Resource Allocation Under Channel Uncertainties for Relay-Aided Device-to-Device Communication Underlaying LTE-A Cellular Networks

2.7 MB·

by  Monowar Hasan

1
Resource Allocation Under Channel Uncertainties
for Relay-Aided Device-to-Device Communication
Underlaying LTE-A Cellular Networks
Monowar Hasan, Ekram Hossain, and Dong In Kim
Abstract—Device-to-device (D2D) communication in cellular In the context of D2D communication, it becomes a crucial
networksallowsdirecttransmissionbetweentwocellulardevices issue to set up direct links between the D2D UEs while
with local communication needs. Due to the increasing number
4 satisfying the quality-of-service (QoS) requirements of tradi-
ofautonomousheterogeneousdevicesinfuturemobilenetworks,
1 tional cellular UEs (CUEs) and the D2D UEs in the network.
an efficient resource allocation scheme is required to maximize
0 networkthroughputandachievehigherspectralefficiency.Inthis In practice, the advantages of D2D communication may be
2
paper, performance of network-integrated D2D communication limiteddueto:i)longerdistance:thepotentialD2DUEsmay

n under channel uncertainties is investigated where D2D traffic not be in near proximity; ii) poor propagation medium: the
a is carried through relay nodes. Considering a multi-user and
link condition between two D2D UEs may not be favourable;
J multi-relay network, we propose a robust distributed solution
iii) interference to and from CUEs: in a spectrum underlay
6 for resource allocation with a view to maximizing network sum-
2 rate when the interference from other relay nodes and the link system, D2D transmitters can cause severe interference to
gains are uncertain. An optimization problem is formulated for other receiving nodes in the network and also the D2D

] allocating radio resources at the relays to maximize end-to- receivers may experience interference from other transmitting
I end rate as well as satisfy the quality-of-service (QoS) require-
N nodes.PartitioningtheavailablespectrumforitsusebyCUEs
ments for cellular and D2D user equipments under total power
and D2D UEs in a non-overlapping manner (i.e., overlay
. constraint. Each of the uncertain parameters is modeled by a
cs bounded distance between its estimated and bounded values. D2D communication) could be an alternative; however, this
[ We show that the robust problem is convex and a gradient- would significantly reduce spectrum utilization [4], [5]. In
aided dual decomposition algorithm is applied to allocate radio suchcases,network-assistedtransmissionthroughrelayscould
1 resources in a distributed manner. Finally, to reduce the cost
enhance the performance of D2D communication when D2D
v of robustness defined as the reduction of achievable sum-rate,
3 we utilize the chance constraint approach to achieve a trade- UEs are far away from each other and/or the quality of
8 off between robustness and optimality. The numerical results D2D communication channel is not good enough for direct
6 showthatthereisadistancethresholdbeyondwhichrelay-aided communication.
6 D2Dcommunicationsignificantlyimprovesnetworkperformance Unlike most of the existing work on D2D communication,
. when compared to direct communication between D2D peers.
1 in this paper, we consider relay-assisted D2D communication
0 Index Terms—D2D communication, LTE-Advanced Layer 3 in LTE-A cellular networks where D2D pairs are served by
4 (L3)relay,robustworst-caseresourceallocation,uncertainchan- the relay nodes. In particular, we consider LTE-A Layer-
1 nel gain, ellipsoidal uncertainty set, chance constraint. 3 (L3) relays1. We concentrate on scenarios in which the
:
v proximity and link condition between the potential D2D UEs
i
X I. INTRODUCTION may not be favorable for direct communication. Therefore,
they may communicate via relays. The radio resources at the
r Device-to-device (D2D) communication enables wireless
a relays(e.g.,resourceblocks[RBs]andtransmissionpower)are
peer-to-peer services directly between user equipments (UEs)
shared among the D2D communication links and the two-hop
to facilitate high data rate local service as well as offload the
cellular links using these relays. An use-case for such relay-
traffic of cellular base station (i.e., Evolved Node B [eNB]
aided D2D communication could be the machine-to-machine
in an LTE-Advanced [LTE-A] network). By reusing the LTE-
(M2M)communication[6]forsmartcities.Insuchacommu-
A cellular resources, D2D communication enhances spectrum
nication scenario, automated sensors (i.e., UEs) are deployed
utilizationandimprovescellularcoverage.Inconjunctionwith
within a macro-cell ranging a few city blocks; however, the
traditional local voice and data services, D2D communication
link condition and/or proximity between devices may not be
opens up new opportunities for commercial applications, such
favorable. Due to the nature of applications, these UEs are
as proximity-based services, in particular social networking
required to periodically transmit data [7]. Relay-aided D2D
applications with content sharing features (i.e., exchanging
communicationcouldbeanelegantsolutiontoprovidereliable
photos, videos or documents through smart phones), local
transmissionaswellasimproveoverallnetworkthroughputin
advertisement, multi-player gaming and data flooding [1]–[3].
M. Hasan and E. Hossain are with the Department of Electrical and 1An L3 relay with self-backhauling configuration performs the same
Computer Engineering, University of Manitoba, Winnipeg, Canada (emails: operationasaneNBexceptthatithasalowertransmitpowerandasmaller
monowar [email protected], [email protected]). D. I. Kim cellsize.Itcontrolscell(s)andeachcellhasitsowncellidentity.Therelay
is with the School of Information and Communication Engineering at the transmitsitsowncontrolsignalsandtheUEsareabletoreceivescheduling
SungkyunkwanUniversity(SKKU),Korea(email:[email protected]). informationdirectlyfromtherelaynode[27].
2
TABLEI
such a scenario.
