Base Station Selection in Uplink Macro Diversity Cellular Systems with Hybrid ARQ

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 6, JUNE 2011 1249

Base Station Selection in Uplink Macro DiversityCellular Systems with Hybrid ARQDavide Zennaro, Student Member, IEEE, Stefano Tomasin, Member, IEEE, and

Lorenzo Vangelista Senior Member, IEEE

Abstract—In a cooperative multi-cell network the uplink signalcoming from each mobile terminal (MT) is simultaneouslydemodulated by multiple base stations (BSs). Both backhaulcapacity and BS processing capabilities limit the number ofdemodulating BSs. In order to fit the information exchangeamong BSs to backhaul resources we minimize the averagenumber of demodulating BSs, under a constraint on the averageoutage probability. The BS selection problem becomes morecomplicated when error control configurations as automaticrepeat request (ARQ) and hybrid ARQ (HARQ) with chasecombining or incremental redundancy are considered. Multi-cellprocessing is implemented both by decoding the packet at eachBS and by a joint decoding among BSs. We first derive the outageprobability as a function of the number of cooperating BSs andthe error control strategy. A heuristic approach for BS selectionis then developed, to find at each frame which BSs performdemodulation and share the information in the backhaul. Lastly,we show that backhaul usage can be reduced up to 64% withrespect to an unoptimized solution.

Index Terms—ARQ, cellular systems, distributed MIMO,macro diversity, multi-cell processing.

I. INTRODUCTION

COOPERATION among base stations (BSs) of wirelesscellular systems is attracting attention as a means of im-

proving the network throughput by implementing a distributedmultiple input multiple output (MIMO) system, also denotedas macro diversity. For uplink communications, this approachcomes also under the name of multi-cell processing (MCP) andprovides that multiple BSs demodulate the signal transmittedfrom a single mobile terminal (MT) then performing jointdecoding at the serving BS [1]. After being initially consideredfrom an information theoretic perspective [2], MCP has beenstudied in more realistic scenarios with correlated Rayleighfading among antennas [3] and for possible application to3GPP long term evolution (LTE) advanced [4]. Scheduling andpower allocation have been first studied for downlink MCPin [5], then for orthogonal transmissions in [6] and for non-orthogonal downlink in [7]. MCP in the uplink requires a high-capacity backhaul to exchange information among the BSs andthe capacity of this system with limited backhaul throughputhas been investigated in [8] and [9]. In [10] the maximization

Manuscript received 1 June 2010; revised 1 December 2010. Part of thematerial of this paper has been presented at the IEEE Global CommunicationsConference (GLOBECOM), in Miami (FL), Dec. 2010.D. Zennaro, S. Tomasin and L. Vangelista are with the Department

of Information Engineering, DEI, University of Padova, via Gradenigo6/B - 35131, Padova, Italy (e-mail: {zennarod, tomasin}@dei.unipd.it,[email protected]).Digital Object Identifier 10.1109/JSAC.2011.110612.

of user throughput is addressed under the constraint of alimited backhaul capacity. In [11] cooperation among BSs hasbeen studied for application to a frequency division duplexing(FDD) LTE network where both demodulation and decodingare performed at the serving BS, while in [12] demodulationtakes place at each receiving BS and only soft information onbits is forwarded to the serving BS. For the asymptotic case ofa large number of users per cell the spectral efficiencies of theoptimum and linear minimum mean-squared error (MMSE)joint multi-cell receivers are investigated in [13], showing asignificant advantage over non-cooperative BS receivers.

The major issue in the implementation of uplink MCP is thehigh-capacity backhaul network required to exchange soft in-formation on the demodulated signal. Selecting the appropriateBSs for cooperation has the potential of significantly reducingthe backhaul usage. In this respect, [14] proposes to select theBSs that maximize the network throughput for a given userdeployment under backhaul capacity constraints. Performanceof handoff for uplink systems with macro diversity and BSselection is provided in [15] and [16]. For multi antenna BSsusing linear receive beamforming, in [17] cooperating BSs areselected through a dynamic greedy algorithm considering thesimultaneous presence of multiple MTs.

The overhead on the backhaul network is even moresignificant in cellular networks using an automatic repeatrequest (ARQ) or hybrid ARQ (HARQ) protocol for errorcontrol purposes. In fact, retransmissions by MTs must bedemodulated and shared by the cooperating BSs until thepacket is decoded.

