IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019 859

Network Dimensioning, QoE Maximization,and Power Control for Multi-TierMachine-Type Communications

Dong Han , Student Member, IEEE, Hlaing Minn , Fellow, IEEE, Utku Tefek ,

and Teng Joon Lim , Fellow, IEEE

Abstract— For a radio resource-limited multi-tier machine-type communication (MTC) network, controlling random accesscongestion while satisfying the unique requirements of eachtier (type) and guaranteeing fairness among nodes is always achallenge. In this paper, we study the network dimensioningand radio resource partitioning for the uplink of an MTCnetwork with the signal-to-interference ratio-based clusteringand relaying, where the MTC gateways (MTCGs) capture andforward the packets sent from MTC devices (MTCDs) to the basestation (BS). Specifically, under transmission outage probabilityconstraints, we investigate the trade-off between network utility(in terms of transmission capacity and revenue) and resourceallocation fairness. With both outage probability constraints andminimum MTCD density constraints, we propose approaches tomaximize the weighted sum of quality of experience of differenttiers of MTCDs. Furthermore, a transmit power control strategyfor MTCG-to-BS link is proposed to achieve a constant data rate.

Index Terms— Multi-tier MTC, data aggregation, networkdimensioning, resource allocation, QoE, power control.

I. INTRODUCTION

A. Background and Motivation

AS PART of the Internet-of-things (IoT), MTC devices(MTCDs) are recording, sensing and generating a huge

amount of data every second. However, without timely gath-ering and processing of the data, useful information cannotbe extracted and acted upon. For instance, future applicationscenarios include automobiles and unmanned aerial vehicles(UAVs) reporting traffic conditions to a data processing center,sensors transmitting video data to a server, computation-limited devices offloading tasks to a cloud server, etc. Theseapplications may require massive numbers of simultaneousconnections between devices and base stations (BS’s), whichleads to severe congestion on the random access channel

Manuscript received April 13, 2018; revised August 14, 2018; acceptedOctober 4, 2018. Date of publication October 12, 2018; date of current versionJanuary 15, 2019. The associate editor coordinating the review of this paperand approving it for publication was Y. J. Zhang. (Corresponding author:Dong Han.)

D. Han and H. Minn are with the Department of Electrical and ComputerEngineering, The University of Texas at Dallas, Richardson, TX 75080 USA(e-mail: dong.han6@utdallas.edu; hlaing.minn@utdallas.edu).

U. Tefek and T. J. Lim are with the Department of Electrical and ComputerEngineering, National University of Singapore, Singapore 117583 (e-mail:utku@u.nus.edu; eleltj@nus.edu.sg).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2018.2875735

(RACH) and impairs the reliability of both the MTC andHuman-to-Human (H2H) communications networks [1].

At the same time, the coexistence of diverse devices requir-ing varying Quality of Service (QoS) in terms of latency [2],transmission rate and outage, connection security [3], etc, is akey desired feature of M2M networks. How to efficientlyaggregate the huge amount of transmissions while satisfy-ing various QoS requirements and mitigating the negativeeffects on H2H communication is still a problem. SinceMTC data can be efficiently aggregated by a hierarchical net-work, a number of clustering and resource allocation methodshave been proposed to solve this problem [4]–[8]. For instance,in [7] and [8], SIR and location based clustering anddecode-and-forward relaying schemes for single-tier1 MTCDswere proposed to maximize the transmission capacity of theMTC network under an outage probability constraint.

In general, orthogonal resource allocation such as timedivision [9]–[11] significantly reduces the interference but haslow spectrum efficiency and requires more demanding syn-chronization. On the other hand, using nonorthogonal resourceallocation such as the nonorthogonal multiple access (NOMA)[12], [13] enhances spectrum efficiency but introduces inter-ference, leading to a more complicated decoding scheme [14].In [7], [8], and [15] and this paper, the combination oforthogonal and nonorthogonal resource allocation approachesis used, where every H2H user, MTCG and MTCD tierare assigned orthogonal channel resources. MTCDs of thesame tier share the same band in a nonorthogonal manner.In this way, the conventional H2H communication is free ofMTC caused interference and the MTC achieves a higherspectrum efficiency through spectrum reuse.

For large-scale MTC networks involving different tiers ofMTCDs, fair resource allocation is always a challenge inthat 1) the measurement of fairness is not unique, such asα-fairness [16], bargaining achievable fairness [17], and 2) thefairness maximization problem may have no global optimalsolution. On the other hand, quality of experience (QoE)has recently received more interest in the network designand optimization of many applications such as 5G [18]–[21]and cloud computing [22]. As defined by the InternationalTelecommunication Union (ITU), QoE is the overall accept-

1I.e., all MTCDs have the same QoS requirements.

0090-6778 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

860 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

ability of an application or service as perceived subjectivelyby the end-user [23]. Thus, improving the user experience ofdata-hungry applications under given QoS requirements is aninteresting problem.

Motivated by the massive data and connection demands andchallenges mentioned above, the main goal of this paper isto realize a cluster-based framework for efficient and flexi-ble MTC data aggregation under multiple QoS constraints.We consider two data aggregation optimization problems,namely 1) network dimensioning and utility maximizationwith fairness constraints, where we measure fairness by howequally the radio resource is partitioned to different MTCDtiers and maximize the network utility2 and 2) QoE maximiza-tion. Next, to support the data aggregation scheme, we proposea power control strategy for the randomly distributed MTCGson the MTCG-to-BS link. We note that since the MTCD-to-MTCG links accommodate large numbers of devices basedon grant-free random access, power control for MTCDs isimpractical and hence is not considered in this paper.

B. Related Works and Our Contributions

In the following, we distinguish our work from several state-of-the-art existing works. Regarding the MTC data aggrega-tion, [11] proposed a multi-hop data aggregation scheme andfound the tradeoffs between the energy density and the cover-age characteristics. They assumed TDMA for the users withineach Voronoi cell for the data aggregation. In contrast, in thispaper we investigate a random access approach for single-hop MTC data aggregation so that the devices with differentQoS constraints can simultaneously transmit their packets.Guo et al. [24] designed a two-phase MTC data aggregationscheme for a single tier of devices and solved the problem ofresource scheduling between different phases. However, all theaggregators were assumed to have a fixed disc-shaped servingzone (i.e., the MTCD locations are modeled as Matern clusterpoint process with the aggregator locations being the parentpoint process), which may limit the efficiency or flexibility ofthe aggregator when the device locations are uniformly distrib-uted. In contrast, in our data aggregation scheme, the locationsof different tiers of devices are assumed to form independenthomogeneous PPPs without transmission boundaries and theaggregators (MTCGs) will successfully capture the packetsfrom any MTCD if the SIR permits. While the energy-efficientdata aggregation scheme proposed in [25] requires the MTCDs(smart meters) to know the aggregator positions, the MTCDsin our scheme can transmit without any knowledge of theMTCG positions. Other papers such as [26] also deal with theenergy-optimal routing issues in MTC data aggregation, butwe focus on the maximization of the network utility and theweighted sum QoE, and the MTCG power control problem.

