRC CONNECTED state, but, after an inac-tivity period, it begins to perform sleep cycles, fromwhich it can return to the RRC CONNECTED state with-out performing the contention-based RACH proce-dure.

Since the above-described access mechanism isbased on a multichannel slotted Aloha, each PS rep-resenting an Aloha channel, its performance degradebeyond the threshold of 1 request/slot per PS. Hence,in dense scenarios, congestion can happen and theRACH procedure can become a system bottleneck.To alleviate congestion, state of the art solutionsadopt the Access Class Barring (ACB) mechanism,which segments devices in several classes [10]. De-vices within each class are managed through two pa-rameters: the access barring probability and the bar-ring time. With ACB, devices that are ready to at-tempt a random access are probabilistically barred,and barred devices wait for a barring time beforemaking another barring decision, i.e., a device canbe barred multiple times in a row. ACB is effectivein smoothing peaks of access requests, but it doesnot change the RACH load under steady-state condi-tions. Moreover, ACB introduces a stochastic delay.

4. Analytical Model

We model the operations of n end-user terminaldevices located in the same cell, under the coverageof one BS. The notation used in this paper is sum-marized in Table 1.

Each device generates uplink transmission requestsaccording to the 3GPP contention-based RACH pro-cedure briefly described in Section 3 to obtain a trans-

Table 1: Notation and Cell Parameters used in Section 6

Quantity Notation Value

Number of devices (or clusters) nBS capacity C 150–1500 [Mb/s]Network max accepted requests M 200Number of Random Access preambles N 54Slot time τ 0.01 [s]Backoff time RACH B0 av. 0.15 [s]Backoff time Network B1 av. 1 [s]ACB access probability pa 0.05–0.95ACB barring time Ba av. 4-512 [s]Transmitted data volume FS av. 1.5 [MB]Transmission time SThink time TTH av. 30 [s]Device uplink speed limit RProbability to skip RACH procedure pJ ≤ 0.5Access delay ATThinking subsystem throughput λNetwork subsystem throughput ξRandom Access subsystem input γArrival rate at Network subsystem σCollision probability pCRejection Probability pB

mission grant from the BS. We account for the factthat the establishment of downlink flows might pro-vide the devices with extra opportunities to obtaintransmission grants, skipping contention through thecontention-free RACH procedure.

In the following, we derive a simple model for ac-cess requests and service operation in the cell, andshow how to compute network utilization, access de-lay, and in general how to assess the behavior of thesystem as a function of the number of devices in thecell, for a given BS configuration (in terms of capac-ity, number of RACH channels, RACH slot duration,backoffs experienced upon failed RACH procedures,etc.).

4.1. Closed representation of the system

The BS has uplink capacity C, in bits per sec-ond, and can share its capacity among at most Mdevices at a time (i.e., there can be up to M de-vices in state RRC CONNECTED). The number of RACHchannels (i.e., orthogonal preamble signatures - PS)available for Random Access is N and the intervalbetween two consecutive Random Access Opportuni-ties (RAOs) is τ seconds. If during τ a single deviceselects a given RACH channel, then the RACH pro-cedure is successful, otherwise the RACH channel is

4

either unused or a collision happens with multiple de-vices attempting to use the same PS.

A RACH collision results in a random backoff B0,after which a RACH retry follows. In case of suc-cessful RACH procedure, the device is granted trans-mission only if there are less than M devices underservice at the BS, otherwise the device goes througha random backoff B1 followed by another RACH pro-cedure. The model also considers ACB with uniformaccess probability pa for all classes, and barring timewith average duration E[Ba] seconds.

For what concerns the traffic generated by end-user terminals, we consider human-operated wirelessdevices, and assume that each device produces a newtransmission request, with random data volume FS ,only after its previous request has been served. Morespecifically, upon service completion, we assume thatthe devices enters a “think time” period with randomduration TTH before generating the next request.Unless otherwise specified, the average service timeE [S] only depends on C, M and the average valueE [FS ], i.e., we assume that the serving speed is fixed

and equal to C/M , so that E [S] = M ·E[FS ]C . We do

so because our analysis concentrates on finding therange of values for the user population n that allowsto use most, if not all, transmission resources whileincurring low delay in the attempts to access the net-work. Therefore, we particularly focus on the net-work behavior under (quasi-) saturation conditions.However, we will also show how to extend the modelto capture the behavior of a non-saturated system,and in particular we will show how to keep the modeltractable while improving the approximation on E [S]by account for: (a) equal sharing of the BS capacityamong the actual number of devices under service inthe system, and (b) service speeds limited by a deviceuplink speed R.

The resulting system model is depicted in Fig. 1.The model comprises 6 main components: i) Think,representing the end-user think time between theend of a service and the generation of a newaccess request; this is modeled with an infiniteserver queue with random (not necessarily exponen-tial) service time with average E [TTH ] seconds; ii)Random Access, representing the RACH contention-

NetworkThink

Network Backoff

RACH Backoff

Barring Time

Random Access

σ

ξ

pJ

λ 1−pJ γ γ = paγ

Figure 1: Closed queueing network model of a cell.

based procedure; this is modeled as a set of N parallelslotted Aloha channels, receiving each 1

N of the totalload offered to the RACH; the slot duration for any ofthe N slotted Aloha channels is τ ; iii) Barring Time,which models ACB operation as an infinite serverwith average service time E[Ba] seconds, affecting aportion 1 − pa of the flow directed to the RandomAccess; iv) Network, representing the BS resources,modeled as an M/G/M/0 queue with average servicetime E [S] seconds. The Network queue is fed bythe output of the Random Access subsystem and bythe requests that skip the Random Access becauseof transmission opportunities generated by downlinktraffic requests; these are modeled by means of the“jump probability” pJ , which is the probability toskip the contention-based RACH procedure, and ac-cess directly the BS resources. v) Network Backoff,and vi) RACH Backoff, representing the two back-offs, which are modeled by means of infinite serverqueues with random service times (the assumptionof an exponential pdf for service times is not neces-sary, but may be an adequate choice to represent thebehavior of real systems), with averages E [B0] andE [B1] seconds, respectively.

Fig. 1 also shows that the system is closed, i.e., thepopulation is finite, with the number of customersfixed to n. We denote by λ the output of the Thinksubsystem, and by ξ the output of the Network sub-system. Because of the closed structure of the system,λ = ξ. We indicate with γ the total arrival rate at theN RACH channels in the Random Access subsystem,and we assume that RACH requests follow N paral-lel and i.i.d. Poisson arrival processes with intensityγN arrivals per second. Although devices decide tosend RACH requests asynchronously, such requests

5

are cumulated over τ seconds and physically sent atthe same time over the same frequency band. Thus,the successful output of each of the N RACH chan-nels is that of a slotted Aloha system with γτ

N arrivals

per slot, which is given by γτN e− γτN successes per slot,

as known from the standard analysis of multichannelslotted Aloha [11]. The maximum throughput perslot of such multichannel slotted Aloha system is N

e ,which is achieved for γτ = N .

With the above, the arrival rate at the networkservice is σ = γe−

γτN + pJλ, the arrival rate at the

RACH backoff B0 is γ(1− e−

γτN

), and the one at the

Network backoff B1 is pBσ, where pB is the block-ing probability, given by the Erlang-B formula withM servers and load ρ = E [S]σ. The load acceptedand served by the network service is ξ = (1− pB)σ.For analytical tractability, we introduce the simpli-fying assumption that all arrival processes are ho-mogeneous and independent Poisson processes. Theimpact of such Poisson assumptions will be assessedwith simulations.