In this paper we consider an uplink macro diversity systemand aim at adaptively selecting the number of cooperatingBSs for a given (H)ARQ protocol in order to minimize therequired number of demodulating BS – thus reducing backhauloccupation – but still ensuring a predefined outage probabilityafter a given number of (H)ARQ retransmissions. Instead ofimposing, as most of the literature does, a constraint on thebackhaul usage we aim at minimizing its occupation whilesatisfying a constraint on the outage probability. This approachis particularly useful when HARQ is used and the duration of apacket transmission is not fixed: the backhaul can therefore beexploited by each MT in a stochastic fashion. A second novelaspect of this paper is the variety of considered configurations:we analyze and optimize MCP with ARQ, HARQ with chasecombining and incremental redundancy. Moreover, we con-sider both a) joint decoding where cooperating BSs exchangesoft information at the end of each frame and decoding is

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1250 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 6, JUNE 2011

performed centrally at the serving BS, and b) local decoding,where each cooperating BS performs decoding independently.We propose a heuristic algorithm and compare its performancewith the optimal (computationally expensive) approach, show-ing the merits of the proposed solution. Although in this paperwe assume to perfectly know the statistical behavior of thechannel, the proposed approaches may provide insight forpossible new implementations of macro diversity solutions.The rest of the paper is organized as follows. In Section

II we introduce the MCP system model and formalize thecell selection problem. In Section III we derive the outageprobability for various transmit and receive configurations.Section IV presents the solution for BS optimization. Nu-merical results are presented in Section V, while conclusionsfollow in Section VI.

II. SYSTEM MODEL

We consider the uplink of a cellular system where a MTis linked to a serving BS, which is supported by cooperatingBSs in a macro diversity approach. A cooperating BS mayalso be serving other MTs and cooperating with other BSs.In the following we assume that a first selection process hasbeen performed, e.g. by the technique proposed in [6] andwe focus on the case of a single MT. Moreover, instead ofusing BSs for cooperation, dedicated devices (relays) couldbe used thus implementing a distributed broadband wirelesscommunication (BWC) system. The issue addressed in thispaper is whether the available BSs should be effectively usedin a cooperative fashion (i.e. be active) or they are not neededby the serving BS to meet certain quality of service targets.Decoding failures are due to channel fading and error

control is performed by using either ARQ or HARQ protocol,the latter either with chase combining (HARQ-CC) or incre-mental redundancy (HARQ-IR). Transmissions from MTs areorganized in frames. Considering a time division duplexingsystem, each frame comprises two slots. The first slot is usedby the MT to transmit data packets at fixed spectral efficiencyR (in bit/s/Hz). The second slot is used by the serving BS totransmit acknowledge (ACK) or not ACK (NACK) packets tothe MT, in order to signal successful or failed packet decoding,respectively. In general, the second slot is much shorter thanthe first, being the size of (N)ACK packets generally smallerthan that of the data packet. We also assume that (N)ACKpackets are transmitted at low rate and are always correctlydecoded at the MT. (H)ARQ frames are transmitted until eitherthe serving BS decodes the packet or a maximum of N framesis reached. In this latter case, if the packet is still not decodedthen we say that outage occurs.We assume that each BS is aware of the average channel

conditions, i.e. of the path loss of the channel with respectto the MT. Fading conditions are instead rapidly time-varyingand not known at the BSs. Based on the path loss conditions,the serving BS selects 1 ≤ xn ≤ M BSs – denoted activeBSs – that will cooperate in decoding the packet from theMT at frame n. Note that xn ≥ 1 since it includes the servingBS. We envision two cooperation methods among active BSsfor packet decoding: local decoding (LD) and joint decod-ing (JD). With LD, each active BS locally attempts packet

decoding after each frame and only exchanges informationon the decoding outcome (either success or failure). WithJD, active BSs exchange soft information at the end of eachframe and packet decoding is performed only at the servingBS. Let Γm be the average signal to noise ratio (SNR) ofthe channel between the MT and BS m. In particular, allBSs surrounding the MT are ordered with increasing pathloss of the BS-MT link, i.e. Γ1 ≥ Γ2 ≥ . . . ≥ ΓM , andthe xn BSs with lower path-loss are selected as active atframe n = 1, 2, . . . , N . Let pn(x,Γ, R) be the probabilityof decoding failure at the serving BS before frame n, withn = 1, . . . , N ; we also set p1(x,Γ, R) = 1. Note that wemade explicit the dependence of pn on the number of activeBSs at each frame x � [x1, . . . , xN ], the SNRs of the BSs-MTlinks Γ � [Γ1, . . . , ΓM ] and the required spectral efficiency R.The averaged number of active BSs considering the decodingprobability after each frame is therefore

f(x) �N∑

n=1

xnpn(x,Γ, R) . (1)

