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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. ?, NO. ?, ????? 2017 1

Understanding the Impacts of Limited ChannelState Information on Massive MIMO Cellular

Network OptimizationJia Liu, Senior Member, IEEE, Atilla Eryilmaz,, Senior Member, IEEE, Ness B. Shroff, Fellow, IEEE,

Elizabeth S. Bentley, Member, IEEE

Abstract—To support the multi-Gigabit per second data ratesof 5G wireless networks, there have been significant efforts onthe research and development of Massive MIMO (M-MIMO)technologies at the physical layer. So far, however, the under-standing of how M-MIMO could affect the performance ofnetwork control and optimization algorithms remains ratherlimited. In this paper, we focus on analyzing the performance ofthe queue-length-based joint congestion control and schedulingframework (QCS) over M-MIMO cellular networks with limitedchannel state information (CSI). Our contributions in this paperare two-fold: i) We characterize the scaling performance of thequeue-lengths and show that there exists a phase transitioningphenomenon in the steady-state queue-length deviation withrespect to the CSI quality (reflected in the number of bits Bthat represent CSI); and ii) We characterize the congestioncontrol rate scaling performance and show that there also existsa phase transitioning phenomenon in steady-state congestioncontrol rate deviation respect to the CSI quality. Collectively, thefindings in this paper advance our understanding of the trade-offs between delay, throughput, and the accuracy/complexity ofCSI acquisition in M-MIMO cellular network systems.

Index Terms—Massive MIMO, channel state information,throughput-delay trade-off, network utility optimization.

I. INTRODUCTION

To allow 5G wireless networks to support multi-Gigabitper second data rates, there have been significant recentefforts on the research and development of massive multiple-input multiple-output systems, or simply being referred toas Massive MIMO (M-MIMO). In contrast to conventional

Manuscript received December 15, 2016 and revised May 1, 2017. Thiswork is supported by NSF grants CCF 1618318, CNS 1527078, 1446582,1314538, 1518829, 1421576, WiFiUS 1456806, CMMI 1562065, ECCS1444026; ONR grant N00014-15-1-2166; ARO grant W911NF-14-1-0368;DTRA grants HDTRA 1-14-1-0058, 1-15-1-0003, AFRL VFRP and SFFPawards; DARPA grant HROOll-15-C-0097, and QNRF grant NPRP 7-923-2-344. This paper was presented in part at ACM MobiHoc’16, Paderborn,Germany, July 2016.

Jia Liu, Atilla Eryilmaz, and Ness B. Shroff are with the Department ofElectrical and Computer Engineering, The Ohio State University, Columbus,OH, 43210 USA (e-mail: {liu.1736, eryilmaz.2, shroff.11}@osu.edu).

Elizabeth S. Bentley is with the Air Force Research Laboratory, InformationDirectorate, Rome, NY, 13441 USA (e-mail: [email protected]).

DISTRIBUTION STATEMENT A: Approved for Public Release; distribu-tion unlimited 88ABW-2017-1861 on 19 April 2017. ACKNOWLEDGMENTOF SUPPORT AND DISCLAIMER (a) Contractor acknowledges Govern-ment’s support in the publication of this paper. This material is based uponwork funded by AFRL under AFRL Contract No. FA8750-16-3-6003. (b)Any opinions, findings and conclusions or recommendations expressed in thismaterial are those of the author(s) and do not necessarily reflect the views ofAFRL.

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

multi-antenna technologies, the number of antennas in M-MIMO is on the order of hundreds or even thousands. Also, inanother key 5G technology called millimeter-wave (mmWave)communication, one can easily pack a large antenna arrayinto small form factors thanks to the short wavelengths,leading to a perfect marriage between M-MIMO and mmWavecommunications. To date, various promising theoretical resultson M-MIMO capacity gain and transmit power efficiencyhave been established (see, e.g., [1]–[3] for comprehensiveoverviews). Also, some lab-scale M-MIMO prototypes havebeen built and favorable field test results have been reported(e.g., [4], [5]). However, in spite of all of this progress, theexisting research efforts on M-MIMO are mostly concernedwith problems at the physical layer or signal processingaspects. The understanding of how M-MIMO could affectthe performance of network control, scheduling, and resourceallocation algorithms remains limited in the literature. In thispaper, our goal is to fill this gap by conducting an in-depththeoretical study on the interactions between M-MIMO phys-ical layer and network control and optimization algorithms athigher layers, as well as their impacts on queueing delay andthroughput performances.

To this end, in this paper, we focus on the performanceanalysis of the celebrated queue-length-based congestion con-trol and scheduling framework (QCS) (see, e.g., [6]–[9], and[10] for a survey) in M-MIMO-based cellular systems, wherethe M-MIMO data transmissions can rely only on limitedchannel state information (CSI). The fundamental rationaleof our work is that, as noted by many researchers [1], [2],CSI acquisition has become one of the most fundamentallimiting factors in the design of M-MIMO-based cellularsystems. Generally speaking, to leverage the MIMO spatialmultiplexing benefits, the transmitter must obtain CSI to per-form spatial beamforming so that independent data streams canbe simultaneously transmitted. In conventional MIMO-basednetworks, such CSI is usually learned at each mobile stationbased on pilot symbols and fed back to the base station (BS).However, due to the constraints on feedback channel capacityand channel coherence time, this traditional CSI feedbackapproach scales poorly with the increase of antennas in M-MIMO. An alternative CSI acquisition strategy is to let thesystem operate in time-division duplexing mode and, basedon channel reciprocity, use the uplink CSI measured at the BSfor downlink transmissions. However, the uplink CSI accuracycould still be limited in practice due to multiple reasons: 1)

2 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. ?, NO. ?, ????? 2017

As indicated in the original Introduction, it has been observedin [1], [11] that the channel reciprocity assumption may notperfectly hold in practice due to the magnetic properties of thechannel environment and transceiver hardware chains; 2) Inmillimeter-wave (mmWave) Massive MIMO systems (wherea large number of antennas can be easily packed into asmall form factor due to the short wavelength), the channelcoherence time is around an order of magnitude lower than thatof microwave bands since Doppler shifts scale linearly withfrequencies. This short channel coherence time implies thatuplink CSI estimation in mmWave Massive MIMO systems ischallenging; 3) Due to the limited transmit power at the mobilestation and the lack of beamforming gains for uplink pilotsymbols, the accuracy of TDD-based CSI estimation throughchannel reciprocity is limited.

In this paper, we accept the reality that CSI inaccuracyis unavoidable and we do not require full CSI at the M-MIMO BS. Instead, we assume that the CSI at the BS islimited and accurate only to a certain degree. Such limitedCSI can be obtained by a small amount of feedback fromeach mobile device using a limited number of bits to approxi-mately represent its channel instantiation. Alternatively, the BScould approximately measure the downlink CSI based on thechannel reciprocity assumption. In such cases, one interestingquestion naturally arises: How does the limited CSI affectthe performance of the QCS framework? In particular, it iswell-known that the QCS framework is throughput-optimalunder full CSI and achieves an [O(1/K), O(K)] utility-delaytrade-off, where K > 0 is a system parameter [8]. Also,the average queue-length deviation and the congestion controlrate optimality gap scale as O(

√K) [12] and O(1/

√K) [7],

respectively. However, when the QCS framework is adoptedin M-MIMO cellular networks with limited CSI, it begs thefollowing question: Will the same utility and delay perfor-mance scaling laws continue to hold? As will be seen later, dueto the complex cross-layer interactions (e.g., precoder design,choice of channel quantization codebook, power allocations,etc.) in M-MIMO cellular systems, answering this question ischallenging.

The main contribution of this paper is that we theoreticallycharacterize the queueing delay and network utility-optimalityperformance of the QCS framework in M-MIMO cellular net-works with B-bit limited CSI. Our main results and technicalcontributions are as follows:• We show that the queues in the network remain stable

under QCS under imperfect CSI; and the steady-state av-erage queue-lengths still follow an O(K) linear scaling.For an imperfect CSI scheme is accurate up to the B mostsignificant bit, the slope (i.e., the hidden constant in Big-O)is affected by B: The larger the value of B (more accurateCSI), the more gradual the slope becomes. Moreover, thesteady-state queue-length deviation from the mean exhibits aphase transition phenomenon: There exists a critical valueBcr such that: i) For all 0 < B < Bcr, the steady-statequeue-length deviation is bounded by O(D(B)K), whereD(B) > 0 is a quantity that depends on the specific channelquantization codebook design; and ii) For all B ≥ Bcr, thesteady-state queue-length deviation scales as O(

√K), i.e.,

...

...

...