While the effects of channel inversion power control inthe stochastic geometry modeling of cellular networks havebeen well studied e.g. [27]–[29], our proposed MTCG powercontrol strategy focuses on a different aspect from those

2The network utility is defined as the weighted sum of network capacity.When the weights are regarded as the price per unit of capacity, the networkutility will represent the economic revenue of the network.

papers. Specifically, [27] and [29] investigated the channelinversion power control for different uplink transmission mod-els in cellular networks and found the coverage probabilitybased on the complementary cumulative distribution function(CCDF) of the signal-to-interference-plus-noise ratio (SINR).Addationally, using channel inversion to compensate the pathloss was considered in [28] to study different spectrum sharingschemes (i.e., overlay and underlay) and transmission mode forD2D communications in cellular networks. In our proposedMTCG power control strategy, we apply channel inversionto compensate both the path loss and the Rayleigh fading.Since the MTCGs are allocated orthogonal radio resources,we derive the CCDF of the received signal-to-noise ratio(SNR) at the BS instead of the SINRs that were consideredin [27] and [29], and hence obtain an accurate result withoutany approximation or modification in the system model.

The main contributions of this paper are summarized asfollows:

1) We propose a multi-tier MTC data aggregation schemeunder different QoS constraints. The proposed schemecauses no interference to the conventional H2H commu-nications due to orthogonal resource allocation betweenM2M and H2H users.

2) Defining the network utility as the weighted sum ofnetwork capacity (throughput), we develop three spe-cific resource partitioning approaches and calculate thecorresponding MTCD transmission densities of differenttiers.

3) We develop a generalized resource partitioning approachto achieve different degrees of tradeoffs between thenetwork utility and fairness of radio resource allocation.

4) Using an increasingly popular performance metric,namely the weighted sum of QoE as a measure ofthe satisfaction of network throughput, we also developglobal optimal resource partitioning solutions.

5) Furthermore, we propose an MTCG transmit powercontrol strategy to enhance the proposed MTC dataaggregation scheme. The effect of the aggregatedMTCG transmit power on the resource partitioningbetween MTCGs and MTCDs is analyzed.

The feasibility of this work can be justified as follows. Theproposed network dimensioning is applied during the systemdesign phase for the deployment of multi-tier MTCDs andno real-time processing is required. The proposed power con-trol achieves the maximum MTCG-to-BS spectral efficiencywith limited aggregated MTCG transmit power, providedthat the channel state information (CSI) and the distancebetween the MTCG and BS are known to the transmit-ting MTCG. Operator installed devices as well as registeredwireless users such as cell phones, laptops and WiFi accesspoints, can serve as MTCGs. Thus, the proposed methods arepractical.

A part of the materials related to the resource allocationmethods of the network utility maximization in items 1) and2) were presented in GLOBECOM 2017 [15]. The rest ofthis paper is organized as follows. The MTC data aggregationmodel is proposed in Section II, where the end-to-end outageprobability is derived and the background of MTCG power

HAN et al.: NETWORK DIMENSIONING, QoE MAXIMIZATION, AND POWER CONTROL FOR MTC 861

Fig. 1. Network Structure (MTCGs capture and relay packets to the BS fromMTCDs belonging to different tiers.)

control is introduced. In Section III, we propose the networkdimensioning and network utility maximization problem andanalyze the tradeoff between resource partitioning fairnessand network utility. Then, the QoE maximization problemunder transmission outage probability constraints and mini-mum MTCD density constraints is solved in Section IV. Next,we investigate the MTCG power control strategies to achievea constant MTCG-to-BS data rate in Section V. Finally,Section VI concludes this paper.

II. SYSTEM MODEL

A. Communication System and Data Aggregation Model

We consider a single-hop MTC uplink relay scenario asshown in Fig. 1. A group of packets sent from MTCDs aregathered by nearby MTCGs and then relayed (in a decode-and-forward way) to the BS so that random access requeststo the BS are moderated. The MTCDs are classified intodifferent tiers according to their required transmission outageprobabilities {ζj}. We assume that there are N tiers of devicesand the locations of the tier j MTCDs form a homogeneousspatial PPP, ΦDj = {Xj,k} with density λj , ∀j = 1, . . . , N .In compact form, we use vector λλλ = (λ1, . . . , λN )T torepresent the densities. Similarly the MTCG locations forma homogeneous spatial PPP, ΦG = {Yi} with density λG.ΦDj ’s are assumed to be pairwise independent from each otherand from ΦG.

The total amount of radio spectrum resources for MTC islimited and it should be partitioned among MTCGs anddifferent tiers of MTCDs. Assuming that there are QM

resource blocks (RBs) reserved for M2M communications andQ1 = γQM RBs allocated for MTCD-to-MTCG link (link 1),then Q2 = (1 − γ)QM RBs are allocated for MTCG-to-BSlink (link 2), where γ ∈ [0, 1]. Next, Qj

1 = βjQ1 RBs areallocated for the jth tier of MTCDs, where 0 ≤ βj and∑N

j=1 βj = 1. The considered resource partitioning is shownin Fig. 2. We denote the spectrum partitioning factors incompact form with vector βββ = (β1, . . . , βN )T. Since [7]stated equal resource partitioning among MTCGs is opti-mal by showing that all MTCGs capture a packet with thesame probability, Q2/G RBs are allocated for each MTCG,

Fig. 2. Resource partitioning among MTCDs and MTCGs.

where constant G is the number of MTCGs in the coverageof BS’s.3

Furthermore, the MTCDs use the same modulation andcoding scheme to send fixed-length packets with a constanttransmit power. The MTCGs decode the captured MTCDpackets and forward the message on orthogonal channels to theBSs under cellular standards. In this scheme, the channel stateinformation (CSI) of link 1 is not required by MTCDs unlessadaptive modulation schemes are used, making the systemsimple and scalable. The CSI of link 2 is assumed known to theMTCGs since they are direct users in the cellular networks.Since link 1 and link 2 use different wireless technologies,transmitting the same message through link 1 and link 2 mayrequire different amount of resources. We assume each MTCDneeds δ1 RBs to transmit one packet, and each MTCGneeds δ2 RBs in order to relay a MTCD’s packet to the BS.Therefore, each MTCG can have at most U2 = �Q2/(Gδ2)�relay channels, while the jth tier of MTCDs has U j

1 =�βjQ1/δ1� orthogonal data channels and the total number oflink 1 channels is U1 =

∑Nj=1 U j

1 ≈ �Q1/δ1�. We also assumeindependent and identically distributed (i.i.d.) Rayleigh fadingwith unit average power gain on each channel, and path losswith exponent α over all channels, i.e., the path loss from thetransmitter to the receiver at a distance of r is r−α.

The data aggregation process is designed as follows. First,each active MTCD randomly selects a channel among thosethat are allocated for its tier. Since two or more MTCDscan potentially transmit on the same channel, they may inter-fere with each other. Second, each MTCG listens to all theMTCD-to-MTCG channels, but only chooses the MTCD withthe highest average received power (i.e., the nearest MTCD)on each channel. Also, the MTCGs can successfully decodea packet only when their received SIR is higher than athreshold η. Third, the MTCGs treat packets from all MTCDsequally in the sense that if no greater than U2 packets aredecoded, all of them will be relayed; otherwise, the MTCGwill randomly choose to relay U2 packets and drop the rest.

B. End-to-End Outage Probability and Capacity

In the data aggregation process, we note that the resourcepartitioning between link 1 and link 2 determines the numberof channels U1 and U2 for the two links, and hence affectsthe end-to-end outage probability. Intuitively, allocating moreresources to link 1 leads to greater transmission density and

3While [7] considers only one BS, we consider multiple BS’s to analyzethe power control problem. The total resource QM is then from all theconsidered BS’s.

862 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

less interference, but higher packet drop rate at the gateways.Therefore, there would be an optimal operating point in theresource partitioning between link 1 and link 2 as we can seein the next section.