In the described system, quantities λ, σ, and ξ (andtherefore also ρ and pB) are functions of γ. It is pos-sible to write a recursive equation in γ by consider-ing that γ is γ minus what enters the Barring Timeblock. γ results from the sum of four arrival rates:λ (1− pJ) from the Think subsystem, the output ofbackoffs B0 and B1, plus the recycle caused by ACB:

γ=λ(1−pJ)+γ(1−e−

γτN

)+pB

(γe−

γτN+pJλ

)+(1−pa)γ,

which, combined with γ = paγ, yields a recursiveexpression for γ, which does not depend on ACB op-eration at all:

γ = λ (1−pJ) + γ(

1−e−γτN

)+ pB

(γe−

γτN +pJλ

).

(1)

The recursive expression Equation (1) has two un-knowns: γ and λ (note that pB can be written asfunction of ξ, and ξ = λ). Unfortunately, this ex-pression is not enough to identify the operating pointof the system, because it contains no dependence onthe population size n. However, to introduce n inthe loop, and remove λ, we can apply Little’s law todifferent blocks in the modeled system, as presentedin the following.

Solving system equations requires iteration, whoseproof of convergence is provided in Section 4.6.

4.2. Dependence on the population size n

From the model described in the previous subsec-tion, we can easily derive the expressions for the net-work utilization, the number of devices under serviceand in any of the system blocks depicted in Fig. 1,the time of a complete cycle between two transmis-sions, and the delay to access the service. All thesequantities can be expressed as function of γ, and γcan be expressed as function of the population size n.

Utilization and distribution of devices. Thenetwork utilization ξ is equal to σ (1− pB) =(γe−

γτN + pJλ

)(1− pB). Therefore, since ξ = λ, it

is immediate to obtain the following expressions forξ, λ, σ and ρ:

ξ = λ =γe−

γτN (1− pB)

1− pJ (1− pB); (2)

σ =ξ

1− pB=

γe−γτN

1− pJ (1− pB); (3)

ρ = E [S]σ =E [S] γe−

γτN

1− pJ (1− pB). (4)

Note that, since ρ in Equation (4) only depends onγ and pB , we have that pB actually depends only onγ. Thus, all the quantities representing arrival ratesin the system model are functions of γ only, for fixedvalues of the other system parameters.

The average number of devices under service, thatcannot exceed M , is computed by applying Little’slaw at the Network, i.e., nS = ξE [S] ≤ M , whichalso implies that utilization cannot exceed M/E[S].Similarly, the average number of devices in Think isproportional to the average number of devices under

service, i.e., nTH = ξE [TTH ] = nSE[TTH ]E[S] .

The rest of the devices n−nS−nTH are attemptingaccess, either waiting for the next RACH opportunity(including after a barring event) or in one of the back-off queues, so applying again Little’s law we obtain:

6

n−nS−nTH =γ

(τ

2+

1−papa

E[Ba]

)+

+γ(

1− e−γτN

)E [B0]+

pBγe− γτN

1−pJ (1−pB)E [B1] ,

where the average delay incurred in a RACH attemptis computed as half of the slot duration because ofthe Poisson arrival assumption. The total number ofdevices in the network can therefore be expressed asa function of γ:

n=γ

(τ

2+

1−papa

E[Ba]

)+γ(

1−e−γτN

)E[B0]+E[B1]

· pB γe−γτN

1−pJ (1−pB)︸ ︷︷ ︸pB

1−pBξ

+(E [S]+E [TTH ])γe−

γτN (1−pB)

1−pJ (1−pB)︸ ︷︷ ︸ξ

.

(5)

This is a monotonic relation between n and γ, whichcan be inverted (although not in closed form) to ex-press γ as a function of n. However, we have seen thatall quantities of interest in the system are functions ofγ, so that we can conclude that they are eventuallyfunctions of n only, i.e., of the device population’ssize.

Cycle duration. The average time for a completecycle in the system (e.g., the cycle between two con-secutive service completions) is denoted by E [Tcycle]and can be easily computed from the model of Fig. 1,by considering that: i) the probability to collide ona slotted Aloha representing the RACH channel withPoisson arrivals of intensity γτ

N arrivals per slot is

pC = 1 − e−γτN , and ii) collisions are assumed to be

independent. Hence, we can write that:

E[Tcycle]=pB

1− pBE[B1]+E[S]+E[TTH ]+

(1

1−pB−pJ

)·[eγτN

(1−papa

E[Ba]+τ

2+E [B0]

)−E[B0]

].

(6)

The term in brackets in Equation (6) is the aver-age time spent in the loop formed by the RACH andthe RACH backoff blocks, which has to be counted

11−pB times on average (i.e., the average number ofBernoulli trials before a success, including the success

that occurs when a device finds the Network avail-able), except for the case in which a request skips theRACH, which occurs with probability pJ . The quan-tity 1−pa

paE[Ba] + τ

2 + E [B0] is the time to completeone of such RACH loops—which includes, on average,1−papa

passages through the ACB backoff—and there

are, on average, pC1−pC = e

γτN −1 collisions before a

successful RACH attempt (in which case the RACHbackoff does not occur). The network backoff is tra-versed only after a failed network access (i.e., pB

1−pBconsecutive times, on average), whilst the Networkand Think subsystems are traversed only once percycle. E [Tcycle] depends on γ since we have shownthat pB also depends on γ. So, using Equation (5) weconclude that E [Tcycle] can be written as function ofn.

Access delay. The access delay, indicated asE [AT ], is the time spent in a cycle, excluding thethink time and the service, and is therefore easilyobtained from Equation (6):

E [AT ] = E [Tcycle]− E[S]− E [TTH ] . (7)

An alternative expression for E [AT ] is obtained byapplying Little’s law to the part of the system thatexcludes network service and think time:

E [AT ] =n− nS − nTH

λ. (8)

Since λ=ξ, Equation (8) reveals that the access delayis (practically) linear with the population size if ξis (roughly) constant in a range of n, so that alsonS and nTH are constant. As we will show later,such range exists if the Network saturates before theRandom Access. That range is very relevant, becauseany point in it leads to maximal utilization.

4.3. QoE indexes

We use two indexes to express the quality of ex-perience (QoE) for the end-user. The first index ηScompares the service time with the time spent wait-ing before service starts, and it decreases with theaccess delay:

ηS :=E[S]

E[S] + E [AT ]. (9)

7

The second index is ηA, which is inversely pro-portional to the service time and fades exponentiallywith the access delay. Service time and access delayused in ηA are normalized to their values obtainedwith the smallest population n that causes the pres-ence of M devices under service (denoted by n′):

ηA :=E[S]|n=n′

E[S]e− E[AT ]E[AT ]|n=n′ . (10)

Differently from ηS , index ηA is very sensitive to rel-ative increases of delay rather than to absolute in-creases.

4.4. Analysis with D2D support

When D2D is used to alleviate RACH contentionproblems, terminal clusters come into play, each ofthem behaving as a single device. Thus, we can usethe same formulas as above, with n, nS , nTH de-noting the number of clusters in the system, underservice and in think time, respectively. Similarly, allarrivals and services refer to clusters. The main ef-fect of clusters is the reduced load to the RandomAccess. The impact is non-linear because γ does notscale linearly with n.

Cluster formation. Clusters form either sponta-neously, when a device announces its willingness towait for other users in its close proximity to join ina random access attempt, or under the control of theBS, when RACH collision probability becomes prob-lematic. In the latter case, the BS can oversee theformation of clusters of users located close to one an-other.

Service time with clusters. If k is the aver-age cluster size, i.e., the average number of devices ina cluster, the average service time per network ac-cess request becomes k times higher than for the casewithout clusters, so as to be able to serve k transmis-sions with one request.