......

qN [t]

q2[t]

q1[t]

Scheduler

a1[t]

a2[t]

aN [t]

CongestionControl

Massive MIMO Base Station

User 1

User 2

User N

Ant. 1

Ant. 2

Ant. 3

Ant. M

s2[t]

s1[t]

sN [t]

B-Bit CSI per User

Fig. 1. A Massive MIMO cellular downlink with M antennas and N users,with M � N .

recovering the same scaling law under full CSI.• For any given B-bit limited CSI collection scheme, weshow that the steady-state average congestion control ratesunder the QCS framework increase as B increases. Interest-ingly, the same phase transition phenomenon also happensin the congestion control rates in the following sense: Thereexists the same critical value Bcr such that: i) For all0 < B < Bcr, the steady-state congestion control ratedeviation scales as O(D(B)) and independent of K; andii) For all B ≥ Bcr, the steady-state congestion controldeviation scales as O(1/

√K), also recovering the same

scaling law under the full CSI.• Collectively, all queue-length and congestion control ratescaling results and their phase transition effects advance ourunderstanding of the trade-offs between delay, throughput,and the accuracy/complexity of CSI acquisition. Also, ourresults suggest that delay and throughput scalings couldpotentially be employed as useful proxies to control CSIquality and acquisition complexity in M-MIMO networks.More importantly, our work establishes a unifying theo-retical framework as well as design guidelines in practicethat enable the development of effective CSI quantizationschemes for M-MIMO cellular networks.The remainder of this paper is organized as follows: In

Section II, we introduce network model and the problem for-mulation. In Section III, we introduce the queue-length-basedcongestion control and scheduling framework and present themain results of this work. Section IV presents the numericalresults and Section V concludes this paper.

II. NETWORK MODEL AND PROBLEMFORMULATION

Notation: We use boldface to denote matrices/vectors. Welet A> and A† denote the transpose and conjugate transpose ofA, respectively. We let v1 ≥ v2 denote entry-wise inequalitybetween vectors. We let vm represent the m-th entry of vectorv. We use ‖ · ‖ and ‖ · ‖1 to denote L2- and L1-norms,respectively. We use R, C, and Z to denote real, complex,and integer spaces, respectively.

1) Massive MIMO Downlink Channel: As shown inFig. 1, we consider an M-MIMO cellular downlink system,where the BS has M antennas and serves N simultaneouslyactive single-antenna users. In this paper, we focus on the caseswhere M � N (e.g., M is in hundreds or even thousands,while N could be well less than tens). Thanks to such excess

LIU et al.: UNDERSTANDING THE IMPACTS OF LIMITED CHANNEL STATE INFORMATION ON MASSIVE MIMO CELLULAR NETWORK OPTIMIZATION 3

degrees of freedom at the BS, it is possible for the BS to serveall N users by simultaneously forming N spatial beams. Wenote that the number of users in a cell usually exceed thenumber of antennas. However, the number of simultaneouslyactive users could be less than the number of antennas in aMassive MIMO system, especially in small cells (a key featurein 5G mmWave M-MIMO wireless networks) and non-peakhours. In fact, exploiting the small number of simultaneouslyactive users is the foundation of most statistical multiplexingschemes. In this paper, we focus on the case where thenumber of users is less than the number of antennas and theassociated RF chains, so that the system can afford to serveall users simultaneously. This would allow us to first simplifythe problem and fully understand the effects of imperfect CSI(caused by numerous factors as outlined in Sec. I) in MassiveMIMO cellular systems.

We assume that the system operates under a time-slottedfashion and time is indexed by t ∈ {0, 1, 2, . . .}. We letH[t] ∈ CN×M denote the channel gain matrix in time-slot tbetween the BS and the users. We assume independent quasi-static block fading, i.e., each entry in H[t] is constant inone time-slot and independently varies in the next time-slot.Moreover, one important property of M-MIMO channels withM � N is that, under favorable propagation conditions, therow vectors of H[t] are asymptotically orthogonal as M →∞[2]. This property enables the use of simple matched-filter(MF) beamforming strategy to approach the MIMO broadcastchannels [2]1. Thus, in what follows, we will briefly introducesome related preliminaries of MF beamforming for M-MIMO.

2) Matched-Filter Beamforming: For the M-MIMO cel-lular downlink in Fig. 1, the received signal of user n in time-slot t can be written as: yn[t] = xn[t]

√pn[t]h†n[t]wn[t] +∑N

j=1,6=n xj [t]√pj [t]h

†n[t]wj [t] + vn[t], where hn[t] ∈ CM

is the channel gain vector seen at user n in time-slot t, i.e.,the n-th row in H[t]; pn[t] is the power allocated to user n intime-slot t; xn[t] represents a unit-power data symbol intendedfor user n in time-slot t; wn[t] ∈ CN is a unit-norm linearprecoding vector for user n in time-slot t; and vn[t] is the whitecomplex Gaussian noise at user n in time-slot t with powerN0. Under MF beamforming, one simply let wn[t] = hn[t],i.e., the n-th row in H[t]. In this setting, the achievable rateunder MF beamforming can be computed as:

rn[t] = log2

(1 +

pn[t]|h†n[t]wn[t]|2

N0 +∑Nj=1,6=n pj [t]|h

†n[t]wj [t]|2

)(a)≈ log2

(1 +

pn[t]

N0‖hn[t]‖2

), (1)

where (a) follows from the fact that the rows of H[t] inM-MIMO channels are nearly orthogonal, i.e., h†n[t]wj [t] =h†n[t]hj [t] ≈ 0, ∀n 6= j, when M is sufficiently large.

We assume that the channel fading can be characterized by atotal of F states H(1), . . . ,H(F ), where each H(f) ∈ CN×M ,f = 1, . . . , F , corresponds to the channel qualities between

1For MIMO broadcast channels, it is known that dirty paper coding (DPC)is capacity-achieving [13]. However, DPC is a nonlinear precoding schemethat is difficult to implement. In contrast, the capacity loss of the simple MFcompared to DPC is negligible in the high signal-to-noise ratio regime [2].

the M antennas and N users in state f . For each H(f), welet CH(f) denote the achievable MF rate region, which is theconvex hull of all achievable MF rate vectors in state f underall feasible power allocations:

CH(f) ,CH

{r(f)n , 1≤n≤N

∣∣∣∣∣ r(f)n =log2

(1+ pn

N0‖h(f)

n ‖2)

pn ≥ 0,∀n,∑Nn=1 pn ≤ Pmax

},

where CH{·} represents the convex hull operation, pn denotesthe power allocated of user n, and Pmax denotes the maximumtransmission power at the BS. Clearly, due to the maximumpower constraint, there exists an rmax < ∞ such that r(f)

n ≤rmax, ∀n, f . We let r(f) = [r

(f)1 , . . . , r

(f)N ]> ∈ RN denote the

feasible MF rates in state f . We let πf , Pr{H[t] = H(f)} bethe stationary distribution of the channel state process being instate f . We let C denote the mean MF achievable rate region,which can be written as:

C ,

r

∣∣∣∣∣∣r =

F∑f=1

πfr(f), ∀r(f) ∈ CH(f)

. (2)

We note that, in this paper, neither the channel state statisticsnor C is assumed to be known at the BS under the QCSalgorithm, which will be introduced in Section III.

3) B-Bit Limited CSI: The use of MF beamforming (i.e.,wn[t] = hn[t]) means that the BS requires full CSI H[t], ∀t.However, as mentioned in Section I, it becomes increasinglydifficult to acquire full CSI as M gets large. One way toaddress this challenge is to use limited CSI by quantizing thechannel (e.g., [14]–[18]). As shown in Fig. 1, such limited CSIcan be obtained from a small amount of feedback by each userusing a small number of bits to represent a quantized channelstate, which is accurate up to the B most significant bits.Note that in the M-MIMO and millimeter-wave (mmWave)systems literature, some hybrid beamforming/precoding archi-tecture has been proposed for frequency division duplex (FDD)systems with a reduced number of RF chains and reducedCSI feedback based on second-order statistics of the channelvectors, i.e., the channel covariance matrices (see, e.g., [19],[20]). In these hybrid beamforming systems, due to the timespent on analog beamforming, the time used for estimatingthe effective channel CSI for digital beamforming is furtherreduced, although the dimension of the effective channels isreduced. This reduced estimation time in effective channel alsointroduce inaccuracy in effective channel CSI.

Alternatively, in time-division duplex (TDD) mode, Fig. 1represents that the BS measures the uplink CSI, which isaccurate up to the B most significant bits and will be usedfor downlink transmissions2. In both cases, the value of B can

2Under the TDD mode, there is no need for quantized CSI feedback.However, uplink CSI inaccuracy may still be unavoidable due to the followingreasons: 1) Due to the limited transmit power at the mobile station and thelack of beamforming gains for uplink pilot symbols, the accuracy of TDD-based CSI estimation through channel reciprocity is limited; 2) In mmWaveM-MIMO systems (a large number of antennas can be easily packed into asmall form factor due to the short wavelength), the channel coherence timeis around an order of magnitude lower than that of microwave bands sinceDoppler shifts scale linearly with frequencies. This short channel coherencetime implies that uplink CSI estimation is challenging. Therefore, it remainsvery relevant to investigate the impacts of imperfect CSI on M-MIMO cellularsystems even in the TDD mode with channel reciprocity.


be viewed as a means to balance the trade-off between CSIaccuracy and acquisition costs. The B-bit limited CSI for eachuser n can be modeled based on a vector quantization code-book Bn , {c1

n, . . . , c2B

n }, where cin ∈ CM , i = 1, . . . , 2B ,denotes a codeword. With the CSI hn[t] in time t, a codewordfor each user n is chosen by:

i∗n[t] = arg maxj∈{1,...,2B}

∣∣h†n[t]cjn∣∣ = arg min

j∈{1,...,2B}sin2(∠(hn[t], cjn)),(3)

where i∗n[t] denotes the index of the chosen codeword. We letH[t] ∈ CN×M denote the corresponding channel gain matrixin time-slot t by aggregating all codewords i∗n[t], ∀n. Then, bytreating H[t] as if it is the accurate CSI, the BS performs MFbeamforming to construct N spatial channels. However, dueto the inaccuracy of H[t], multi-user interference may becomenon-negligible. Clearly, the impact of multi-user interferencedepends heavily on the codebook size 2B and the design ofthe quantization codebook.