MATHEMATICALNOTATIONS
Duetotime-varyingandrandomnatureofwirelesschannel,
we formulate a robust resource allocation problem with an Notation Physicalinterpretation
objective to maximizing the end-to-end rate (i.e., minimum N ={1,2,...,N} SetofavailableRBs
achievable rate over two hops) for the UEs while maintaining L={1,2,...,L} Setofrelays
the QoS (i.e., rate) requirements for cellular and D2D UEs ul AUEservedbyrelayl
under total power constraint at the relay node. The link gains, Ul,|Ul| SetofUEsandtotalnumberofUEsservedby
the interference among relay nodes and interference at the relayl,respectively
receiving D2D UEs are not exactly known (i.e., estimated h(n) Directlinkgainbetweenthenodeiandj over
i,j
RBn
with an additive error). The robust problem formulation is
observed to be convex, and therefore, we apply a gradient- Ru(nl) End-to-enddatarateforul overRBn
bnaosdeedwmitehthpoodlytonosmolivael cthoempprloebxlietym. Wdiestdriebmutoivnesltyraatetethaacthirnetlraoy- x(unl),Su(nl,)l RpoBwearllfoocrautilonoveinrdRicBatonr,raenspdecaticvteulayl transmit
ducingrobustnesstodealwithchanneluncertaintiesaffectsthe Iu(nl,)l ARBggnregatedinterferenceexperiencedbyul over
achievablenetworksum-rate.Toreducethecostofrobustness g(n) NominallinkgainvectoroverRBninhopi
defined as the corresponding reduction of achievable sum- l,i
g¯(n),gˆ(n) Estimatedanduncertain(i.e.,theboundederror)
rate, we utilize the chance constraint approach to achieve l,i l,i
linkgainvector,respectively,overRBninhop
a trade-off between robustness and optimality by adjusting i
some protection functions. We compare the performance of (cid:60)(gnl,)i,∆(gnl,)i Uncertaintysetandprotectionfunction,respec-
our proposed method with an underlay D2D communication tively,forlinkgainoverRBninhopi
sthceheamsseistwanhceereofthreelaDy2sD. TUheEsnucmoemrimcaulnriceasuteltsdisrheocwtlythwatitahfotuert (cid:60)(Inu)l,l,∆(Inu)l,l Ufenrecnecrteailnetvyels,erteasnpdecptirvoetleyc,tifoonrfuulncotvieornRoBfinnter-
Ψ(n) Boundofuncertaintyinlinkgainforhopiover
a distance threshold for the D2D UEs, relaying D2D traffic l,i
RBn
provides significant gain in achievable data rate. The main
contributions of this paper can be summarized as follows: Υ(unl) BovoeurnRdBofnuncertaintyininterferencelevelforul
• We analyze the performance of relay-assisted D2D com- (cid:107)y(cid:107)α Linearnormofvectorywithorderα
municationunderuncertainsystemparameters.Theprob- (cid:107)y(cid:107)∗ Dualnormof(cid:107)y(cid:107)
lem of RB and power allocation at the relay nodes abs{y} Absolutevalueofy
for the CUEs and D2D UEs is formulated and solved A(j,:) j-throwofmatrixA
for the globally optimal solution when perfect channel Λ(κt) Stepsizeforvariableκatiterationt
gain information for the different links is available. As R∆ Reductionofachievablesum-rateduetouncer-
tainty
opposedtomostoftheresourceallocationschemesinthe
Θ(n) Threshold probability of violating interference
literaturewhereonlyasingleD2Dlinkisconsidered,we l,i
constraintforRBninhopi
consider multiple D2D links along with multiple cellular
links that are supported by relay nodes. SΘl(,ni)(R∆) SensitivityofR∆ inhopioverRBn
• Assumingthattheperfectchannelinformationisunavail-
able, we formulate a robust resource allocation problem
for relay-assisted D2D communication under uncertain II. RELATEDWORKANDMOTIVATION
channel information in both the hops and show that the
AlthoughresourceallocationforD2Dcommunicationinfu-
convexity of the robust formulation is maintained. We
ture generation orthogonal frequency-division multiple access
propose a distributed algorithm with a polynomial time
(OFDMA)-based wireless networks is one of the active areas
complexity.
ofresearch,thereareveryfewworkwhichconsiderrelaysfor
• The cost of robust resource allocation is analyzed. In
D2D communication. A resource allocation scheme based on
order to achieve a balance between the network perfor-
a column generation method is proposed in [4] to maximize
manceandrobustness,weprovideatrade-offmechanism.
the spectrum utilization by finding the minimum transmission
The rest of this paper is organized as follows. A review of length (i.e., time slots) for D2D links while protecting the
the related work and motivation of this work are presented in cellularusersfrominterferenceandguaranteeingQoS.In[8],a
Section II. In Section III, we present the system model and greedyheuristic-basedresourceallocationschemeisproposed
assumptions. In Section IV, we formulate the RB and power for both uplink and downlink scenarios where a D2D pair
allocationproblemforthenominal(i.e.,non-robust)case.The sharesthesameresourceswithCUEonlyiftheachievedSINR
robustresourceallocationproblemisformulatedinSectionV. is greater than a given SINR requirement. A new spectrum
In order to allocate resources efficiently, we propose a robust sharingprotocolforD2Dcommunicationoverlayingacellular
distributed algorithm and discuss the robustness-optimality network is proposed in [9], which allows the D2D users
trade-off in Section VI. The performance evaluation results to communicate bi-directionally while assisting the two-way
arepresentedinSectionVIIandfinallyweconcludethepaper communicationsbetweentheeNBandtheCUE.Theresource
in Section VIII. The key mathematical notations used in the allocationproblemforD2Dcommunicationunderlayingcellu-
paper are listed in Table I. larnetworksisaddressedin[10].In[11],theauthorsconsider
3
relayselectionandresourceallocationforuplinktransmission calculations to solve. To avoid this difficulty, the robust
inLTE-Advanced(LTE-A)networkswithtwoclassesofusers problem is converted to a convex optimization problem and
having different (i.e., specific and flexible) rate requirements. solved in a traditional way. Although not in the context of
The objective is to maximize system throughput by satisfying D2D communication, there have beena few work considering
the rate requirements for the rate-constrained users while resource allocation under uncertainties in the radio links. One
confining the transmit power within a power-budget. of the first contributions dealing with channel uncertainties is
Although D2D communication was initially proposed to [21]wheretheauthormodelsthetime-varyingcommunication
relayusertraffic[12],notmanyworkconsiderusingrelaysin for single-access and multiple-access channels without feed-
D2D communication. To the best of our knowledge, relay- back.ForanOFDMAsystem,theresourceallocationproblem
assisted D2D communication was first introduced in [13] under channel uncertainty for a cognitive radio (CR) base
where the relay selection problem for D2D communication station communicating with multiple CR mobile stations is
underlayingcellularnetworkwasstudied.Theauthorspropose considered in [22] for downlink communication. Two robust
a distributed relay selection method for relay assisted D2D power control schemes are developed in [23] for a CR net-
communication system which firstly coordinates the interfer- work with cooperative relays. In [24], a robust power control
ence caused by the coexistence of D2D system and cellular algorithm is proposed for a CR network to maximize the
network and eliminates improper relays correspondingly. Af- social utility defined as the network sum rate. A robust worst-
terwards, the best relay is chosen among the optional relays case interference control mechanism is provided in [25] to
using a distributed method. In [14], the authors consider maximize rate while keeping the interference to primary user
D2D communication for relaying UE traffic toward the eNB below a threshold.
and deduce a relay selection rule based on the interference Taking the advantage of L3 relays supported by the 3GPP
constraints. In [15], [16], the maximum ergodic capacity and standard, in our earlier work [26], we studied the perfor-
outage probability of cooperative relaying are investigated in mance of network-assisted D2D communications assuming
relay-assisted D2D communication considering power con- the availability of perfect CSI and showed that relay-aided
straintsattheeNB.Thenumericalresultsshowthatmulti-hop D2D communication provides significant performance gain
relaying lowers the outage probability and improves cell edge for long distance D2D links. In this paper, we extend the
throughputcapacitybyreducingtheeffectofinterferencefrom work utilizing the theory of worst-case robust optimization to
the CUE. maximize the end-to-end data rate under link uncertainties for
In all of the above cited work, it has generally been theUEswithminimumQoSrequirementswhileprotectingthe
assumed that complete system information (e.g., channel state other receiving relay nodes and D2D UEs from interference.
information [CSI]) is available to the network nodes, which is To make the robust formulation more tractable and obtain a
unrealistic for a practical system. Uncertainty in the CSI (in near-optimal solution for satisfying all the constraints in the
particularthechannelqualityindicator[CQI]inanLTE-Asys- nominal problem, we apply the notion of protection function
tem) can be modeled by sum of estimated CSI (i.e., the nom- instead of uncertainty set.
inal value) and some additive error (the uncertain element).
