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Hybrid-Beamforming-Based Millimeter-WaveCellular Network Optimization

Jia Liu, Senior Member, IEEE, and Elizabeth S. Bentley, Member, IEEE

Abstract—Massive MIMO and millimeter-wave communica-tion (mmWave) have recently emerged as two key technologiesfor building 5G wireless networks and beyond. To reconcile theconflict between the large antenna arrays and the limited amountof radio-frequency (RF) chains in mmWave systems, the so-called hybrid beamforming becomes a promising solution andhas received a great deal of attention in recent years. However,existing research on hybrid beamforming focused mostly on thephysical layer or signal processing aspects. So far, there is alack of theoretical understanding of how hybrid beamformingcould affect mmWave network optimization. In this paper, weconsider the impacts of hybrid beamforming on utility-optimalityand queueing delay in mmWave cellular network optimization.Our contributions in this paper are three-fold: i) we developa joint hybrid beamforming and congestion control algorithmicframework for mmWave network utility maximization; ii) wereveal a pseudoconvexity structure in the hybrid beamformingscheduling problem, which leads to simplified analog beamform-ing protocol design; and iii) we theoretically characterize thescalings of utility-optimality and delay with respect to channelstate information (CSI) accuracy in digital beamforming.

I. INTRODUCTION

In recent years, millimeter wave communication (mmWave)has emerged as a promising technology for building 5Gwireless networks and beyond. The excitements of mmWavecommunications are primarily due to: i) the rich unlicensedspectrum resources in 60 GHz bands; ii) the ease of packinglarge antenna arrays into small form factors (a consequence ofthe short wavelengths); and iii) a much simplified interferencemanagement thanks to the highly directional “pencil-beam-like” mmWave signals. Moreover, recent field tests (see, e.g.,[1], [2], etc.) have shown that the large directivity gainsof mmWave transceivers can offset the high atmosphericattenuation in mmWave bands, dispelling the common concernthat mmWave is not suitable for outdoor communications. Thepotential of mmWave networks has also stimulated many stan-dardization activities (e.g., IEEE 802.15.3 wireless personal

Manuscript received May 22, 2019, revised September 10, 2019, andaccepted October 4, 2019. This work is supported by NSF grants ECCS-1818791, CCF-1758736, CNS-1758757, CNS-1446582; ONR grant N00014-17-1-2417, and AFRL grant FA8750-18-1-0107. This paper was presented inpart at IEEE/IFIP WiOpt, Paris, France, May 2017. Any opinions, findingsand conclusions or recommendations expressed in this material are those ofthe author(s) and do not necessarily reflect the views of AFRL and ONR.

Jia Liu is with the Department of Computer Science, Iowa State University,Ames, IA, 50011 USA (e-mail: jialiu@iastate.edu).

Elizabeth S. Bentley is with the Air Force Research Laboratory, InformationDirectorate, Rome, NY, 13441 USA (e-mail: elizabeth.bentley.3@us.af.mil).

DISTRIBUTION STATEMENT A: Approved for Public Release; distribu-tion unlimited 88ABW-2019-4902 on 09 October 2019. Other requests shallbe referred to AFRL/RIT 525 Brooks Rd Rome, NY 13441.

Digital Object Identifier xx.xxxx/XXXX.xxxx.xxxxx.

area networks, 802.11ad wireless local area networks, and fast-growing interests in mmWave cellular networks [3]).

However, the highly directional propagation of mmWavesignals and the special mmWave hardware requirements alsointroduce several unique technical challenges for networkingsystems. One major problem in mmWave networking is itsvulnerability to blockage, which is due to the weak diffractionability of mmWave communications [3], [4]. Mitigating block-age in mobile cellular networks requires a frequent search fornew unblocked directed spatial paths, which entails a largecommunication overhead and complicates the scheduling andcongestion control algorithmic designs at higher layers. An-other main technical challenge is energy-efficient beamformingarchitecture design, which lies at the heart of mmWave direc-tional networking. Although large antenna arrays can be easilydeployed in mmWave systems, the high power consumptionof mixed mmWave signal components significantly limits thenumber of radio-frequency chains (RF chains), rendering fulldigital beamforming (requiring one RF chain per antenna)impractical [5]. Moreover, most of the digital beamformingschemes in traditional MIMO systems require full chan-nel state information (CSI), which is difficult to acquire inmmWave systems due to the fast fading in mmWave spectrumand the low signal-to-noise ratio (SNR) before beamforming[6]. Because of the RF chain limitations in mmWave systems,analog beamforming approaches have been proposed (see, e.g.,[7], [8]). The basic idea of analog beamforming is to controlthe phase shifters of antenna elements, so that the energyof the transmitted data stream is concentrated in a singledirection to obtain a high directivity gain. Compared to digitalbeamforming, analog beamforming can be achieved by onlyone RF chain without requiring any CSI at the transmitter.However, analog beamforming can only transmit in a singlebeam direction and cannot leverage any spatial multiplexingcapability of the large mmWave antenna array.

In light of the limitations of analog and digital beamform-ings, there is a growing consensus that the more suitablearchitecture for mmWave cellular networks is the hybridbeamforming architecture, which exploits the large mmWaveantenna arrays and yet only requires a limited number of RFchains [6], [9]–[12]. Hybrid beamforming enjoys the best ofboth worlds: On one hand, it uses analog beamforming to offerspatial division and directivity gains to combat large mmWavechannel attenuations. On the other hand, digital beamformingprovides multiplexing gains for the lower dimensional effectivechannels, for which the CSI is relatively easier to acquire.It has been shown in [6], [13] that hybrid beamformingachieves a data rate performance comparable to full digital

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beamforming with 8 to 16 times fewer RF chains.So far, however, the existing works on mmWave hybrid

beamforming are mostly concerned with problems at thephysical layer or signal processing aspects. To date, thereremains a lack of theoretical understanding on how hybridbeamforming could affect mmWave networking performancesin terms of congestion control, scheduling, and resourceoptimization algorithms. In this paper, our goal is to fillthis gap by conducting an in-depth study on the impacts ofhybrid beamforming on throughput and delay performancesin mmWave cellular network optimization.

Specifically, in this paper, we focus on the algorithmicdesign and the throughput-delay analysis for the celebratedqueue-length-based congestion control and scheduling frame-work (QCS) (see, e.g., [14], [15], and [16] for a survey) inhybrid-beamforming-based mmWave cellular networks. Ourmain results and technical contributions are as follows:

• We develop an accurate analytical model that captures theessence of hybrid beamforming in mmWave cellular net-works, while being tractable enough to enable network-levelunderstanding and analysis. Based on this analytical model,we investigate the problem of joint hybrid beamformingand congestion control for network utility maximization.We show that the joint hybrid beamforming and congestioncontrol optimization is non-convex by nature, which createschallenges for the algorithmic designs in the MaxWeightscheduling component in the QCS framework.• By exploiting the special problem structure of the mmWaveMaxWeight scheduling component, we show that the non-convex scheduling subproblem admits a pseudoconvex ap-proximation under a wide range of hybrid beamformingparameters of practical interests. Moreover, our analysisreveals that, to solve the scheduling subproblem, one onlyneeds to adjust the analog beamwidth at the base station(BS), while the analog beamwidth adjustment at the mo-bile station (MS) side is unnecessary. This insight greatlysimplifies the analog beamforming training protocol design.• We investigate the impact of CSI inaccuracy on networkperformance with hybrid beamforming, where we assumethat the true CSI is quantized by Q bits. We reveal a pair ofinteresting phase transition phenomena in utility-optimalityand delay in the following sense: There exists a criticalvalue Q] such that: i) if 0 < Q < Q], then the deviations ofsteady-state queue-length grows linearly and the congestioncontrol rate is bounded by a constant; ii) If Q ≥ Q], thedeviations of queue-lengths and congestion control rateshave the same scaling laws as in the full CSI case.

Collectively, these results not only deepen our theoreticalunderstanding of mmWave network optimization with hybridbeamforming, but also provide insights for low-complexityanalog beam training and effective CSI quantization in prac-tice. The remainder of this paper is organized as follows: InSection III, we introduce network models and the problemformulation. Section IV presents the mmWave congestioncontrol and scheduling framework, as well as the algorithmicdesign for analog beam training. Section V studies the impactsof inaccurate CSI on digital beamforming. Section VI provides

(a) (b)

Fig. 1. Traditional beamforming receiver architectures: (a) Analog beam-forming using phase shifters along the signal path, and (b) and conventionaldigital beamformer using a separate ADC for each signal path.

numerical results and Section VII concludes this paper.

II. HYBRID BEAMFORMING: BACKGROUND ANDMOTIVATION

In this section, we provide a brief overview on the basicsand the current state-of-knowledge related to MIMO beam-forming techniques to further motivate the significance ofhybrid beamformng for mmWave communications systems.Simply speaking, beamforming (also known as spatial fil-tering) is the ability for an antenna system to adaptivelyand electronically steer its beam along a desired directionwhile suppressing the reception of potential interferers fromother directions. Depending on the “level of intelligence,”beamforming can be classified into two major categories,namely digital and anlalog beamforming. Due to the symmetry(with reversed signal paths) between transmitter and receiverarchitecture, our discussions in this section are mostly focusedon receiver design to avoid repetition. We note that most ofthe characteristics of receiver beamforming techniques applysimilarly to the transmitter side.