Note that backhaul occupancy is related to the number ofcooperating BSs. Note also that if our goal was uniquely tominimize the outage probability pN+1(x,Γ, R), we shoulduse all the available BSs, i.e. xn = M , for n = 1, 2, . . . , N ;on the other hand, this would increase the occupancy of thebackhaul. In this paper instead we aim at minimizing f(x)ensuring that the outage probability remains below a pre-setthreshold θ. In other words we address the problem

minx

f(x), (2a)

subject topN+1(x,Γ, R) ≤ θ, (2b)

xn ∈ {0, 1, . . . , M}, n = 1, 2, . . . , N. (2c)

The outage probability depends on channel conditions,cooperation methods between the BSs and ARQ type.

III. OUTAGE PROBABILITY COMPUTATION

In order to enforce the outage constraint (2b) we computethe outage probability after n frames for all (H)ARQ protocolsand for all cooperation methods. In this Section, we introducequantities and notations useful for the forthcoming analysisand we compute the outage probability in the case of atransmission on a flat Rayleigh fading channel including pathloss. We also focus our attention on the computation ofthe outage probability when the MT is at the edge of allsurrounding cells, i.e. Γm = Γ form = 1, 2, . . . , M . This caseis of interest since it provides the highest outage probabilityand must be considered for the evaluation of cell coverage asthe point at the furthest distance from BS at which receptionis possible.We analyze the case in which either JD or LD are used

together with either HARQ-IR or HARQ-CC protocols. Forcomparison purposes we also study the combination of LDand ARQ, while JD with ARQ is not analyzed since it turnsout to be equivalent to LD with the only difference of anincreased data exchange between BSs due to joint decoding.

ZENNARO et al.: BASE STATION SELECTION IN UPLINK MACRO DIVERSITY CELLULAR SYSTEMS WITH HYBRID ARQ 1251

In this section we do not consider co-user interference inorder to obtain analytical results. This scenario applies whenonly one MT is transmitting inside the sector at that specifictime and can be achieved by various techniques, includingboth use of multiple antenna/receivers in each BS and time-multiplexing among cells, which could either increase thecosts or decrease the spectral efficiency. In Section V wewill consider both inter-cell and intra-cell interference for thenumerical results.When JD and HARQ are considered, the number of active

BSs translates also into a higher required backhaul occupancy,since information on the received signal must be shared amongactive BSs. Instead, with LD no soft information sharing isneeded. Let us define the following indicator function relativeto BS m at frame n

ym(n) �{

0 no cooperation

1 cooperation.(3)

Remember that BSs are ordered with increasing path losswith respect to the MT therefore ym(n) can be univocallydetermined from xn. Furthermore, we define the number offrames in which BS m has cooperated until frame n as

Ym(n) �n∑

j=1

ym(j), (4)

for m = 1, 2, . . . , M , and for n = 1, 2, . . . , N .

A. Joint Decoding with HARQ-IR (JD-HARQ-IR)

For HARQ-IR, the serving BS accumulates information ateach frame, so that the resulting spectral efficiency for a givenchannel realization is

ΦJD−IR(x,Γ) �n∑

j=1

log2

(1 +

M∑m=1

ym(j)Γm|hm(j)|2)

,

(5)where hm(j) are the i.i.d. normalized complex Gaussianrandom variables of the Rayleigh fading channel from the MTto BS m at frame j. Path loss is taken into account by Γm.The outage probability after n frames can therefore be writtenas

pJD−IRn+1 (x,Γ, R) = P

[ΦJD−IR(x,Γ) < R

], (6)

where P[·] denotes the probability of the event in squaredbrackets.In general, a closed-form expression of (6) can not be

derived, even for the simple case of large n and Γm = Γfor all m, i.e. when the MT is at the same distance from allBSs. Indeed, in this case a Gaussian approximation of (5) hasa variance σ2

Φ for which, again, a closed-form expression isnot available, as discussed in Appendix A.