Note that the end-to-end outage probability for single-tier SIR-based clustering and relaying MTC data aggregationscheme was analyzed in [7]. Therein, it was clarified thatsuccessful transmission is equivalent to the joint occurrenceof three events, namely a) the typical MTCD is the nearestMTCD on a randomly selected channel u to an MTCG locatedat Yi, denoted by N u

X0,Yi, b) the typical MTCD’s packet on

channel u is successfully captured by an MTCG located at Yi,denoted by Cu

X0,Yi, c) the typical MTCD’s packet received on u

is successfully relayed by an MTCG located at Yi, denotedby Ru

X0,Yi, where X0 represents the nearest MTCD to Yi

transmitting on channel u. Provided that the MTCD densityis λD, the (conditional) probabilities of these three events are

Pr(N uX0,Yi

) = exp(

−θλD

U1‖X0 − Yi‖2

)

(1)

Pr(CuX0,Yi

|N uX0,Yi

) = exp(

−θλD

U1η

2α ‖X0 − Yi‖2 Kα,η

)

(2)

Pr(RuX0,Yi

|CuX0,Yi

) = 1− ε

= 1−U1−1∑

n=U2

(U1 − 1

n

)

pnc,SIR

×(1− pc,SIR)U1−1−n

(

1− U2

1 + n

)

(3)

where Kα,η =∫∞

η− 2α

dt1+tα/2 , pc,SIR =

(1 + η

2α Kα,η

)−1

is the probability of an MTCG capturing a packet on anylink 1 channel and ε represents the link 2 outage probability.We note that if U2 ≥ U1, then ε = 0.

To be specific, Eq. (1) is the probability of there being noMTCDs in the circular region B(Yi, ‖X0 − Yi‖) [30], whereB(xc, ρ) represents a circle centered at xc with radius ρ.Eq. (2) is achieved by deriving the Laplace transform of theprobability density function (PDF) of the sum interferenceobserved at Yi from all MTCDs transmitting on channel u.Then, the expression of pc,SIR is found by de-conditioningEq. (2) over the distance ‖X0−Yi‖. Since the capture eventsare independent and equally probable across all channels withprobability pc,SIR, the total number of the packets captured bythe MTCG at Yi is Binomial distributed as Bin(U1, pc,SIR).Finally, by considering the failure due to random packetdropping at the MTCG when U2 < U1, Eq. (3) is obtained.

Next, applying the chain rule on (conditional) probabilitiesin (1) - (3), the end-to-end outage probability of a typicalMTCD transmission is expressed as

�(λD, γ) = E

∏

Yi∈ΦG

[1− Pr(Ru

X0,Yi|Cu

X0,Yi)

×Pr(CuX0,Yi

|N uX0,Yi

) Pr(N uX0,Yi

)]

= exp

(

− λGU1

λD

(1 + η2/αKα,η

) (1 − ε)

)

, (4)

where in the derivation of Eq. (4), probability generatingfunctional (PGFL) [31] and variable change t ← r2 (r is theradial coordinate of a two-dimensional polar coordinate) areapplied. The details of the derivation for (1)-(4) can be foundin [7].

For the multi-tier scenario, we apply the same data aggre-gation scheme, and the gateways treat all the packets equally.Thus, the (conditional) probabilities of the events, N uj

X0,Yiand

Cuj

X0,Yi|N uj

X0,Yiare only modified by λj ← λD and U j

1 ← U1

for the jth tier, while Pr(Ruj

X0,Yi|Cuj

X0,Yi) is the same as (3)

for all tiers because each MTCG listens to the U1 channelsbut not some specific U j

1 channels. Therefore, the end-to-endoutage probabilities can be easily extended from (4) as

�(λj , γ) = exp

(

− λGU j1

λj

(1 + η2/αKα,η

) (1− ε)

)

≈ exp(

−βjφ(γ)λj

)

, (5)

where we define

φ(γ) Δ=λGU1

1 + η2/αKα,η(1− ε) (6)

as a function4 of γ. We note that the approximation in (5) isdue to the approximation in allocating the resource blocks,i.e., U j

1 = �βjQ1/δ1� ≈ βj�Q1/δ1� ≈ βjU1, and it isasymptotically close to the exact value of �(λj , γ) whenQ1/δ1 is large.

According to [32], the transmission capacity or area spectralefficiency for a single type of device is the density of thesimultaneously transmitting MTCDs multiplied with their end-to-end transmission success probability. Thus, for an N-tierMTC network, the sum network capacity can be expressed by∑N

j=1 λj(1−�j), where �j = �(λj , γ) is the end-to-end outageprobability of the jth tier of MTCDs. We may also consider theeconomic revenue of this MTC network as

∑Nj=1 λjθj(1−�j),

where θj denotes the price or revenue per unit capacity oftier j.

C. System Model for MTCG Power Control

We analyze the MTCG transmit power control strategyunder an aggregated MTCG power constraint. Recall that toanalyze the worst-case interference at MTCGs and the end-to-end MTC outage probabilities, the MTCD and MTCGlocations are assumed to form homogeneous PPPs in aninfinite region. However, investigating the aggregated MTCGtransmit power consumption in an infinite region will result inunrealistically unbounded results. Therefore, we assume theMTCG locations form a PPP with constant density λG withina finite circular region B(0, ρ), where B(xc, ρ) represents acircle centered at xc with radius ρ, and the BS locations formanother PPP with density λB within this region. We notethat as long as the area of the finite region is large enoughto contain a reasonable number of devices, the analytical

4φ(γ) is also a function of λG and max φ(γ) determines whether theMTCG deployment is able to support the required MTCD densities with theoutage probability constraint.

HAN et al.: NETWORK DIMENSIONING, QoE MAXIMIZATION, AND POWER CONTROL FOR MTC 863

results based on the infinite region scenario can be closeto the results based on the finite region scenario, since theinterference to a typical receiver is dominated by the nearbytransmitters [33]. To be more convincing, [7] shows thatthe single-tier transmission capacity simulated with MonteCarlo method in a finite region is close to the correspondinganalytical capacity based on an infinite region.

We consider that the transmissions between MTCGs andthe BS encounter both Rayleigh fading and path loss. Thepath loss model is represented as L(r) = r−α, where r isthe distance between the MTCG and the BS. Practically,L(r) holds for r ≥ 1, meaning that the path loss is normalizedto unity at unit distance. We note that some papers use thepath loss model L′(r) = 1

1+rα to avoid the singularity atthe origin [34]. Our analysis applies to both L(r) and L′(r)and the numerical results based on both path loss modelsare similar since for most of the distances we have r � 1.Therefore, to be consistent with the data aggregation model,we chose L(r) as the path loss model to analyze MTCG powercontrol strategies.

III. NETWORK DIMENSIONING AND

UTILITY MAXIMIZATION

In this section, we consider the scenario of N tiersof MTCDs for which the required outage probabilities areζ1, . . . , ζN . In particular, assuming the MTCGs treat all thepackets equally, for the jth tier, the outage probability �(λj , γ)is upper bounded by ζj . Accordingly, the network utilitymaximization problem can be formulated as

(P1) : maxλλλ,βββ,γ

N∑

j=1

λjθj

(

1− exp(

−βjφ(γ)λj

))

(7a)

s.t. 0 ≤ λj ≤ βjφ(γ)ln(1/ζj)

, (7b)

N∑

j=1

βj = 1, 0 ≤ βj (7c)

0 ≤ γ ≤ 1 (7d)

where the right-hand-side inequality in (7b) comes fromthe requirement that �(λj , γ) ≈ exp (−βjφ(γ)/λj) ≤ ζj .