Let CH[t]|H[t] denote the actual MF rate region achievedunder H[t] based on the belief that H[t] is the accurate CSI.Also, let H1[t] and H2[t] represent two estimated CSI valuesobtained by using B1 and B2 bits, respectively. Further, welet CH[t] denote the original MF achievable rate region underfull CSI H[t] . Then, one can show the following inclusionresult of the MF achievable rate regions under limited CSI inM-MIMO networks (the proof is relegated to Appendix A):

Lemma 1 (MF Rate Region Inclusion). If B1 ≤ B2, then thereexists a CSI quantization scheme under which CH[t]|H1[t] ⊆CH[t]|H2[t]. Further, CH[t]|H[t]→CH[t] as B→∞.

4) Queueing Model: As illustrated in Fig. 1, the BSmaintains a separate queue for each user. Let an[t] denotethe number of packets injected into queue n in time-slot t.As shown in Fig. 1, the arrival processes {an[t]}, ∀n, arecontrolled by a congestion controller. Also, we assume thatthere exists a finite constant Amax such that an[t] ≤ Amax,∀n, t. Let sB [t] , [sB,1[t], . . . , sB,N [t]]> denote the scheduledservice rate vector in time-slot t based on the belief that thecurrent B-bit limited CSI is accurate (the scheduling algorithmthat determines sB [t] will be presented in Section III). Then,the queue-length of each user evolves as follows: qn[t+ 1] =(qn[t] − sB,n[t] + an[t]

)+, ∀n, where (·)+ , max(0, ·).

Let q[t] = [q1[t]], . . . , qN [t]]>. In this paper, we adopt thefollowing notion of queue-stability (same as in [7], [8]): Wesay that a network is stable if the steady-state total queue-length is finite, i.e.,

lim supt→∞

E {‖q[t]‖1} <∞. (4)

5) Problem Statement: Let an , limT→∞1T

∑T−1t=0 an[t]

denote the average controlled arrival rate of user n. Eachuser n is associated with a utility function Un(an), whichrepresents the utility gained by user n when data is injectedat rate an. We assume that Un(·), ∀n, is strictly concave,monotonically increasing, and twice continuously differen-tiable. We assume that Un(·) satisfies the following strongconcavity condition: there exist constants 0 < φ < Φ < ∞such that φ ≤ −U ′′n (an) ≤ Φ, ∀an ∈ [0, Amax], where Amax

is the maximum arrival rates for burst control. For example,log(an + ε) is strongly concave for ε > 0. Our goal isto maximize

∑Nn=1 Un(an), subject to the MF beamforming

rate region CH[t]|H[t] due to limited CSI in each time-slotand the queue-stability constraint. Putting together the modelspresented above yields the following joint congestion controland scheduling (JCCS) optimization problem:

JCCS: MaximizeN∑n=1

Un(an)

subject to Queue-length stability constraint in (4),sB,n[t] ∈ CH[t]|H[t], an[t] ∈ [0, Amax] ∀n, t.

Note that, when perfect CSI is available (B →∞), the well-known QCS algorithmic framework [6]–[9] optimally solvesProblem JCCR in the following sense: The a , [a1, . . . , aN ]>

obtained from the QCS algorithm achieves a utility optimalitygap O(1/K) at the expense of queue-length scaling as O(K),where K > 0 is a system parameter. It will be clear inSection III-B that, from optimization theory perspective, thisparameter K can be interpreted as the inverse of the step-sizein a stochastic subgradient descent method. Hence, the utilityoptimality gap can be made arbitrarily small by increasingK asymptotically (implying an asymptotically large queueingdelay). However, in M-MIMO cellular networks, it is notclear whether or not the QCS algorithmic framework will beoptimal based on B-bit limited CSI. This constitutes the maindiscussions in the next section.

III. PERFORMANCE ANALYSIS OF THE QCS ALGORITHMWITH LIMITED CSI

In this section, we first present a variant of the QCSalgorithm adapted for M-MIMO with B-bit limited CSI inSection III-A. Then, we will examine a deterministic problemrelated to Problem JCCS in Section III-B to facilitate ourdiscussions. The main theoretical results and their proofs willbe presented in Sections III-C and III-D, respectively.

A. The QCS Algorithm with Limited CSI

Algorithm 1: Queue-Length-Based Congestion Control andScheduling for M-MIMO Cellular Networks with B-Bit CSI.

Initialization:1. Select an appropriate K > 0 and an appropriate B > 0.

Main Loop:2. Queue-Length-Based MaxWeight Scheduler: In time-

slot t ≥ 1, given the queue-length vector q[t] ,[q1[t], . . . , qN [t]]> and the B-bit estimated CSI H[t],the scheduler chooses a power allocation p[t] =[p1[t], . . . , pN [t]]> such that the (believed) MF achiev-able rates r[t] = arg maxx∈C

H[t](q[t])>x. As a result,

the actual MF achievable service rates sB,n[t], ∀n, underp[t] are:

sB,n[t]=log2

(1+

pn[t]∣∣h†n[t]hn[t]

∣∣2N0 +

∑Nj=1,6=n pj [t]

∣∣h†n[t]hj [t]∣∣2).(5)


3. Congestion Controller: Given the queue-length vectorq[t] , [q1[t], . . . , qN [t]]>, the congestion controllerchooses data inject rates an[t], ∀n, which are integer-valued random variables satisfying:

E{an[t]|qn[t]} = min

{U′−1n

(qn[t]

K

), Amax

}, (6)

E{a2n[t]|qn[t]} ≤ Amax

2 <∞, ∀qn[t], (7)

where U′−1n (·) represents the inverse function of first-

order derivative of Un(·). In (6) and (7), Amax and Amax2

are positive constants.4. Queue-Length Updates: Update the queue-lengths as:

qn[t+ 1] = (qn[t]− sB,n[t] + an[t])+, ∀n. (8)

Let t = t + 1. Go to Step 2 and repeat the schedulingand congestion control processes.

Some remarks on Algorithm 1 are in order: First, as men-tioned in Sec. II, we focus on the case where M � N in thispaper so that the M-MIMO BS has sufficient spatial degrees offreedom to serve all users simultaneously. This would allow usto first simplify the problem and fully understand the effectsof imperfect CSI. Once we gain a fundamental understandingon the impacts of imperfect CSI on M-MIMO cellular networkoptimization, we can extend these results to include userselection/grouping for the case with M < N by imposing aconstraint that only N0 out of N users are allowed to be servedsimultaneously and N0 � M . Clearly, adding this discreteuser selection scheduling constraint makes the MaxWeightscheduling subproblem in Step 2 in Algorithm 1 non-convexand NP-hard. Fortunately, there exists a large body of literatureon user selection scheduling schemes (see, e.g., greedy maxi-mal matching (see, e.g., [21]–[23] and references therein) forthe QCS framework that have strong performance guaranteesand can be applied in our problem setting. We note that alow-complexity scheduling scheme was recently proposed in[24], where the channel hardening effect in M-MIMO wasexploited to yield a polynomial-time user selection algorithm.However, this user selection algorithm cannot be extended toour problem setting since our work differs [24] in the followingaspects: i) Perfect CSI was assumed in [24] to enable theuse of the channel hardening effect to approximate the ratecalculation, while we do not assume perfect CSI in this paper.Because of this relaxation on perfect CSI assumption, the inter-user interference cannot be eliminated due to the CSI error,and hence it is unclear how well the channel hardening resultwould continue to hold. ii) The beamforming scheme used in[24] is the Linear Zero Forcing Beamforming (LZFBF), whilewe use the Matched-Filter (MF) Beamforming. As a result,the simplified approximate rate expression [24, Eq. (39)] doesnot apply in our problem setting.

Second, since user selection is not needed in our problemsetting, now the main challenge is reflected in the limitedand inaccurate CSI, which leads to suboptimal service ratesin (5). This incurs service rate losses compared to the fullCSI case, where the MaxWeight scheduler is of the forms[t] = arg maxx∈CH[t]

(q[t])>x. In what follows, we will

focus on the impact of this inaccurate MaxWeight schedulingsolution due to the B-bit limited CSI.

B. A Deterministic Problem

To facilitate the presentation of our theoretical results inSection III-C, we first introduce a K-parameterized deter-ministic problem, where we assume that the channel stateprocess is not random and fixed at its mean level. That is, themean achievable rate region CB , {r|r =

∑Ff=1 πf r

(f),∀r(f)

∈ CH(f)|H(f)

B

}, where CH(f)|H(f)

B

represents the actual MF rate

region achieved under H(f) based on the belief that the CSIis H

(f)B , i.e., the B-bit quantized CSI for state f . Also, the

congestion control and scheduling variables are time-invariant,which are denoted as an and sB,n, ∀n, respectively. Then, thedeterministic congestion control and scheduling problem (K-DJCCS) can be written as:

K-DJCCS: Maximize K

N∑n=1

Un(an)

subject to an − sB,n ≤ 0, ∀n,sB,n ∈ CB , an ∈ [0, amax], ∀n.