Device think time with clusters. Clustersare ephemeral entities, i.e., they do not persist af-ter entering the Think subsystem; they are ratherdismantled, and devices are re-shuffled to form newclusters built at random from the set of devices inthe Think subsystem. Hence, there is no per-clusterthink time but rather a per-device think time that

depends on cluster formation rules. In the case ofclustered RACH access, the average think time ex-perienced by each device increases as well. Indeed,within each cluster, the effective think time of thedevice that initiates the cluster corresponds to itsown think time, plus the time needed for the othermembers to join, because a cluster-cumulative RACHmessage is sent only after all cluster members havecompleted the thinking procedure. If a cluster ofsize k is formed, the device that initiates the clus-ter suffers the highest additional think time, due tothe wait for k − 1 devices to join; the second mem-ber of the cluster must wait for k− 2 devices to join,and so on. Only the last (k-th) device to join doesnot suffer any increase in the think time. Consider-ing the first device, and assuming a very high den-sity of devices that are in the condition of joiningthe cluster (the scenario is very crowded), and as-suming exponentially distributed think times, form-ing a cluster of a few units is very quick. The worstcase average additional think time is computed forthe first device in a cluster of k formed from a popu-lation of m � k devices that can join the cluster as∑k−1i=1

E[TTH ]m−i ' E [TTH ]

(k−1m

). A similar conclusion

can be reached in the case of generally distributedthink times, by considering an asymptotic normal ap-proximation (thanks to the large number of potentialcluster members) and the (k − 1)-st order statisticsdistribution. In practice, the per-device think timeincrease due to clustering is negligible in crowded en-vironments, and so we neglect it in the model andassume that for each cluster that enters the Thinksubsystem, another cluster leaves after a think timeequal to the one of a single device.

4.5. Impact of resource sharing under non-saturatedconditions

If we consider that the BS resources can be sharedby active connections, it is obvious that underloadedsystems offer higher rates to the active devices.1

1 We remark that a cluster is seen as a single device andthat the order in which cluster members’ transmissions arescheduled is not important for our model, and is anyway outof the scope of the article. A cluster here is associated with

8

Therefore, the analysis proposed so far is valid in theregion in which the Network subsystem is fully loaded(which is the focus of this paper), while it containsan approximation elsewhere. To fix this approxima-tion, let us consider a Network subsystem that sharesequally its resources among the connected devices, upto a rate R that can be interpreted as the maximumrate achievable by a device or as the maximum ratespecified in the user’s service level agreement. In suchcase, the service time conditional to j active connec-tions becomes:

E [S|j] = E [FS ] max

{1

R,

1

C/j

}.

Such a dependence on the number of active con-nections requires the use of a processor sharing (PS)model serving up to a maximum number of users, of-fering each user up to a maximum service rate. Whilethe resulting analytical model remains tractable us-ing the class of Whittle Networks [12], the addedcomplexity can be seen to have a marginal impact,especially in the operating regions that are the fo-cus of this paper. Thus, we only examine this casethrough simulation, and we use an approximation tohandle the case in which the Network subsystem isnot fully loaded, letting:

E [S] = E [FS ] max

{1

R,

1

C/nS

}. (11)

That is, we associate with every user a capacity equalto the total capacity divided by the average numberof users in the Network subsystem.

Note that E [S] is equal to E[FS ]R when the num-

ber of devices under service is not enough to saturatethe Network subsystem. The adaptation of E[S] tothe number of devices under service introduces a fur-ther element of dependence on n, and a non-linearity.Although its impact on system performance is quitelimited in most of the cases, the use of (11) is helpfulwhen the Network subsystem approaches saturation,so we will use it in the rest of the paper.

a message that consists in the concatenation of its members’messages.

4.6. Convergence

With constant E[S]. For a given user populationn, we find iteratively γ, pB and λ, and then computeall other quantities. More specifically, we use twonested iterations. First, we note that we can eval-uate the value of n that corresponds to an input γby using Equation (5) and compare it with the tar-get. As we will show later in Equation (19), γ and nare asymptotically proportional for large values of thepopulation, with γ being upper-bounded by n times aconstant. Thus, for fixed n, the search range for γ is[0, ωn], with ω = τ

2 + 1−papa

E[Ba] +E[B0]. However,

the evaluation of Equation (5) requires to know thevalue of pB , which is not a monotonic function of γ.For fixed γ, the value of pB can be computed with anested iteration. Specifically, we use the Erlang for-mula for pB and Equation (2) for λ, thus yielding thefollowing system of equations to find pB (and λ atthe same time, as a byproduct):pB =

[E[S]

(γe−

γτN +pJλ

)]M/M !∑M

j=0

[E[S]

(γe−

γτN +pJλ

)]j/j!

;

λ = γe−γτN (1−pB)

1−pJ (1−pB) .

(12)

On the one hand, increases of λ correspond to in-creases of pB in the first expression in Equation (12)because, for fixed γ, λ is proportional to the load ofthe Network subsystem ρ = E[S]

(γe−

γτN + pJλ

). On

the other hand, increases of pB correspond to mono-tonic decreases of λ computed with the second ex-pression in Equation (12). With the above propertyin mind, the value of λ can be found by using a di-chotomic search in the interval

[0, γe−

γτN /(1− pJ)

],

which is the range of λ when pB ∈ [0, 1] and γ isfixed. Once λ is computed, also pB is determined,which means that the value of pB is univocally de-termined by the value imposed for γ in the currentiteration. At that point, we can go back to adjustγ by comparing the target value of n with Equation(5) computed with the values of γ and pB of the cur-rent iteration. This comparison tells us if we have toincrease or decrease γ for the next iteration. Specifi-cally, if the difference is above a specified tolerance, γwill be adjusted proportionally to the difference be-tween the target value of n and its value resulting at

9

the current iteration. Moreover, if an increase (or de-crease) of γ is required, then the lower (resp. upper)limit for the search range for future iterations can beset to the current value of γ. This guarantees con-vergence, since the search interval keeps shrinking atany iteration.

With E[S] depending on nS. In this case, to theiteration on γ with nested iteration on pB , we needto add another nested iteration on E[S], Specifically,E[S] has to satisfy the following recursive equationobtained from Equation (11) with a number of de-vices under service computed with Little’s result asnS = λE[S]:

E[S] = E [FS ] max

{1

R,λE[S]

C

}. (13)

Therefore, for fixed γ, we can start with the worstcase for E[S], which is E[FS ]/(C/M) under fully sat-urated network conditions, and then proceed to solveEquation (12) as in the case for fixed E[S], so asto update pB and λ, and hence iterate on E[S] us-ing Equation (13). After the first iteration, sinceλE[S] = ns ≤ M , the value of E[S] is either thesame as before—and hence no further iteration isrequired—or less, which means that pB decreases,and thereby λ increases in the next iteration, thusleading to a further decrease of E[S] to compensateλ, because the average population nS cannot increasewith respect to previous iteration (because the ser-vice time was taken as the worst case). The sameapplies at every iteration, and since E[S] is bounded,the iterations converge.

Remarks. The convergence of the described it-erations is typically quite fast. For instance, con-sider that, for the numerical results shown later, wehave set the convergence threshold to one part permillion with respect to increments of variables overwhich we iterate, and all results contained in a fig-ure were obtained in a few tens of seconds using alaptop equipped with a dual core 3 GHz Intel i7 pro-cessor and 16 GB of RAM). However, there are pointsat which the variation of n with γ can be very fast,which leads to possible numerical errors in the com-putation of E[S], when it is not fixed, and thereforeof pB and λ. This happens unless the resolution used

for γ is made very fine (i.e., using 12 digits for ex-pressing its decimal part).