Accordingly,byusingrobustoptimizationtheory,thenominal
III. SYSTEMMODELANDASSUMPTIONS
optimization problem (i.e., the optimization problem without
A. Network Model
considering uncertainty) is mapped to another optimization
problem (i.e., the robust problem). To tackle uncertainty, two A relay node in LTE-A is connected to the radio ac-
approaches have commonly been used in robust optimization cess network (RAN) through a donor eNB with a wireless
theory. First, the Bayesian approach (Chapter 6.4 in [17]) connection and serves both the cellular and D2D UEs. Let
considers the statistical knowledge of errors and satisfies the L={1,2,...,L}denotethesetoffixed-locationLayer3(L3)
optimizationconstraintsinaprobabilisticmanner.Second,the relays in the network as shown in Fig. 1. The system band-
worst-case approach (Chapter 6.4 in [17], [18]) assumes that width is divided into N RBs denoted by N = {1,2,...,N}
theerror(i.e.,uncertainty)isboundedinaclosedsetcalledthe which are used by all the relays in a spectrum underlay
uncertainty set and satisfies the constraints for all realizations fashion. When the link condition between two D2D peers is
oftheuncertaintyinthatset.AlthoughtheBayesianapproach too poor for direct communication, scheduling and resource
has been widely used in the literature (e.g., in [19], [20]), the allocation for the D2D UEs can be done in a relay node (i.e.,
worst-case approach is more appropriate due to the fact that L3 relay) and the D2D traffic can be transmitted through that
it satisfies the constraints in all error instances. By applying relay. We refer to this as relay-aided D2D communication
the worst-case approach, the size of the uncertainty set can whichcanbeanefficientapproachtoprovidebetterquality-of-
be obtained from the statistics of error. As an example, the service (QoS) for communication between distant D2D UEs.
uncertainty set can be defined by a probability distribution The CUEs and D2D pairs constitute set C ={1,2,...,C}
function of uncertainty in such a way that all realizations and D = {1,2,...,D}, respectively, where the D2D pairs
of uncertainty remain within the uncertainty set with a given are discovered during the D2D session setup. We assume that
probability. the CUEs are outside the coverage region of eNB and/or
Applying robustness brings in new variables in the opti- having bad channel condition, and therefore, the CUE-eNB
mizationproblem,whichmaychangethenominalformulation communications need to be supported by the relays. Besides,
to a non-convex optimization problem and require excessive direct communication between two D2D UEs requires the
4
The unit power SINR for the link between relay l and eNB
for CUE (i.e., u ∈{C∩U }) in the second hop is as follows:
l l
h(n)
γ(n) = l,eNB . (2)
L3 relay l,ul,2 (cid:88) P(n)g(n) +σ2
j,uj j,eNB
∀uj∈{D∩Uj},j(cid:54)=l,j∈L
eNB L3 relay Similarly, the unit power SINR for the link between relay l
and receiving D2D UE for the D2D-pair (i.e., u ∈{D∩U })
l l
in the second hop can be written as
L3 relay
h(n)
γ(n) = l,ul . (3)
Cellular UE l,ul,2 (cid:88) P(n)g(n) +σ2
j,uj j,ul
D2D UE ∀uj∈Uj,j(cid:54)=l,j∈L
Fig.1. Asinglecellwithmultiplerelaynodes.WeassumethattheCUE-eNB In (1)–(3), P(n) is the transmit power in the link between i
i,j
linksareunfavourablefordirectcommunicationandtheyneedtheassistance and j over RB n, σ2 =N B where B is bandwidth of
of relays. The D2D UEs are also supported by the relay nodes due to long 0 RB RB
distanceand/orpoorlinkconditionbetweenpeers. anRB,andN0 denotesthermalnoise.h(l,ne)NB isthegaininthe
relay-eNBlinkandh(n) isthegaininthelinkbetweenrelayl
l,ul
and receiving D2D UE corresponding to the D2D transmitter
assistance of a relay node. We assume that association of
UE u .
l
the UEs (both cellular and D2D) to the corresponding relays
The achievable data rate for u in the first hop can be
areperformedbeforeresourceallocation.TheUEsassistedby expressed as r(n) = B log (cid:16)1l+P(n)γ(n) (cid:17). Note that,
relay l are denoted by ul. The set of UEs assisted by relay l ul,1 RB 2 ul,l ul,l,1
(cid:83) this rate expression is valid under the assumption of Gaussian
is U such that U ⊆{C∪D},∀l ∈L, U ={C∪D}, and
(cid:84) Ul =∅. l l l (and spectrally white) interference which holds for a large
l l number of interferers. Similarly, the achievable data rate in
We assume that during a certain time instance, only one (cid:16) (cid:17)
the second hop is r(n) = B log 1+P(n)γ(n) . Since
relay-eNB link is active in the second hop to carry CUEs’ ul,2 RB 2 l,ul l,ul,2
data (i.e., transmissions of relays to the eNB in the second we are considering a two-hop communication, the end-to-end
hop are orthogonal in time). Scheduling of the relays for data rate for ul on RB n is half of the minimum achievable
transmissioninthesecondhopisdonebytheeNB.2 However, data rate over two hops [28], i.e.,
multiple relays can transmit to their corresponding D2D UEs
1 (cid:110) (cid:111)
in the second hop. Note that, in the first hop, the transmission R(n) = min r(n) ,r(n) . (4)
ul 2 ul,1 ul,2
between a UE (i.e., either CUE or D2D UE) and relay can
be considered an uplink communication. In second hop, the
transmission between a relay and the eNB can be considered IV. RESOURCEBLOCK(RB)ANDPOWERALLOCATIONIN
an uplink communication from the perspective of the eNB RELAYNODES
whereas the transmission from a relay to a D2D UE can
A. Formulation of the Nominal Resource Allocation Problem
be considered as a downlink communication. In our system
model, taking advantage of the capabilities of L3 relays, For each relay, the objective of radio resource (i.e., RB
scheduling and resource allocation for the UEs is performed and transmit power) allocation is to obtain the assignment of
in the relay nodes to reduce the computation load at the eNB. RB and power level to the UEs that maximizes the system
capacity,whichisdefinedastheminimumachievabledatarate
overtwohops.Letthemaximumallowabletransmitpowerfor
B. Achievable Data Rate UE (relay) is Pmax (Pmax). The RB allocation indicator is a
ul l
We denote by h(i,nj) the direct link gain between node i and binary decision variable x(unl) ∈{0,1}, where
j over RB n. The interference link gain between relay (UE) (cid:40)
i(realnady)UjEis(rneloatya)sjsoocviaetredRBwinthirseldaeyno(UteEd)biy. Tgih(,nje)uwnhiterpeowUeEr xu(nl) = 10,, iofthReBrwniseis. assigned to UE ul (5)
SINR for the link between UE u ∈U and relay l using RB
l l
n in the first hop is given by (cid:88)N
Let R = x(n)R(n) denotes the achievable sum-rate over