Since Problem K-DJCCS is strictly convex, an optimal solu-tion exists and is unique. Further, we associate dual variablesqB,n ≥ 0, ∀n with the constraints an−sB,n ≤ 0, ∀n, to obtainthe Lagrangian as follows:

ΘK(qB), maxa,sB∈CB

{K

N∑n=1

Un(an)+

N∑n=1

qB,n(sB,n−an)

},(9)

where the vector qB , [qB,1, . . . , qB,N ]> ∈ RN+ contains alldual variables. Then, the Lagrangian dual problem of ProblemK-DJCCS can be written as:

K-LD-JCCS: Minimize ΘK(qB)

subject to qB ∈ RN+ .

It can be verified that Problem K-DJCCS satisfies the Slatercondition [25]. Hence, the optimal value of Problem K-LD-JCCS is equal to that of Problem K-DJCCS. Let (a∗B , s

∗B) and

q∗B,(K) be a pair of optimal primal and dual solutions. Then,q∗B,(K) can be shown to have the following properties:

Lemma 2 (Optimal dual solution scaling of the deterministicproblem). For a given K, q∗B,(K) = Kq∗B,(1), or equivalently,q∗B,(K) scales linearly as O(K) and the slope is determined bythe entries in q∗B,(1). Further, q∗B1,(1) ≥ q∗B2,(1) if B1 ≤ B2.

Lemma 2 can be proved by examining the Karush-Kuhn-Tucker (KKT) conditions [25] of Problem K-DJCCS (seeAppendix B). Also, by noting the fact that K is just a scalingfactor in the objective function and a∗B = s∗B at optimality (byKKT conditions), we immediately have the following result fora∗B :

Lemma 3 (Optimal primal solution of the deterministic prob-lem). The optimal congestion control rate a∗B is independentof K and equal to the optimal service rate s∗B over CB .


C. Main Theoretical Results

In this section, we present the main performance analysisresults of Algorithm 1. Our first result says that the steady-state queue-lengths q∞ stay in a neighborhood of q∗B,(K) (theexistence of steady-state will also be proved later). Further, thescaling of the steady-state queue-length deviation from q∗B,(K)

exhibits a phase-transition phenomenon:

Theorem 1 (Phase Transition Phenomenon of Queue-LengthScaling). For any B-bit CSI quantization scheme in Algo-rithm 1 with parameter K, there exists a critical value Bcr

that is independent of K, such that the following hold:• For all 0 < B < Bcr, E{‖q∞ − q∗B,(K)‖} = O(D(B)K),

where the parameter D(B) ≥ 0 depends on the quantizationcodebook design and shrinks as B increases;

• For all B ≥ Bcr, E{‖q∞ − q∗B,(K)‖} = O(√K).

Collectively, Theorem 1 and Lemma 2 describe the steady-state queue-length behaviors. In particular, they show that ifB ≥ Bcr, the steady state queue-length deviation is upperbounded by O(

√K), which is small compared to the magni-

tude of q∗B,(K), which grows linearly as O(K) and the slopeis affected by B: the larger the value of B, the more gradualthe slope. Note that the scaling of the queue-length deviationfor B ≥ Bcr is the same as the classical result under full CSI[12]. This implies an interesting and surprising insight thatfull CSI is not necessary to induce certain desirable queueingbehaviors in M-MIMO cellular networks.

Now, let a∞B,n , E{min{U ′−1n (q∞n /K), amax}}, ∀n, be the

steady-state congestion control rates under some B-bit CSIquantization and let a∞B , [a∞B,1, . . . , a

∞B,N ]>. Our second

main result is on the scaling of a∞B ’s deviation from a∗B :

Theorem 2 (Phase Transition Phenomenon of CongestionControl Rate Scaling). For any B-bit CSI quantization schemein Algorithm 1 with parameter K, there exists a critical valueBcr (same as in Theorem 1) such that the following hold:• For all 0 < B < Bcr, ‖a∞B − a∗B‖ = O(D(B)), where the

parameter D(B) ≥ 0 is the same as in Theorem 1;• For all B ≥ Bcr, ‖a∞B − a∗B‖ = O(1/

√K).

Similar to the results in Theorem 1, Theorem 2 combinedwith Lemma 3 suggest that a phase transition phenomenonalso exists in a∞B : When B < Bcr, parameter K becomesineffective in the control of a∞B ’s deviation from a∗B . On theother hand, when B ≥ Bcr, a∞B ’s deviation from a∗B scalesas O(1/

√K) and can be made arbitrarily small by increasing

K. Since this O(1/√K) scaling is the same as that under full

CSI [7], [8], Bcr represents the smallest codebook size of thegiven CSI quantization scheme that recovers the performancecontrol functionality of parameter K.

D. Proofs of the Main Theorems

In this subsection, we provide proofs for Theorems 1 and 2.To this end, we first show a positive Harris-recurrence resultof the queue-length process, which implies the existence ofsteady-state and will be useful for proving Theorems 1 and 2later. Let 1A(x) be the indicator function that takes value 1if x ∈ A and 0 otherwise. The queue-length positive Harris-recurrence result can be stated as follows:

Theorem 3 (Queue-Length Positive Recurrence). Consider aLyapunov function V (q[t]) , 1

2K ‖q[t]−q∗B,(K)‖2 for a given

K. For the scheduler (5) and congestion controller (6)–(7),there exist constants δ, η > 0, both independent of K, 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]}

≤ − δ

ΦK

∥∥q[t]− q∗B,(K)

∥∥1Bc(q[t]) + η1B(q[t]), (10)

where B , {q ∈ ZN+ |‖q−q∗B,(K)‖ ≤ βK} for some constantβ > 0 and Bc denotes the complement of B in ZN+ .

We relegate the proof details of Theorem 3 to Appendix C.The inequality in (10) shows that the conditional mean driftis negative when the deviation of the queue-length vectorq[t] away from q∗B,(K) is sufficiently large. Since (10) isexactly the Foster-Lyapunove criterion [26, Proposition I.5.3],{q[t]}∞t=0 is positive recurrent, a steady-state distribution ofqueue-lengths exists. We denote the queue-length vector insteady-state as q∞. With Theorem 3, we are now in a positionto prove Theorem 1.

Proof of Theorem 1. To prove Theorem 1, we use an α-parameterized quadratic Lyapunov function: Vα(q[t]) =

12Kα ‖q[t] − q∗B,(K)‖

2, where the parameter α ∈ {0, 1} andits value will be specified later. Different choices of theα value would lead to different Lyapunov functions, whichsubsequently lead to different scaling laws for different CSIaccuracy levels. Following similar steps in the proof of Theo-rrem 3 (see Appendix C), we can bound the conditional meanLyapunov drift as follows:

E{Vα(q[t+ 1])− Vα(q[t])|q[t]}(a)

≤ 1

Kα(q[t]− q∗B,(K))

>(E{a[t]|q[t]} − s∗B)+

1

KαE{(q[t]− q∗B,(K))

>(s∗B − sB [t])|q[t]}+D0

Kα,

(b)

≤ 1

Kα

[− 1

ΦK

∥∥q[t]− q∗B,(K)

∥∥2+D0

]+

1

KαE{

(q[t]− q∗B,(K))>(s∗B − sB [t])

∣∣q[t]}

(c)

≤ 1

Kα

[− 1

ΦK

∥∥q[t]− q∗B,(K)

∥∥2+D0

]+

1

KαE{

(q[t])>(s∗ − sB [t])∣∣q[t]

}, (11)

where D0 , N2 (Amax

2 + (smax)2) and s∗ , limB→∞ s∗B . In(11), (a) follows from adding and subtracting s∗B ; (b) followsfrom (36); and (c) follows from s∗B ≤ s∗ (by Lemma 1)and the scheduler design, which implies (q∗B,(K))

>sB [t] ≤(q∗B,(K))

>s∗B . 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

}]


(b)

≤T−1∑t=0

∑q∈ZN+

Pr(q[t]=q|q[0]){ 1

Kα

[ −1

ΦK

∥∥q[t]−q∗B,(K)

∥∥2+D0

]}

+

T−1∑t=0

∑q∈ZN+

Pr(q[t]=q|q[0]){ 1

KαE{q>(s∗−sB [t])

}}, (12)

where (a) follows from the fact that q[t] is a discrete stateMarkov chain in ZN+ and (b) follows from (11). 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}, (13)

where J , limT→∞1T

∑T−1t=0

∑q∈ZN+

Pr(q[t] =

q|q[0]){ 1Kα [ −1

ΦK ‖q[t] − q∗(K)‖2 + D0]}, s∞B ,

arg maxx∈CH[∞]|H[∞]

(q∞)>x represents the steady-stateservice rates with B-bit CSI.

Next, consider the term E{

(q∞)>(s∗ − s∞B )}

in (13). Forany 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∞. (14)

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

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

based on the positivity of E{

(q∞)>(s∗ − s∞B )}

as follows:

Case I): B ≥ Bcr such that E{

(q∞)>(s∗ − s∞B )}≤ 0: In

this case, it follows from (13) that

0 ≤ limT→∞

1

T

∑T−1

t=0

∑q∈ZN+

Pr(q[t] = q|q[0])×{ 1

Kα

[− 1

ΦK

∥∥q[t]− q∗B,(K)

∥∥2+D0

]}. (15)

We now consider the term in the second line in (15) by settingα = 0. Similar to the proof of Theorem 3, suppose that

∥∥q[t]−q∗B,(K)

∥∥ ≥ β√K, where β will be specified shortly. This

implies that 1‖q[t]−q∗

B,(K)‖ ≤

1β . Then, the second line in (15)

can be upper bounded as:− 1

ΦK

∥∥q[t]− q∗B,(K)

∥∥2+D0 = − 1

Φ√K

∥∥q[t]−q∗B,(K)

∥∥×(‖q[t]− q∗B,(K)‖√K

+D0Φ√K

‖q[t]− q∗B,(K)‖

)≤ − 1

Φ√K

∥∥q[t]− q∗B,(K)

∥∥(β − D0Φ

β

). (16)

Hence, by choosing β >√D0Φ, we have

− 1

ΦK

∥∥q[t]−q∗B,(K)

∥∥2+D0 ≤ −

δ

Φ√K

∥∥q[t]−q∗B,(K)

∥∥,(17)

where δ = β − D0Φβ > 0. Plugging in β >

√D0Φ to define

a ball B , {q : ‖q − q∗B,(K)‖ ≤√D0ΦK}, we have that if

q[t] ∈ Bc, the following holds:

− 1

ΦK

∥∥q[t]−q∗B,(K)

∥∥2+D0≤−

δ√K‖q[t]−q∗B,(K)‖.