5. System Behavior

Here we study the bottlenecks of the system, pointout some notable points in the performance curves,and analyze how performance is affected by the num-ber of devices present in the cell and by the introduc-tion of D2D-based clusters.

5.1. Bottlenecks

The model depicted in Fig. 1 has two potential bot-tlenecks: the Random Access and the Network sub-systems. The former filters network access attempts,and asymptotically prevents any network request asγ grows with the population n. The Network subsys-tem has finite capacity, and therefore cannot servemore than M simultaneous requests.

Fig. 2 shows a typical case in which the maximumthroughput of the Random Access is below the ca-pacity of the Network subsystem, and thus is theonly bottleneck, for all population sizes. In this case,the Network subsystem throughput ξ and the in-put σ of the Network subsystem are equal, since theblocking probability pB is negligible. From Equation(7), the access delay becomes a linear affine func-tion of e

γτN , and therefore grows with en. However,

as shown in Fig. 2, a system in which the RandomAccess saturates before the Network subsystem doesnot suffer high delay. The range of device populationsthat roughly maximizes network utilization is quitenarrow, and corresponds to a rather small intervalaround the peak efficiency of a multichannel slottedAloha system, i.e., to values of n close to the one thatyields γτ = N (about 400 in the figure). This is thecontext that was previously analysed in the literaturefor the case of machine to machine (M2M) communi-cations [13], with a system model similar to ours, andstudied by means of a Markov chain. Here we focuson the more complex two-bottleneck case, in whichthe Network subsystem saturates before the RandomAccess, which is typical for the stadium scenario.

Fig. 3 shows an example of the model behaviourwhen both the Random Access and the Network

10

Figure 2: Random Access-limited model behaviour. Left scalefor σ and ξ, right scale for access delay.

Figure 3: Model behavior with Network subsystem saturation.Left scale for σ and ξ, right scale for access delay. Dashedlines are computed with the worst case value for E[S], while forgenerating the solid lines we have used the approximation (11).

subsystem can become the system bottleneck. In-deed, the Network subsystem is a bottleneck for lowervalues of population size, until the Random Accessreaches success probabilities too low to starve theNetwork subsystem. In the figure we can identifythree operational regions. In the first region (lownumber of devices and low load: roughly below 550devices for the specific example), pB and pC are close

to zero, σ ' ξ, and the delay is practically negligible.In the second region (shaded in the figure, roughlyfrom 550 to 7500 users), the throughput of the Net-work subsystem is constant, while σ follows the fa-miliar bell-shaped curve of slotted Aloha, and thedelay grows linearly with the user population, as vis-ible from Equation (8) (note the logarithmic verticalscale on the right). In the third region, pB is negligi-ble again, so that σ ' ξ like in the first region, but thedelay now grows exponentially with γ, and thereforewith n. Out of such three regions, only the secondone is desirable for system operation, since the Net-work subsystem resources are not wasted, and delayscales linearly with the number of devices in the cell.

5.2. Notable operational points

Random Access saturates first.2 In this case,

pB ' 0, so that ξ ' γe−γτN

1−pJ , the average number of

devices in service is nS = ξE[S] < M , and the av-

erage service time E [S] must be equal to E[FS ]R since

the Network is not saturated and every device underservice obtains its maximum allowable rate R. In thisscenario, the Network throughput is maximal whenthe output of the Random Access is maximal. Thisoccurs for a number n∗ of users that results in γ = N

τ .From Equation (5) we obtain the approximation (lin-ear in N):

n∗ ' N

2+N

τ

[e−1

E[FS ]R + E[TTH ]

1−pJ

+ (1− e−1)E [B0] +1− papa

E[Ba]

].

With Network saturation.3 In this case, tocharacterize the behavior of the system in the threeoperational regions shown in Fig. 3, in addition to

2In this example, we use very few preamble signatures anda very high transmission capacity, so that the Random Accesswill saturate well before the Network. The configuration usedis as follow: N = 5, C = 600 Mbps, pa = 1 and pJ = 0.

3This example uses a relatively high number of preamblesignatures and limited capacity, so that the Network will sat-urate before the Random Access. Specifically, in this case weuse N = 54, C = 150 Mbps, pa = 1 and pJ = 0.

11

n∗ we characterize n′ and n′′, i.e., the values of nthat correspond to the first and the second knee ofthe curve representing ξ vs. n. The figure re-ports throughputs and access delay computed withthe model using the constant worst case approxima-tion for the service time E[S] (dashed lines) or theiterative approximation (11) (solid lines). The differ-ences between the two cases are minimal. However,the first knee of the throughput curves is slightly dif-ferent, and using the constant approximation leadsto overestimate the point at which the Network sub-system saturates. A similar effect, although less pro-nounced, can be noticed for the value of n′′,

Note that n′ ≤ n∗ ≤ n′′, and the throughput of theNetwork subsystem is constant and equal to C

E[FS ] for

all values in the interval [n′, n′′]. Therefore, Equation(2) reduces to:

γe−γτN =

C

E [FS ]

1− pJ (1− pB)

1− pB, ∀γ | n ∈ [n′, n′′].

At the extremes of the considered interval [n′, n′′],the Network subsystem has exactly enough resourcesto satisfy the demand, so that we can consider pB '0:

γe−γτN ' C

E [FS ](1− pJ) , γ | n ∈ {n′, n′′}. (14)

Considering that the L.H.S. of Equation (14) is a non-negative continuous function of γ that starts from 0,grows until it reaches the value N

eτ at γ = Nτ and then

decreases asymptotically to 0, expression Equation(14) admits two (possibly coinciding) real solutionsonly if C

E[FS ] (1− pJ) ≤ Neτ . So, a range of values of n

such that the throughput of the Network subsystemis constant and maximal exists if and only if

N ≥ eτC

E [FS ](1− pJ) . (15)

The distance between the zeros of γ in Equation (14)decreases logarithmically with C increasing (and withpJ decreasing). Since γ is monotonic with respectto n, this means that the interval [n′, n′′] becomessmaller with larger capacities C (and with smallerprobabilities pJ), and n′ = n′′ = n∗ when Equation

(15) holds as equality. If Equation (15) does not hold,the Network subsystem cannot saturate, and we fallback to the Random Access-limited scenario of Fig. 2.

The above condition also tells that the number ofRACH channels needed to allow network saturationscales linearly with the capacity of the network andwith (1− pJ).

The notable points described above and the asymp-totic behavior of γ vs. n can be approximated bymeans of the following closed form expressions thatcan be readily derived, as shown next.

Approximated values for the notable points.The value of n′ can be approximated in closed form ifthe network saturation throughput is much less thanthe maximum Random Access throughput. In thiscase, there is neither network blocking nor collisions(i.e., pB ' 0 and pC ' 0) at n′ and ξ = λ = σ = C

E[FS ]

while σ ' γ + pJλ. Considering that in this case the

service time becomes E[S] = M ·E[FS ]C , the result is

that Equation (5) reduces to:

n′'M +C

E [FS ]

[E [TTH ]

+ (1− pJ)

(τ

2+

1− papa

E[Ba]

)]; (16)

Therefore, n′ scales linearly with network capacity,slot duration, and probability of skipping the Ran-dom Access.