Note that λj

(1− exp

(−βjφ(γ)

λj

))is the effective network

throughput of tier-j MTCDs, and thus when θj’s denote thepricing strategy (i.e., revenue per unit capacity) of tier j,the objective can be regarded as the economic revenue of theMTC network, and when θj’s are not given specific physicalmeanings, the objective is the weighted sum network capacityof the MTC network.

Lemma 1: The network utility maximization problem (P1)described by (7a) - (7d) can be simplified to

(P1-1) : maxβββ

∑Nj=1

βjπj(1−ζj)ln(1/ζj)

φ(γ) (8a)

s.t.∑N

j=1 βj = 1, 0 ≤ βj , (8b)

where γ = arg maxφ(γ) is the optimal resource partitioningparameter that maximizes the objective functions of (P1-1)and (P1). Correspondingly, the optimal MTCD density of

the jth tier relates to its bandwidth proportion as

λj =

φ(γ)βj

ln(1/ζj). (9)

Proof: Given βββ and γ, objective function in (7a) increasesmonotonically with λj , ∀j. Thus, according to the constrainton λj in (7b), the MTCD density of the jth tier should relateto its bandwidth proportion as

λj =φ(γ)βj

ln(1/ζj)(10)

to achieve maximum network utility. Substituting λj in (10)into (7a), we simplify the original problem to

maxβββ,γ

∑Nj=1

βjπj(1−ζj)ln(1/ζj)

φ(γ) (11a)

s.t.∑N

j=1 βj = 1, 0 ≤ βj , 0 ≤ γ ≤ 1, (11b)

where we can recognize that to maximize the objectivefunction (11a), we should have γ = argmax φ(γ).Then, the corresponding optimal λ

j is achieved accordingto (10). Thus the problem is further simplified to (P1-1) andLemma 1 is proved. �

In the rest of this section, we will discuss network dimen-sioning and resource partitioning methods that solve thenetwork utility maximization problem (P1) while satisfyingdifferent fairness requirements in resource allocation.

A. Utility-Optimal Solution

According to Lemma 1, we solve (P1) with the simplifiedproblem (P1-1). As we can always find a tier n∗ such thatn∗ = argmaxj

πj(1−ζj)ln(1/ζj)

, allocating all resources to this tier,

i.e., βββ∗ = eeen∗5 and thereby λλλ∗ = φ(γ�)ln(1/ζn∗)eeen∗ , will achieve

the maximum network utility

R∗ =θn∗(1− ζn∗)

ln(1/ζn∗)φ(γ). (12)

For the convenience of analysis, let z = (z1, z2, ..., zN)T

where zj = πj(1−ζj)ln(1/ζj) ≥ 0, and we rewrite (12) as

R∗ = ‖z‖∞φ(γ) (13)

where ‖ · ‖∞ represents the infinity norm. We note thisresource allocation method strongly favors the tier with theleast stringent outage constraint, or the one paying the highestprice. We also notice that zj monotonically increases with ζj

and θj . Intuitively, zj reflects the favorability of the jth tierfrom the perspective of the network operator, since a greater zj

indicates either a higher price θj or a less stringent (a greatervalue of) outage probability constraint ζj . In the followingsubsections, we will provide other resource allocation methodsfocusing on the fairness issue.

5We define eeei = (0, . . . , 0, 1, 0, . . . , 0)T where only the ith element is 1.

864 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

B. Geometric Mean (GM) Based Resource Allocation

Instead of maximizing weighted sum network capacity,we replace the objective function of (P1) in (7a) with thegeometric mean of the weighted capacities of all tiers as

1N

log Rtotal =1N

log

⎛

⎝N∏

j=1

Rj

⎞

⎠ =1N

N∑

j=1

log Rj (14)

where the tier-utility, Rj = λjθj

(1− exp

[−βjφ(γ)

λj

]), rep-

resents the weighted capacity or economic revenue of thejth tier. The motivation of using this logarithmic productfunction comes from a branch of cooperative game theory,namely bargaining theory [35] and the notion of proportionalfairness [36]. Further justification of this bargaining model isbeyond the scope of this paper, and interested readers can referto section 7.1.3 of [35] and [36], [37] for more details. Similarto the proof of Lemma 1, we notice that the objective increasesmonotonically with λj . Thus, to satisfy the first constraintof (P1) by equality, optimal MTCD density of the jth tiershould also be the λ

j specified in (9). With this knowledge,we obtain the following simplified problem,

(P1-2) : maxβββ

∑Nj=1 log

(βjπj(1−ζj)

ln(1/ζj) φ(γ))

(15a)

s.t.∑N

j=1 βj = 1, 0 ≤ βj . (15b)

The objective function in (15a) can be rewritten as

maxβββ

N∑

j=1

log βj +N∑

j=1

log(

θj(1 − ζj)ln(1/ζj)

φ(γ))

(16)

which allows us to maximize the objective function over βj .Due to the concavity of log function, equal partition isthe optimal GM-based resource allocation among MTCDs,i.e., βGM

j = 1N . Thus, the corresponding MTCD density is

λGMj = φ(γ�)

N ln(1/ζj)according to (9). Substituting λj , βj and

γ in the objective function of (P1) with λGMj , βGM

j and γ

respectively, the network utility achieved by the GM-basedmethod is

RGM =1N

N∑

j=1

θj(1− ζj)ln(1/ζj)

φ(γ) =1N‖z‖1φ(γ) (17)

where ‖·‖1 represents the l1-norm. Different from the methodintroduced in section III-A, this GM-based method achievesabsolute fairness in terms of an equal resource allocationacross all tiers of MTCDs, regardless of the outage probabilityconstraints.

Comparing the network utility achieved by the GM-basedmethod in (17) with the maximum value in (13), we haveRGM ≤ R∗, where equality holds when all the elements inz are equal. Clearly, the fairness achieved by the GM-basedmethod is at the cost of network utility. On the other hand,if the price values (weights) θj’s are adjusted to make thevalue of elements in z more uniform, the difference betweenRGM and R∗ will be reduced.

C. Cauchy-Schwarz (CS) Based Resource Allocation

While the GM-based method results in equal partitioning,and the utility-optimal allocation yields maximum networkutility, we are also interested in a trade-off between fairnessand efficiency, which motivated us to propose the CS-basedresource allocation.

We consider the optimization problem (P1-1). WithCauchy-Schwarz inequality, we have

(βββT z)2 ≤ ‖βββ‖22‖z‖22 (18)

where ‖ · ‖2 represents l2-norm and the equality holds whenβ1/z1 = . . . = βN/zN . Specifically, with this equality and thefact that the summation of all the nonnegative βj is 1, we have

βCSj =

zj

‖z‖1 =πj(1−ζj)ln(1/ζj)

∑Nl=1

πl(1−ζl)ln(1/ζl)

. (19)

We can see that this solution relates the resource partitioningparameter to the outage probability constraints and prices. Foreach tier, the stricter the outage constraint is, less resourceis allocated; and the higher the price is, more resources areallocated. Therefore, we set βCS

j as the resource partition-ing parameter and name this method as CS-based resourceallocation.