B. Local Decoding with HARQ-IR (LD-HARQ-IR)

When no soft information exchange is allowed betweenthe BSs and HARQ-IR is still considered, information isaccumulated at each BS. The resulting spectral efficiency canbe written as

ΦLD−IR(x, Γm) �n∑

j=1

ym(j) log2

(1 + Γm|hm(j)|2) (7)

and the outage probability after n frames is

pLD−IRn+1 (x,Γ, R) =

M∏m=1

P[ΦLD−IR(x, Γm) < R

]. (8)

In other words, BS m fails to decode the packet when thesum of Ym(N) random variables of the kind log2 (1 + ΓmZ)is less than R. As in [18], we approximate ΦLD−IR(x, Γm)with a Gaussian random variable having mean μLD−tot(m) �Ym(N)μ(Γm) and variance σ2

LD−tot(m) � Ym(N)σ2(Γm),where

μ(Γm) � E[log2(1 + ΓmZ)]= log2(e)e1/ΓmE1(1/Γm)

(9)

and

σ2(Γm) � E[(log2(1 + ΓmZ) − μ(Γm))2]= 2

Γmlog2

2(e)e1/ΓmG4,03,4

(1/Γm

∣∣0,0,0

0,−1,−1,−1

)−

−μ2(Γm),(10)

where E1(x) �∫∞1 t−1e−xtdt and Gm,n

p,q

(z∣∣a1,...,ap

b1,...,bq

)is the

Meijer G function. We therefore have

pLD−IRn+1 (x,Γ, R) ≈

M∏m=1

Q[μLD−tot(m) − R

σLD−tot(m)

], (11)

where Q[·] is the tail probability of the standard Gaussiandistribution.

C. Joint Decoding with HARQ-CC (JD-HARQ-CC)

For JD-HARQ-CC we define the random variable of thespectral efficiency

ΦJD−CC(x,Γ) � log2

⎛⎝1 +

n∑j=1

xj∑m=1

Γm|hm(j)|2⎞⎠ (12)

and the outage probability after n frames

pJD−CCn+1 (x,Γ, R) = P

[ΦJD−CC(x,Γ) < R

]. (13)

A closed-form expression for probability (13) is derived inAppendix B, but it turns out to be computationally intractable.Therefore we focus on the worst case scenario in the absenceof interference, where all BSs experience the same path losswith respect to the MT, i.e. the MT is at the edge of allsurrounding sectors and Γm = Γ for m = 1, 2, . . . , M . In thisspecial case we have that

∑nj=1

∑xj

m=1 |hm(j)|2 is gammadistributed with scale 1 and shape kn, where kn �

∑nj=1 xj

and its cumulative distribution function (CDF) is

FC(kn, z) =1

(kn − 1)!γ(kn, z), z ≥ 0 , (14)

where

γ(kn, z) =∞∑

l=0

(−1)l

l!zkn+l

kn + l, z ≥ 0. (15)

The relative average outage probability after n frames istherefore

pJD−CCn+1 (x,Γ, R) = P[Γ‖hn‖2 < 2R − 1]

= FC

(kn, 2R−1

Γ

).

(16)


In Section V we show the performance of a system with MTspositioned at random in the sector, also including interference,thus providing a broader picture than the special case whereΓm = Γ.

D. Local Decoding with HARQ-CC (LD-HARQ-CC)

For LD-HARQ-CC the average outage probability is theproduct of the partial outage probabilities at each BS. Bydefining the random variable of the spectral efficiency at BSm as

ΦLD−CCm (x, Γm) � log2

⎛⎝1 +

n∑j=1

ym(j)Γm|hm(j)|2⎞⎠ ,

(17)the probability of being in outage after n frames is

pLD−CCn+1 (x,Γ, R) =

M∏m=1

P[ΦLD−CC

m (x, Γm) < R]. (18)

Probability (18) can be rearranged as

M∏m=1

P

⎡⎣ n∑

j=1

ym(j)Γm|hm(j)|2 < 2R − 1

⎤⎦ =

M∏m=1

FC

(Ym(n),

2R − 1Γm

).