ul ul ul
h(n) n=1
γ(n) = ul,l . (1) allocated RB(s) and let the QoS (i.e., rate) requirements for
ul,l,1 (cid:88) P(n)g(n) +σ2 UEu isdenotedbyQ .ConsideringthatthesameRB(s)will
uj,j uj,l l ul
∀uj∈Uj,j(cid:54)=l,j∈L be used by the relay in both the hops (i.e., for communication
betweenrelayandeNBandbetweenrelayandD2DUEs),the
2SchedulingofrelaynodesbytheeNBisoutofthescopeofthispaper. resourceallocationproblemforeachrelayl∈Lcanbestated
5
as follows: interference threshold in the first hop, each UE u associated
l
(P1) max (cid:88) (cid:88)N x(n)R(n) twhiethotrheelaryrenloadyesalnodbttahienscothrreersepfoenrednincgeuchsearnnue∗llagsasioncgia(tne)dwfiothr
x(unl),Pu(nl,)l,Pl(,nu)l ul∈Uln=1 ul ul ∀n according to the following equation: u∗l,l,1
(cid:88)
subject to x(n) ≤ 1, ∀n∈N (6a)
ul
ul∈Ul
N
n(cid:88)=1x(unl)Pu(nl,)l ≤ Pumlax, ∀ul ∈Ul (6b) u∗l =argmj ax gu(nl,)j, ul ∈Ul,j (cid:54)=l,j ∈L. (7)
N
(cid:88) (cid:88)x(n)P(n) ≤ Pmax (6c)
ul l,ul l
ul∈Uln=1
(cid:88) x(n)P(n)g(n) ≤ I(n), ∀n∈N (6d) Similarly, in the second hop, for each relay l, the transmit
ul ul,l u∗l,l,1 th,1 power will be adjusted accordingly considering interference
ul∈Ul
(cid:88) x(unl)Pl(,nu)lgl(,nu)∗l,2 ≤ It(hn,)2, ∀n∈N (6e) irneltaroyds)ucceodnstiodetrhiengretcheeivcionrgreDsp2oDndUinEgsc(haasnsnoecliagtaeidnwgl(i,tnuh)∗,o2thfoerr
ul∈Ul ∀n where the reference user is obtained by l
R ≥ Q , ∀u ∈U (6f)
ul ul l l
P(n) ≥ 0, P(n) ≥ 0, ∀n∈N,u ∈U (6g)
ul,l l,ul l l
where the rate of u over RB n
l
R(n) = 1minBRBlog2(cid:16)1+Pu(nl,)lγu(nl,)l,1(cid:17), u∗l =argumjax gl(,nu)j, j (cid:54)=l,j ∈L,uj ∈{D∩Uj}. (8)
ul 2 BRBlog2(cid:16)1+Pl(,nu)lγl(,nu)l,2(cid:17)
and the unit power SINR for the first hop,
From (4), the maximum data rate for UE u over RB n is
l
γ(n) = h(unl,)l asecchoienvdedhowph,etnhePup(nlo,)lwγeu(nrl,)l,a1llo=catPedl(,nu)lfγolr(,nu)Ul,2E. Tuh,erPefore,ciannthbee
ul,l,1 I(n) +σ2 l l,ul
ul,l,1 expressed as a function of power allocated for transmission
and the unit power SINR for the second hop, in the first hop, Pul,l as follows: Pl(,nu)l = γγu((nnl,))l,1Pu(nl,)l. Hence
 h(n) the data rate for u over RB n can be exprels,suel,d2 as
 I(n)l,eN+Bσ2, ul ∈{C∩Ul} l
γ(n) = l,ul,2
l,ul,2  I(n)h(l,nu+)l σ2, ul ∈{D∩Ul}. R(n) = 1B log (cid:16)1+P(n)γ(n) (cid:17). (9)
l,ul,2 ul 2 RB 2 ul,l ul,l,1
IntheaboveI(n) andI(n) denotetheinterferencereceived
ul,l,1 l,ul,2
by u over RB n in the first and second hop, respectively, and
l
are given as follows: I(n) = (cid:88) x(n)P(n)g(n)
ul,l,1 uj uj,j uj,l
∀uj∈Uj,j(cid:54)=l,j∈L
I(n) =  ∀uj∈{D∩(cid:88)Uj},j(cid:54)=l,j∈Lx(unj)Pj(,nu)jgj(,ne)NB, ul ∈{C∩Ul}B. Continuous Relaxation and Reformulation
l,ul,2  (cid:88) x(unj)Pj(,nu)jgj(,nu)l, ul ∈{D∩Ul}.
∀uj∈Uj,j(cid:54)=l,j∈L
Withtheconstraintin(6a),eachRBisassignedtoonlyone TheoptimizationproblemP1isamixed-integernon-linear
UE. With the constraints in (6b) and (6c), the transmit power program (MINLP) which is computationally intractable. A
is limited by the maximum power budget. The constraints in common approach to tackle this problem is to relax the
(6d) and (6e) limit the amount of interference introduced to constraint that an RB is used by only one UE by using the
the other relays and the receiving D2D UEs in the first and time-sharing factor [30]. Thus x(unl) ∈ (0,1] is represented as
second hop, respectively, to be less than some threshold. The thesharingfactorwhereeachx(unl) denotestheportionoftime
constraint in (6f) ensures the minimum QoS requirements for that RB n is assigned to UE ul and satisfies the constraint
(cid:88)
the CUE and D2D UEs. The constraint in (6g) is the non- x(n) ≤ 1, ∀n. Besides, we introduce a new variable
ul
negativity condition for transmit power. ul∈Ul
Similar to [29], the concept of reference node is adopted S(n) =x(n)P(n) which denotes the actual transmit power of
ul,l ul ul,l
here. For example, to allocate the power level considering the UE u on RB n [31]. Then the relaxed problem can be stated
l
6
as(Pfo2ll)ows: where θu(nl,)l = (cid:16)x(unl)ωu(nlS)u(+nlS,)lu(γnlu(,)nll,γ)lu,(1nl,)l,1(cid:17)ln2.
Proof: See Appendix A.
max (cid:88) (cid:88)N 1x(n)B log (cid:32)1+ Su(nl,)lh(unl,)l,1(cid:33) Proposition1. ThepowerandRBallocationobtainedby(11)
x(unl),Su(nl,)l,ωu(nl) ul∈Uln=12 ul RB 2 x(unl)ωu(nl) and(12)isagloballyoptimalsolutiontotheoriginalproblem
subject to (cid:88) x(n) ≤ 1, ∀n (10a) P1.
ul
ul∈Ul Proof:SinceP2isaconstraint-relaxedversionofP1,the
(cid:88)N S(n) ≤ Pmax,∀u (10b) solution(cid:16)x(unl)∗,Pu(nl,)l∗(cid:17)givesanupperboundtotheobjective
ul,l ul l of P1. Besides, since x(n)∗ satisfies the binary constraints in
(cid:88) (cid:88)N hh(u(nnnl,))=l,11Su(nl,)l ≤ Plmax (10c) aPls1o, g(cid:16)ixv(uenls)∗a,lPouw(nl,e)lr∗(cid:17)bosuantidusl.fies all constraints in P1 and hence
ul∈Uln=1 l,ul,2 In the above problem formulation it is assumed that each
(cid:88) Sn g(n) ≤ I(n), ∀n (10d) of the relays and D2D UEs has the perfect information about
ul,l u∗l,l,1 th,1 the experienced interference. Also, the channel gains between
ul∈Ul
(cid:88) h(unl,)l,1S(n)g(n) ≤ I(n), ∀n (10e) trheleayrse)laayreaknndowthnetooththeer rUelEasy.(Hasoswoceivaetre,deswtiimthatninegigthhbeouexriancgt
ul∈Ul h(l,nu)l,2 ul,l l,u∗l,2 th,2 values of link gains is not easy in practice. To deal with the
(cid:88)N 1x(n)B log (cid:32)1+ Su(nl,)lh(unl,)l,1(cid:33) ≥ Q , ∀u (10f) uronbcuesrttaoinpttiiemsiiznattihoenemsteitmhoatded[3v3a]l.ues, we apply the worst-case
2 ul RB 2 x(n)ω(n) ul l
n=1 ul ul
S(n) ≥ 0, ∀n,u (10g) V. ROBUSTRESOURCEALLOCATION
ul,l l A. Formulation of Robust Problem
I(n) +σ2 ≤ ω(n),∀n,u (10h)
ul,l ul l Let the vector of link gains between relay l and other
where ω(n) is an auxiliary variable for u over RB n and transmitting UEs (associated with other relays, i.e., for ∀j ∈
let I(n) u=l max(cid:110)I(n) ,I(n) (cid:111). The dulality gap of any L,j (cid:54)= l) in the first hop over RB n be denoted by gl(,n1) =
optimuilz,lation problemuls,al,t1isfyl,iunlg,2the time sharing condition is (cid:104)g1(n∗,)l,1, g2(n∗,)l,1,··· , g|(Unl)|∗,l,1(cid:105),where|Ul|isthetotalnumber
negligible as the number of RB becomes significantly large. of UEs associated with relay l. Similarly, the vector of link
Our optimization problem satisfies the time-sharing condition gains between relay l and receiving D2D UEs (associated
andhencethesolutionoftherelaxedproblemisasymptotically with other relays) in the second hop over RB n is given by
(cid:104) (cid:105)
optimal [32]. Since the objective function is concave, the g(n) = g(n) , g(n) ,··· , g(n) .