By substituting βj with βCSj in (9), the maximum transmis-

sion density for tier j is

λCSj =

πj(1−ζj)[ln(1/ζj)]2

∑Nl=1

πl(1−ζl)ln(1/ζl)

φ(γ) (20)

which results in the network utility

RCS =

∑Nj=1

(πj(1−ζj)ln(1/ζj)

)2

∑Nj=1

πj(1−ζj)ln(1/ζj)

φ(γ) =‖z‖22‖z‖1 φ(γ). (21)

Compared with the GM-based method that provides equalresource partitioning for all tiers, the CS-based method pro-vides less uniform resource allocation for different tiers.Furthermore, since ‖z‖21 ≤ N‖z‖22 for z � 0, we can concludeRGM ≤ RCS. The equality holds when all the elements inz are equal. In other words, the equality condition is

θj =ln(1/ζj)1− ζj

a, j = 1, ..., N (22)

where a is a nonnegative constant. Equation (22) indicates thatthe GM-based resource allocation can be regarded as a specialcase of the CS-based method and that, this pricing (weighting)strategy serves as a reference to evaluate fairness level in theCS-based resource allocation process. In general, the closerto this pricing strategy, the fairer resource allocation will be.We notice that the relationship between the three resourceallocation methods is

RGM ≤ RCS ≤ R∗ (23)

where both inequalities are satisfied with equality when (22)holds. Besides, the GM-based method and the utility-optimalallocation method achieve absolute fairness and absoluteunfairness respectively, while the CS-based method offers atrade-off between the two extremes.

HAN et al.: NETWORK DIMENSIONING, QoE MAXIMIZATION, AND POWER CONTROL FOR MTC 865

Fig. 3. Comparison of resource partitioning parameter.

D. Generalized Expression and Utility-Fairness Tradeoff

Recalling the three proposed approaches to allocate theradio resource in the MTC data aggregation scheme, we foundthe utility-optimal solution by directly solving (P1-1) but itresults in all resource being monopolized by one specific tier.We then change the linear summation into the sum-log formwhich is the same to the objective in a bargaining processand an equal resource partitioning is achieved. A heuristicCS-based resource allocation is then proposed to trade offthe needs of maximal network utility for a fair resourceallocation. Nevertheless, whether the CS-based method keepsa good balance between utility and fairness given the specificQoS requirements, i.e., the outage probability constraints,and the pricing strategy, is not clarified. In this subsection,we generalize an uniform expression (24) from the above threeresource partitioning parameters βββ∗, βββGM and βββCS.

βj(k)=zk

j∑N

i=1 zki

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

1‖z‖0

, k = 0 (GM-based)...

zj

‖z‖1, k = 1 (CS-based)

...z∞

j�Ni=1 z∞

i

, k =∞ (Utility-optimal),

(24)

∀j ∈ {1, . . . , N}. This generic resource partitioning based onk-norm can achieve different fairness levels by varying thevalue of a single factor k ∈ [0, +∞]. Specifically, the fairnesslevel increases monotonically with decreasing k. For instance,when k = 0, 1 and ∞, the generic resource allocationparameter βj(k) corresponds to the GM-based solution βGM

j ,CS-based solution βCS

j and utility-optimal solution β∗j , respec-

tively. Furthermore, with (9), the network utility achieved bythe generic resource partitioning is

N∑

i=1

λiθi(1− ζi) =∑N

i=1 zk+1i

∑Ni=1 zk

i

φ(γ) (25)

where k is the parameter as in (24).

E. Numerical Performance Comparisons

Comparisons among different resource partitioning methodsare shown in Fig. 3-6 respectively. To be specific, we considerN = 10 tiers of MTCDs with outage probability constraints,ζ1, . . . , ζN , ranging from 0.05 to 0.5 (the tier with a larger

Fig. 4. Comparison of MTCD density.

Fig. 5. Comparison of capacity per tier.

Fig. 6. Network utilities achieved by different resource partitioning schemes.

number has a larger outage probability) and assume thatφ(γ) = 1, θj = 1, ∀j. As can be seen from Fig. 3, whileGM based method (when k = 0) allocates the resourcesequally, the increase of the value of k tends to allocateless resources to the tiers with more stringent outage con-straints as the green and yellow bars show. Under our defin-ition of fairness, the GM-based method provides the fairestallocation, the utility-optimal resource allocation method(when k = +∞) is the most unfair one, and the methods whenk ∈ (0, +∞) achieves different degrees of fairness betweenthe two extremes. The differences in the focus on fairness leadto the differences in MTCD density and transmission capacitywhich are shown in Fig. 4 and Fig. 5 respectively. Comparingthe resource partitioning methods for different values of kin Fig. 4 and Fig. 5, it is clear that a larger k results in higherdensities and per-tier utilities of the tiers with more stringentoutage constraints, than those of the other two methods. On theother hand, as shown in Fig. 6, the network utility increaseswith the increase of the k value and it converges to around 0.72when all the resources are monopolized by a singletier.

866 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

IV. RESOURCE PARTITIONING WITH MINIMUM DENSITY

REQUIREMENTS AND QOE MAXIMIZATION

A. Problem Formulation

The network dimensioning solution provides an approachto allocate the radio resources among the gateways and dif-ferent MTCD tiers under QoS constraints in terms of outageprobability such that the densities of the MTCDs of differenttiers are found to maximize the network utility. The essentialof the proposed network dimensioning approach is that it canachieve different trade-off levels between the network utilityand the fairness of resource partitioning.

On the other hand, the minimum MTCD transmissiondensity (or equivalent capacity) requirement might be specifiedin some practical scenarios such that each tier of MTCDscan be guaranteed to conduct some basic tasks and achievea basic level of satisfaction. In practice, the user’s satisfactiontowards a certain tier-utility, e.g., capacity, delay, etc., is alsoregarded as quality of experience (QoE). The QoE is nota linear function of the corresponding tier-utility but has alogarithmic relationship to it according to the Weber-FechnerLaw (WFL). Intuitively, the QoE increases rapidly when thecurrent experience is very poor but increases slightly when thecurrent experience is already good enough.

In this section, our objective is to maximize the weightedsum of the QoEs of different tiers instead of the linearlyweighted sum of the capacities. The QoE is estimated bymeans of how satisfiable the tier-utility Rj is to the jth tier.The QoE factor for the jth tier is represented by

Γj = 1− exp

(

−τRj

Rj

)

, j = 1, . . . , N (26)

according to [21], where Rj = λj(1−ζj), ∀j, is the minimumtier-utility requirement (which also serves as a reference pointfor the satisfaction of tier-utility Rj), λj is the correspondingminimum density requirement and τ is a factor indicatingthe steepness of the QoE curve. For instance, when the tier-utility Rj is greater than its reference value Rj , the QoEfactor of the jth tier increases slowly. On the contrary, whenRj is lower than Rj , the QoE factor decreases dramatically.Considering the QoS requirements in both the maximumoutage probability and the minimum transmission density,we formulate a weighted sum QoE maximization problem as

(P2) : maxλλλ,βββ

N∑

j=1

ωj

[

1− exp

(

−τRj

Rj

)]

(27a)

s.t. Rj = λjθj

[

1− exp(

−βjφ(γ)λj

)]

(27b)

λj ≤ λj ≤ βjφ(γ)ln(1/ζj)

(27c)

N∑

j=1

βj = 1, 0 ≤ βj (27d)

where Rj = λj(1− ζj), ∀j. We note that in (P2) we directlyuse γ = argmax φ(γ) as the optimal resource partitioningbetween link 1 and link 2, since it maximizes not only the

objective function but also the size of feasible regions ofboth λλλ and βββ. To solve (P2), we have the following twopropositions.

Proposition 1: The solution of (P2) can be represented by

βj = βj + βj (28)

λj =βjφ(γ)ln(1/ζj)

(29)

where βj = λj ln(1/ζj)φ(γ�) and βj is the solution of

(P3) : min{βj}

N∑

j=1

ωj exp

(

−τβjzjφ(γ)

Rj

)

(30a)

s.t.N∑

j=1

βj ≤ Δβ, 0 ≤ βj (30b)

where Δβ = 1−�

λj ln(1/ζj)φ(γ�) .