For what concerns the value of n∗, we can againuse Equation (5) with γτ = N , ξ = C

E[FS ] , ρ =

E[S](Neτ + pJ

CE[FS ]

). Considering that ρ � M if

the network saturates well before the Random Ac-cess, then pB ' 1 − M

ρ and nS ' M . In conclusion,the following approximation holds:

n∗'M+E [TTH ]C

E [FS ]+N

2+N

τ

(1−e−1

)E [B0]

+N

τ

1− papa

E[Ba] +

[N

eτ− C

E [FS ](1−pJ)

]E [B1] .

(17)

Therefore, the value of n∗ scales linearly not onlywith C, but also with M , N , pJ and τ−1.

The value of n′′ has to be computed numerically.From Equation (5) computed for pB ' 0 and with

12

the product γe−γτN given by Equation (14), it results

that:

n′′ ' n′ +[γ′′ − C

E [FS ](1− pJ)

]·(τ

2+

1−papa

E[Ba] + E [B0]

), (18)

where γ′′ is the largest root of Equation (14). Onecan notice that while n′ increases linearly with C and1 − pJ , the distance n′′ − n′ decreases logarithmi-cally with the same parameters, due to Equation (14).Therefore, n′′ decreases with C and 1− pJ for smallvalues of such parameters, where the log decrease issuperlinear, and then increases when the logarithmbecomes sublinear. Moreover, n′′ grows with N , be-cause N increases the L.H.S. of Equation (14), andtherefore it has the effect of spacing apart the zerosof that equation, while n′ is not affected by N , asnoticed before with Equation (16).

The accuracy of the approximations for n′ n∗ andn′′ can be appreciated by looking at Fig. 3. It canbe clearly seen that the approximate values shownon the x axis match very well the model behaviorobserved in the three curves.

The behavior of n′ and n′′ vs. C is shown in Fig. 4for the same example discussed in Fig. 3 (in therethe cell capacity was fixed to C = 150 Mb/s, whilenow we consider more values). The behavior of n′

is clearly linear, whereas n′′ initially decreases andafterwards grows. Overall, n′′ exhibits a quadraticbehavior. In all cases, the distance n′′−n′ diminisheswith C.

Asymptotic behaviour of γ vs. n. For n > n′′,the Network throughput starts to vanish exponen-tially, and the result is that the device populationprogressively moves to the Random Access block.Asymptotically, there are no devices either in serviceor in think time, so that all devices loop between theACB block, the Random Access subsystem and itsbackoff:

limn→∞

n

γ=τ

2+

1− papa

E[Ba] + E[B0]. (19)

Fig. 5 shows the behavior of γ vs. n for the ex-ample of Fig. 3. In the figure, γ grows very slowly

Figure 4: Behavior of n′ and n′′ versus the cell capacity (com-puted without ACB).

Figure 5: Monotonic relation between γ and n (computed with-out ACB).

for n < n′ (access requests do not collide and thereis practically no blocking at the Network). Betweenn′ and n′′, the value of γ grows faster and faster,especially in the zone in which the RACH through-put has a negative slope (this is due to high collisionprobability and high Network blocking). However,as soon as pB decreases again, due to excessive colli-

13

sions, the curve of γ vs. n changes towards a linearrelation (right before n′′, where pB ' 0). After n′′ alldevices move to the Random Access and backoff B0

subsystems (no device will be under service, asymp-totically), and eventually the relation between γ andn approximates Equation (19).

With clusters. As explained in Section 4.4, clus-tering k devices results in transferring kE [FS ] bitsper network access, hence the cluster service timeE[S] becomes k times longer. So, n′ decreases withincreasing cluster size. However, the number ofdevices within clusters becomes kn′. Denoting byE[S|1] the service time without clusters, we have:

kn′' C

E [FS ]

[kE[S|1]+E [TTH ]

+(1−pJ)

(τ

2+

1− papa

E[Ba]

)], (20)

which includes (k− 1)CE[S|1]E[FS ] more devices w.r.t. the

case without clusters. Similarly, we can observe thatkn∗ grows by M plus a number of devices propor-tional to N for each increase of 1 in the cluster sizek.

The interval n′′−n′ increases with the cluster size,because a factor k appears in the denominator ofthe R.H.S. of Equation (14) when clusters are used.Therefore, the increase of the size of the network satu-ration region, in terms of devices, becomes k(n′′−n′),which is more than a k-fold increase.

We can conclude that the beneficial impact of clus-tering is larger than the one obtained by increasingcell capacity, which is linear, and it comes at a muchlower deployment cost.

5.3. Delay

The access delay E [AT ] is negligible when the Ran-dom Access saturates first, and for n < n′ when theNetwork subsystem also saturates, unless ACB intro-duces high delay by using low values for pa and/orhigh values for E[Ba]. When the Network subsystemis saturated, we know from Equation (8) that E [AT ]

is proportional to n with coefficient 1λ = E[FS ]

C . Forn > n′′, the delay explodes exponentially. Therefore,

Figure 6: Model validation for a cell with 150 Mb/s capacityand pa =1.0. Left scale for σ and ξ, right scale for access delay.

the desirable range of population sizes goes from n′

to n′+ ∆n, where ∆n is such that the delay ∆nE[FS ]C

is bearable by the applications running at the devicesin the network.

So, in practice, the study of n′ and its approxi-mation are key to tune system parameters properlyduring network design.

5.4. Validation through packet-level simulation

In order to validate the simplifying assumptionsthat we had to introduce for the analytical tractabil-ity of the model, we developed an event-based packet-level simulator that reproduces the behaviour of theclosed model in Fig. 1. However, in the simulator weused uniformly distributed (rather than exponential)file sizes; the arrival processes are not Poisson, andthe output of the Random Access subsystem is animpulsive process in which all successful RACH at-tempts are brought at the Network subsystem ingressat the same time. In addition, the simulator accu-rately implements the processor sharing of the Net-work processor capacity among active connections,i.e., it equally allocates resources to active jobs (i.e.,packets under service) based on the actual number ofjobs, with a maximum per-job rate equal to R.

Fig. 6 reports an example of the simulated resultsfor σ and ξ, together with the analytical results.

14

Specifically, we report numerical results for a cell withC = 150 Mb/s, R = 10 Mb/s, N = 54, M = 200,pJ = 0.3 and τ = 0.01 s, which are typical valuesfor LTE BSs. Moreover, we used E [TTH ] = 30 s,E [B0] = 0.15 s, E [B1] = 1 s, E [FS ] = 1.5 MB,pa = 1 and E [Ba] = 4.0 s to account for typicalupload of pictures and small videos during crowdedevents by using applications like WhatsApp, with au-tomatic file upload retry. To ease the reader, suchconfiguration will be used through the rest of the pa-per and, only when required, we will point out theparameters that deviate from the configuration pre-sented above. We run each experiment a sufficientnumber of times to obtain small 95% confidence in-tervals. The figure clearly shows that the model is ex-tremely accurate. The access delay computed withthe model for values smaller than n′ and greater thann′′ underestimates the one computed via simulationby a small quantity. This is due to the fact that theservice time is overestimated in those regions, so thatthe number of users in the RACH is (slightly) under-estimated. Similarly, we can notice that the RACHthroughput and the throughput of the Network sub-system are a bit underestimated for values close toand greater than n′′, where the service rate is actu-ally better than what used in the model.

We tested a wide range of values for all relevant pa-rameters, and found very similar model accuracy inall cases. The extremely good match between modelpredictions and simulation results is a clear indicationof the model validity beyond the simplifying assump-tions introduced for tractability. The figure alsoshows that the approximations used to compute thenotable operational points in close form are quite ac-curate. Some minor error can only be noticed in n′,whose approximated value is slightly higher than theone obtained with the full model (which approximatethe processor sharing operation in the computation ofE[S]) and confirmed by the simulations (which usesa pure processor sharing). In fact, since the approxi-mated expression for n′ is derived by using the worstcase service time E[S], which in turn leads to overes-timate the size of the population that saturates theNetwork subsystem.