constraintin(10f)isconvex,andalltheremainingconstraints l,2 l,1∗,2 l,2∗,2 l,|Ul|∗,2
We assume that the link gains and the aggregated interfer-
are affine, the optimization problem P2 is convex. Due to ence (i.e., I(n), ∀n,u and elements of g(n),g(n), ∀n) are
convexity of the optimization problem P2, there exists a ul,l l l,1 l,2
unknown but are bounded in a region (i.e., uncertainty set)
unique optimal solution.
with a given probability. For example, the channel gain in
Statement 1. (a) The power allocation for UE ul over RB n the first hop is bounded in (cid:60)(gnl,)1, with estimated value g¯l(,n1)
is given by and the bounded error gˆ(n), i.e., g(n) = g¯(n) + gˆ(n), and
l,1 l,1 l,1 l,1
Pu(nl,)l∗ = Sxu((nnl,))l∗∗ =(cid:34)δu(nl,)l− hω(nu(n)l) (cid:35)+ (11) ggll((,,nn11)).∈Sim(cid:60)(ginlla,)1r,ly∀,nle∈t(cid:60)N(gnl,,)2w, h∀enrebe(cid:60)t(ghnle,)1uinscethrteaiunntycesretatifnotrythseetlifnokr
ul ul,l,1 gains in the second hop and (cid:60)(n) , ∀n,u be the uncertainty
where δ(n) = 12BRB(1+lnλ2ul) and set for interference level. Iul,l l
[(cid:15)]+ =mualx,l{(cid:15),0}.ρul+hh(l(u,nnlu)),ll,,21νl+gu(n∗l),l,1ψn+hh(u(l,nnlu)),ll,,12gl(,nu)∗l,2ϕn rnaotmIenientxhaplerpefrososrbimolenumla[tiif.oeon.r,moeufqlaurtoaibtoiunos.ntAp(l9rto)h]boluaegsmh,thdweeealoiunntegiliwzuesiethadssiiinmmiitllhaaerr
utilityfunction(i.e.,rateequation)forbothnominalandrobust
(b) The RB allocation is determined as follows: problems is quite common in literature (e.g., in [22], [23],
 [25]), when perfect channel information is not available to
x(n)∗ =1, µn ≤χ(unl,)l (12) receiver nodes, the rate obtained by (9) actually approximates
ul 0, µ >χ(n) the achievable rate 3 The solution to P2 is robust against
n ul,l uncertainties if and only if for any realization of g(n) ∈
l,1
and χ(n) is defined as
ul,l 3Accordingtoinformation-theoreticcapacityanalysis,inpresenceofchan-
χ(unl,)l = 12(1+λul)BRB(cid:20)log2(cid:18)1+ Sxu(nl(un,)ll)hω(unu(lnl),l),1(cid:19)−θu(nl,)l(cid:21) namreealthugenimvceeanrtitcabainyltiteerqsacutaaattbioitlhnietsy,r(e4wc6ee)ivraeenrs,dortth(4et9o)lo(iw9n)er[t2oa1n]cd,alrcueupslppaeetecrtitbvhoeeluyna.dcHshioeowvfaebtvhleeer,drafaottear
(13) rateinboththenominalandrobustproblemformulations.
7
(cid:60)(n),g(n) ∈ (cid:60)(n), and I(n) ∈ (cid:60)(n) , the optimal solution B. Uncertainty Set and Protection Function
sagtil,s1fiesl,t2he congslt,r2aints inu(1l,0ld), (1I0uel),l, and (10h). Therefore, From P3, the optimization problem is impacted by the
the robust counterpart of P2 is represented as uncertainty sets (cid:60)(n),(cid:60)(n), and (cid:60)(n) . To obtain the robust
formulation, we cgol,n1sidegrl,2that theIuul,nlcertainty sets for the
(P3) uncertainparametersarebasedonthedifferencesbetweenthe
max (cid:88) (cid:88)N 1x(n)B log (cid:32)1+ Su(nl,)lh(unl,)l,1(cid:33) aucntcuearlta(iin.ety.,)uvnaclueretsa.inT)haenseddnioffmerinenalce(si.ec.a,nwbitehomuattchoenmsaidtiecrailnlgy
x(unl),Su(nl,)l,ωu(nl) ul∈Uln=12 ul RB 2 x(unl)ωu(nl) represented by general norms [34]. For example, the uncer-
tainty sets for channel gain in the first and second hops for
subject to (10a), (10b), (10c), (10d),
∀n∈N are given by
(10e), (10f), (10g), (10h)
and g(n) ∈(cid:60)(n), g(n) ∈ (cid:60)(n), ∀n (14a) (cid:60)(n) = (cid:26)g(n)|(cid:107)M(n) ·(cid:16)g(n)−g¯(n)(cid:17)T (cid:107)≤Ψ(n)(cid:27) (15a)
l,1 gl,1 l,2 gl,2 gl,1 l,1 gl,1 l,1 l,1 l,1
Iu(nl,)l ∈ (cid:60)(Inul),l, ∀n,∀ul (cid:60)(n) = (cid:26)g(n)|(cid:107)M(n) ·(cid:16)g(n)−g¯(n)(cid:17)T (cid:107)≤Ψ(n)(cid:27) (15b)
(14b) gl,2 l,2 gl,2 l,2 l,2 l,2
wheretheconstraintsin(14a)and(14b)representtherequire- where (cid:107) · (cid:107) denotes the general norm, Ψ(n) and Ψ(n)
l,1 l,2
ments for the robustness of the solution. represent the bound on the uncertainty set; g(n), g(n) are the
l,1 l,2
Proposition2. When(cid:60)(gnl,)1,(cid:60)(gnl,)2,and(cid:60)Iul,l arecompactand actual and g¯l(,n1), g¯l(,n2) are the estimated (i.e., nominal) channel
convex sets, P3 is a convex optimization problem. gain vectors; M(gnl,)1 and M(gnl,)2 are the invertible R|Ul|×|Ul|
Proof: The uncertainty constraints in (10d), (10e), and weight matrices for the first and second hop, respectively.