On the other hand, when ‖q[t] − q∗B,(K)‖ ≤√D0ΦK, it is

clear that −(1/ΦK)∥∥q[t] − q∗B,(K)

∥∥2+ D0 ≤ η for some

η > 0. Combining these facts, we have

− 1

ΦK

∥∥q[t]− q∗B,(K)

∥∥2+D0

≤ − δ

K‖q[t]− q∗B,(K)‖1Bc(q[t]) + η1B(q[t]). (18)

Substituting (18) into (15) yields:

0 ≤ limT→∞

1

T

T−1∑t=0

∑q∈ZN+

Pr(q[t] = q|q[0])×

(− δ

K‖q[t]− q∗B,(K)‖1Bc(q) + η1B(q)

)= η

∑q∈B

π∞q −δ√K

∑q∈Bc

‖q− q∗B,(K)‖π∞q . (19)

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√K

∑q∈ZN+

‖q−q∗B,(K)‖π∞q ≤

∑q∈B

(η +

δ√K‖q− q∗B,(K)‖

)π∞q

≤ (η + δβ)∑q∈B

π∞q ≤ (η + δβ), (20)

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

KE{‖q∞ −

q∗B,(K)‖}. Thus, multiplying both sides by√K/δ, we have:

E{‖q∞ − q∗B,(K)‖} ≤(β +

η

δ

)√K = O(

√K). (21)

Case II): B ≤ Bcr such that E{

(q∞)>(s∗ − s∞B )}> 0: In

this case, we set α = 1. It thus follows from (11) that:

E{

∆V1(q[t])|q[t]}≤ − 1

ΦK2

∥∥∥q[t]−q∗B,(K)

∥∥∥2

+

1

K‖q[t]− q∗B,(K))

>‖D(B) +D0

K, (22)

where D(B) is defined in the proof of Theorem 3 (cf. Eq. (33)).Note that (22) is identical to (37). Then, following exactly thesame steps as in the proof of Theorem 3, we have:

E{∆V1(q[t])|q[t]=q}≤− δ1K‖q−q∗B,(K)‖1Bc1(q)+η11B1

(q).

where δ1, η1, and B1 are the same as in the proof ofTheorem 3. Then, it follows from (12) that

E{V1(q[T ]|q[0])}−V1(q[0]) ≤ η1

∑q∈B1

T−1∑t=0

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

− δ1K

∑q∈Bc1

‖q− q∗(K)‖T−1∑t=0

Pr{q[t] = q|q[0]}. (23)


Following similar steps as in Case I to divide T on both sideson (23) and let T → ∞, we have 0 ≤ η1

∑q∈B1

π∞q −δ1K

∑q∈Bc1

‖q−q∗B,(K)‖π∞q . Re-arranging the terms and with

some manipulations, the above inequality can be written as:δ1K

∑q∈ZN+

‖q− q∗B,(K)‖π∞q

≤∑q∈B1

(η1 +

δ1K‖q− q∗B,(K)‖

)π∞q

≤ (η1 + δ1β1)∑q∈B

π∞q ≤ (η1 + δ1β1),

where β1 is the same as in the proof of Theorem 3. Note thatthe left-hand-side is δ1

KE{‖q∞ − q∗B,(K)‖}. Multiplying bothsides by K

δ1, we have:

E{‖q∞ − q∗B,(K)‖} ≤(β1 +

η1

δ1

)K

=([

(D(B)Φ) +√

(D(B)Φ)2 + 4D0Φ]

+η

δ

)K

= O(D(B)K).

This completes the proof of Theorem 1.

One important remark regarding Bcr is in order: From theproof of Theorem 1, we can see that Bcr is determined by thecondition E{(q∞)>(s∗ − s∞B )} = 0. Note that

E{(q∞)>(s∗ − s∞B )} = E{(q∞)>(s∗ − s∞ + s∞ − s∞B )}= E{(q∞)>(s∗ − s∞)}︸︷︷︸

< 0

+E{(q∞)>(s∞ − s∞B )}︸︷︷︸↓ 0 as B→ 0

From the analysis in the proof of Theorem 1, we know thatE{(q∞)>(s∗ − s∞)} < 0 and E{(q∞)>(s∞ − s∞B )} ↓ 0as B → ∞, both of which are due to the MaxWeightscheduling property. Therefore, Bcr depends on the quantitiesq∞, s∗, and s∞, which determine how negative the termE{(q∞)>(s∗ − s∞)} is. Note that the quantities q∞, s∗,and s∞ are determined by the specific problem instancefactors, such as the utility functions Un(·), ∀n, the mean MFachievable rate region C, etc.

Proof of Theorem 2. To show the results in Theorem 2, wefirst note that E{an[t]|qn[t]} = min{U ′−1

n ( qn[t]K , Amax)} and

a∗n = U′−1n (

q∗nK ), ∀n. Thus, we have:

‖a∞B − a∗B‖ ≤ ‖a∞B − a∗B‖1

=

N∑n=1

∣∣∣∣E{min{U′−1n

(q∞nK,Amax

)}}− U

′−1n

(q∗B,(K),n

K

)∣∣∣∣(a)

≤N∑n=1

E{∣∣∣min

{U′−1n

(q∞nK,Amax

)}− U

′−1n

(q∗B,(K),n

K

)∣∣∣}(b)

≤N∑n=1

E{∣∣∣U ′−1

n

(q∞nK

)− U

′−1n

(q∗B,(K),n

K

)∣∣∣}(c)=

N∑n=1

E{∣∣∣[U ′−1

n

( qnK

)]′(q∞nK−q∗B,(K),n

K

)∣∣∣}

(d)

≤N∑n=1

E{∣∣∣ 1

U ′′n ( qnK )

∣∣∣∣∣∣q∞nK−q∗B,(K),n

K

∣∣∣}

≤N∑n=1

E{

1

φK

∣∣q∞n −q∗B,(K),n

∣∣} =1

φKE{∥∥q∞−q∗B,(K)

∥∥1

}≤√N

φKE{∥∥q∞−q∗B,(K)

∥∥}, (24)

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 1 (cf. (13)), we have 0≤J+

∑q∈ZN+

π∞q (q)>(s∗−s∞B )=J+E

{(q∞)>(s∗−s∞B )

}. Again, based on the positivity

of the term E{

(q∞)>(s∗−s∞B )}

, we consider two cases:

Case I): B > Bcr such that E{

(q∞)>(s∗ − s∞B )}≤ 0: In

this case, we can again discard E{

(q∞)>(s∗ − s∞B )}

in (13)and let α = 0 to obtain:

0 ≤ limT→∞

1

T

∑T−1

t=0

∑q∈ZN+

Pr(q[t] = q|q[0])×{− 1

ΦK

∥∥q[t]− q∗B,(K)

∥∥2}

+D0.

By re-arranging, multiplying both sides by ΦK, and notingthat limT→∞

1T

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

E{‖q∞ − q∗B,(K)‖

2}≤ D0ΦK. (25)

It then follows from (24) that

‖a∞B − a∗B‖2 ≤(√

N

φKE{∥∥q∞ − q∗B,(K)

∥∥})2

(a)

≤ N

φ2K2E{∥∥q∞−q∗B,(K)

∥∥2} (b)

≤ N

φ2K2D0ΦK=

ND0

φ2K,(26)

where (a) follows from Jensen’s inequality; and (b) followsfrom (25). Taking square root on both sides of (26) yields‖a∞B − a∗B‖ = O(1/

√K).

Case II): B ≤ Bcr such that E{

(q∞)>(s∗ − s∞B )}> 0: In

this case, we set α = 1 and it follows from (11) that:E{

∆V1(q[t])|q[t]}≤ − 1

ΦK2

∥∥∥q[t]−q∗B,(K)

∥∥∥2

+

D(B)

K‖q[t]− q∗B,(K))

>‖+D0

K

= − 1

ΦK2

(∥∥q[t]− q∗B,(K)

∥∥− D(B)ΦK

2

)2

+D, (27)

where D(B) is defined in the proof of Theorem 3 (cf. Eq. (33))and D ,

D(B)

4 + D0

ΦK . Telescoping the inequality in (27) fromt = 0 to T − 1 yields:

E{V1(q[T ]|q[0])}−V1(q[0])≤− 1

ΦK2

T−1∑t=0

∑q∈ZN+

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

×(∥∥q[t]− q∗B,(K)

∥∥− D(B)ΦK

2

)2

+DT. (28)

Dividing both sides of (28) by TK2 , letting T →∞, and noting

that limT→∞1T

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

we have that:


E{(∥∥q∞ − q∗B,(K)

∥∥− D(B)ΦK

2

)2}≤ DΦK2.