Figure 7: Throughput (left) and access delay (right): impactof ACB with [pa = 0.95, E[Ba]=4] (solid lines) and clustering(where k is specified) in the stadium scenario.

6. Stadium: Numerical Results

We consider a stadium covered by a set of LTEcells. The system parameters are as reported in Table1 (right-most column). Fig. 7 illustrates the impactof ACB and clustering in the specified scenario. Weonly report the results obtained with the ACB con-figuration that causes less delay, and one example ofclustering (k=2). The figure shows that either ACBor clustering make it possible to significantly increasethe number of users in the system. In particular,clustering groups of as few as 2 users is very effectivein increasing n′. ACB suffers large delays, so as tomake it quite undesirable even for limited user pop-ulation sizes. However, Fig. 7 also shows that ACBand clustering in combination achieve low delay andguarantee access to very large user populations.

Fig. 8 reports the values for the two QoE indexesηS and ηA we defined in Section 4.3. Both indexescapture the user satisfaction, combining the servicetime and the access delay. In the first case we justcompute the ratio between the service time and thesum access delay plus service time. In the secondcase we define a more elaborate parameter. It is in-versely proportional to the service time, normalizedto the service time when M users are under service.

15

Figure 8: End-user QoE indicators ηS and ηA. Quality de-grades with ACB [pa = 0.95, E[Ba] = 4] (solid lines) w.r.t.scenario without ACB (dashed lines), because of the additionaldelay it causes.

It further fades exponentially with the access delay,normalized to the access delay value at n′. Thus, thissecond metric is very sensitive to relative increases ofdelay rather than to absolute increases.

The curves of the QoE parameters show qualita-tively similar trends. As regards ηS , with a low pop-ulation of UEs, the network access time E[TA] is verylow and mostly depends on RACH transit and ACBoperation. Each device in service is guaranteed a rateequal to R, keeping ηS close to 1, unless ACB is usedand E[AT ] cannot be neglected. When the BS canno longer provide the maximum rate R to each oneof the nS devices in service, E[S] starts to increase,while E[AT ] is practically constant (without ACB)or slowly increasing (with ACB), so that its weighin ηS diminishes as the population increases. How-ever, when the number of devices reaches the valuen′, the access delay E[AT ] starts increasing fast (andlinearly) causing a hyperbolic decrease of ηS towardszero. The QoE parameter starts dropping around550 devices in the cell. In general, the figure showsthat using ACB is detrimental in terms of quality ofexperience in steady state conditions, especially withsmall populations, when the ACB delay is the mostprominent component of the access delay.

For what concerns ηA, the figure shows that, with-

Figure 9: Access Delay for variable cell capacity without ACB(solid lines; 4 cases with capacity 150, 300, 900 and 1500Mbps), and with ACB [pa = 0.95, E[Ba] = 4] (2 dashed linescorresponding to extreme case capacities equal to 150 and 1500Mbps).

out ACB, it starts from the value 10/0.75 = 13.33.This is the ratio between the data rate cap for eachindividual device, and the data rate given by the BSto each user once the maximum number of users (200)is reached (150 Mb/s divided by 200 users means 0.75Mb/s per user). With ACB, the additional delay dueto barring decreases the initial value of ηA. In allcases, the curve stays close to the initial value aslong as the access delay remains negligible, then itrapidly drops. Also in this case, the QoE parameterstarts dropping around 500 devices. Note that thismeans that a coverage of the 50, 000 users in the sta-dium with good QoE would require about 100 cells,if each user carries just one device, 200 cells if eachuser carries two devices, and so on.

Of course, one possibility to improve performanceis to use cells with higher capacity. In Fig. 9 we plotcurves of E[AT ] for cell capacities in the range 150-1, 500 Mb/s. The critical element for QoE is givenby the points where the access delay starts increas-ing significantly. This means about 550 devices withcapacity 150 Mb/s and about 4, 000 devices with ca-pacity 1, 500 Mb/s. The latter translates into 12 cellsfor 50, 000 devices, 25 in the case each spectator car-ries 2 devices.

16

Figure 10: Values of n′ versus the cell capacity, for variablecluster sizes. ACB curves are practically superposed to curveswithout ACB.

In addition, Fig. 9 clearly shows that the operat-ing area where both end-users and network operatorswish “to be” is just before the curve’s first knee. Insuch neighbourhood, ACB does not play any signif-icant role, and E[AT ] is a fraction of E[S], beforestarting to rapidly move to bigger values. It is im-portant to recall that this “change of phase” in theaccess delay is pinpointed by n′. The second knee ofthe curves corresponds to n′′, and both knees changewith the cell capacity. It is very important to noticethat in the whole interval [n′, n′′] the system bot-tleneck is the Network due to the limitation of MRRC CONNECTED devices. When the number of devicesin the cell becomes larger than n′′, we see a switchin the bottlenecks, and only from this point on theRACH subsystem becomes unstable and the accesstime explodes, going asymptotically to infinity.

Increasing the cell capacity or the number of cellsis quite costly, and may not be the most desirable so-lution to achieve good QoE in crowded environments.A much simpler option can be to allow users to co-alesce in their network access attempts through theformation of clusters. Fig. 10 shows the values of n′

as a function of the cell capacity, for variable clus-ter sizes. We immediately appreciate the advantagesof clusters: the adoption of coalitions brings a gaincomparable to the one obtained increasing C with a

Figure 11: Impact of skipping the contention-based RACHprocedure, σ and ξ solid and dashed lines respectively.

negligible cost (if any) to the network provider. In-deed, a gain equal to or larger than that obtained bydoubling the cell capacity can be achieved by adopt-ing a cluster size k = 3.

Finally, to evaluate the importance of reducing theload of the Random Access in presence of downlinktraffic, we repeat our tests with different values of pJ .Skipping the contention-based RACH procedure in-troduces a small improvement in terms of the heightof the point at n∗, allowing the RACH to sustain aslightly higher arrival frequency. However, as can beseen in Fig. 11 for the case with no ACB, the mainimpact of pJ on the performance of the system is re-flected in the value of n′′, which is moved towardslarger values of n. It must be noted that the increasein height at n∗ is so small not to be visible on thegraphs, and that the increase in the value of n′′ isnot relevant from the point of view of applications,because at those numbers of users per cell, perfor-mance (e.g., in terms of access delay) is intolerablybad.

7. Practical Validity of the Modeland its Limitations

The analysis presented in this paper accounts forthe main parameters of a 3GPP-compliant wireless

17

Table 2: Configuration parameters used to evaluate the prac-ticality of our simple model (with pa = pj = 0)

Quantity Notation Value

Max number of RACH attempts kmax {10, 20, 30}Timeout of RRC CONNECTED RRCTO {1, 2, 3, 5, 10} [s]

Application-related timeout (patience) ARTO {15, 30, 60} [ms]

access network. However, there are some parame-ters in the configuration of the network and in theapplication generating data to transmit that are notaccounted for in the model and that are object of re-search and technical proposals [14, 15]. Those param-eters can impose practical limitations to the behaviorof the cellular access system. In particular, there arethree main parameters that might be relevant andare not considered in the simple model presented anddiscussed so far: (i) the maximum number or RACHretries that a request can go through before beingdropped; (ii) the timeout used in RRC CONNECTED

state, which guarantees that resources allocated toa device remain available beyond the last packet istransmitted, and which is useful to avoid incurringin a new RACH request procedure for customers re-turning within a few seconds; and (iii) the patienceof a user, i.e., the maximum delay allowed by the ap-plication running at the user’s device after which atransmission request is dropped and a new request isissued after some application-specific backoff.