1) Analog Beamforming: The basic idea of analog beam-forming is to control the phase shifters of all antenna elements,so that the correlated transmitted and received signal energyis constructively combined at some desirable direction. Amajor advantage of analog beamforming at the RF front-endis the use of a single mixer to perform frequency translation.As such, the combined RF signal is down-converted to anintermediate frequency (IF) to be digitized by a single ADCfor post-processing. This implies low energy consumption.However, due to the limited phase tuning, analog phase shifterscan only sense a single spatial direction at a time [17].That is, beams from multiple directions cannot be formedsimultaneously, limiting the capability of the beamformer,particularly for MIMO applications. We note that this problemwill be overcome by the hybrid beamforming architecture inSection III. Also, phase shifters suffer from high losses andbandwidth limitations. This is even more exacerbated for largeantenna arrays as they require a large number of phase shifters.As a result, traditional analog beamformers suffer from largesize, weight, and complexity of the array, and not to mentionpower consumption.

For wideband operations in mmWave, bandwidth limitationcan be overcome by local oscillator (LO) phase-shifting, wherea tunable oscillator is used to sweep the bandwidth [18].However, the demand for fine phase resolution, necessary inreliable scanning, implies extreme hardware complexity. In ad-dition, traditional analog beamformers have other drawbacks.Among them are: i) beamforming performance suffers fromquantized levels of phase increments, ii) retraining of the phase

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shifting network implies processing overhead and sacrificingdata throughput, iii) have significant hardware complexity thatimpacts size, power, and cost, and iv) do not accommodatespatial multiplexing unless a hybrid architecture is used.

2) Digital Beamforming: The aforementioned pitfalls ofanalog beamforming motivate the design of digital beamform-ing. For wideband operations, digital beamforming approachesoffer more flexibility [19], [20] since beamforming and relatedadaptive algorithms are carried out at the back-end of thetransceiver using FPGAs or other digital processing units. Indigital beamforming, the RF signal is processed and digitizedprior to amplitude scaling and phase shifting, as depicted inFig. 1(b). To perform beamforming, the digitized basebandsignal yn is multiplied by a complex weight wn for the n-thsignal path. Therefore, the output is yout =

∑n ynwn, where

the complex weight can be written as wn = aneφn . Digitaladaptive beamformers can achieve more accurate main beams,null steering, side lobes levels control, simultaneous multi-directional beams, and spatial multiplexing. However, existingdigital beamformers at baseband have extensive hardwarerequirements as they employ separate ADCs for each signalpath, as shown in Fig. 1(b). The large number of high-cost andpower-hungry ADCs results in excessive power consumptionin the back-end circuitry, making such approaches limited tosmall arrays. Further, most digital beamforming techniques fortraditional MIMO require the full knowledge of channel stateinformation (CSI), which is difficult to acquire for the largeantenna array deployed in mmWave platforms.

To address the limitations of analog and digital beamform-ing, in the next section, we will introduce the analytic model ofthe cellular hybrid beamforming architecture, which achievesthe best of both worlds of analog and digital beamforming.

III. NETWORK MODEL AND PROBLEM FORMULATION

Notation: We use boldface to denote matrices/vectors. A†denotes the conjugate transpose of A. We use ‖ · ‖ and ‖ · ‖1to denote `2- and `1-norms, respectively. We let I denote theidentity matrix, whose dimension is conformal to the context.We let R and C denote real and complex spaces, respectively.

1) Hybrid-Beamforming-Based mmWave Downlink: Asshown in Fig. 2, we consider a mmWave cellular downlinksystem with N users. The BS and each user have MBS andMMS antennas, respectively. The mmWave downlink adoptsa hybrid beamforming architecture with MBRF and M

MRF RF

chains at the BS and each user’s MS, respectively (see Fig. 3).The system operates in a time-slotted mode. The time-slots areindexed by t ∈ {0, 1, 2, . . .}. As shown in Fig. 4, each time-slot is of period T and contains two phases. The first phase isfurther divided into N mini-slots corresponding to the N users.Each mini-slot contains two parts τAn and τ

Dn . In τ

An , both the

BS and a user n perform analog beam search to refresh theirbeam directions to mitigate link breakage caused by user n’smovements [3], [4]. In τDn , the BS estimates the CSI of user nfor digital beamforming. In the data transmission phase, basedon the analog beam and digital CSI training results, the BSpicks one of the N users and steers analog beams to this user.Likewise, the scheduled user also steers analog beams toward

...

...

......

...

qN [t]

q2[t]

q1[t]a1[t]

a2[t]

aN [t]

CongestionControl

Ant. 1

Ant. 2

Ant. 3

User 1

User 2

User N

Ant. MBS

sn[t]

MMS antennas per user

Scheduling

Decisions

CSI (Based on Feedback or Channel Reciprocity)

mmWave Cellular Base Station

Fig. 2. A mmWave cellular downlink with a MBS-antenna base station andN MMS-antenna users.

...

...

...

...

...

......

...... .

..

......

......

RF Chain

RF ChainBaseband

(Digital

Beam-

forming)

MBRF Chains

MBSAntennas RF Chain

RF Chain

forming)

Baseband

(Digital

Beam-Antennas

MMS

Base Station User n’s Mobile Station

F(n)D [t]

RF analog RF analog

beamformer F(n)A [t] beamformer W

(n)A [t]

Beamformer BeamformerChannel Hn[t]

Actual

MMRF Chains

Effective

W(n)D [t]

Channel H(n)E [t]Streams

un[t]

xn[t]

Streams

yn[t]

K Data K Data

Fig. 3. Block diagram of a mmWave cellular network with hybrid beamform-ing.

...Beam searching & CSI learning

τDτA1 τAN Data Transmission

TUser 1 User N

τD

Fig. 4. Frame structure of a time-slot in mmWave cellular networks withhybrid beamforming.

the BS. Further, by leveraging the learned CSI to performspatial multiplexing, the BS and a scheduled user communicatevia K data streams. For mmWave systems in practice, weusually have: i) K ≤ MBRF ≤ MBS; ii) K ≤ MMRF ≤ MMS;iii) MMRF ≤MBRF; and iv) MMS ≤MBS.

a) Analog beamforming process: In time-slot t, the analogbeamformers on the BS and user sides are determined bya beam training process, during which the BS and user nsearch over all possible direction combinations within theircorresponding sectors1, as shown in Fig. 5 (this exhaustivebeam training process has been adopted in IEEE 802.11ad andIEEE 802.15.3c standards). Let Tp denote the time requiredfor transmitting and receiving a pilot symbol. Let ψBn andψMn denote the sector-level beamwidth at the BS and user n,respectively. Also, let θB [t] and θn[t] denote the beam-levelbeamwidth at the BS and user n’s MS, respectively. Then, thebeam search time τAn can be computed as: τ

An =

ψBnθB [t]

ψMnθn[t]

Tp.In this paper, we adopt a widely used sectored antenna

pattern model (see, e.g., [21]–[23]): We assume that the gainsare a constant for all angles within the main lobe and equal toa smaller constant in the side lobes. As shown in Fig. 5, we letωBn and ω

Mn represent the angles deviating from the strongest

path between the BS and user n, respectively (the strongestpath needs not be line-of-sight and Fig. 5 is only for illustrativepurposes). Let gBn (ω

Bn , θB [t]) and g

Mn (ω

Mn , θn[t]) denote the

transmission and reception gains at the BS and user n. In thispaper, we adopt the following widely used antenna radiation

1In this paper, we assume that both the BS and user know the sectors of eachother’s location in each time-slot. This is a reasonable assumption because thesector information can be inferred with high accuracy from the beam directionin the previous time-slot and the mobility/trajectory information of the user.

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steering

Target

sector ψBn

Target

sector ψMn

BS Beamwidth θB[t] Beamwidth θn[t]

BS beam

steering

MS beamAngle fromAngle from

strongest path ωBn strongest path ωMn

User n

Fig. 5. The analog beamforming trainingprocedure.

Side lobegain η

Beamwidth θ[t]

Main lobe gain2π−(2π−θ[t])η

θ[t]

Fig. 6. The simplified antennaradiation pattern.

pattern model [21]–[23] (see Fig. 6):

gBn (ωBn , θB [t]) =

{2π−(2π−θB [t])η

θB [t], if |ωBn | ≤

θB [t]2 ,

η, otherwise,(1)

gMn (ωMn , θn[t]) =

{2π−(2π−θM [t])η

θn[t], if |ωMn | ≤

θn[t]2 ,

η, otherwise,(2)

where η ∈ [0, 1) is the side lobe gain. In practice, η � 1 fornarrow beams (i.e., θB [t] and θn[t] are small). This model cap-tures the essential features of antenna patterns (e.g., directivegains, front-to-back ratio, half-power beamwidth, etc. [23]).Once the optimal directions for transmission and receptionhave been determined, the communication link can be estab-lished, and data transmission phase starts. The beam trainingis finished when the BS and the user’s beams are aligned withthe strongest path, i.e., the conditions |ωBn | ≤

θB [t]2 in (1) and

|ωMn | ≤θn[t]

2 in (2) are satisfied.b) Digital beamforming process: Once the analog beam

search is completed, the analog beamformers are known.Therefore, we can estimate the CSI of the effective channelH

(n)E [t], which is assumed to take τ

D = βTp amount of time(cf. Fig. 4), where β > 0 is some constant. With the learnedCSI, the BS and user n jointly choose baseband beamform-ers based on some digital beamforming strategies, such assingular value decomposition (SVD), zero-forcing (ZF), etc.One particularly interesting case arises when MBRF �MMRF.In this case, the row vectors in the effective channel H(n)E [t]are asymptotically orthogonal to each other as MBRF gets large.Thanks to this nice property, one can use the so-called conju-gate beamforming, which has been shown to be asymptoticallycapacity-achieving in the high SNR regime [24]. We willfurther discuss conjugate beamforming in Section V.