Taking square root on both sides yields:[E{(∥∥q∞ − q∗B,(K)

∥∥− D(B)ΦK

2

)2}] 12

≤ K√DΦ.

(29)

Moreover, examining the left-hand-side of (29), we have[E{(∥∥q∞ − q∗B,(K)

∥∥− D(B)ΦK

2

)2}] 12

(a)

≥ E{[(∥∥q∞ − q∗B,(K)

∥∥− D(B)ΦK

2

)2] 12}

= E{∣∣∣∣∥∥q∞ − q∗B,(K)

∥∥− D(B)ΦK

2

∣∣∣∣}≥ E

{∥∥q∞ − q∗B,(K)

∥∥− D(B)ΦK

2

}= E

{∥∥q∞ − q∗B,(K)

∥∥}− D(B)ΦK

2, (30)

where (a) follows from Jensen’s inequality. Combining (24),(29), and (30) yields:

‖a∞B − a∗B‖ ≤√N

φKE{∥∥q∞ − q∗B,(K)

∥∥}=

√N

φK

(D(B)ΦK

2+K√DΦ

)= O(D(B)).

Note that Cases I and II are exactly the same results as statedin Theorem 2. This completes the proof.

IV. NUMERICAL RESULTS

In this section, we conduct numerical experiments to verifythe theoretical results presented in Section III. In our sim-ulations, we use a 128-antenna M-MIMO base station withMF precoding to serve four users. Each user’s channel isi.i.d. Rayleigh faded. The maximum total signal-to-noise ratio(SNR) of the BS is set to 30dB. We use log(· + 0.001)as the utility function for each user, i.e., the proportionalfairness metric [10]. In our numerical studies, the channelstates (i.e., the channel gain matrix H[t] in each time-slot t)are randomly generated in MATLAB. In the perfect CSI case,we directly use H[t] in each time slot t as CSI input in Step2 in Algorithm 1. We adopt the random vector quantization(RVQ) scheme, which has been widely used in the MIMOlimited CSI feedback literature [14], [15], [18]. The value ofB is set to be 1, 2, 4, 8, 16, 32, 53, and 64, covering casesfrom the simplest two-state channel quantization to channelquantizations with high granularity.

We first study the impacts of B on the delay performance.The results of average queue-length deviations with respect tothe changes of B are illustrated in Fig. 2. In Fig. 2, each solidcurve represents the average queue-length deviation scalingwith respect to K under a certain CSI accuracy parameterizedby B. For each scaling curve, we also plot an accompanyingline (the red dash lines). The bottom 4 accompanying linesrepresent (starting from bottom) 0.44

√K, 0.5

√K, 0.58

√K,

0.58√K, and 0.64

√K. They correspond to B = 64, 53, 32,

and 16, respectively. Fig. 2 shows that the average queue-length deviation curves sit below each accompanying linewhen B ≥ 16, confirming that the average queue-lengthdeviations are bounded by the O(

√K) scaling. On the other

hand, when B < 16, there is no O(√K) accompanying

lines that can dominate the average queue-length deviationcurves, which means that the average queue-length deviationgrows faster than the square root law and approximatelyexhibits a linear growth with respect to K. To see this, inFig. 2, we plot 3 straight accompanying lines. We can see thatthe average queue-length deviation curves hovering aroundthese linear scaling accompanying lines, confirming the linearscaling growth. Finally, combining two parts, we can see thatB = 16 is the critical point, where the phase transition fromO(K) scaling to O(

√K) happens.

Also, when B = 64, we can see that the queue-lengthdeviations almost coincide with that in the full CSI case,showing that the 64-bit RVQ scheme is almost as accurateas full CSI. Note that, as discussed earlier, obtaining full CSIis costly or even unrealistic in practice: On one hand, if CSIis measured at the mobile station, then the mobile stationneeds to have long enough time to measure the channelsfrom all transmit antennas. This is unrealistic in MassiveMIMO systems because the channel measurement time growswith the number of antennas, while the channel coherencetime is essentially a constant. On the other hand, if CSIis measured at the base station under the TDD mode andbased on channel reciprocity, then the base station needs tohave enough computing resources to measure the channelsbetween transmit-receive antenna pairs simultaneously, whichincurs high hardware costs. Also, the massive antenna arraysneed to be carefully calibrated to compensate the reciprocityimpairments in practice, which complicates the base stationhardware design.

The results of average queue-lengths’ growth with respectto K under different values of B are illustrated in Fig. 3. Wecan see from Fig. 3 that the average queue-lengths increaselinearly with respect to K under all B values, agreeing withLemma 2. Also, the value of B plays an important role inthe slope of the linear scaling: the large the B value, the moregradual the slope, again confirming Theorem 1. Also, the slopeof B=64 is almost the same as that of full CSI.

Next, we study the impacts of B on the congestion controlperformance and the results are illustrated in Fig. 4. When Bis small, we can observe in Fig. 4 that a∞B are independentof K and only affected by B. The congestion control ratesapproach that under full CSI as B increases. This confirms thefirst part of Theorem 2 and Lemma 3. Similar to the growthsof queue-length deviations, we can also observe that B =16 is the critical point, beyond which the congestion controlrates start to exhibit an O(1/

√K) shrinking gap to a∗B . All of

these observations agree with the phase transitioning results inTheorem 2.

V. CONCLUSION

In this paper, we conducted an in-depth theoretical studyon the impact of limited CSI on the performances of the


200 300 400 500 600 700 800 900 1000

K

5

10

15

20

25

30

35

AverageQueu

e-Len

gth

DeviationE{‖

q∞−

q∗ B,(K)‖}

B = 1B = 2

B = 4

B = 8Critical Point:B = 16 : 0.64

√K

B = 32 : 0.58√K

B = 64 and Full CSI: 0.44√K

B = 53 : 0.50√K

Fig. 2. Average queue-length deviation E{‖q∞−q∗B,(K)

‖} with respect toK for B=1, 2, 4, 8, 16, 32, 53, 64.

200 300 400 500 600 700 800 900 1000

K

0

200

400

600

800

1000

1200

AverageQueu

e-Len

gth

E{‖q∞ B,(K)‖}

B = 1

B = 2

B = 4

B = 8

B = 16

B = 32

B = 53

B = 64 and Full CSI

Fig. 3. The growths of average queue-lengths with respect to K for B =1, 2, 4, 8, 16, 32, 53, and 64.

40 60 80 100 120 140

K

1

2

3

4

5

6

7

8

CongestionControlRatesE{‖

a∞ B‖}

B = 1

B = 2

B = 4

B = 8

B = 16: Critical Point

B = 32

B = 53

B = 64 and Full CSI

Gap shrinking atO(1/

√K) speed

‖a∗64‖

‖a∗53‖

‖a∗32‖

‖a∗16‖

Fig. 4. The steady-state congestion control rates with respect to K for B =1, 2, 4, 8, 16, 32, 53, and 64.

queue-length-based joint congestion control and schedulingalgorithm in M-MIMO cellular networks. We have theoreti-cally characterized the queueing delay and congestion controlscalings under limited CSI. We showed that there exist phase

transitioning phenomena in the steady-state queue-length andcongestion control rate deviations with respect to CSI quality.Collectively, our theoretical results in this paper advance theunderstanding of the interactions and trade-offs between delay,throughput, and the accuracy/complexity of CSI acquisitionin M-MIMO networks. Our work also establishes a unifyingtheoretical framework as well as practical design guidelinesto enable the development of effective channel quantizationschemes for M-MIMO networks. Similar to various knownschemes (see, e.g., [27]–[29]) that enhance the original QCSframework for traditional wireless networks under perfect CSI,it is highly interesting to consider new algorithmic techniquesto further sharpen the throughput and delay performancesfor Massive MIMO under imperfect CSI. Also, it is veryimportant to further incorporate user selection/grouping intothe MaxWeight scheduling component. To that end, our workserves as an important step toward an exciting M-MIMO net-working research paradigm that explores various new conges-tion control and scheduling algorithmic designs, which couldpotentially offer better throughput and delay performancesunder limited CSI.

APPENDIX APROOF OF LEMMA 1

For ease of exposition, we first show the second part ofLemma 1. Let {pn[t], n= 1, . . . , N} be an arbitrary feasiblepower allocation. Since the BS performs MF beam formingby treating H[t] as if it is the accurate CSI, the receivedsignal can be written as yn[t] = sn[t]pn[t]h>n [t]wn[t] +∑Nj=1,6=nsj [t]pj [t]h

>n [t]wj [t] + vn[t], where wj [t] = hj [t],

1 ≤ j ≤ N , i.e., the j-th row of H[t]. Hence, the MF ratessB,n[t] achieved under H[t] based on the belief that the CSIis B-bit CSI H[t] can be computed as:

sB,n[t] = log2

(1 +

pn[t]∣∣h†n[t]hn[t]

∣∣2N0 +

∑Nj=1,6=n pj [t]

∣∣h†n[t]hj [t]∣∣2)

< log2

(1 +

pn[t]

N0

∥∥hn[t]∥∥2)

= sn[t], ∀n, (31)

where the inequality in (31) holds because∣∣h†n[t]hn[t]

∣∣2 ≤‖hn[t]‖2 and |h†n[t]hj [t]|2 ≥ 0. Thus, for every rate pointsB [t] = [sB,1[t], . . . , sB,N [t]]T ∈ CH[t]|H[t], its correspond-ing power allocation {p1[t], . . . , pN [t]} achieves a rate points[t] = [s1[t], . . . , sN [t]]> ∈ CH[t] that dominates sB [t] inevery coordinate. Hence, CH[t] ⊆ CH[t]. Also, as B → ∞,H[t]→ H[t]. It thus follows from (31) that sB [t] ↑ s[t], whichimplies that CH[t] → CH[t].