(19)

Note that similar results have been obtained in [19] withapplication to multi-relay networks.

E. Local Decoding with ARQ (LD-ARQ)

For local decoding with ARQ the outage probability can bedecomposed into the product – over the active BSs and frames– of the probabilities that each BS m fails decoding duringframe n. By defining the random variable of the spectralefficiency at BS m as

ΦLD−Am (x, Γm) � log2

(1 + ym(j)Γm|hm(j)|2) , (20)

the outage probability can be written as

pLD−An+1 (x,Γ, R) =

n∏j=1

xj∏m=1

P[ΦLD−Am (x, Γm) < R] =

=n∏

j=1

xj∏m=1

am,

(21)

where

am � P[Γm|hm(j)|2 < 2R − 1] == FC

(1, 2R−1

Γm

)=

= 1 − e−2R−1Γm .

(22)

Since BSs are ordered by increasing path loss, we have

0 ≤ a1 ≤ . . . ≤ aM ≤ 1. (23)

IV. BASE STATION SELECTION

Problem (2) is an integer programming problem whosesolution requires an exhaustive search (ES) over all possible(M + 1)N configurations of active BSs at each frame. As thenumber of cooperating BSs M and the number of (H)ARQframes N increase, ES becomes unfeasible. As an alternativewe consider a continuous relaxation of the problem: when theexpression of the outage probability is tractable we can indeedbenefit from this approach, as shown later in Section IV-Afor the LD-ARQ case and BSs placed at the same distancefrom the MT. On the other hand, when continuous relaxationdoes not help we resort to a heuristic recursive search (RS)algorithm that requires a significantly lower computationalcomplexity than that of ES.The RS algorithm starts from a feasible solution that sat-

isfies constraints (2b) and (2c) and at each iteration modifiesthe current solution in a greedy fashion reducing the totalaverage number of active BSs while satisfying the outageprobability constraint. As initial feasible solution we considerstatic cooperation (SC), where the number of cooperating BSsis the same at each frame, i.e.

x(0)n = xmin �min{1 ≤ M ′ ≤ M :

pN+1([M ′, . . . , M ′],Γ, R) ≤ θ}. (24)

The value of x(0)n is obtained by a search procedure over all

possible values of the single variable M ′.The idea behind the update of BS activity in the RS algo-

rithm is that if correct decoding occurs at an early stage, BSsscheduled for cooperation in the next frames are not active.However, early decoding comes at the cost of more activeBSs at the early frames, therefore a trade off between earlydecoding and having more active BSs at the last frames mustbe found. Indeed, as it will be shown in Section IV-A, thisapproach – although in general suboptimum – is optimum forthe ARQ system with equally distant BSs. Therefore, beforeupdating the BS activity we check if the new configurationyields a lower number of active BSs averaged over successfuldecodings after each frame.In details, at iteration i + 1 of the RS algorithm with i =

0, 1, . . ., BS activity is modified by selecting two frame indicess and d, with s < d. Then we decrease by one the numberof active BSs at the early frame s and increase by one thenumber of active BSs at the later frame d, i.e.

x(i+1)s = x(i)

s − 1 , x(i+1)d = x

(i)d + 1 . (25)

Then iff(x(i+1)) ≤ f(x(i)) (26)

and condition (2b) is satisfied the new BS configuration x(i+1)

is saved as the tentative optimum configuration of active BSs(x̄). Therefore at each iteration the average total number ofcooperating BSs is decreased while constraints (2b) and (2c)are satisfied. The choice of s and d is performed by scanningall N(N − 1)/2 possible values satisfying

s, d ∈ {1, 2, . . . , N} and s < d , (27)

starting from s = 1. Moreover, based on the assumption thata later BS cooperation occurs with a reduced probability, for


TABLE IRECURSIVE SEARCH ALGORITHM.