In the following we analyze and discuss the impactof each of such parameters by means of simulations,whose results are compared to the model predictions.The configurations parameters used in what followsare summarized in Table 2. As we will see, our modelcaptures the behavior of the system in a wide spec-trum of configurations and, most important, it al-ways gives accurate results for the regions of oper-ation (i.e., the ranges of the number of users) thathave practical importance.

7.1. Impact of the maximum number of retries on theRACH

Let’s denote the maximum number of RACH re-tries as kmax. This is the maximum number of con-secutive failed RACH attempts, after which an access

request is dropped. If such drop occurs, the devicethat was not granted network access will either giveup or retry after some time (depending on the appli-cation), according to a backoff mechanism that has alength comparable with the think time of our model(tens of seconds). We therefore simulate a networkin which, after a request fails RACH access for kmax

consecutive times, the device goes back to the Thinkstation of Fig. 1 without sending any data. Therefore,in this modified system, differently from the one usedin the performance analysis of Section 6, it is possibleto have failures. Note also that, for kmax → ∞, thesystem tends to the one we have modeled so far.

Fig. 12 shows that the impact of kmax on thethroughput of the RACH (σ) and on the through-put of the Network station (ξ) is important only forpopulation sizes n > n∗. Specifically, the smallerkmax, the larger the distance n′′ − n′ because n′′ in-creases while n′ does not change significantly. Simi-larly, n∗ remains practically unchanged. This can beexplained by considering that the probability to failkmax consecutive times on the RACH is an increas-ing function of the load γ, and for populations belown∗ devices, the probability to fail even as few as fouror five times in a row is low because the probabilityof a single RACH failure is of the order of 1− 1/e orsmaller. The figure also shows that ξ does not changebefore n′′.

Let’s now consider the impact of kmax on the accessdelay. As depicted in Fig. 13, E [AT ] is impacted onlystarting from the value n∗ identified with our model.For larger values of the population of devices, the ac-cess delay decreases, still it remains quite high. Thisdelay reduction is however obtained at the expense ofthe success probability experienced by a network ac-cess request. Indeed, as shown in Fig. 14, the systemsuffers high failure probability starting from n = n∗,which is the point at which the simple model showsa peak in the RACH throughput σ.

By jointly considering access delay and failureprobability, it is clear that the network cannot be ef-ficiently and satisfactorily operated with populationsmuch larger than n′. As a consequence, our modeloffers accurate predictions well beyond the range ofinterest, since it is accurate up to n∗ > n′.

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Figure 12: Effect of the max number of RACH retries kmax onσ and ξ, solid and dashed lines respectively.

Figure 13: Effect of the max number of RACH retries kmax onthe access time

7.2. Impact of timeout for the RRC CONNECTED state

We now consider that, in real 3GPP access net-works, devices that enter the RRC CONNECTED state,leave that state based on an activity timeout, i.e.,only after a time RRCTO has elapsed, during whichno transmission occurred. Otherwise, if after a filetransmission is complete a device wants to initiate anew file transmission before the timeout expires, thatdevice will not need to go through the RACH again.We simulate such system by (i) counting all devicesin RRC CONNECTED state, including the ones that have

Figure 14: Failure probability introduced by the max numberof RACH retries.

transmitted their file, in the set of devices that areunder service, and whose number cannot exceed M ;(ii) if a device exits the Think station and it is still inRRC CONNECTED state, it jumps to the Network stationof our system of Fig. 1. Note that our simple modelcorresponds to the case in which RRCTO = 0. Notealso that, differently from the case of kmax discussedbefore, the use of a timeout for the RRC CONNECTED

state does not lead to failures, although it allows lessdevices to use the transmission channel at the sametime, as commented in what follows.

In this case, the network throughput ξ is practi-cally not affected, as shown in Fig. 15, except n′′

shifts forward due to returning devices sustaining ξeven when σ fades off. The Network in our system isPS, which means that when less than M devices areunder service, they still use all resources, i.e., they areserved faster than with exactly M devices. Therefore,although the use of a timeout RRCTO > 0 imposesthat some devices in the Think station do not freetheir Network allocation before RRCTO time unitsafter their file is transmitted, Network resources arenot wasted. Fig. 15 also shows that the throughput ofthe RACH with RRCTO > 0 is barely affected by thetimeout. However, the longer the timeout, the higherand the sooner σ grows before n′, and the later it fallsafter n′′. This is due to the fact that if devices returnto the Network station before the timeout expires,

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the load of the RACH is alleviated, with less colli-sions experienced. However, in the interval betweenn′ and n′′, σ is simply shifted backward. Indeed, de-noting by r the probability that the timeout does notexpire, when Network is saturated the RACH sees asystem with M(1−r) serving slots instead of M , andour model can be applied to compute the resultingRACH throughput σ.

For what concerns access delay, Fig. 16 shows thatonly minor differences with respect to our model canbe appreciated. Specifically, apart for a backwardshift of n′ and a forward shift of n′′ that we havecommented above, the linear slope of E[AT ] in be-tween is only slightly changed because the value ofλ to be used in this case in Equation (8) is ξ(1 − r)instead of ξ. This effect is due to the increased prob-ability to fail Network access, where rM allocationslots are now reserved for returning customers, andr increases with RRCTO. This is only partially com-pensated by the short access delay experienced bydevices for which the timeout does not expire, result-ing in less delay at n′ and higher delays at n′′. In theextreme case in which RRCTO → ∞, the access de-lay is minimized for returning customers (it reducesto the latency of the Network station), although itbecomes infinite for the rest of users, so that, as soonas n > M , the average access delay diverges.

Fig. 17 shows the average number of devices underservice as a function of RRCTO. The ratio betweenM and the value plotted in the figure is the coefficientr discussed above, which grows with the timeout andcauses increased access delays.

In practical circumstances in which RRCTO is ofthe same order of magnitude as the Think time, theprobability r covers a small percentage of cases, andour model provides a good approximation for all val-ues of the population n.

7.3. Impact of the application timeout

Finally, we consider the impact of the applicationtimeout. We simulate the system of Fig. 1, althoughwe interrupt and drop a service request if its accessdelay exceeds the application timeout ARTO. Oursimple model corresponds to the case ARTO → ∞.When a request is dropped, a failure occurs and thedevice goes back to the Think station.

Figure 15: Effect of several configuration of RRCTO on σ andξ, solid and dashed lines respectively.

Figure 16: Effect of several configuration of RRCTO on theaccess time, using a semi-log scale. The zooms represent E[AT ]in linear scale, for populations in the ranges around n′, withinthe interval [n′, n′′] and around n′′, respectively.

Fig. 18 compares the throughput of our modelwith the one of the simulator using various valuesof ARTO. The figure unveils that no differences canbe appreciated in σ and ξ for population size up ton′ and slightly above that point. Beyond that point,the curves of σ separate. Both n∗ and n′′ increasewith ARTO decreasing. To explain this behavior,

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Figure 17: Number of users in service due to the use of a finiteand not null timeout for the RRC CONNECTED state.

consider that the average access delay in our modelincreases with the population size. So, as soon asthe average access delay increases, the probabilityto trigger an application timeout increases. WithARTO larger than the RACH latency (in the or-der of τ , which is extremely small for an applica-tion timeout), a non-negligible timeout probabilityis possible only when the RACH experiences signif-icant collisions, i.e., starting with some value of nbetween n′ and n∗, which is what we observe in thefigure. From that point on, requests that suffer time-outs take a Think time backoff, which is longer thanthe RACH or the Network backoff, thus resulting inreduced RACH load. This is why n∗ and n′′ moveupwards.