Next, we argue why the first part of Lemma 1 is true.Let B1

n and B2n denote the vector quantization codebooks

corresponding to B1 and B2 bits, respectively. Since B1 ≤ B2,it follows that the codebook sizes |B1

n| ≤ |B2n|. Hence, given

codebook B1n, one can construct B2

n by simply retaining allcodewords in B1

n and adding new code words that are notin B1

n, which implies B1n ⊂ B2

n. As a result, for any givenCSI hn[t], one can always find a codeword in B2

n whosedistance to hn[t] is not larger than that from B1

n in the sense of(3). Hence, the SINR term in (31) becomes larger under B2

n,


implying sB1,n[t] ≤ sB2,n[t]. Since this is true for arbitrarypower allocation, we have CH[t]|H1[t] ⊆ CH[t]|H2[t].

APPENDIX BPROOF OF LEMMA 2

Dividing K on both sides of (9), we have 1KΘK(qB) =

maxa,sB∈CB{∑N

n=1 Un(an) +∑Nn=1 qB,n(sB,n − an)

},

where qB,n = qB,n/K. Note that the right hand side isprecisely Θ1(qB), for which the maximizer is q = q∗B,(1).Hence, we have ΘK(q) is maximized at Kq∗B,(1). Thisproves the first part of Lemma 2.

To show the second part of Lemma 2, we first note from theKKT complementary slackness condition and the monotonic-ity of Un(·) that, at optimality, a∗n = s∗B,n, ∀n. We let a∗n(B1)and a∗n(B2) denote the optimal congestion control rates underB1 and B2, respectively. If B1 ≤ B2, we have from Lemma 1that s∗B1,n

≤ s∗B2,n, which further implies a∗n(B1) ≤ a∗n(B2).

On the other hand, from the KKT stationarity condition, wehave U ′n(a∗n(B)) − q∗(B),n = 0. Since a∗n(B1) ≤ a∗n(B2), itfollows from the concavity of Un(·) that q∗(B1),n ≥ q∗(B2),n.This completes the proof.

APPENDIX CPROOF OF THEOREM 3

Consider the quadratic Lyapunov function defined in Theo-rem 3: V (q[t]) = 1

2K ‖q[t]− q∗B,(K)‖2, where q[t] represents

the queue-length vector in time-slot t under parameters K andB; and q∗B,(K) denotes the optimal dual solution for the staticversion of Problem JCCR under parameter K. Then, the one-slot mean Lyapunov drift of VK(q[t]), which can computedas:

E{V (q[t+ 1])− V (q[t])|q[t]}

=1

2KE{

(q[t+ 1]− q[t])>(q[t+ 1] + q[t]− 2q∗B,(K))∣∣∣q[t]

}(a)

≤ 1

2KE{

(−sB [t]+a[t])>(2q[t]−2q∗B,(K)−sB [t]+a[t])∣∣∣q[t]

}=

1

K(q[t]−q∗B,(K))

>(−sB [t]+a[t])+1

2KE{‖− sB [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]} < Amax

2 N . Also, sincesB,n[t] falls in a bounded instantaneous capacity region CH[t],∀n, we must have sB,n[t] ≤ smax for some smax > 0. Hence,by defining D0 , N

2 (Amax2 + (smax)2), which is a constant

upper bound of the second moments of the arrival and serviceprocesses, we have

E {∆V (q[t])|q[t]} ≤ 1

K(q[t]− q∗B,(K))

>E {a[t]− sB [t]}+D0

K(a)=

1

K(q[t]− q∗B,(K))

>(E{a[t]|q[t]} − s∗B)+

1

KE{(q[t]− q∗B,(K))

>(s∗B − sB [t])|q[t]}+D0

K,

(b)

≤ 1

K(q[t]− q∗B,(K))

>(E{a[t]|q[t]} − s∗B)+

1

K‖q[t]− q∗B,(K))

>‖ × E{‖s∗B − sB [t]‖|q[t]

}+D0

K,

(32)

where s∗B is such that (s∗B ,q∗B,(K)) is a pair of optimal primal

and dual solutions to Problem K-DJCCS under parameterK. In (32), (a) follows from adding and subtracting s∗B aswell as the fact that a[t] is independent of the channel stateand determined solely by q[t]; and (b) follows from Cauchy-Schwarz inequality.

Note from Lemma 3 that s∗B is independent of K andsB,n[t] ∈ CH[t] is upper-bounded. Thus, we have

E{‖s∗B − sB [t]‖|q[t]

}≤ D(B) , max

q:‖q‖=1E{‖s∗B − sB‖q},

(33)where D(B) is a constant depending only on the CSI accuracyB and independent of K, which bounds the cross term in (32).Hence, we can further upper bound (32) as:

E {∆V (q[t])|q[t]} ≤ 1

K(q[t]−q∗B,(K))

>(E{a[t]|q[t]}−s∗B)+

1

K‖q[t]− q∗B,(K))

>‖D(B) +D0

K, (34)

Now, let us consider the first term on the right hand side in(34), i.e., 1

K (q[t]− q∗B,(K))>(E{a[t]|q[t]}− s∗). Since Un(·)

is concave and increasing, ∀n, we have(qn[t]− q∗B,(K),n

)> [U′−1n

(qn[t]

K

)− U

′−1n

(q∗B,(K),n

K

)]≤ 0.

Thus, by Cauchy-Schwatz inequality, we have:

(q[t]− q∗B,(K))>(E{a[t]|q[t]} − s∗B) =

N∑n=1

(qn[t]− q∗B,(K),n

)>×[U′−1n

(qn[t]

K

)− U

′−1n

(q∗B,(K),n

K

)]≤ −

N∑n=1

∣∣qn[t]−

q∗B,(K),n

∣∣ ∣∣∣∣U ′−1n

(qn[t]

K

)−U

′−1n

(q∗B,(K),n

K

)∣∣∣∣ . (35)

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]

K−q∗B,(K),n

K

∣∣∣∣≤ ∣∣∣∣U ′−1n

(qn[t]

K

)−U

′−1n

(q∗B,(K),n

K

)∣∣∣∣ .Hence, we can further upper-bound (35) as:

(q[t]− q∗B,(K))>(E{a[t]|q[t]} − s∗B) ≤ − 1

ΦK

N∑n=1

(qn[t]−

q∗B,(K),n

)2=− 1

ΦK

∥∥∥q[t]−q∗B,(K)

∥∥∥2

. (36)

Substituting (36) into (34), we have

E {∆V (q[t])|q[t]} ≤ − 1

ΦK2

∥∥∥q[t]−q∗B,(K)

∥∥∥2

+

1

K‖q[t]− q∗B,(K))

>‖D(B) +D0

K. (37)

Now, suppose that∥∥q[t]− q∗B,(K)

∥∥ ≥ β1K, where β1 will


be specified shortly. Note also that K ≥ 1, we have

1

‖q[t]− q∗B,(K)‖≤ 1

β1K≤ 1

β1.

It then follows that (37) can be further upper bounded as:

E{∆V (q[t])|q[t]} = − 1

ΦK

∥∥q[t]− q∗B,(K)

∥∥ · ∥∥q[t]− q∗B,(K)

∥∥K

+1

K‖q[t]−q∗B,(K))

>‖D(B)+∥∥q[t]−q∗B,(K)

∥∥ D0∥∥q[t]−q∗B,(K)

∥∥K≤ − 1

ΦK

∥∥q[t]− q∗B,(K)

∥∥(β1 −D(B)Φ−D0Φ

β1

). (38)

By choosing β1 such that β1 −D1Φ− D0Φβ1

> 0, we have

E{∆V (q[t])|q[t]} ≤ − δ1ΦK

∥∥q[t]− q∗B,(K)

∥∥ (39)

where δ1 = β1−D(B)Φ−D0Φβ1

. Solving β1−D(B)Φ−D0Φβ1

= 0

and plugging in the obtained β1 to define a ball B1 , {q :∥∥q − q∗B,(K)

∥∥ ≤ K2 [(D(B)Φ) +

√(D(B)Φ)2 + 4D0Φ]}, we

have

E{∆V (q[t])|q[t]} ≤ − δ1K

∥∥q[t]− q∗B,(K)

∥∥, if q[t] ∈ Bc1,(40)

where δ1 , δ1Φ . On the other hand, when q[t] ∈ B1, it is

clearly 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}≤− δ1K‖q−q∗B,(K)‖1Bc1(q)+η11B1(q).

This completes the proof of Theorem 3.

REFERENCES

[1] E. G. Larsson, O. Edfors, and T. L. Marzetta, “Massive MIMO for nextgeneration wireless systems,” IEEE Commun. Mag., vol. 52, no. 2, pp.186–195, Feb. 2014.

[2] F. Rusek, D. Persson, B. K. Lau, E. G. Larsson, T. L. Marzetta,O. Edfors, and F. Tufvesson, “Scaling up MIMO: Opportunities andchallenges with very large arrays,” IEEE Signal Process. Mag., vol. 30,no. 1, pp. 40–46, Jan. 2013.