Regardless the choice of digital beamforming schemes,the digital beamforming process converts H(n)E [t] into K ≤min{MBRF,MMRF} spatial channels (depending on the rank ofH

(n)E [t]). We let g

(k)n [t] denote the effective gain of the k-th

spatial channel. Based on the models of hybrid analog/digitalbeamforming, we have that the hybrid beamforming achiev-able rate of user n can be computed as:2

rn(θB [t], θn[t])=

(1− τ

A+NτD

T

) K∑k=1

log2

(1+

PmaxKN0

gBn (ωBn , θB [t])g

Mn (ω

Mn , θM [t])g

(k)n [t]

), (3)

2In this paper, equal power allocation is used for lower rate evaluationcomplexity in the effective MIMO channel. This is because it has been shownthat the rate loss of equal power allocation is negligible under moderate andhigh SNR regimes. Also, equal power allocation is asymptotically capacity-achieving in high SNR regime [25].

where τA ,∑Nn=1 τ

An and Pmax denotes the maximum

transmission power at the BS. Then, for a given channel statein time-slot t, we let Cn[t] denote the instantaneous achievablerate region under a chosen digital beamforming scheme:

Cn[t] ,{rn(θB [t], θn[t])

∣∣∣∣ θB [t]∈(0, ψBn ],θM [t] ∈ (0, ψMn ]}. (4)

It can be seen from (3) that the beamwidths θB [t] and θn[t]need to be chosen judiciously: On one hand, from (1) and (2),gBn (ω

Bn , θB [t]) and g

Mn (ω

Mn , θn[t]) increase as θB [t] and θn[t]

decrease, leading to a higher SNR and hence a higher datarate. However, the smaller the beamwidths θB [t] and θn[t],the shorter the transmission phase, i.e., there exists a trade-offbetween data rate and transmission time.

2) Queueing Model: As shown in Fig. 2, the BS maintainsa separate queue for each user. Let an[t] denote the numberof packets injected into queue n in time-slot t. The arrivalprocesses {an[t]}, ∀n, are controlled by a congestion con-troller. We assume that there exists a finite constant Amax

such that an[t] ≤ Amax, ∀n, t. Let s[t] , [s1[t], . . . , sN [t]]>denote the scheduled service rate vector in time-slot t (thescheduling algorithm that determines s[t] will be presentedin Section IV). Then, the queue-length of user n evolves as:qn[t + 1] =

(qn[t] − sn[t] + an[t]

)+, ∀n, where (·)+ ,

max(0, ·). Let q[t] = [q1[t]], . . . , qN [t]]>. In this paper, weadopt the following notion of queue-stability (same as in [14],[15]): We say that a network is stable if the steady-state totalqueue-length is finite, i.e., lim supt→∞ E {‖q[t]‖1} 0 controls the utility-optimality gap. Hence, the utility-optimality gap can be madearbitrarily small by decreasing �. We note that the queue-length-based congestion control mechanism is different andnot to be confused with conventional TCP congestion con-trol mechanism. Based on this insight, let us consider thefollowing QCS algorithm specialized for hybrid beamformingin mmWave networks:

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A. The QCS Algorithm Specialized for Hybrid Beamforming

Algorithm 1: Queue-Length-Based Joint Congestion Controland Scheduling for mmWave-Based Cellular Networks.

Initialization: Choose parameters � > 0. Set t = 0.Main Loop:1. MaxWeight Scheduler: In time t≥ 1, given queue-lengths

q[t] and CSI H[t], the scheduler chooses a service ratevector s[t] from Cn[t] by hybrid beamforming such that:

s[t] = arg maxrn∈Cn[t],∀n

{ N∑n=1

qn[t]rn

}, (5)

where Cn[t] is defined in (4).2. Congestion Controller: Given queue-lengths q[t], the con-

gestion controller chooses data injection rates an[t], ∀n,which are integer-valued random variables satisfying:

E{an[t]|qn[t]} = min{U′−1n (�qn[t]) , A

max}, (6)

E{a2n[t]|qn[t]} ≤ Amax2 (x2 − x1) ≥ 0 implies f(x2) ≥ f(x1)or equivalently ∇f(x2)>(x2 − x1) ≥ 0. The function f is said to bepseudoconcave if −f is pseudoconvex.

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methods); iii) It can be seen from the proof of Lemma 1that we have defined θ̃[t] , θB [t]θM[t]. Note that the optimalobjective value of Problem (10) is only a function of θ̃∗[t] anddoes not depend on the specific values of θB [t] and θM[t], aslong as their product is equal to θ̃∗[t]. This implies that wecan simply set θM[t] to some appropriate fixed value and onlyadjust θB[t] at the BS side. In other words, there is no need tojointly adjust θB[t] and θM[t]. This insight greatly simplifiesthe protocol designs in the analog beamforming phase.

Collectively, the results in this section provide an algorith-mic solution to Problem JCS assuming that the CSI learned inτD (hence the digital beamforming gains g(k)n [t]) is accurate.However, it remains unclear how the network utility and delayperformance of Algorithm 1 will be affected if the CSI isinaccurate. This problem will be addressed in the next section.

V. THE IMPACTS OF INACCURATE CSI ON THE QCSALGORITHM WITH HYBRID BEAMFORMING

Generally speaking, in traditional multi-antenna networks,CSI is measured at each MS based on pilot symbol training andthen fed back to the BS. However, due to the short coherencetime of mmWave channels (around an order of magnitudelower than that of microwave bands since Doppler shifts scalelinearly with frequencies [3]), this traditional CSI feedbackapproach is not suitable for mmWave-based cellular networks.Another CSI acquisition method is to have the system run intime-division duplexing (TDD) mode. Based on the channelreciprocity, the uplink CSI measured at the BS will be used fordownlink transmissions. However, the limited transmit powerat the MSs and the lack of beamforming gains for the uplinkpilot symbols limit the accuracy of TDD-based CSI estimation.Further, the short coherence time of mmWave bands impliesthat the channel reciprocity assumption is only valid for low-mobility scenarios. Given these CSI estimation challenges inmmWave cellular systems, it is likely that the CSI learnedduring the τD period (cf. Fig. 4) is inaccurate.

In studying the impacts of CSI inaccuracy, we are interestedin the case where the number of RF chains at the BS is muchgreater than that at the MSs (e.g., 10 times larger). This settingis particularly interesting because it is the relevant case formmWave cellular networks in practice: First, as mentioned inSection I, large antenna arrays can be easily deployed at theBS due to the short wavelengths of mmWave bands. Also,because of the physical size and power constraints, the MSsusually accommodate much fewer RF chains compared to thatat the BS side. Moreover, as noted in many studies [24], CSIacquisition is one of the most fundamental limiting factorsin the system designs of large-scale antenna cellular systems.In what follows, we start with the digital beamforming foreffective mmWave channels with a large number of RF chainsat the BS and its operations under a limited CSI model.

1) Digital Beamforming for Effective Channels: Asmentioned in Section III, if MBRF � MMRF, due to thenear orthogonality between the rows in the effective chan-nel in this case, the simple conjugate digital beamformingtechnique from the Massive MIMO literature and related

queueing network techniques [27], [28] can be adopted4.Recall that the received signal of user n can be written as:y[t] = W

(n)†D [t]H

(n)E [t]F

(n)D [t]un[t] + ñ[t], where H

(n)E [t] ∈

CMMRF×MBRF is the effective channel by taking into accountthe effects of analog beamforming; and F(n)D [t] and W

(n)†D [t]

are the transmit and receive digital beamformers, respectively.Under conjugate beamforming, we have W(n)†D [t] = I andF

(n)D [t] = H

(n)E [t]

†, i.e., the conjugate transpose of H(n)E [t].Also, we assume that the effective channel H(n)E [t] is of fullrow rank so that we can let K = MMRF (i.e., all receiver RFchains are utilized). Then, the achievable rate under digitalconjugate beamforming can be approximately computed as:

rn[t]≈(

1− τA+τD

T

) K∑k=1

log2

(1+

PmaxKN0

‖h(n)E,k[t]‖2), (11)

where h(n)E,k[t] denotes the k-th row of H(n)E [t]. In (11),

the approximation holds because the rows of H(n)E [t] arenearly orthogonal as MBRF gets large. Note that in h

(n)E,k[t]

in (11), we have absorbed the array gains gBn (ωBn , θB [t]) and

gMn (ωMn , θn[t]) (cf. Eq. (3)) achieved by analog beamforming.

2) CSI Inaccuracy Modeling: Given the inevitable CSIerrors and to alleviate the CSI estimation burden for digitalbeamforming, we adopt the limited CSI model in the literature(see, e.g., [25] and references therein). Such limited CSI can beobtained by Q bits of feedback from each user. Alternatively,based on the channel reciprocity, the BS could use Q bits torapidly quantize the uplink CSI (see Fig. 2). In either case, thevalue of Q depends on the CSI learning time τD and efficiencyof the specific CSI learning algorithm. The Q-bit limited CSIfor each RF chain k can be determined by a vector quantizationcodebook Bk , {c1k, . . . , c2

Q

k }, where cik ∈ CMBRF , i =

1, . . . , 2Q, represents a codeword. Given an effective channelH

(n)E [t], the codeword for its k-th row vector h

(n)E,k[t] is chosen

by picking the one that is closest to h(n)E,k[t] in the followingsense [25]: i∗k[t] = arg minj∈{1,...,2Q} sin

2(∠(h(n)E,k[t], cjn)),

where i∗k[t] denotes the index of the chosen codeword in time-slot t. Let Ĥ(n)E [t]∈CM

MRF×M

BRF denote the estimated channel

matrix by collecting all codewords i∗k[t], ∀k. Then, based onĤ

(n)E [t], the BS performs conjugate beamforming to construct

K spatial channels. However, due to the errors in Ĥ(n)E [t],inter-channel interference is not negligible under conjugatebeamforming, and the amount of interference depends on thecodebook size 2Q and the choice of the quantization scheme.