For what concerns the impact on access delay,Fig. 19 shows significant differences from the pointat which timeouts start occurring. Interestingly, theaccess delay with ARTO grows very slowly after thecurves in the figure split, and this is due to the hardbound imposed by the timeout. Therefore, using anapplication timeout could allow to use the system wellbeyond n′ and even beyond n∗. However, as shownin Fig. 20, the failure probability becomes relevantwell before n∗. So we conclude that using the systemwith populations much higher than n′ is not a goodidea even in this case.

Figure 18: Effect of several configuration of the applicationtimeout on σ and ξ, solid and dashed lines respectively.

Figure 19: Effect of several configuration of application time-out on the access time

In conclusion, the predictions of our model are veryaccurate for the range of population sizes in which thefailure probability is negligible or bearable (below afew percentage).

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Figure 20: Failure probability introduced by the applicationtimeout.

8. Related Work

In [16], 3GPP has identified the random accessmechanism as a possible problem when the numberof connected devices rises to tens of thousands. Forthis reason, MAC overload control has been inves-tigated, and a broad literature exists on this topic.See [17] for a comprehensive overview. Simple mod-els to estimate the probability of preamble collisionin the PRACH channel are presented in a few 3GPPstandard documents (e.g., [16]), and in the literature(e.g. [13], [17], [18], [19]). The conclusions of most ofthese studies point out that for Machine-Type Com-munications (MTC) applications, the Random Ac-cess procedure can drastically limit network perfor-mance. Possible approaches to modify the PRACHaccess procedure have been proposed in [20, 21, 22].

Most of the previous studies on dense cellular en-vironments have focused on MTC scenarios, and [17]shows that the differences between the human-basedand the MTC scenarios are substantial. Nevertheless,the PRACH access mechanism, and its interactionswith the other phases of the network usage cycle playan important role also in case of human-based scenar-ios. This was shown in [1], through a measurement-based study of cellular network performance duringcrowded events, showing that network access failuresbecome orders of magnitude higher than those ob-served on routine days, and the interaction between

access and transmission phases generates behaviorsdifficult to predict. Recent works have shed light onthe characteristics of the load of the Random Accesssubsystem. For instance, the authors of [23] derivedthe joint distribution of collisions and successes, so tobe able to estimate the number of users based on ob-served events. Before that, the authors of [24] haveshown how to model RACH successes in the pres-ence of bursty arrivals. In [25], the authors proposea model of Random Access system with limits on thenumber of acknowledged requests per RAO, whichis somehow equivalent to introduce network capacityconstraints, although in a much simplified and roughmanner. The simple analytical model presented inthis paper is oblivious to traffic statistics and goesbeyond previous models because it provides a toolto understand the root causes of the behaviors mea-sured, e.g., in [1], and to quantify the impact of thecrowd size on RACH and network performance as atandem system. Moreover, our analysis permitted usto identify possible approaches to correctly dimen-sion the network and—with the help of D2D sidelinksand BS-orchestrated clusters—to mitigate the nega-tive impacts of crowds.

ACB and its extensions have been proposed andanalyzed beyond MTC and crowded scenarios, so tobe able to provide class-based priority in the Ran-dom Access [23, 26, 27]. These mechanisms promiseto re-shape the RACH traffic load and even to adaptACB parameters dynamically. However, this kind ofapproaches can only impact per-class delay statistics,not steady-state flows. Indeed, our model shows thatthe volume of flows at the entrance of the RandomAccess subsystem does not depend on the presenceof ACB, see (1). Thus, ACB schemes are useful inscenarios in which the network operator needs to in-troduce priorities, but they do not improve systemthroughput.

Lately, radio access network limitations have be-come a hot topic, due to the start of deploymentof 5G technology, where ultra-low latency and ex-tremely dense scenarios are included in the standardoperational framework. To cope with such challeng-ing requirements, several proposals have been devel-oped, such as for example reported in [28]. In or-der to reduce latency in 5G access networks, 3GPP

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proposes a new unit of scheduling called a mini-slot,which can be flexibly configured to last between 1 and6 orthogonal frequency-division multiplexed symbols.Furthermore, 5GPPP in [14] pinpoints group basedRACH, that is coalescing access requests, as a solu-tion aimed at handling the initial access bottlenecksdue to massive connectivity.

9. Conclusions

This paper presents a model to capture the key as-pects of the behaviour of cellular networks in crowdedenvironments. The main merit of the model lies inthe insight that it brings on cellular system opera-tions in very crowded environments, and in the possi-bility to use it to drive the correct dimensioning of thecellular system in very crowded environments. As anexample, the model allows the assessment of the ben-efits achievable through the adoption of D2D commu-nications to reduce the congestion on the RACH moreeffectively than with ACB, thus significantly improv-ing performance and QoE. For example, our modelshows that, instead of serving 50,000 terminals with100 cells of capacity 150 Mb/s each, it is possible touse 25 cells, each of capacity 300 Mb/s, provided thatclusters of 5 devices are formed to access the RACH.

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Paolo Castagno is Post-Doc at the ComputerScience Department, University of Torino, Italy. Hereceived his Master and Ph.D. degrees from the Uni-versity of Torino, in 2014 and 2018, respectively. Hisresearch focus is on performance evaluation of com-puter systems and communication networks, with aspecific interest on wireless networks.

Vincenzo Mancuso is Research Associate Profes-sor at IMDEA Networks, Madrid, Spain, and recipi-ent of a Ramon y Cajal research grant of the SpanishMinistry of Science and Innovation. Previously, hewas with INRIA (France), Rice University (USA) andUniversity of Palermo (Italy), from where he obtainedhis Ph.D. in 2005. His research focus is on analysis,design, and experimental evaluation of opportunisticwireless architectures and mobile broadband services.

Matteo Sereno was born in Nocera Inferiore,Italy. He received the Laurea degree in ComputerScience from the University of Salerno, in 1987 andthe Ph.D. degree in Computer Science from the Uni-versity of Torino, in 1992. He is currently Full Profes-sor at the Computer Science Department, Universityof Torino. His current research interests are in thearea of performance evaluation of computer systems,communication networks, peer-to-peer systems, com-pressive sensing and coding techniques in distributedapplications, game theory, queueing networks, andstochastic Petri net models.

Marco Ajmone Marsan is full professor at theElectronics and Telecommunications Department ofthe Politecnico di Torino in Italy, and part-time re-search professor at IMDEA Networks Institute inLeganes, Spain. Marco Ajmone Marsan obtained de-grees in EE from the Politecnico di Torino in 1974 andthe University of California, Los Angeles (UCLA)in 1978. He received a honorary doctoral degreefrom the Budapest University of Technology and Eco-nomics in 2002. Since 1974 he has been at Politec-nico di Torino, in the different roles of an academiccareer, with an interruption from 1987 to 1990, whenhe was a full professor at the Computer Science De-partment of the University of Milan. Marco Ajmone

Marsan has been doing research in the fields of digi-tal transmission, distributed systems and networking.He has been a member of the editorial board and ofthe steering committee of the ACM/IEEE Transac-tions on Networking. He is a member of the edito-rial boards of the journals Computer NetworksandPerformance Evaluationof Elsevier, and of the ACMTransactions on Modeling and Performance Evalua-tion of Computer Systems. He served in the organiz-ing committee of several leading networking confer-ences, and he was general chair of INFOCOM 2013.Marco Ajmone Marsan is a Fellow of the IEEE, amember of the Academy of Sciences of Torino, and amember of Academia Europaea.

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