Set x(0)n = xmin from (24), for all n = 1, . . . , N ;

Set i = 0for s = 1 to �N

2�

for d = N down to s + 1while x

(i)s > 0

x(i+1)s = x

(i)s − 1;

if x(i)d < M

then x(i+1)d ← x

(i)d + 1

otherwise break;end ifif pN+1(x∗ ,Γ, R) ≤ θ and if f(x(i+1)) < f(x(i))

then x̄← x(i+1);end if

i← i + 1end

endendReturn x̄ as the set of active BSs.

a given value of s we check all values of d > s starting fromd = N and then reduce d, thus first activating BSs at the laterstages, in order to have the sharper decrease of f(xi+1). Thepseudocode of the resulting RS algorithm is reported in TableI. Note that in the RS algorithm we always keep unchangedthe total number of active BSs, i.e. we enforce

N∑n=1

x(i)n = xminN . (28)

When the MT is at the same distance from all BSs and ARQis used, condition (28) ensures that the outage probabilityconstraint is always satisfied. In fact, from (21) we observe thatin this case the outage probability only depends on the totalnumber of active BSs across all frames. However, in generalthe outage probability depends on the configuration of activeBSs in each frame and we must check the outage probabilityconstraint at each iteration.The performance of the RS algorithm closely approaches

that of ES, as we will see in Section V. Other schemes, such asgenetic or sphere decoding algorithms, may also lead to goodresults but at the cost of a higher computational complexity[14]. In fact, by representing an extremely straightforwardapproach to the problem of minimizing f(x), one of the majoradvantages of RS is actually its low computational complexity,as it converges in O

(MN2

)iterations, i.e. in polynomial time.

On the other hand, the exhaustive search of the optimum pointrequires the evaluation of (1) for each configuration of activeBSs at each frame, yielding a complexity O

((M + 1)N

), i.e.

exponential on N .

A. Optimization in the ARQ Case

In this section we derive the closed-form solution for thecontinuous relaxation of problem (2) in the LD-ARQ case. Wefirst observe the following property:

Theorem 1.1. Vector x solving the minimization problem (2)for LD-ARQ satisfies

xn ≤ xn+1, n = 1, 2, . . . , N − 1. (29)

Proof: See Appendix C

In Appendix D we derived the closed-form solution of therelaxed version of problem (2) in the worst case when theMT is at the same distance from all BSs, i.e. Γm = Γ form = 1, . . . , M , which provides

xn =

{− loga(1 − xn+1 log a) 1 ≤ n ≤ N − 1− loga

δθ n = N ,

(30)

with δ being a solution of the implicit equation

EN (δ) = 1 , (31)

where

E1(δ) � δ

E2(δ) � δ[1 + log δ

θ

]En(δ) = En−1(δ)

[1 + log En−1(δ)

En−2(δ)

]n = 3, . . . , N .

(32)It turns out that (31) is transcendent therefore it has to be

solved using numerical methods; on the other hand, its solutionis unique, as proved in Appendix D.Finally, to obtain a solution in Z

N we approximate thesolution x provided by (30) with the vertex in Z

N of thehypercube around x satisfying the outage constraint and min-imizing the average number of active BSs. This discretizationprocess is a source of sub-optimality for the analytical solutionbut it provides close to optimal performance, as corroboratedby numerical results, here omitted due to lack of space.

V. NUMERICAL RESULTS

We consider a cellular system and we focus on the per-formance of a single MT supported by M = 3 surroundingBSs arranged on three vertices of an hexagon of unitary sidedividing the cell into three sectors. The maximum number of(H)ARQ frames is set at N = 3. The considered antennaattenuation for out of sector signals is 20 dB [20]. Thechannel is assumed Rayleigh fading and includes path losswith coefficient 3.4, so that the SNR of the link between BSm and the MT at a distance d is

Γm = Γ0 (d)−3.4 |ξ|2 , (33)

where Γ0 is the average SNR at unitary distance and ξ isa complex Gaussian random variable with zero mean andunitary variance representing fading. We have considered aninterference scenario where one MT is active in each neigh-boring sector with interference probability q, where q = 0corresponds to the absence of interference and q = 1 refersto the scenario where all MTs in neighboring sectors aretransmitting. Location of interfering MTs is random withintheir sectors. Hence, with this scenario we take into accountboth inter-sector interference and outer-cell interference byincluding a ring of cells around the considered cell, with anactive MT for each sector of the cell ring.Fig. 1 shows the outage probability as a function of the

MT position in the sector where the serving BS is located atcoordinates (0,0) when R = 2 bit/s/Hz, LD-HARQ-IR is used,all BSs are active and in the absence of interference (q = 0).We observe that the position at the edge of all cells is the worstcase scenario yielding the highest outage probability as theMT is further away from any of the surrounding BSs. Similar


Fig. 1. pN+1 as a function of the position of the MT within a sector usingLD-HARQ-IR in the absence of interference (q = 0). The serving BS islocated at coordinates (0,0) in the plot.