Let r̂Qn [t] denote the actual conjugate beamforming achiev-able rate under the true CSI H(n)E [t] while the system istreating the Q-bit limited CSI Ĥ(n)E [t] as if it is accurate. Also,let Ĥ(n)E,1[t] and Ĥ

(n)E,2[t] represent two estimated CSI values

obtained by using Q1 and Q2 bits, respectively. Then, we canshow that the following monotonicity result of the conjugate

4Due to this fact, the effective channel H(n)E [t] ∈ CMMRF×MBRF under

hybrid beamforming is usually low-rank (MMRF and MBRF are the numbers

of RF chains at MS and BS, respectively). However, as long as H(n)E [t] isnot rank one (i.e., having two or more non-zero singular values), then sinceMMRF �M

BRF under the large antenna array adopted in mmWave transmitter

at the base station, then the rows in H(n)E [t] are close to orthogonal to eachother from random matrix theory (see [29]).

7

beamforming achievable rate holds under limited CSI, whichwill be used in our subsequent analysis (the proof is relegatedto [30] due to space limitation):

Lemma 2 (Monotonicity of beamforming achievable rate). IfQ1 ≤ Q2, then there exists a CSI quantization scheme underwhich r̂Q1n [t] ≤ r̂Q2n [t]. Further, r̂Qn [t] ↑ rn[t] as Q→∞.

3) Algorithmic Changes to the QCS Framework: Due tothe use of Q-bit limited CSI in mmWave hybrid beamforming,we modify the QCS algorithmic framework in Algorithm 1accordingly as follows:

Algorithm 2: Queue-Length-Based Congestion Control andScheduling in mmWave Cellular Network with Q-Bit CSI.

Initialization: Choose parameters � > 0. Set t = 0.Main Loop:1. MaxWeight Scheduler: In time-slot t ≥ 1, given the queue-

length vector q[t] and the Q-bit estimated CSI Ĥ(n)E [t],∀n, we let r̃n[t] be the presumed conjugate beamformingachievable rate under Ĥ(n)E [t], ∀n. Then, the schedulerchooses a user n such that n = arg maxn′∈{1,...,N}{qn′ [t]r̃n′ [t]}. Thus, the actual achievable service rates aresQ,n[t] = r̂

Qn [t] and sQ,n′ [t] = 0, ∀n′ 6= n.

2. Congestion Controller: Same as in Algorithm 1.3. Queue-Length Updates: Same as in Algorithm 1.

4) Performance Analysis: For better readability, we struc-ture our performance analysis into the following key steps:

Step 1) A Deterministic Joint Congestion Control andScheduling Problem: To describe our main theoretical results,we first need the following deterministic problem, where weassume that the channel state process is not random and fixedat its mean. We let C̄Q , {rQn ,∀n : rQn = E{r̂Qn [t]}} denotethe mean achievable rate region. Also, the congestion controland scheduling variables are time-invariant and denoted as anand sQ,n, ∀n, respectively. Then, the deterministic congestioncontrol and scheduling problem can be written as:

Maximize

1�N∑n=1

Un(an)

∣∣∣∣∣∣an − sQ,n ≤ 0,∀n,sQ,n ∈ C̄Q,∀n,an ∈ [0, amax], ∀n.

. (12)Based on the convex approximation argument in Theorem 1,it is clear that Problem (12) is approximately convex. As-sociating dual variables qQ,n ≥ 0, ∀n with the constraintsan − sQ,n ≤ 0, ∀n, we obtain the Lagrangian as follows:

Θ�(qQ) , maxa,sQ∈C̄Q

{1

�

N∑n=1

Un(an)+

N∑n=1

qQ,n(sQ,n−an)

},

where the notation Θ�(·) signifies the Lagrangian’s depen-dence on � and the vector qQ , [qQ,1, . . . , qQ,N ]> ∈ RN+contains all dual variables. Then, the Lagrangian dual problemof Problem (12) can be written as:

Minimize{

Θ�(qQ)∣∣qQ ∈ RN+ } . (13)

It can be seen that since Problem (13) is unconstrained, theSlater condition [26] trivially holds if the primal problem isfeasible. Therefore, the optimal value of Problem (12) can be

obtained by solving the dual problem in (13) since the primalproblem is approximately convex. Let (a∗Q, s

∗Q) and q

∗Q,(�) be

the optimal primal and dual solutions to Problem (12) andProblem (13), respectively. Then, it can be shown that q∗Q,(�)satisfies the following properties:

Lemma 3 (Dual solution). q∗Q,(�) =1�q∗Q,(1), i.e., q

∗Q,(�)

grows linearly and the slope depends on q∗Q,(1). Further, ifQ1 ≤ Q2, then the slopes satisfy q∗Q1,(1) ≥ q

∗Q2,(1)

.

Lemma 3 can be proved by using the Karush-Kuhn-Tucker(KKT) conditions [26] (the proof is relegated to [30] due tospace limitation). Also, note that � is only a scaling factor inthe objective function in (12). Then, by contradiction, we canshow the following result (see [30] for proof details):

Lemma 4 (Primal solution). The congestion control solutiona∗Q is independent of � and equal to the service rate s

∗Q.

With the results of the deterministic problem, we are in aposition to analyze the delay and congestion control perfor-mance of the original stochastic problem.

Step 2) Delay Performance of the Original Stochastic Prob-lem: With Lemmas 3 and 4, we are now ready to present themain results in this section. Our first result says that the steady-state queue-length vector q∞ lies in a bounded neighborhoodof the dual solution q∗Q,(�) of Problem (13). Further, the sizeof the neighborhood manifests a phase-transition phenomenon(see proof details in Appendix C).

Theorem 2 (Queueing delay phase transition). Under Algo-rithm 2 with any given Q-bit CSI quantization scheme, thereexists a critical value Q] independent of the performancecontrol parameter � of Algorithm 2, such that:

• If 0 < Q < Q], then E{‖q∞ − q∗Q,(�)‖} = O(C(Q)1� ),

where C(Q) ≥ 0 is a constant depending on the quantiza-tion codebook, and C(Q) decreases as Q increases;

• If Q ≥ Q], then E{‖q∞ − q∗Q,(�)‖} = O(1/√�).

Remark 2. Theorem 2 and Lemma 3 characterize the steady-state queue-length scalings: If Q is larger than the critical valueQ], the steady state queue-length deviation grows sublinearly,which is much slower compared to the linear growth whenQ ≤ Q]. Also, the slope of mean queue-length q∗Q,(�) dependson Q: the smaller the value of Q (i.e., poorer CSI accuracy),the steeper the slope. Note that the O(1/

√�)-scaling of queue-

length deviation when Q ≥ Q] is the same as that under thefull CSI case [15]. This shows a somewhat unexpected insightthat full CSI is not necessary to produce (in order sense) theoriginal QCS queue-length scaling behavior.

Step 3) Congestion Control Performance of theOriginal Stochastic Problem: Now, let a∞Q,n ,E{min{U ′−1n (�q∞n ), amax}}, ∀n, be the steady-statecongestion control rates under a given Q-bit CSI quantizationscheme and let a∞Q , [a

∞Q,1, . . . , a

∞Q,N ]

>. The next mainresult characterizes the phase transition of the scaling ofa∞Q ’s deviation from the solution a

∗Q of Problem (12) (see

Appendix D for proof details):

8

0.1 0.2 0.3 0.4 0.5 0.6

θ̃n[t] (normalized by π)

0

5

10

15

20

25

30

35

Achievable

rate

rn[t](bits/sec/Hz)

Approximation

η = 0.1

η = 0.2

η = 0.3

η = 0.4

η = 0.5

(a) Tp/T = 0.01.

0.1 0.2 0.3 0.4 0.5 0.6

θ̃n[t] (normalized by π)

0

5

10

15

20

25

30

35

40

45

Achievable

rate

rn[t](bits/sec/Hz)

Approximation

η = 0.1

η = 0.2η = 0.3

η = 0.4

η = 0.5

(b) Tp/T = 0.001.Fig. 7. The approximation gaps of Problem (10) under different analog slidelobe gains η.

200 400 600 800 1000

5

10

15

20

25

30

35

(a) SNR = 20 dB.

200 400 600 800 1000

5

10

15

20

25

30

35

(b) SNR = 30 dB.Fig. 8. Average queue-length deviation with respect to 1/� for Q =1, 2, 4, 8, 16, 32, 48, 64 bits (MBRF = 80).

200 400 600 800 1000

5

10

15

20

25

30

35

(a) SNR = 20 dB.

200 400 600 800 1000

5

10

15

20

25

30

35

(b) SNR = 30 dB.Fig. 9. Average queue-length deviation with respect to 1/� for Q =1, 2, 4, 8, 16, 32, 48, 64 bits (MBRF = 90).

Theorem 3 (Congestion control phase transition). UnderAlgorithm 2 with any Q-bit CSI quantization scheme, thereexists a critical value Q], same as in Theorem 2, such that:• If 0 < Q < Q], then ‖a∞Q −a∗Q‖ = O(C(Q)), where C(Q) ≥

0 is the same constant as defined in Theorem 2;• If Q ≥ Q], then ‖a∞Q − a∗Q‖ = O(

√�).