Proof: To simplify (P2), we first recognize that theobjective of (P2) monotonically increases with the increaseof λj , ∀j. In addition, the λj has both upper and lower boundsshown in (27c), where the upper bound is linearly proportionalto the resource ratio βj and the lower bound is the mini-mum density constraint. Therefore, given a certain ratio βj ,the upper bound of MTCD density should be achieved. In otherwords, the optimal solution of (P2) should guarantee

λj =βjφ(γ)ln(1/ζj)

. (31)

Generally, these facts motivate us to solve (P2) with thefollowing two steps:

1) allocating minimum resource to each tier to meet theminimum transmission density requirements,

2) maximizing the QoE with the residual resources.Specifically, after replacing λj with βjφ(γ�)

ln(1/ζj)according

to (31), (P2) is simplified as

minβββ

N∑

j=1

ωj exp

(

−τβjzjφ(γ)

Rj

)

(32a)

s.t. λj ≤ βjφ(γ)ln(1/ζj)

(32b)

N∑

j=1

βj = 1, 0 ≤ βj . (32c)

where Rj = λj(1− ζj). Next, we denote

βj =λj ln(1/ζj)

φ(γ)(33)

according to (32b), and

Δβ = 1−N∑

j=1

βj = 1−∑

λj ln(1/ζj)φ(γ)

≥ 0. (34)

We mention that Δβ ≥ 0 since∑N

j=1 βj ≤ 1 as βj ≤ βj

according to (32b) and (33).Furthermore, we represent the original resource ratio βj as

βj = βj + βj , ∀j, where βj ≥ 0 indicates the minimum

HAN et al.: NETWORK DIMENSIONING, QoE MAXIMIZATION, AND POWER CONTROL FOR MTC 867

resource ratio allocated to tier-j to meet the minimum MTCDtransmission density requirements and the βj ≥ 0 is a newoptimization variable representing the resource partitioningratio for the residue amount of resource Δβ to tier-j. Replac-ing βj with βj + βj in (32a) to (32c) and regarding βj as a theoptimization variable, the problem will be further simplifiedto (P3). �

Proposition 2: The solution to (P3) is

βj =[

− 1τ zj

ln(

ν

ωjτ zj

)]+

(35)

with zj = zjφ(γ�)

Rjand ν ≥ 0, such that

∑Nj=1 βj = Δβ.

Proof: (P3) is recognized as a convex optimization prob-lem. The Lagrangian function of (P3) is

L(βj, ν)=N∑

j=1

ωj exp

(

−τβjzjφ(γ)

Rj

)

+ν

⎛

⎝N∑

j=1

βj −Δβ

⎞

⎠.

(36)

The solution of (P3) is then represented as the solution to∂L(βj, ν)/∂βj = 0 where βj’s are positive. By numericallysearching for the proper value of ν such that

∑Nj=1 βj = Δβ,

the solution to (P3) is obtained. �Substituting zj = zjφ(γ�)

Rjand zj = πj(1−ζj)

ln(1/ζj) to (35),

the solution βj can be rewritten as

βj =

[

− λj ln(1/ζj)θjφ(γ)τ

ln

(νλj ln(1/ζj)ωjθjφ(γ)τ

)]+

. (37)

In general, the monotonicity of βj over the minimum densityrequirement λj and the maximum outage probability constraintζj is not clear without the knowledge of other parameters suchas ν, φ(γ), ωj, θj and τ .

B. Numerical Results

In the simulation, we assume there are two tiers of MTCDsand compare the QoE, the network throughput, the achievableMTCD density and the resource allocation ratio under differentoutage probability constraints ζj ∈ [0.002, 0.3], ∀j and thesame minimum MTCD density constraint λj = 0.08, ∀j.Without loss of generality, we assume φ(γ) = 1, τ = 1,θj = 1 and ωj = 1, ∀j. In particular, the sum QoEs fordifferent outage probability constraints are shown in Fig.7,from which we notice that the sum QoE increases with thegrowth of both ζ1 and ζ2. We can also see that the sum QoEincreases more rapidly when both ζ1 and ζ2 are relativelysmall, which indicates the QoE is more sensitive to stringentoutage probability constraints. A similar trend can be seenin Fig. 8 where the QoE of the tier-1 MTCD is shown.We also observe from Fig. 8 that ζ1 has a more noticeableeffect on the QoE of tier-1 than ζ2 does. Next, Fig. 9 andFig. 10 show the throughput and the corresponding MTCDdensity of the tier-1. In line with the relationship between theQoE and outage probability constraints, the throughput andthe MTCD density increase with the growth of ζ1 and ζ2.Finally, the resource allocation ratio is shown in Fig. 11.In contrast to the resource allocation solutions to the network

Fig. 7. Sum QoE under different outage probability constraints.

Fig. 8. QoE of tier-1 under different outage probability constraints.

Fig. 9. Throughput of tier-1 under different outage probability constraints.

utility maximization problem we discussed in section III,we notice that to maximize the sum QoE, more resourcesare allocated to the tier with more stringent constraint. Forinstance, when ζ1 is around zero and ζ2 = 0.3, the resourceratio for tier-1 is more than 0.7 as Fig.11 shows. Thisfact indicates that to achieve a satisfiable QoE under verystringent constraint, a relatively high resource ratio must beguaranteed.

868 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

Fig. 10. Allowable MTCD density of tier-1 under different outage probabilityconstraints.

Fig. 11. Resource allocation ratio of tier-1 under different outage probabilityconstraints.

V. MTCG UPLINK POWER CONTROL

In the proposed MTC scheme, the successful relaying of aMTCD packet captured by a MTCG depends on two factors:1) the amount of packets captured by the MTCG and 2) theavailable channels for the MTCG. We assume that an MTCGneeds one channel to relay one packet so that when the amountof packets captured by a MTCG is larger than the total numberof channels allocated to it, certain packets will be randomlydropped. Since the MTCDs are assumed always in active trans-mission mode with certain density and each packet is of thesame length, the MTCG-to-BS links need to keep a constantdata rate. We assume the BS locations form a PPP with densityλB independent of the MTCG locations and each MTCG onlytransmits to its nearest BS. We represent the MTCG locationin a two dimensional coordinate as x. Since we assume thatthe transmissions between MTCGs and the BS encountersboth Rayleigh fading and the exponential path loss, a strictlyconstant data rate can hardly be achieved as the random fadingcould be significantly deep. Alternatively, we consider to keepa constant outage capacity (spectral efficiency) with truncatedchannel inversion policy [38]. We denote Px as the transmitpower of a MTCG located at x, hx and rx as the fading powergain and the distance between the MTCG located at x and the

target BS respectively. Let L(rx) = r−αx represents the path

loss, σ2 to be the noise power on a MTCG-to-BS channel,and Pout to be the outage probability defined as the probabilitythat the overall channel power gain (including both fading andpath loss) is less than a certain threshold. The outage capacity(spectral efficiency) is represented as

C = log2

(

1 +hxL(rx)Px

σ2

)

(1− Pout) , ∀x. (38)

Recalling that we assume each MTCG has U2 =�Q2/(Gδ2)� channels, where Q2 = (1−γ)QM is the numberof RBs allocated for all MTCG-to-BS links, G is the number ofMTCGs and δ2 is the number of RBs needed for an MTCG torelay a packet. In general, a higher spectral efficiency indicatesa larger number of MTCG-to-BS channels U2 given the sameamount of RBs and the same data rate requirement for eachMTCG packet transmission. This fact connects the spectralefficiency in (38) to parameter φ(γ) in (6) with a commonfactor U2. Thus, the network dimensioning and resourceallocation problems discussed in previous sections are affectedby the spectral efficiency via φ(γ). In particular, we can adapt

the definition of U2 as U2 =⌊

Q2 Cχ1Gδ2χ2

⌋(i.e., in this section,

we use the adapted definition U2 as a replacement of U2 ),where χ1 (Hz/RB) is the bandwidth per RB, χ2 (bps/RB)is the required data rate per RB and G = θρ2λG is theaverage number of MTCGs within the region B(0, ρ). Thus,the numerator Q2 Cχ1 is the aggregated MTCG-to-BS datarate and the term δ2χ2 indicates the required data rate totransmit one packet on a MTCG-to-BS channel for eachMTCG. Then, since φ(γ) defined in (6) is a function of both γand U2 (U2), the optimal resource partitioning γ betweenlink 1 and link 2 can be numerically computed given theMTCG-to-BS spectral efficiency C.