Fig. 2. pN+1 as a function of the position of the MT within a sector usingLD-ARQ in the presence of interference (q = 1). The serving BS is locatedat coordinates (0,0) in the plot.

conclusions can be driven from Fig. 2, which reports, for thecase of interference (q = 1), the average (with respect to theinterfering nodes’ positions) outage probability as a functionof the MT position in the sector for the LD-ARQ technique.Note that similar results are obtained for the other cooperationmethods, where the point at the maximum distance fromsurrounding BSs represents always the worst case: they arenot shown due to lack of space.In the following two macro diversity configurations are

considered: 1) adaptive cooperation (AC), where the activeBSs are selected according to the methods proposed in thispaper, and 2) static cooperation (SC), where BSs are activeaccording to (24). For AC, we report the choice of activeBSs obtained with both the RS and the ES methods. Thetarget outage probability at the end of three frames is set atθ = 10−2.

A. Cell Coverage

1) No Interference Scenario: In order to evaluate the cellcoverage of our system using macro diversity and in theabsence of interference, i.e. q = 0, we place the MT at thecenter of the hexagon having the BSs on three equally spacedvertices, i.e. the MT is at the edge of its cell, equally distantfrom all surrounding BSs. In this situation we can establishthe minimum value of the SNR Γ0 that allows to fulfill theoutage constraint with the various cooperation approaches.We consider all BSs active, hence x = [M, . . . , M ] and the

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−20

−15

−10

−5

0

5

10

15

R [bit/s/Hz]

Γm

in [d

B]

JDLDNCLD−HARQ−IR−Ga

Fig. 3. Γmin vs the spectral efficiency of the first transmission for variousdecoding strategies. Solid line: HARQ IR; dashed line: HARQ-CC; dottedline: ARQ.

minimum value of Γ0 satisfying (2b) is

Γmin = minΓ0

{Γ0 : pN+1(x,Γ, R) ≤ θ} . (34)

Fig. 3 shows the value of Γmin as a function of the normal-ized spectral efficiency R using a) macro diversity with JD,b) macro diversity with LD and c) no cooperation (NC), i.e.only the serving BS is active. As we are considering the worstcase of a MT positioned at the edge of the cell, in JD and LDsystems all BSs cooperate and all BS selection methods yieldthe same performance. Therefore in the figure we show onlyone line for each error protection and cooperation scheme.We observe that JD significantly outperforms both macrodiversity with LD and NC with a reduction of the requiredΓmin by about 3.5 dB and 9 dB, respectively, for R = 0.4bit/s/Hz. Moreover, in Fig. 3 we also report the performanceof LD-HARQ-IR evaluated with the Gaussian approximationdiscussed in Section III-B, indicated with label LD-HARQ-IR-Ga and we observe that it provides a very good approximationof the simulated performance.On the other hand, Γmin can be converted into distance

values and we can compute the coverage achieved in theabsence of interference by the MCP system. Assuming thatNC achieves a spectral efficiency of 0.4 bit/s/Hz at the edgeof a hexagonal cell with unitary side, we obtain that the samespectral efficiency is achieved by JD and LD on cells with aside of 1.3 and 1.8, respectively.We can also read Fig. 3 as a plot of the achieved spec-

tral efficiency with respect to the SNR therefore obtaininginformation about diversity and multiplexing gains [21]. Inparticular, we note that HARQ-CC (dashed lines) has a highermultiplexing gain with respect to ARQ (dotted lines), whileHARQ-IR (solid lines) exhibits also a diversity gain, as canbe seen from the slope at high SNRs.

2) Interference Scenario: We now consider an interferencescenario with the MT in the sector of interest positioned atthe edge of three sectors (worst case scenario). Fig. 4 shows


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R [bit/s/Hz]

q max

JDLDNC

Fig. 4. Maximum interference probability qmax vs the spectral efficiency ofthe first transmission for various decoding strategies. Solid line: HARQ IR;dashed line: HARQ-CC; dotted line: ARQ.

the maximum interference probability

qmax = maxq

{q : pN+1(x,Γ(q), R) ≤ θ} , (35)

at which a given spectral efficiency R of the first transmissioncan be still reached by the MT of interest. Notice that theoutage probability pN+1(x,Γ(q), R) is also a function of theinterference probability q which determines the SINR vectorΓ(q). Also in this case we notice that JD approaches providethe highest immunity to interference for a fixed spectralefficiency or equivalently that by letting BSs cooperate theachievable spectral efficiency is increased for a given level ofinterference.