(10h) are satisfied if and only if Likewise, the uncertainty set for the experienced interference
is expressed as
max (cid:88) S(n)g(n) ≤I(n), ∀n
gl(,n1)∈(cid:60)(gnl,)1ul∈Ul ul,l u∗l,l,1 th,1 (cid:60)(Inul),l =(cid:110)Iu(nl,)l|(cid:107)MI(unl),l ·(cid:16)Iu(nl,)l−I¯u(nl,)l(cid:17)(cid:107)≤Υ(unl)(cid:111) (16)
max (cid:88) h(unl,)l,1S(n)g(n) ≤I(n), ∀n where I(n) and I¯(n) are the actual and estimated interference
gl(,n2)∈(cid:60)(gnl,)2ul∈IU(ln)hm∈(l,an(cid:60)ux)(ln,2) Iuu(lnl,l,)ll+,u∗lσ,22 ≤ωtu(hnl,2), ∀n,ul lΥev(unel)ls,isrutehls,epleucptipveerluybl;,oluthnedvoanritahbeleunMceI(runtla),lintdyensoett.es weight and
ul,l Iul,l In the proof of Proposition 2, the terms
which is equivalent to ∆(n) = max (cid:88) S(n)(cid:16)g(n) −g¯(n) (cid:17) (17a)
u(cid:88)l∈UlSgu(l(n,nl1m,))l∈g¯a(cid:60)u(xn∗l(gn),ll,)1,1u(cid:88)l∈+UlSu(nl,)l(cid:16)gu(n∗l),l,1−g¯u(n∗l),l,1(cid:17) ≤It(hn,)1, ∀n ∆∆(ng(gnll),,)12 == ggll((,,nn12m))m∈∈a(cid:60)(cid:60)axxg(g(nnll,,))12uu(cid:88)(cid:16)ll∈∈IUU(lln)hhl(u(u−,nnlul)),,lllI,,¯21(Sn)uu((cid:17)nl∗l,),ll,(cid:16)1gl(,nu)∗lu,2∗l,−l,1g¯l(,nu)∗l,2(((cid:17)1177bc))
(cid:88) h(unl,)l,1S(n)g¯(n) + Iul,l Iu(nl,)l∈(cid:60)(Inu)l,l ul,l ul,l
ul∈Ul h(l,nu)l,2 ul,l l,u∗l,2 are called protection functions for constraint (10d), (10e),
max (cid:88) h(unl,)l,1S(n)(cid:16)g(n) −g¯(n) (cid:17) ≤I(n), ∀n adnedpen(1d0sh)o,nretshpeecutnivceelryt,aiwnhpoasreamvaeltueres.(iU.es.,inpgrottheectipornotvecatliuoen)
gl(,n2)∈(cid:60)(gnl,)2ul∈Ul h(l,nu)l,2 ul,l l,u∗l,2 l,u∗l,2 th,2 function, the optimization problem can be rewritten as
(cid:16) (cid:17)
I¯u(nl,)l+Iu(nl,)ml∈a(cid:60)xIu(nl,)l Iu(nl,)l−I¯u(nl,)l +σ2 ≤ωu(nl), ∀n,ul. (P4) max (cid:88) (cid:88)N 1x(n)B log (cid:32)1+ Su(nl,)lh(unl,)l,1(cid:33)
funScitniocen (tSheectmioanx3.f2u.n4citnio[n17o]v),ercoanvceoxnitvyexofstehte ipsroablecmonvPe3x xu(nl),Su(nl,)l,ωu(nl) ul∈Uln=12 ul RB 2 x(unl)ωu(nl)
subject to (10a), (10b), (10c), (10f), (10g) and
is conserved.
The problem P2 is the nominal problem of P3 where it is (cid:88) Su(nl,)lg¯u(n∗l),l,1+∆(gnl,)1 ≤ It(hn,)1, ∀n (18a)
assumedthattheperfectchannelstateinformationisavailable, ul∈Ul
it.hee.,itnhceluessitoimnaotefduvnacleuretsaianrteycionns(i1d0edr)e,d(a1s0ee)x,acatndval(u1e0sh.)W, tihthe (cid:88) hh((unnl,))l,1Su(nl,)lg¯l(,nu)∗l,2+∆(gnl,)2 ≤ It(hn,)2, ∀n (18b)
constraints in the optimization problem P3 are still affine. In ul∈Ul l,ul,2
order to express the constraints in closed-form (i.e., to avoid I¯(n) +∆(n) +σ2 ≤ ω(n),∀n,u (18c)
using the uncertainty set), in the following, we utilize the ul,l Iul,l ul l
nseott.ion of protection function [33], [34] instead of uncertainty awndhe(r1e7c∆),(gnrl,e)1s,p∆ec(gntli,)v2,elayn.d ∆(Inul),l are defined by (17a), (17b),
8
Proposition 3. The protection functions for the uncertainty 1 . In addition, since the channel uncertainties are random,
ajj
sets represented by general norms [i.e., by (15a), (15b), and a commonly used approach is to represent the uncertainty set
(16)] are by an ellipsoid, i.e., the linear norm with α = 2 so that the
∆(gnl,)1 =Ψ(l,n1) (cid:107)M(gnl,)1−1·(cid:16)S(l,n1)(cid:17)T (cid:107)∗ (19a) dpuroabllenmormP5istuarnlsinteoarancoonrmic qwuiathdraβtic=pr2og[r3a7m],m[i3n8g].prHoebnlecme,
∆(gnl,)2 =Ψ(l,n2) (cid:107)M(gnl,)2−1·(cid:16)H(ln)·S(l,n1)(cid:17)T (cid:107)∗ (19b) [a3id9e].dIanlgoorrditehrmtoissodlevveelPop5edefifincitehnetlfyo,llaowdiisntgribsuectetidong.radient-
∆(n) =Υ(n) (cid:107)M(n) −1·I(n) (cid:107)∗ (19c)
Iul,l ul Iul,l ul,l VI. ROBUSTDISTRIBUTEDALGORITHM
(cid:104) (cid:105)
where S(n) = S(n), S(n),··· , S(n) , H(n) = A. Algorithm Development
(cid:20)h(1n,l),1, h(2n,l)l,1,1,··· , h(|Unl)|,l1,1,l(cid:21) an2d,l(cid:107) · (cid:107)∗ is|Utlh|e,l dual nlorm of Statement 2. (a) The optimal power allocation for ul over
h(n) h(n) h(n) RB n is given by the following water-filling equation:
l,1,2 l,2,2 l,|Ul|,2
(cid:107)·(cid:107).
the uPncroeortfa:inUtysinsegtt(h1e5ae)xpbreecsosmioenswl(,n1) = M(gnl,)1·(cid:16)gΨ¯l((l,n,n11))−gl(,n1)(cid:17)T, Pu(nl,)l∗ = Sxu(u(nnll,))l∗∗ =(cid:34)δu(nl,)l− hω(unu(ln,)ll),1(cid:35)+ (23)
(cid:110) (cid:111) where δ(n) is found by (24).
(cid:60)(n) = w(n)|(cid:107)w¯(n)−w(n) (cid:107)≤1 , ∀n. (20) ul,l
gl,1 l,1 l,1 l,1 (b)TheRBallocationforul overRBnisobtainedby(12).
Besides, the protection function (17a) can be rewritten as Proof: See Appendix B.
max (cid:88) S(n)(cid:16)g(n) −g¯(n) (cid:17) Based on Statement 2, we utilize a gradient-based method
gl(,n1)∈(cid:60)(gnl,)1 ul∈Ul ul,l u∗l,l,1 u∗l,l,1 (ingdiveepnenidnenAtlpypepnerdfioxrmCs)thtoe urepsdoautercethaellvoacraitaibolnesa.nEdaaclhlocrealtaeys
(cid:16) (cid:17)T
= max S(n)· g(n)−g¯(n) resources to the associated UEs. For completeness, the dis-
l,1 l,1 l,1
gl(,n1)∈(cid:60)(gnl,)1 tributed joint RB and power allocation algorithm is summa-
= max S(n)·(cid:16)M(n)−1·w(n)(cid:17). (21) rized in Algorithm 1.
l,1 gl,1 l,1
gl(,n1)∈(cid:60)(gnl,)1
Algorithm 1 Joint RB and power allocation algorithm
Note that, given a norm (cid:107) y (cid:107) for a vector y, its dual
norm induced over the dual space of linear functionals z is 1: Each relay l ∈L estimates the reference gain g¯u(n∗),l,1 and
l
(cid:107) z (cid:107)∗= max zTy [34]. Since the protection function in g¯(n) from previous time slot ∀u ∈U and n∈N.
u∗,l,2 l l
(21) is the(cid:107)dyu(cid:107)a≤l1norm of uncertainty region in (15a), the proof 2: Iniltialize Lagrange multipliers to some positive value and
follows. The protection functions for the uncertainity sets in set t:=0, Su(nl,)l := PumNlax ∀ul,n.