Remark 3. Theorem 3 also indicates a phase transition fora∞Q : When Q < Q

], the performance control parameter �of Algorithm 2 has no effect on the deviation ‖a∞Q −a∗Q‖.However, when Q ≥ Q], a∞Q ’s deviation from a∗Q grows asO(√�), which is the same as the full CSI case [14], [15].

Another way to interpret this phase transition phenomenonis that Q] represents the minimum codebook size for a CSIquantization scheme, such that it can resurrect the performancetuning capability of parameter � in the QCS algorithm. BothTheorems 2 and 3 can be proved by Lyapunov stabilityanalysis, and the details are relegated to the appendix.

VI. NUMERICAL RESULTSIn this section, we conduct simulations to demonstrate the

theoretical results in Sections IV and V. We first verify the

30 40 50 60 70 80 90 100

2

3

4

5

6

7

(a) SNR = 20 dB.

40 50 60 70 80 90 100

2

3

4

5

6

7

8

(b) SNR = 30 dB.Fig. 10. The congestion control rates with respect to 1/� for Q =1, 2, 4, 8, 16, 32, 48, 64 bits. (MBRF = 80).

50 60 70 80 90 100

2

4

6

8

10

(a) SNR = 20 dB.

50 60 70 80 90 100

2

4

6

8

10

(b) SNR = 30 dB.Fig. 11. The congestion control rates with respect to 1/� for Q =1, 2, 4, 8, 16, 32, 48, 64 bits. (MBRF = 90).

200 400 600 800 1000

5

10

15

20

25

30

35

40

45

50

(a) SNR = 40 dB.

40 50 60 70 80 90 100

0

2

4

6

8

10

(b) SNR = 20 dB.Fig. 12. The queue-length deviation and congestion control rates with respectto 1/� for Q = 1, 2, 4, 8, 16, 32, 48, 64 bits. (MBRF = 100, 10 users).

approximation accuracy and the pseudoconvexity of Prob-lem (10). We set SNR to 30 dB and set the Tp/T ratios to0.01 and 0.001. We vary the side lobe gain η from 0.1 to 0.5and the results are shown in Figs. 7(a) and 7(b). We can seethat, under both Tp/T ratios, the approximation gaps shrinksas η decreases. In these examples, the gaps under η = 0.1 arealmost negligible. Moreover, we note that the approximationfunction is indeed pseudoconcave, as predicted by Theorem 1.

Next, we examine the impacts of CSI quality, characterizedby the number of quantization bits Q, on the queue-lengths andthe results are shown in Figs. 8 to 11. The number of users isset to five. In Figs. 8(a) and 8(b), we suppose that the BS andeach MS have 80 and 2 RF chains, respectively, and we setthe total SNR to be 20 dB and 30 dB, respectively. We uselog(·) as the utility function for each user (i.e., the proportionalfairness metric [16]) and adopt random vector quantization(RVQ) as our Q-bit CSI quantization codebook [25]. We setthe value of Q to be 1, 2, 4, 8, 16, 32, 48, and 64. We alsodraw an accompanying dash line to show the scaling trend ofeach curve in Fig. 8(a). For small Q values, we can see that themean queue-length deviation increases faster than the square

9

root law, roughly displaying a linear scaling with respect to� as indicated in Theorem 2. For this example, the criticalvalue of Q turns out to be 8. Once Q ≥ 8, the queue-lengthdeviations scale as O(1/

√�), also confirming Theorem 2. In

Figs. 9(a) and 9(b), we increase the number of RF chains atthe BS to 90 and run the experiments under total SNR 20 dBand 30 dB again, respectively. Compared to Fig. 8, we cansee that the same trends still hold under larger number of RFchains and the queue-length deviation is slightly smaller. Thisshows that the system delay performance is mostly limitedby the number of RF chains at the MSs, hence the minorimprovements. Also, when SNR is increased from 20 dB to30 dB, we can see that the queue-deviation (i.e., delay) slightlydecreases.

In Figs. 10(a) and 10(b), we study the impacts of Q-bit CSIon the congestion control rates. In Figs. 10(a) and 10(b), wesuppose that the BS and each MS have 80 and 2 RF chains,respectively, and we set the total SNR to be 20 dB and 30dB, respectively. For small Q values, we can see that a∞Q isonly affected by Q and is a constant independent of �. Also,a∞Q ’s gap to the full CSI case shrinks as Q increases, whichconfirms Lemma 4 and Theorem 3. Again, we can observethat the critical value of Q is 8: When Q ≥ 8, a∞Q displays anO(√�) diminishing gap to a∗Q, which agrees with Theorem 3.

In Figs. 11(a) and 11(b), we increase the number of RF chainsat the BS to 90 and run the experiments under total SNR20 dB and 30 dB again, respectively. Again, compared toFig. 10, we can see that the same trends still hold underlarger number of RF chains and the queue-length deviation isslightly smaller. This shows that the system delay performanceis mostly limited by the number of RF chains at the MSs,hence the minor improvements. Also, when SNR is increasedfrom 20 dB to 30 dB, we can see that the congestion controlrate increases, which is due to the increase of channel capacity.

Lastly, we change the number of users to ten and repeatthe same set of experiments as in Figs. 8 to 11, and theresults are illustrated in Figs. 12(a) and 12(b). We can seefrom Figs. 12(a) and 12(b) that the same trends of queue-length deviation and congestion control rates continue to holdunder the setting with 10 users. In Fig. 12(a), the queue-length deviation is larger compared to the settings with fiveusers when Q is small, which shows that delay performance ismore sensitive to CSI quality as the number of users increasesunder mmWave hybrid beamforming. This is mainly due to thefact that under hybrid-beamforming, as the number of usersincreases, the duration of digital beamforming decreases asthe number of users increases, hence smaller service ratescompared to settings with fewer users. In Fig. 12(b), thecongestion control rates (i.e., system throughput) is smallercompared to settings with five users when Q is small, whichshows that the throughput performance is also more sensitiveto CSI quality as the number of users increases under mmWavehybrid beamforming.

VII. CONCLUSION

In this paper, we studied the impacts of hybrid beamformingon the delay and network utility performance in mmWave

cellular network optimization. We proposed a queue-length-based hybrid beamforming scheduling and congestion controlframework for mmWave network utility maximization. We firstshowed that the hybrid beamforming scheduling subproblemin this framework enjoys a hidden pseudoconvexity structure,which leads to simplified analog beam training design. We thencharacterized two phase transition phenomena in throughputand delay with respect to CSI accuracy in digital beam-forming. Collectively, these results deepen our understandingof mmWave networking performances. Hybrid beamformingin mmWave networking is an exciting and under-exploredresearch area. Our future directions include, e.g., multi-cellmmWave networks with hybrid beamforming, the impacts ofCSI inaccuracy on limited RF chains at the BS side, etc.

APPENDIX APROOF OF LEMMA 1

For simplicity, we let rn[t] denote the objective functionof Problem (9). Substituting (1) and (2) into rn[t] and usingthe defined constants, we can rewrite the objective function ofProblem (9) as:

rn[t] =

(b0 −

b1θB [t]θM[t]

) K∑k=1

log2

(1+

2π − (2π − θB [t])ηθB [t]

· 2π − (2π − θM[t])ηθM[t]︸ ︷︷ ︸

(∆)

c(k)n

). (14)

Note that the term (∆) can be further written as:

(∆) =

(2π(1− η)θB [t]

+ η

)(2π(1− η)θM[t]

+ η

)(a)=

(4π2(1− η)2

θ̃[t]+

2π(1− η)(θB [t] + θM[t])θ̃[t]

+ η2),

where in (a) we define θ̃[t] , θB [t]θM[t]. Now, we claim that

4π2(1− η)2

θ̃[t]� 2π(1− η)(θB [t] + θM[t])

θ̃[t]η (15)

is true if η� 13 . To see this, we first note that η�13 implies

4π� 2π(1−η)η . Also, since θB [t], θM[t]∈(0, 2π], we have

θB [t] + θM[t] ≤ 4π �2π(1− η)

η,

which implies that (15) is true. Hence, it follows from (15)and η � 13 < 1 that (∆) ≈ (

4π2

θ̃[t]+ η2), which further implies

rn[t] = (14) ≈(b0 −

b1

θ̃[t]

) K∑k=1

log2

(1 +

4π2c(k)n

θ̃[t]

),

i.e., the objective function in (10). This completes the proof.

APPENDIX BPROOF OF THEOREM 1

As mentioned earlier, verifying the pseudoconvexity ofProblem (10) means verifying the pseudoconcavity of theobjective function. Toward this end, we let f(θ̃[t]) denote the

10

negative objective function and our goal is to show that f(θ̃[t])is pseudoconvex, which means that for any θ̃1[t] and θ̃2[t] inthe feasible interval, iff ′(θ̃1[t])(θ̃2[t] − θ̃1[t]) ≥ 0, we mustalso have f ′(θ̃2[t])(θ̃2[t]− θ̃1[t]) ≥ 0.