[3] L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “Anoverview of massive MIMO: Benefits and challenges,” IEEE Journal ofSelected Topics in Signal Processing, vol. 8, no. 5, pp. 742–758, Oct.2014.

[4] Argos: Practical many-antenna base stations. [Online]. Available:argos.rice.edu

[5] E. G. Larsson and F. Tufvesson, “Massive MIMO systemstutorial,” 2013. [Online]. Available: http://www.commsys.isy.liu.se/vlm/icc tutorial P2.pdf

[6] X. Lin and N. B. Shroff, “The impact of imperfect scheduling on cross-layer congestion control in wireless networks,” IEEE/ACM Trans. Netw.,vol. 14, no. 2, pp. 302–315, Apr. 2006.

[7] A. Eryilmaz and R. Srikant, “Joint congestion control, routing, andMAC for stability and fairness in wireless networks,” IEEE J. Sel. AreasCommun., vol. 24, no. 8, pp. 1514–1524, Aug. 2006.

[8] M. J. Neely, E. Modiano, and C.-P. Li, “Faireness and optimal stochasticcontrol for heterogeneous networks,” IEEE/ACM Trans. Netw., vol. 16,no. 2, pp. 396–409, Apr. 2008.

[9] A. Eryilmaz and R. Srikant, “Fair resource allocation in wirelessnetworks using queue-length-based scheduling and congestion control,”IEEE/ACM Trans. Netw., vol. 15, no. 6, pp. 1333–1344, Dec. 2007.

[10] X. Lin, N. B. Shroff, and R. Srikant, “A tutorial on cross-layer opti-mization in wireless networks,” IEEE J. Sel. Areas Commun., vol. 24,no. 8, pp. 1452–1463, Aug. 2006.

[11] C. Shepard, H. Yu, N. Anand, L. E. Li, T. L. Marzetta, R. Yang, andL. Zhong, “Argos: Practical many-antenna base stations,” in Proc. ACMMobiCom, Istanbul, Turkey, August 2012, pp. 53–64.

[12] A. Eryilmaz and R. Srikant, “Fair resource allocation in wirelessnetworks using queue-length-based scheduling and congestion control,”in Proc. IEEE INFOCOM, Miami, FL, Mar. 2005, pp. 1804–1814.

[13] H. Weingarten, Y. Steinberg, and S. Shamai (Shitz), “The capacity regionof the Gaussian multiple-input multiple-output broadcast channel,” IEEETrans. Inf. Theory, vol. 52, no. 9, pp. 3936–3964, Sep. 2006.

[14] N. Jindal, “MIMO broadcast channels with finite rate feedback,” IEEETrans. Inf. Theory, vol. 52, no. 11, pp. 5045–5059, Nov. 2006.

[15] W. Santipach and M. Honig, “Signature optimiation for CDMA withlimited feedback,” IEEE Trans. Inf. Theory, vol. 51, no. 10, pp. 3475–3492, Oct. 2005.

[16] W. Santipach and M. L. Honig, “Asymptotic performance of MIMOwireless channels with limited feedback,” in Proc. IEEE Mil. Commun.Conf., vol. 1, Oct. 2003, pp. 141–146.

[17] ——, “Asymptotic capacity of beamforming with limited feedback,” inProc. IEEE ISIT, Jul. 2004, p. 290.

[18] C. K. Au-Yeung and D. J. Love, “On the performance of random vectorquantization limited feedback beamforming in a MISO system,” IEEETrans. Wireless Commun., vol. 6, no. 2, pp. 458–462, Feb. 2007.

[19] A. F. Molisch, V. V. Ratnam, S. Han, Z. Li, S. L. H. Nguyen, L. Li,and K. Haneda, “Hybrid Beamforming for Massive MIMO – A Survey,”arXiv preprint arXiv:1609.05078, 2016.

[20] J. Y. A. A. Adhikary, J. Nam and G. Caire, “Joint spatial division andmultiplexing – the large-scale array regime,” IEEE Trans. Inf. Theory,vol. 59, no. 10, pp. 6441 – 6463, Oct. 2013.

[21] C. Joo, X. Lin, J. Ryu, and N. B. Shroff, “Distributed greedy approxima-tion to maximum weighted independent set for scheduling with fadingchannels,” IEEE/ACM Transactions on Networking, vol. 24, no. 3, pp.1476 – 1488, June 2016.

[22] B. Ji, G. R. Gupta, M. Sharma, X. Lin, and N. B. Shroff, “Achievingoptimal throughput and near-optimal asymptotic delay performancein multi-channel wireless networks with low complexity: A practicalgreedy scheduling policy,” IEEE/ACM Transactions on Networking,vol. 23, no. 3, pp. 880 – 893, June 2015.

[23] C. Joo and N. B. Shroff, “Local greedy approximation for scheduling inmulti-hop wireless networks,” IEEE Transaction on Mobile Computing,vol. 11, no. 3, pp. 414 – 426, March 2012.

[24] D. Bethanabhotla, G. Caire, and M. J. Neely, “WiFlix: Adaptive videostreaming in massive MU-MIMO wireless networks,” IEEE Trans.Wireless Commun., vol. 15, no. 6, pp. 4088–4103, Jun. 2016.

[25] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming:Theory and Algorithms, 3rd ed. New York, NY: John Wiley & SonsInc., 2006.

[26] S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability,2nd ed. Cambridge, UK: Cambridge University Press, 2009.

[27] J. Liu, A. Eryilmaz, N. B. Shroff, and E. Bentley, “Heavy-ball: Anew approach to tame delay and convergence in wireless networkoptimization,” in Proc. IEEE INFOCOM, April 10-15, 2016.

[28] M. J. Neely, “Super-fast delay tradeoffs for utility optimal fair schedulingin wireless networks,” IEEE J. Sel. Areas Commun., vol. 24, no. 8, pp.1489–1501, Aug. 2006.

[29] L. Huang and M. J. Neely, “Delay reduction via lagrange multipliers instochastic network optimization,” IEEE Trans. Autom. Control, vol. 56,no. 4, pp. 842–857, Apr. 2011.


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 joined the Ohio State University asa postdoctoral researcher afterwards. He is currentlya Research Assistant Professor in the Departmentof Electrical and Computer Engineering at the OhioState University. His research focus is in the areasof optimization of communication systems and net-works, theoretical foundations of cross-layer opti-mization on wireless networks, design of algorithms,

and information theory. Dr. Liu is a senior member of IEEE. His work hasreceived numerous awards at top venues, including IEEE INFOCOM 2016Best Paper Award, IEEE INFOCOM 2013 Best Paper Runner-up Award,IEEE INFOCOM 2011 Best Paper Runner-up Award, and IEEE ICC 2008Best Paper Award. He is a recipient of the Bell Labs President Gold Award in2001 and Chinese Government Award for Outstanding Ph.D. Students Abroadin 2008. He has served as a TPC member for IEEE INFOCOM since 2010.

Atilla Eryilmaz (S’00–M’06–SM’17) received hisM.S. and Ph.D. degrees in Electrical and Com-puter Engineering from the University of Illinois atUrbana-Champaign in 2001 and 2005, respectively.Between 2005 and 2007, he worked as a Postdoc-toral Associate at the Laboratory for Information andDecision Systems at the Massachusetts Institute ofTechnology. He is currently an Associate Professorof Electrical and Computer Engineering at The OhioState University. Dr. Eryilmaz’s research interestsinclude design and analysis for communication net-

works, optimal control of stochastic networks, optimization theory, distributedalgorithms, pricing in networked systems, and information theory. He receivedthe NSF-CAREER Award in 2010 and two Lumley Research Awards forResearch Excellence in 2010 and 2015. He is a co-author of the 2012 IEEEWiOpt Conference Best Student Paper, the 2016 IEEE Infocom Best Paper,and 2017 IEEE WiOpt Conference Best Paper. He has served as TPC co-chairof IEEE WiOpt in 2014 and of ACM Mobihoc in 2017, and is an AssociateEditor of IEEE/ACM Transactions on Networking since 2015, and of IEEETransactions on Network Science and Engineering since 2017.

Ness B. Shroff (S’91–M’93–SM’01–F’07) receivedhis Ph.D. degree in Electrical Engineering fromColumbia University in 1994. He joined Purdueuniversity immediately thereafter as an AssistantProfessor in the school of ECE. At Purdue, hebecame Full Professor of ECE in 2003 and directorof CWSA in 2004, a university-wide center onwireless systems and applications. In July 2007, hejoined The Ohio State University, where he holds theOhio Eminent Scholar endowed chair in Networkingand Communications, in the departments of ECE and

CSE. He holds or has held visiting (chaired) professor positions at TsinghuaUniversity, Beijing, China, Shanghai Jiaotong University, Shanghai, China,and the Indian Institute of Technology, Bombay, India. Dr. Shroff is currentlyan editor at large of IEEE/ACM Trans. on Networking, senior editor of IEEETransactions on Control of Networked Systems, and technical editor for theIEEE Network Magazine. He has received numerous best paper awards for hisresearch and listed Thomson Reuters Book on The World’s Most InfluentialScientific Minds as well as noted as a highly cited researcher by ThomsonReuters. He also received the IEEE INFOCOM achievement award for seminalcontributions to scheduling and resource allocation in wireless networks.

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.

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