(15b) and (16) are obtained in a similar way. 3: repeat
Since the dual norm is a convex function, the convexity of 4: Set t:=t+1.
P4 is preserved. In addition, when the uncertainty set for any 5: Calculate x(unl) and Su(nl,)l for ∀ul,n using (12) and (23).
vectoryisalinearnormdefinedby(cid:107)y(cid:107) =((cid:80)abs{y}α)α1 6: Update the Lagrange multipliers by (C.2a)–(C.2h) and
α
with order α ≥ 2, where abs{y} is the absolute value of calculate the aggregated achievable network rate as
(cid:88)
y and the dual norm is a linear norm with order β = 1+ Rl(t):= Rul(t).
1 . In such cases, the protection function can be defined as ul∈Ul
aα−li1near norm of order β. Therefore, the protection function 7: until t = Tmax or the convergence criterion met (i.e.,
abs{R (t)−R (t−1)}<ε, where ε is the tolerance for
becomesadeterministicfunctionoftheoptimizationvariables l l
convergence).
(i.e.,x(n),S(n),andω(n)),andthenon-linearmaxfunctionis
ul ul,l ul 8: Allocate resources (i.e., RB and transmit power) to as-
eliminated from the protection functions [i.e., from constraint
sociated UEs for each relay and calculate the average
(18a),(18b),and(18c)].Consequently,theresourceallocation
achievable data rate.
problemturnsouttobeastandardformofconvexoptimization
problem P5, where ∆(n) = Υ(n)(cid:107)M(n) −1·I(n) (cid:107) and
Iul,l ul Iul,l ul,l β Note that, the L3 relays are able to perform their own
A(j,:) denotes the j-th row of matrix A.
scheduling (unlike L1 and L2 relays in [27]) as an eNB.
IntheLTE-Asystem,whichexploitsorthogonalfrequency-
These relays can obtain information such as the transmission
division multiplexing (OFDM) for radio access, fading can
powerallocationattheotherrelays,channelgaininformation,
be considered uncorrelated across RBs (Chapter 1 in [35]);
etc. by using the X2 interface (Section 7 in [40]) defined
hence, it can be assumed that uncertainty and channel gain in
in the 3GPP specifications. In particular, a separate load
eachelementofgl(,n1) andgl(,n2) arei.i.d.randomvariables[36]. indicationprocedureisusedovertheX2interfaceforinterface
Therefore, M(n) and M(n) become a diagonal matrix. Note management (for details refer to [40] and references therein).
gl,1 gl,2
that for any diagonal matrix A with j-th diagonal element Asaresult,therelayscanobtainthechannelstateinformation
a , the vector A−1(j,:) contains only non-zero elements without increasing signaling overhead at the eNB.
jj
9
(P5) max (cid:88) (cid:88)N 1x(n)B log (cid:32)1+ Su(nl,)lh(unl,)l,1(cid:33)
x(unl),Su(nl,)l,ωu(nl) ul∈Uln=12 ul RB 2 x(unl)ωu(nl)
(cid:88)
subject to x(n) ≤ 1, ∀n (22a)
ul
ul∈Ul
N
(cid:88)S(n) ≤ Pmax, ∀u (22b)
ul,l ul l
n=1
(cid:88) (cid:88)N h(unl,)l,1S(n) ≤ Pmax (22c)
h(n) ul,l l
ul∈Uln=1 l,ul,2
 1
(cid:88) Su(nl,)lg¯u(n∗l),l,1+Ψ(l,n1)(cid:88)|Ul|(cid:16)M(gnl,)1−1(k,:)·Sl(,n1)(cid:17)ββ ≤ It(hn,)1, ∀n (22d)
ul∈Ul k=1
 1
(cid:88) hh((unnl,))l,1Su(nl,)lg¯l(,nu)∗l,2+Ψ(l,n2)(cid:88)|Ul|(cid:16)M(gnl,)2−1(k,:)·(cid:16)H(ln)·Sl(,n1)(cid:17)(cid:17)ββ ≤ It(hn,)2, ∀n (22e)
ul∈Ul l,ul,2 k=1
(cid:88)N 1x(n)B log (cid:32)1+ Su(nl,)lh(unl,)l,1(cid:33) ≥ Q , ∀u (22f)
2 ul RB 2 x(n)ω(n) ul l
n=1 ul ul
S(n) ≥ 0, ∀n,u (22g)
ul,l l
I¯(n) +∆(n) +σ2 ≤ ω(n), ∀n,u (22h)
ul,l Iul,l ul l
δ(n) = 12BRB(1+lnλ2ul) (24)
ul,l ρul +νlhh(l(u,nnlu)),ll,,21 +ψn(cid:16)g¯u(n∗l),l,1+Ψ(l,n1)m(unlu)lgl,1(cid:17)+ϕnhh(l(u,nnlu)),ll,,12 (cid:16)g¯l(,nu)∗l,2+Ψ(l,n2)m(unlu)lgl,2(cid:17)
B. Complexity Analysis are O(cid:0) 1 (cid:1) iterations required for convergence to have the
ε2
boundlessthanε[41].Hence,thecomplexityoftheproposed
Proposition 4. Using a small step size in gradient-based algorithm is O(cid:0)(cid:0)1+ 1 (cid:1)|U |N(cid:1).
ε2 l
updating, the proposed algorithm achieves a sum-rate such
that the difference in the sum rate in successive iterations is C. Cost of Robust Resource Allocation
less than an arbitrary ε > 0 with a polynomial computation
An important issue in robust resource allocation is the
complexity in |U | and N.
l substantial reduction in the achievable network sum-rate. Re-
ductionofachievablesum-rateduetointroducingrobustnessis
Proof: It is easy to verify that the computational com-
measuredbyR =(cid:107)R∗−R∗ (cid:107) ,whereR∗ andR∗ arethe
plexity at each iteration of variable updating in (C.2a)–(C.2h) ∆ ∆ 2 ∆
optimal achievable sum-rates obtained by solving the nominal
is polynomial in |U | and N. There are |U |N computations
l l
and the robust problem, respectively.
which are required to obtain the reference gains and if T
iterationsarerequiredforconvergence,theoverallcomplexity Proposition 5. Let ψ∗, ϕ∗, (cid:37)∗ be the optimal values of
of the algorithm is O(|U |N +T|U |N). Lagrange multipliers for constraint (10d), (10e), and (10h)
l l
intFerovralan[0y,κLagra]n,gtheemduislttaipnlcieerbκet,wiefewneκ(c0h)ooasnedκκ(∗0i)siunpptheer tihnePr2ed,urcetsipoenctoifvealcyh.iFeovraballelsvuamluersatoefc∆an(gnl,b)1e,∆ap(gnlp,)2r,oxainmda∆ted(Inual),sl
max
boundedbyκ .Thenitcanbeshownthatatiterationt,the
max N N N
distance between the current best objective and the optimum R ≈ (cid:88)ψ∗∆(n) +(cid:88)ϕ∗∆(n) + (cid:88) (cid:88)(cid:37)n∗∆(n) .
t ∆ n gl,1 n gl,2 ul Iul,l
κ2max+κ(t)2(cid:88)Λ(κi)2 n=1 n=1 ul∈Uln=1 (25)
objectiveisupperboundedby i=i .Ifwetake
t Proof: See Appendix D.