Provided that the channel state information (CSI) andthe distance between the MTCG and BS are known tothe transmitting MTCG but the average aggregated transmitpower for the MTCGs is bounded, we propose a powerallocation strategy for any MTCG within the coverage of theBS to realize the constant outage capacity. Remember thatthe BS locations and the MTCG locations form independenthomogeneous spatial PPPs within B(0, ρ) with density λB

and λG respectively. Similar to the MTCD-to-MTCG channel,we assume i.i.d. Rayleigh fading with average power gain of μon each MTCG-to-BS channel, and path loss with exponent αover all channels. In order to maintain a constant transmissionrate, each MTCG applies truncated channel inversion to adaptits transmit power. Specifically, two power control strategiesbased on truncated channel inversion are considered. The twostrategies compensate for both fading and path loss but theydiffer in the criteria of when the compensation is performed.In the first proposed strategy (S1), the MTCG transmits onlyif the overall channel power gain (including both fading andpath loss) is above a certain threshold. The second strategy(S2) allows MTCG to transmit only if the fading gain is abovea certain threshold, and it is a baseline strategy proposed tocompare with the first one. The expressions of the transmit

HAN et al.: NETWORK DIMENSIONING, QoE MAXIMIZATION, AND POWER CONTROL FOR MTC 869

power for both strategies are represented as

(S1) : Px =

{P

hxL(rx) , hxL(rx) ≥ h

0, otherwise(39)

(S2) : Px =

{P

hxL(rx) , hx ≥ h

0, otherwise(40)

where P is the constant BS received signal power and h andh are the thresholds of channel inversion for (S1) and (S2)respectively. Given an aggregated power constraint Psum,we are interested in finding proper values of P , h and h so thatthe outage capacity of each MTCG-to-BS link is maximized.Therefore, the problem is formulated as

(P4) : maxg

log2

(

1 +P

σ2

)

P (gx ≥ g) (41a)

s.t. E

⎡

⎣∑

x∈ΦG,x∈B(0,ρ)

Px

⎤

⎦ = Psum (41b)

Px =

{P

hxL(rx) , gx ≥ g0, otherwise

(41c)

where gx = hxL(rx), g = h when applying (S1) andgx = hx, g = h for (S2). The objective function (41a) isobtained by substituting Px to (38), while (41b) and (41c)represent the aggregated transmit power constraint and thepower control strategy.

To solve (P4), both the distribution of gx and the expectationof the aggregated MTCG transmit power, i.e., the left hand sideof (41b) should be properly represented, in order to obtainthe success probability P (gx ≥ g) and the received power P .In the following of this section, both issues will be discussed.

A. Development of Outage Probability

Since the MTCGs only choose the closest BS to transmit,the BS’s form Poisson Voronoi tessellations and each BS onlyreceives the signals from the MTCGs within its Voronoi cell.In this case, the PDF of the distance rx between a MTCGlocated at x and the corresponding BS is [30]

frx(r) = 2θλBre−πλBr2, r ≥ 0. (42)

On the other hand, given the distance rx, the conditionalcumulative density function (CDF) of gx can be written as

Fgx|rx(g) = 1− exp

(

− μg

L(rx)

)

, g ≥ 0 (43)

for the first strategy (S1).Deconditioning (43) with the PDFs of rx, the CDF of gx is

obtained as

Fgx(g) =∫ ρ

0

Fgx|r(g)frx(r)dr

= 1− 2θλB

∫ ∞

0

e−μg

L(r)−πλBr2

rdr. (44)

for the first strategy (S1). For the baseline strategy (S2),the CDF of gx is

Fgx(g) = 1− exp(−μg) (45)

since gx = h ∼ exp(1/μ).

Fig. 12. Comparison of outage capacities at different outage probabilities.(Psum = 65 dBW).

Thus, the successful transmission probability in (41a) canbe represented as

P(gx > g) =

{2θλB

∫∞0 e−

μgL(r)−πλBr2

rdr, for (S1)

exp(−μg), for (S2).(46)

B. Development of Aggregated Transmit Power

To achieve a constant uplink transmission data rate, eachMTCG varies its transmit power according to its location andthe fading gain of the channels. For the first strategy (S1),given a truncated threshold g = h, the aggregated MTCGtransmit power within B(0, ρ) is developed as show in (47),as shown at the bottom of the next page, where Ei(x) =∫∞−x

1t e−tdt is the incomplete exponential integral function,

(a) follows from deriving the expectation of 1/hx condi-tioning on hx ≥ g/L(rx) and (b) holds because the setof distances {rx} is mutually independent. Then (c) followsfrom the Campbell’s theorem [31]. Similarly, for the baselinestrategy (S2), we have

E

⎧⎨

⎩

∑

x∈ΦG∩B(0,ρ)

Px

⎫⎬

⎭

= E{ΦG}

⎧⎨

⎩E{hx,rx}

⎡

⎣∑

x∈ΦG∩B(0,ρ), hx≥h

P

hxL(rx)

⎤

⎦

⎫⎬

⎭

= −2θ2ρ2λBλGμP

∫ ρ

0

Ei(−μh)L(r)

e−πλBr2rdr. (48)

Thus, given the average aggregated MTCG powerconstraint Psum, the received constant signal power P at theBS is shown in (49), as shown at the bottom of the next page.

Therefore, with (46) and (49), the objective function (41a)can be maximized by numerically searching for the optimalthreshold h for (S1) or h for (S2).

C. Numerical Results and Discussions

In the simulation, we use the following default settings forFig. 12 - Fig. 14, ρ = 100, λB = 0.008, λG = 0.005,α = 3, σ2 = 1 and μ = 1. Other specific settings are

870 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

Fig. 13. Comparison of outage capacities at different aggregated powerlevels.

Fig. 14. Comparison of φ(γ) at different γ and aggregated power levelsusing (S1). (QM = 48000, δ1 = 30, δ2 = 5, η = 3, χ1/χ2 = 1).

mentioned below the corresponding figures. Fig. 12 comparesour proposed MTCG power control strategy (S1) and the base-line strategy (S2) in terms of the achievable outage capacityunder different outage probabilities. It can be observed that

the (S1) outperforms (S2) for all possible outage probabilitiesand that the outage probability of (S1) is slightly larger thanthe outage probability of (S2) when maximum outage capacityis achieved. For instance, (S1) achieves the maximum capacitywhen Pout = 0.20, while the maximum capacity of (S2)is obtained when Pout = 0.14. Next, Fig. 13 shows that(S1) achieves higher outage capacity than that of (S2) fordifferent aggregated powers. We also notice that the differencebetween the two capacities is proportionally more obviouswhen the aggregated power is small, which indicates that (S1)can allocate power more efficiently in a power-limit condition.