First, let us consider the case where θ̃2[t] ≥ θ̃1[t]. Then,showing f ′(θ̃2[t])(θ̃2[t] − θ̃1[t]) ≥ 0 is equivalent to show-ing f ′(θ̃2[t]) ≥ 0. Note that, in this case, the conditionf ′(θ̃1[t])(θ̃2[t]− θ̃1[t]) ≥ 0 simply means f ′(θ̃1[t]) ≥ 0, i.e.,

f ′(θ̃1[t]) =

K∑k=1

1

θ̃21

[4π2c

(k)n

ln(2)· b0θ̃1[t]− b1θ̃1[t] + 4π2c

(k)n︸ ︷︷ ︸

(P1)

−

b1 log2

(1 +

4π2c(k)n

θ̃1[t]

)︸ ︷︷ ︸

(P2)

]≥ 0. (16)

It is obvious that the term (P2) is an increasing function ofθ̃[t]. Now, consider the fractional term b0θ̃1[t]−b1

θ̃1[t]+4π2c(k)n

in (P1),which is negative-valued according to the definitions of b0, b1,and the feasible interval. Also, from the definition of b0, wehave b0 < 1, implying that the absolute value of the nominatoris increasing at a slower rate than that of the denominator. Thismeans that (P1) is also an increasing function of θ̃[t]. Hence,f ′(θ̃[t]) is increasing since both (P1) and (P2) are increasing.As a result, f ′(θ̃1[t]) ≥ 0 and θ̃2[t] ≥ θ̃1[t] imply f ′(θ̃2[t]) ≥ 0and thus the case of θ̃2[t] ≥ θ̃1[t] is proved. The other casewhere θ̃2[t] ≤ θ̃1[t] can also be proved by similar argumentsand we omit the details in here for brevity.

To show that the optimal solution is unique and achievedin the interior of the feasible interval, it suffices to showthat ∂rn[t]

∂θ̃[t]’s values at two end points of the interval have

opposite signs. Then, from the decreasing derivative propertyof rn[t] (rn[t] = −f(θ̃[t])), ∂rn[t]∂θ̃[t] must have exactly one zero-crossing point in the interior of the feasible interval. Also, thepseudoconcavity of rn[t] means that the zero-crossing point isthe global maximum. First, if θ̃[t] = b1b0 , we have (P1) = 0.Hence, ∂rn[t]

∂θ̃[t]> 0 since −(P2) > 0 (because b1, θ̃2[t], and

the log(·) rate expressions are positive). On the other hand,when θ̃[t] ↑ ψBnψMn , it follows from Tp � T that

rn[t] =

[1−

(Nβ +

∑Nn′=1 ψ

Bn′ψ

Mn′

ψBnψMn

)TpT

]×

K∑k=1

log2

(1 +

4π2c(k)n

θ̃[t]

)

≈K∑k=1

log2

(1 +

4π2c(k)n

θ̃[t]

),

which is decreasing in θ̃[t] and must have a negative derivativeat θ̃[t] = ψBnψ

Mn . This completes the proof.

APPENDIX CPROOF OF THEOREM 2

To prove Theorem 2, we first show the existence of steady-state by proving a positive Harris-recurrence result of the

queue-length process. This result implies the existence ofsteady-state, which lays the foundation for proving Theo-rems 2. We let 1A(x) denote the indicator function, whichtakes value 1 if x ∈ A and 0 otherwise. We state the queue-length positive Harris-recurrence result as follows:

Proposition 1 (Queue-Length Positive Recurrence). Considera Lyapunov function V (q[t]) , �2‖q[t]− q

∗Q,(�)‖

2 for a given�. For the scheduler (5) and congestion controller (6)–(7),there exist constants δ, η > 0, both independent of �, suchthat the queue-length process {q[t]}∞t=0 satisfies the followingconditional mean drift condition:

E{∆V (q[t]|q[t])} , E{V (q[t+ 1])− V (q[t])|q[t]}

≤ −�δΦ

∥∥q[t]− q∗Q,(�)∥∥1Qc(q[t]) + η1Q(q[t]), (17)where Q , {q ∈ ZN+ |‖q− q∗Q,(�)‖ ≤ γ/�} for some constantγ > 0 and Bc denotes the complement of Q in ZN+ .

Proof. Consider the quadratic Lyapunov function defined inProposition 1: V (q[t]) = �2‖q[t]−q

∗Q,(�)‖

2, where q[t] repre-sents the queue-length vector in time-slot t under parameters� and Q; and q∗Q,(�) denotes the optimal dual solution for thestatic version of Problem JCS under parameter �. Then, theone-slot mean Lyapunov drift of V (q[t]) can computed as:

E{V (q[t+ 1])− V (q[t])|q[t]}

= E{ �

2‖q[t+ 1]− q∗(�)‖

2 − �2‖q[t]− q∗Q,(�)‖

2∣∣∣q[t]}

=�

2E{

(q[t+ 1]− q[t])>(q[t+ 1] + q[t]− 2q∗Q,(�))∣∣∣q[t]}

(a)

≤ �2E{

(−sQ[t]+a[t])>(2q[t]−2q∗Q,(�)−sQ[t]+a[t])∣∣∣q[t]}

=�

2E{−sQ[t]+a[t]‖2+2(q[t]−q∗Q,(�))

>(−sQ[t]+a[t])∣∣∣q[t]}

= �(q[t]−q∗Q,(�))>(−sQ[t]+a[t])+

�

2E{‖ − sQ[t]+a[t]‖2

},

where (a) follows from the non-expansive property of themax{0, ·} operation. Note that, from the definition of Al-gorithm 1, we have E{‖a[t]‖2|q[t]} < Amax2 N . Also, sincesQ,n[t] falls in a bounded instantaneous capacity region CĤ[t],∀n, we must have sQ,n[t] ≤ smax for some smax > 0. Hence,by defining D0 , N2 (A

max2 + (s

max)2), we have

E {∆V (q[t])|q[t]} ≤ �(q[t]− q∗Q,(�))>E {a[t]− sQ[t]}+�D0

(a)= �(q[t]− q∗Q,(�))

>(E{a[t]|q[t]} − s∗Q)+�E{(q[t]− q∗Q,(�))

>(s∗Q − sQ[t])|q[t]}+ �D0,(b)

≤ �(q[t]− q∗Q,(�))>(E{a[t]|q[t]} − s∗Q)+

�‖q[t]− q∗Q,(�)‖E{‖s∗Q − sQ[t]‖|q[t]

}+ �D0, (18)

where s∗Q is such that (s∗Q,q

∗Q,(�)) is a pair of optimal primal

and dual solutions to Problem (13) under parameter �. In(18), (a) follows from adding and subtracting s∗Q as wellas the fact that a[t] is independent of the channel state anddetermined solely by q[t]; and (b) follows from Cauchy-Schwarz inequality.

11

Note from Lemma 4 that s∗Q is independent of � andsQ,n[t] ∈ CĤ[t] is upper-bounded. Thus, we have

E{‖s∗Q−sQ[t]‖|q[t]

}≤C(Q), max

q:‖q‖=1E{‖s∗Q−sQ‖q}, (19)

where C(Q) signifies that its value depends on Q. Hence, wecan further upper bound (18) as:

E {∆V (q[t])|q[t]} ≤ �(q[t]− q∗Q,(�))>×

(E{a[t]|q[t]} − s∗Q) + �‖q[t]− q∗Q,(�)‖C(Q) + �D0, (20)

Now, let us consider the first term on the right hand side in(20), i.e., �(q[t]−q∗Q,(�))

>(E{a[t]|q[t]}− s∗). Since Un(·) isconcave and increasing, ∀n, we have(qn[t]− q∗Q,(�),n

)> [U′−1n (�qn[t])− U

′−1n

(�q∗Q,(�),n

)]≤ 0.

Thus, by the Cauchy-Schwatz inequality, we have:

(q[t]− q∗Q,(�))>(E{a[t]|q[t]} − s∗Q)

=

N∑n=1

(qn[t]−q∗Q,(�),n

)> [U′−1n (�qn[t])−U

′−1n

(�q∗Q,(�),n

)]≤−

N∑n=1

∣∣qn[t]−q∗Q,(�),n∣∣∣∣∣U ′−1n (�qn[t])−U ′−1n (�q∗Q,(�),n)∣∣∣.(21)By the strong convexity of −Un(·) and the Lipschitz continuityof U ′n(·), we have

|U ′n (an,1)− U ′n (an,2)| ≤ Φ |an,1 − an,2| .

Therefore, by the inverse function lemma, we have

1

Φ

∣∣∣�qn[t]− �q∗Q,(�),n∣∣∣ ≤ ∣∣∣U ′−1n (�qn[t])− U ′−1n (�q∗Q,(�),n)∣∣∣ .Hence, we can further upper-bound (21) as:

(q[t]− q∗Q,(�))>(E{a[t]|q[t]} − s∗Q)

≤ − �Φ

N∑n=1

(qn[t]−q∗Q,(�),n

)2=− 1

Φ�

∥∥∥q[t]−q∗Q,(�)∥∥∥2 . (22)Substituting (22) into (20), we have

E {∆V (q[t])|q[t]} ≤ − �2

Φ

∥∥∥q[t]− q∗Q,(�)∥∥∥2+�‖q[t]−q∗Q,(�)‖C(Q)+�D0. (23)

Now, suppose that∥∥q[t] − q∗Q,(�)∥∥ ≥ γ1/�, where β1 will

be specified shortly. Note also that we can choose � ≥ 1, wehave

1

‖q[t]− q∗Q,(�)‖≤ �γ1≤ 1γ1.