(cid:88)
2 Λ(κi) From Proposition 5, the value of R∆ depends on the
the step size Λ(κi) = √ai, where a isi=aismall constant, there Run∆cerctaaninbtyesceotntarnodllebdy. adjusting the size of ∆(gnl,)1 and ∆(gnl,)2,
10
D. Trade-off Between Robustness and Achievable Sum-rate and τ ≥ 0 are used for safe approximation of chance
Pj
constraints and depend on the probability distribution P .
The robust worst-case resource allocation dealing with j
For a fixed value of P the values of these parameters are
channel uncertainties is very conservative and often leads j
listed in Table III (see Appendix E). The constraints in (28a)
to inefficient utilization of resources. In practice, uncertainty
and (28b) turn the resource allocation problem into a conic
does not always correspond to its worst-case and in many
quadratic programming problem [39] and using the inequality
instances the robust worst-case resource allocation may not
(cid:107)y(cid:107) ≤(cid:107)y(cid:107) , the optimal RB and power allocation can be
be necessary. In such cases, it is desirable to achieve a trade- 2 1
obtained in a distributed manner similar to that in Algorithm
off between robustness and network sum-rate. This can be
1.Notethatin(29a)and(29b),theprotectionfunctionsdepend
achieved through modifying the worst-case approach, where
on Θ(n) and Θ(n). By adjusting Θ(n) and Θ(n), a trade-off
theuncertaintysetischoseninsuchawaythattheprobability l,1 l,2 l,1 l,2
between rate and robustness can be achieved.
of violating the interference threshold in both the hops is kept
below a predefined level, and the network sum-rate is kept
close to optimal value of nominal case. Therefore, we modify E. Sensitivity Analysis
the constraints (10d) and (10e) in P2 as
In the previous section we have seen that the protection
(cid:32) (cid:33) functions depend on Θ(n) and Θ(n). In the following, we
P (cid:88) S(n)g(n) ≥I(n) ≤Θ(n), ∀n (26a) analyze the sensitivity olf,1R to tlh,2e values of the trade-off
ul,l u∗l,l,1 th,1 l,1 ∆
ul∈Ul parameters. Using the protections functions (29a) and (29b),
P(cid:32) (cid:88) h(unl,)l,1S(n)g(n) ≥I(n)(cid:33)≤Θ(n), ∀n (26b) R∆ is given by
ul∈Ul h(l,nu)l,2 ul,l l,u∗l,2 th,2 l,2 R∆ ≈ (cid:88)N ψn∗∆˜(gnl,)1 +(cid:88)N ϕ∗n∆˜(gnl,)2 + (cid:88) (cid:88)N (cid:37)nul∗∆(Inul),l.
where Θ(n) and Θ(n) are given probabilities of violation of n=1 n=1 ul∈Uln=1
l,1 l,2 (30)
constraints (10d) and (10e) for any n in the first hop and
Differentiating (30) with respect the to trade-off parameters
second hop, respectively. By changing Θ(l,n1) and Θ(l,n2), the Θ(n) and Θ(n), the sensitivity of R , i.e., S (R ) =
trade-off between robustness and optimality will be achieved. l,1 l,2 ∆ Θ(n) ∆
ByreducingΘ(l,n1) andΘ(l,n2),thenetworkbecomesmorerobust ∂∂ΘR(∆n) is obtained as follows: l,i
l,i
against uncertainty, while by increasing Θ(n) and Θ(n), the
l,1 l,2 1
naeptTpwroooadrkcehas.luWwmi-htrheanttehthiisseitcnraocdnreesa-trosaefifdn.twsearueseaftfihneecfhuanncctieoncso,nfsotrraii.ni.edd. S (R )=−ψn∗u(cid:88)l∈UlτP2u(nl,)l,1(cid:16)Su(nl,)lgˆu(n∗l),l,1(cid:17)22 (31a)
values of uncertain parameters, (10d) and (10e) can be re- Θ(l,n1) ∆ Θ(n)(cid:118)(cid:117)(cid:117)(cid:116)2ln(cid:32) 1 (cid:33)
placed by convex functions as their safe approximations [33]. l,1 Θl(,n1)
Applying this approach we obtain 1
(cid:88) Su(nl,)lgu(n∗l),l,1 = (cid:88) Su(nl,)lg¯u(n∗l),l,1+ (cid:88) ξu(nl,)l,1Su(nl,)lgˆu(n∗l),l,1 S (R )=−ϕ∗nu(cid:88)l∈UlτP2l(,nu)l,2(cid:18)hhl(u(,nnlu)),ll,,21Su(nl,)lgˆl(,nu)∗l,2(cid:19)22.
uu(cid:88)ll∈∈UUll hh(u(l,nnlu)),ll,,12Su(nl,)lgl(,nuu)∗ll,∈2U=l ul∈Ul Θ(l,n2) ∆ Θl(,n2)(cid:118)(cid:117)(cid:117)(cid:116)2ln(cid:32)Θ(l1,n2)(cid:33) (31b)
(cid:88) h(unl),l,1S(n)g¯(n) + (cid:88) ξ(n) h(unl),l,1S(n)gˆ(n)
ul∈Ul h(l,nu)l,2 ul,l l,u∗l,2 ul∈Ul l,ul,2h(l,nu)l,2 ul,l l,u∗l,2 VII. PERFORMANCEEVALUATION
A. Simulation parameters and assumptions
where ξ(n) = gj(n)−g¯j(n),∀n is varied within the range Inordertoobtaintheperformanceevaluationresultsforthe
j gˆ(n)
[−1,+1]. Under the ajssumption of uncorrelated fading chan- proposed resource allocation scheme we use an event-driven
nels, all values of ξ(n) and ξ(n) are independent of each simulator in MATLAB. For propagation modeling, we con-
ul,l,1 l,ul,2 sider distance-dependent path-loss, shadow fading, and multi-
other and belong to a specific class of probability distribution
P(n) and P(n) , respectively. Now the constraints in (10d) path Rayleigh fading. In particular, we consider a realistic
ul,l,1 l,ul,2 3GPP propagation environment4 presented in [42]. For exam-
and (10e) can be replaced by Bernstein approximations of
ple, propagation in UE-to-relay and relay-to-D2D UE links
chance constraints [33] as follows:
follows the following path-loss equation: PL ((cid:96)) =
(cid:88) S(n)g¯(n) +∆˜(n) ≤I(n), ∀n (28a) 103.8 + 20.9log((cid:96)) + L + 10log(φ), whereu(cid:96)l,lis t[hdeB]link
ul∈Ul ul,l u∗l,l,1 gl,1 th,1 distance in kilometer; Lssuu accounts for shadow fading and
(cid:88) h(unl),l,1S(n)g¯(n) +∆˜(n) ≤I(n), ∀n (28b) is modelled as a log-normal random variable, and φ is an
ul∈Ul h(l,nu)l,2 ul,l l,u∗l,2 gl,2 th,2 ethxepoRnaeynlteiaigllhy fdaidstirnigbuctehdanrnaenldopmowevrargiaabinle. wFohricah sraempreeselinntks
where the protection functions ∆˜(n) and ∆˜(n) are given by
gl,1 gl,2 4AnyotherpropagationmodelforD2Dcommunicationcanbeusedforthe
(29a) and (29a), respectively. The variables −1 ≤ ηP+j ≤ +1 proposedresourceallocationmethod.