With strategy (S1), the effect of aggregated power con-straints on the resource partitioning between MTCGs andMTCDs is shown in Fig, 14, where the values of φ(γ) are com-pared. In Fig. 14, we can see that a greater aggregated powerresults in a larger φ(γ). Additionally, a greater aggregatedpower also corresponds to a greater γ. Since the spectralefficiency of the MTCG-to-BS link increases with Psum andif the MTCG-to-BS links have higher spectral efficiency, moreresources can be allocated the MTCD-to-MTCG links (i.e., γ

will be greater).

VI. CONCLUSION

We proposed network dimensioning and resource alloca-tion approaches for the multi-tier MTC data aggregationscheme to trade off between the network utility and the fairresource allocation. First, we analyzed three specific scenar-ios of resource partitioning by means of the utility-optimalmethod, the GM-based method, and the CS-based method,under maximum transmission outage probability constraints.Then, we developed a generalized resource partitioning forvarious levels of fairness. Next, considering both the maximumoutage probability constraint and the minimum MTCD densityrequirements, we investigated the overall QoE maximizationproblem. In contrast to the resource allocation solution tothe network utility maximization problem, we found it ispreferable to allocate more resource to the tier with morestringent outage probability constraint. Finally, we investi-gated two different truncated channel inversion power control

E{ΦG}

⎡

⎣∑

x∈ΦG∩B(0,ρ)

Px

⎤

⎦ = E{ΦG}

⎧⎨

⎩E{hx,rx}

⎡

⎣∑

x∈ΦG∩B(0,ρ), hx≥h/L(rx)

P

hxL(rx)

⎤

⎦

⎫⎬

⎭

(a)= E{ΦG}

⎧⎨

⎩

∑

x∈ΦG∩B(0,ρ)

E{rx}

[P

L(rx)

(

−μEi

(

− μh

L(rx)

))]⎫⎬

⎭

(b)= E{ΦG}

⎧⎨

⎩

∑

x∈ΦG∩B(0,ρ)

∫ ∞

0

P

L(r)

[

−μEi

(

− μh

L(r)

)]

frx(r)dr

⎫⎬

⎭

(c)= − 2θ2ρ2λBλGμP

∫ ρ

0

r

L(r)Ei

(

− μh

L(r)

)

e−πλBr2dr

(47)

P =

⎧⎪⎨

⎪⎩

−Psum

[2θ2ρ2λBλGμ

∫ ρ

0r

L(r)Ei(− μh

L(r)

)e−πλBr2

dr]−1

, for (S1)

−Psum

[2θ2ρ2λBλGμ

∫ ρ

0r

L(r)Ei(−μh)e−πλBr2dr]−1

, for (S2).(49)

HAN et al.: NETWORK DIMENSIONING, QoE MAXIMIZATION, AND POWER CONTROL FOR MTC 871

strategies (S1 and S2) for the MTCG. Our proposed strategy(S1) outperforms the baseline strategy (S2) in terms of achiev-able outage capacity since it takes both the randomnesses oflocation and the fading factor into account.

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Dong Han (S’17) received the B.E. degree in com-munication engineering from the University of Elec-tronic Science and Technology of China, Chengdu,in 2015, and the M.S. degree in electrical engi-neering from the National University of Singaporein 2016. He is currently pursuing the Ph.D. degreewith the Department of Electrical and ComputerEngineering, University of Texas at Dallas, Richard-son, TX, USA. He is also a Graduate Teaching andResearch Assistant with the University of Texas atDallas. His research interests include machine-to-

machine communications, signal processing, and optimization.

872 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 1, JANUARY 2019

Hlaing Minn (S’99–M’01–SM’07–F’16) receivedthe B.E. degree in electrical electronic engineeringfrom the Yangon Institute of Technology, Yangon,Myanmar, in 1995, the M.E. degree in telecommu-nications from the Asian Institute of Technology,Thailand, in 1997, and the Ph.D. degree in elec-trical engineering from the University of Victoria,Victoria, BC, Canada, in 2001. He was a Post-Doctoral Fellow with the University of Victoriain 2002. He has been with The University of Texasat Dallas, Richardson, TX, USA, since 2002, where

he is currently a Full Professor. His research interests include wirelesscommunications, signal processing, signal design, dynamic spectrum access,and sharing, next generation wireless technologies, and bio-medical signalprocessing.

He has served as a Technical Program Committee Member for over30 IEEE conferences. He served as the Technical Program Co-Chair for theWireless Communications Symposium of the IEEE GLOBECOM 2014 andthe Wireless Access Track of the IEEE VTC, Fall 2009. He served as anEditor for the IEEE TRANSACTIONS ON COMMUNICATIONS from 2005 to2016. He was also an Editor of the International Journal of Communicationsand Networks from 2008 to 2015. He has been serving as an Editor-at-Largefor the IEEE TRANSACTIONS ON COMMUNICATIONS since 2016.

Utku Tefek received the B.Sc. degree (Hons.) inelectrical and electronics engineering from BilkentUniversity, Turkey, in 2013, and the Ph.D. degreefrom the National University of Singapore in 2017.From 2017 to 2018, he was a Post-Doctoral Fellowat the National University of Singapore. Since 2018,he has been a Researcher at the Advanced Dig-ital Sciences Center, a Singapore-Based ResearchCenter established by the University of Illinois atUrbana–Champaign. His research interests includethe application of stochastic models to wireless net-

works, machine-to-machine communications, cyber-physical systems security,with a recent focus on the security of urban transportation systems.

Teng Joon Lim (S’92–M’95–SM’02–F’17) wasborn in Singapore. He received the B.Eng. degree(Hons.) in electrical engineering from the NationalUniversity of Singapore in 1992, and the Ph.D.degree from the University of Cambridge in 1996.From 1995 to 2000, he was a Researcher at theCentre for Wireless Communications, Singapore,one of the predecessors of the Institute for InfocommResearch (I2R). From 2000 to 2011, he was anAssistant Professor, an Associate Professor, and aProfessor at The Edward S. Rogers Sr. Department

of Electrical & Computer Engineering, University of Toronto. Since 2011, hehas been a Professor with the Electrical & Computer Engineering Department,National University of Singapore. where he served as the Deputy Headfrom 2014 to 2015. Since 2015, he has served as the Vice-Dean (GraduatePrograms) with the Faculty of Engineering.

Dr. Lim has volunteered on the Organizing Committee of a number ofIEEE conferences, including serving as the TPC Co-Chair of the IEEEGlobecom 2017, and the TPC chair of the IEEE International Conferenceon Communication Systems (ICCS) 2018. He is also the Chair of theSingapore Chapter of the IEEE Communications Society. He was an AreaEditor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS from2013 to 2018, and previously served as an Associate Editor for the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS. He has also servedas an Associate Editor for the IEEE WIRELESS COMMUNICATIONS LET-TERS, the Wiley Transactions on Emerging Telecommunications Technologies,the IEEE SIGNAL PROCESSING LETTERS, and the IEEE TRANSACTIONS ON

VEHICULAR TECHNOLOGY.His research interests span many topics within wireless communications,

including cyber-security in the Internet of Things, heterogeneous networks,cooperative transmission, energy-optimized communication networks, multi-carrier modulation, MIMO, cooperative diversity, cognitive radio, and sto-chastic geometry for wireless networks. He has published widely in theseareas.