It then follows that (23) can be further upper bounded as:

E{∆V (q[t])|q[t]} = − �Φ

∥∥q[t]− q∗Q,(�)∥∥ · �∥∥q[t]− q∗Q,(�)∥∥+ �‖q[t]− q∗Q,(�)‖D1 +

∥∥q[t]− q∗Q,(�)∥∥ �D0∥∥q[t]−q∗Q,(�)∥∥≤ − �

Φ

∥∥q[t]− q∗Q,(�)∥∥(γ1 − C(Q)Φ− D0Φγ1). (24)

By choosing γ1 such that γ1 −D1Φ− D0Φγ1 > 0, we have

E{∆V (q[t])|q[t]} ≤ −�δ̂1Φ

∥∥q[t]− q∗Q,(�)∥∥ (25)where δ̂1 = γ1−C(Q)Φ−D0Φγ1 . Solving β1−C(Q)Φ−

D0Φβ1

= 0

and plugging in the obtained γ1 to define a ball B1 , {q :∥∥q−q∗Q,(�)∥∥ ≤ 12� [(C(Q)Φ)+√(C(Q)Φ)2 + 4D0Φ]}, we haveE{∆V (q[t])|q[t]}≤−�δ1

∥∥q[t]−q∗Q,(�)∥∥, if q[t] ∈ Bc1, (26)where δ1 , δ̂1Φ . On the other hand, when q[t] ∈ B1, it isclearly true that E{∆V (q[t])|q[t]} ≤ η1 for some η1 > 0.Combining these facts yields the following:

E{∆V (q[t])|q[t]=q}≤−�δ1‖q−q∗Q,(�)‖1Bc1(q)+η11B1(q).

This completes the proof of Proposition 1.

The inequality in (17) suggests that the conditional meandrift is negative when the deviation of the queue-length vectorq[t] away from q∗Q,(�) is sufficiently large. Since (17) is just theFoster-Lyapunov criterion [31, Proposition I.5.3], {q[t]}∞t=0 ispositive recurrent, we have that a steady-state distribution ofqueue-lengths exists. Thus, we let q∞ denote the queue-lengthvector in steady-state. With Proposition 1, we are now in aposition to prove Theorem 2.

Next, to prove Theorem 2, we use an α-parameterizedquadratic Lyapunov function: Vα(q[t]) = �

α

2 ‖q[t]− q∗Q,(�)‖

2,where the parameter α ∈ {0, 1} and its value will be specifiedlater. Following similar steps in the proof of Proposition 1, wecan bound the conditional mean Lyapunov drift as follows:

E{Vα(q[t+ 1])− Vα(q[t])|q[t]}(a)

≤ �α(q[t]− q∗Q,(�))>(E{a[t]|q[t]} − s∗Q)+

�αE{(q[t]− q∗Q,(�))>(s∗Q − sQ[t])|q[t]}+ �αD0,

(b)

≤ �α[− �

Φ

∥∥q[t]− q∗Q,(�)∥∥2 +D0]+�αE

{(q[t]− q∗Q,(�))

>(s∗Q − sQ[t])∣∣q[t]}

(c)

≤ �α[− �

Φ

∥∥q[t]− q∗Q,(�)∥∥2 +D0]+�αE

{(q[t])>(s∗ − sQ[t])

∣∣q[t]}, (27)where D0 , N2 (A

max2 + (s

max)2) and s∗ , limQ→∞ s∗Q. In(27), (a) follows from adding and subtracting s∗Q; (b) followsfrom (22); and (c) follows from s∗Q ≤ s∗ (by Lemma 2)and the scheduler design, which implies (q∗Q,(K))

>sQ[t] ≤(q∗Q,(K))

>s∗Q. Next, consider the T -step conditional meanLyapunov drift. For any q[0] ≥ 0, we have that

E{Vα(q[T ])|q[0]} − Vα(q[0])

=

T−1∑t=0

E{V (q[t+ 1])− V (q[t])|q[0]}

(a)=

T−1∑t=0

∑q∈ZN+

[Pr(q[t] = q|q[0])

E{Vα(q[t+ 1])− Vα(q[t])|q[t] = q

}]

12

(b)

≤T−1∑t=0

∑q∈ZN+

Pr(q[t] = q|q[0]){�α[− �

Φ

∥∥q[t]− q∗Q,(�)∥∥2+D0

]}+

T−1∑t=0

∑q∈ZN+

Pr(q[t] = q|q[0])×

{�αE

{q>(s∗ − sQ[t])

}}, (28)

where (a) follows from the fact that q[t] is a discrete stateMarkov chain in ZN+ and (b) follows from (27). Note that forany q[t] ∈ ZN+ , limT→∞ 1T

∑T−1t=0 Pr(q[t] = q|q[0]) = π∞q ,

where π∞q denotes the stationary distribution of the Markovchain q[t]. Moving V (q[0]) to the right hand side, dividingboth sides by T , and letting T →∞ yields:

0≤J+∑q∈ZN+

π∞q (q)>(s∗−s∞B )=J+E

{(q∞)>(s∗−s∞B

}, (29)

where J , limT→∞ 1T∑T−1t=0

∑q∈ZN+

Pr(q[t] =

q|q[0]){�α[ −�Φ ‖q[t] − q∗(�)‖

2 + D0]}, and s∞Q ,arg maxx∈C

H[∞]|Ĥ[∞](q∞)>x represents the steady-state

service rates with Q-bit CSI.Next, consider the term E

{(q∞)>(s∗ − s∞Q )

}in (29). For

any given realization of q∞ in the steady-state, from the designof the MaxWeight scheduler in (5), we have that

(q∞)>s∗ ≤ maxx∈CH[∞]

(q∞)>x = (q∞)>s∞. (30)

where s∞ , limQ→∞ s∞Q and H[∞] represent the full CSI inthe steady state. Hence, for any realization of q∞ such thatq∞ 6= ρs∗ for some ρ ∈ R, if Q is sufficiently large, wemust have (q∞)>s∗ − (q∞)>s∞Q ≤ 0. Hence, there existsa critical value Q] such that for all Q > Q], the averagevalue of (q∞)>s∗−(q∞)>s∞Q can be made non-positive, i.e.,E{

(q∞)>(s∗ − s∞Q )}≤ 0. Hence, we consider two cases

based on the positivity of E{

(q∞)>(s∗ − s∞Q )}

as follows:

Case I): Q ≥ Q] such that E{

(q∞)>(s∗ − s∞Q )}≤ 0: In

this case, it follows from (29) that

0 ≤ limT→∞

1

T

∑T−1t=0

∑q∈ZN+

Pr(q[t] = q|q[0]){�α[− �

Φ

∥∥q[t]− q∗Q,(�)∥∥2 +D0]}. (31)We now consider the term in the second line in (31) by settingα = 0. Similar to the proof of Proposition 1, suppose that∥∥q[t] − q∗Q,(�)∥∥ ≥ γ/√�, where γ will be specified shortly.This implies that 1‖q[t]−q∗

Q,(�)‖ ≤

1γ . Then, the second line in

(31) can be upper bounded as:− �

Φ

∥∥q[t]− q∗Q,(�)∥∥2 +D0= −√�

Φ

∥∥q[t]− q∗Q,(�)∥∥(√�‖q[t]− q∗Q,(�)‖+D0Φ√

�‖q[t]− q∗Q,(�)‖

)≤ −√�

Φ

∥∥q[t]− q∗Q,(�)∥∥(γ − D0Φγ). (32)

Hence, by choosing γ >√D0Φ, we have

− �Φ

∥∥q[t]− q∗Q,(�)∥∥2 +D0 ≤ −√�δ̂Φ ∥∥q[t]− q∗Q,(�)∥∥, (33)

where δ̂ = γ − D0Φγ > 0. Plugging in γ >√D0Φ to define a

ball B , {q : ‖q− q∗Q,(�)‖ ≤√D0Φ/

√�}, we have

− �Φ

∥∥q[t]−q∗Q,(�)∥∥2+D0 ≤−√�δ‖q[t]−q∗Q,(�)‖, if q[t]∈Bc.On the other hand, when ‖q[t] − q∗Q,(�)‖ ≤

√D0Φ/

√�, it is

clear that −(�/Φ)∥∥q[t]− q∗Q,(�)∥∥2 +D0 ≤ η for some η > 0.

Combining these facts, we have

− �Φ

∥∥q[t]− q∗Q,(�)∥∥2 +D0≤ −�δ‖q[t]− q∗Q,(�)‖1Bc(q[t]) + η1B(q[t]). (34)

Substituting (34) into (31) yields:

0 ≤ limT→∞

1

T

T−1∑t=0

∑q∈ZN+

Pr(q[t] = q|q[0])× (35)

(− �δ‖q[t]− q∗Q,(�)‖1Bc(q) + η1B(q)

)=η∑q∈B

π∞q −√�δ∑q∈Bc

‖q− q∗Q,(�)‖π∞q , (36)

where we use the fact that, ∀q ∈ ZN+ ,limT→∞

1T

∑T−1t=0 Pr{q[t] = q|q[0]} = π∞q . Re-arranging

the terms and with some manipulations, the above inequalitycan be written as:√�δ∑

q∈ZN+

‖q−q∗Q,(�)‖π∞q ≤

∑q∈B

(η+√�δ‖q−q∗Q,(�)‖

)π∞q

≤ (η + δγ)∑q∈B

π∞q ≤ (η + δγ), (37)

where the second inequality follows from the definition of B.Note here that the left-hand-side is precisely

√�δE{‖q∞ −

q∗Q,(�)‖}. Thus, multiplying both sides by 1/√�δ, we have:

E{‖q∞ − q∗Q,(�)‖} ≤(γ +

η

δ

) 1√�

= O

(1√�

). (38)

Case II): Q ≤ Q] such that E{

(q∞)>(s∗ − s∞Q )}> 0: In

this case, we set α = 1. It thus follows from (27) that:

E{

∆V1(q[t])|q[t]}≤ −�

2

Φ

∥∥∥q[t]− q∗Q,(�)∥∥∥2 +�‖q[t]− q∗Q,(�)‖C(Q) + �D0, (39)

where C(Q) is defined in the proof of Proposition 1 (cf.Eq. (19)). Note that (39) is identical to (23). Then, followingexactly the same steps as in the proof of Proposition 1, wehave:

E{∆V1(q[t])|q[t] = q} ≤ −�δ1‖q−q∗Q,(�)‖1Bc1(q)+η11B1(q).

where δ1, η1, and B1 are the same as in the proof ofProposition 1. Then, it follows from (28) that

E{V1(q[T ]|q[0])}−V1(q[0])≤η1∑q∈B1

T−1∑t=0

Pr{q[t]=q|q[0]}

− �δ1∑q∈Bc1

‖q− q∗(�)‖T−1∑t=0

Pr{q[t] = q|q[0]}. (40)

Following similar steps as in Case I to divide T on both sideson (40) and let T → ∞, we have 0 ≤ η1

∑q∈B1 π

∞q −

13

�δ1∑

q∈Bc1‖q− q∗Q,(�)‖π

∞q . Re-arranging the terms and with

some manipulations, the above inequality can be written as:�δ1∑

q∈ZN+

‖q− q∗Q,(�)‖π∞q ≤

∑q∈B1

(η1 + �δ1‖q− q∗Q,(�)‖

)π∞q

≤ (η1 + δ1γ1)∑q∈B

π∞q ≤ (η1 + δ1γ1),

where γ1 is the same as in the proof of Proposition 1. Notethat the left-hand-side is �δ1E{‖q∞ − q∗Q,(�)‖}. Multiplyingboth sides by 1�δ1 , we have:

E{‖q∞ − q∗Q,(�)‖} ≤(γ1 +

η1δ1

)1

�

=([

(C(Q)Φ)+√

(C(Q)Φ)2+4D0Φ]

+η

δ

) 1�

= O

(C(Q)

1

�

).

This completes the proof of Theorem 2.

APPENDIX DPROOF OF THEOREM 3

To show the results in Theorem 3, we first note thatE{an[t]|qn[t]} = min{U

′−1n (�qn[t], A

max)} and a∗n =U′−1n (�q

∗n), ∀n. Thus, we have:

‖a∞Q − a∗Q‖ ≤ ‖a∞Q − a∗Q‖1

=

N∑n=1

∣∣∣∣E{min{U ′−1n (�q∞n , Amax)}}− U ′−1n (�q∗Q,(�),n)∣∣∣∣(a)

≤N∑n=1

E{∣∣∣min{U ′−1n (�q∞n , Amax)}− U ′−1n (�q∗Q,(�),n)∣∣∣}

(b)

≤N∑n=1

E{∣∣∣U ′−1n (�q∞n )− U ′−1n (�q∗Q,(�),n)∣∣∣}

(c)=

N∑n=1

E{∣∣∣[U ′−1n (�q̃n)]′(�q∞n − �q∗Q,(�),n)∣∣∣}

(d)

≤N∑n=1

E{∣∣∣ 1U ′′n (�q̃n)

∣∣∣∣∣∣�q∞n − �q∗Q,(�),n∣∣∣}

≤N∑n=1

E{�

φ

∣∣q∞n − q∗Q,(�),n∣∣} = �φE{∥∥q∞ − q∗Q,(�)∥∥1}≤ �√N

φE{∥∥q∞−q∗Q,(�)∥∥}, (41)

where (a) follows from Jensen’s inequality and the convexityof the L1-norm; (b) follows from relaxing the projection onto[0, Amax]; (c) follows from the mean value theorem; and (d)follows from the inverse function lemma. Recall in the proofof Theorem 2 (cf. (29)), we have 0≤J+

∑q∈ZN+

π∞q (q)>(s∗−

s∞Q )=J+E{

(q∞)>(s∗−s∞Q )}

. Again, based on the positivityof the term E

{(q∞)>(s∗−s∞Q )

}, we consider two cases:

Case I): Q > Q] such that E{

(q∞)>(s∗ − s∞Q )}≤ 0: In

this case, we can again discard E{

(q∞)>(s∗ − s∞Q )}

in (29)and let α = 0 to obtain:

0 ≤ limT→∞

1

T

∑T−1t=0

∑q∈ZN+

Pr(q[t] = q|q[0])×{− �

Φ

∥∥q[t]− q∗Q,(�)∥∥2}+D0.

By re-arranging, multiplying both sides by Φ/�, and notingthat limT→∞ 1T

∑T−1t=0 Pr{q[t] = q|q[0]} = π∞q , we have

E{‖q∞ − q∗Q,(K)‖

2}≤ D0Φ/�. (42)

It then follows from (41) that

‖a∞Q − a∗Q‖2 ≤(�√N

φE{∥∥q∞ − q∗Q,(�)∥∥})2 (a)≤

�2N

φ2E{∥∥q∞ − q∗Q,(�)∥∥2} (b)≤ �2Nφ2 D0 Φ� = �ND0Φφ2 , (43)

where (a) follows from Jensen’s inequality; and (b) followsfrom (42). Taking square root on both sides of (43) yields‖a∞Q − a∗Q‖ = O(

√�).

Case II): Q ≤ Q] such that E{

(q∞)>(s∗ − s∞Q )}> 0: In

this case, we set α = 1 and it follows from (27) that:

E{

∆V1(q[t])|q[t]}≤ −�

2

Φ

∥∥∥q[t]− q∗Q,(�)∥∥∥2 +�C(Q)‖q[t]− q∗Q,(�)‖+ �D0

= −�2

Φ

(∥∥q[t]− q∗Q,(�)∥∥− C(Q)Φ2�)2

+D, (44)

where C(Q) is defined in the proof of Proposition 1 (cf.Eq. (19)) and D , C(Q)4 +

�D0Φ . Telescoping the inequality

in (44) from t = 0 to T − 1 yields:

E{V1(q[T ]|q[0])}−V1(q[0])≤−�2

Φ

T−1∑t=0

∑q∈ZN+

Pr{q[t]=q|q[0]}

×(∥∥q[t]− q∗Q,(�)∥∥− C(Q)Φ2�

)2+DT. (45)

Dividing both sides of (45) by �2T , letting T →∞, and notingthat limT→∞ 1T

∑T−1t=0 Pr{q[t] = q|q[0]} = π∞q , ∀q ∈ ZN+ ,

we have that:

E{(∥∥q∞ − q∗Q,(�)∥∥− C(Q)Φ2�

)2}≤ DΦ

�2.

Taking square root on both sides yields:[E{(∥∥q∞ − q∗Q,(�)∥∥− C(Q)Φ2�

)2}] 12≤√DΦ

�. (46)

Moreover, examining the left-hand-side of (46), we have[E{(∥∥q∞ − q∗Q,(�)∥∥− C(Q)Φ2�

)2}] 12(a)

≥ E{[(∥∥q∞ − q∗Q,(�)∥∥− C(Q)Φ2�

)2] 12}= E

{∣∣∣∣∥∥q∞ − q∗Q,(�)∥∥− C(Q)Φ2�∣∣∣∣}

≥ E{∥∥q∞ − q∗Q,(�)∥∥− C(Q)Φ2�

}= E

{∥∥q∞ − q∗Q,(�)∥∥}− C(Q)Φ2� , (47)where (a) follows from Jensen’s inequality. Combining (41),(46), and (47) yields:

‖a∞Q − a∗Q‖ ≤�√N

φE{∥∥q∞ − q∗Q,(�)∥∥}

=�√N

φ

(C(Q)Φ

2�+

√DΦ

�

)= O(C(Q)).

14

Note that Cases I and II are exactly the same results as statedin Theorem 3. This completes the proof.

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Jia Liu (S’03–M’10–SM’16) received his Ph.D.degree in the Bradley Department of Electrical andComputer Engineering at Virginia Tech, Blacksburg,VA in 2010. He then joined the Department ofElectrical and Computer Engineering at The OhioState University as a Postdoctoral Researcher andsubsequently a Research Assistant Professor. He iscurrently an Assistant Professor in the Departmentof Computer Science at Iowa State University, wherehe joined in Aug. 2017. His research areas includetheoretical foundations of control and optimization

for stochastic networked systems, distributed algorithms design, optimizationof cyber-physical systems, data analytics infrastructure, and machine learning.Dr. Liu is a senior member of IEEE, a member of ACM, and a member ofSIAM. His work has received numerous awards at top venues, including IEEEINFOCOM’19 Best Paper Award, IEEE INFOCOM’16 Best Paper Award,IEEE INFOCOM’13 Best Paper Runner-up Award, IEEE INFOCOM’11 BestPaper Runner-up Award, and IEEE ICC’08 Best Paper Award. He is a recipientof Bell Labs President Gold Award in 2001. He has served as a TPC memberfor IEEE INFOCOM since 2010 and a TPC member of ACM MobiHoc since2017. His research has been supported by NSF, AFOSR, AFRL, and ONR.

Elizabeth S. Bentley has a B.S. degree in ElectricalEngineering from Cornell University, a M.S. degreein Electrical Engineering from Lehigh University,and a Ph.D. degree in Electrical Engineering fromUniversity at Buffalo. She was a National ResearchCouncil Post-Doctoral Research Associate at the AirForce Research Laboratory in Rome, NY. Currently,she is employed by the Air Force Research Labora-tory in Rome, NY, performing in-house research anddevelopment in the Networking Technology branch.Her research interests are in cross-layer optimiza-

tion, wireless multiple-access communications, wireless video transmission,modeling and simulation, and directional antennas/directional networking.