B. Performance with Random User Position

We now consider MTs uniformly drop at random in thesector and the number of active BSs is determined at eachposition for the various schemes with the SNR toward eachBS computed with (33). We focus on a spectral efficiency ofthe first transmission R = 0.4 bit/s/Hz.1) Total Active BSs: We first study the total average number

of active BSs (1) in order to compare the performance of thevarious policies. In analogy with the well known soft handoveroverhead, the function f(x) − 1 may be seen as a macrodiversity overhead, which is proportional to the additionalhardware resources required in the cellular system to deployuplink BS cooperation. In fact, active BSs must be able toprocess additional uplink signals in order to cooperate withthe serving BS. If no extra hardware is provided in the activeBSs, cooperation translates into a spectral efficiency reductionsince active BSs will not be able to serve their own MTs orMTs of other BSs requiring cooperation. Moreover, backhaulwill be partially used for the required exchange of informationbetween cooperating BSs, determining an additional overhead,which will be considered in more details in Section V-B2.For HARQ-IR, Fig. 5 shows E[f(x)] as a function of the

interference probability (E denotes the expectation), wherethe average is taken over MT position and Γ0 = 0 dB. JD

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

E[f]

q

RS−HARQ−IRSC−HARQ−IRES−HARQ−IR

Fig. 5. Total average number of active BSs for HARQ-IR as a function ofthe interference probability. Solid line: JD; dashed line: LD.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

2

2.5

3

3.5

4E

[f]

q

RS−HARQ−CCSC−HARQ−CCES−HARQ−CC

Fig. 6. Total average number of active BSs for HARQ-CC as a function ofthe interference probability. Solid line: JD; dashed line: LD.

outperforms LD in the entire range of q (especially for valuesof q close to 1) since sharing information among the BSssignificantly reduces the average number of active slots asthe outage probability is reduced. For both JD and LD wenote that our recursive algorithm performs very close to theoptimum solution and always outperforms SC: for q = 0.5,for example, we save more than 25% of active BSs in the JDcase.Fig. 6 shows E[f(x)] in the HARQ-CC case as a func-

tion of the interference probability. We first note that ourproposed algorithm has only slightly suboptimal performancewith respect to the best configuration of active BSs at eachframe; in particular, for q = 0.5, ES outperforms RS onlyby 4% for the JD case, while having a reduced computationalcomplexity: for example, in our simulations, for RS the searchis completed in less than 5 iterations in all cases, instead of(M + 1)N = 64 iterations of the ES approach. Moreover, RSsignificantly reduces the backhaul usage with respect to SC,


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

2.5

3

3.5

4

4.5

5

5.5

E[f]

q

RS−LD−ARQSC−LD−ARQES−LD−ARQ

Fig. 7. Total average number of active BSs as a function of the interferenceprobability for ARQ.

since for q = 0.5, it requires 24% less total active BSs onaverage for the JD case.Fig. 7 shows E[f(x)] for the LD-ARQ case as a function of

the interference probability. Also in this case RS significantlyoutperforms SC while only slightly under-performing withrespect to ES. For example, the resource saving with respect toSC is well observable at q = 0.5, where RS reduces E[f(x)]by about 20%.2) Backhaul Usage: In order to better assess the resource

usage of JD we assume that the BSs are on a ring, so thateach BS is connected only to the two adjacent BSs via a fiberoptic link. We denote by T the number of backhaul packetsexchanged among active BSs over all the frames. Note that Tcan be evaluated only for JD as both LD and NC do not requireexchange of soft information among the BSs. In JD schemesthe soft information on the received bits must be forwarded tothe serving BS and therefore the number of required backhaulpackets is the sum of the lengths of the paths from each activeBS to the serving BS, i.e.

T =N∑

n=1

[⌊xn − 1

2

⌋(⌊xn − 1

2

⌋+ 1

)+

+Υ(xn)⌊

xn + 12

⌋],

(36)

with

Υ(xn) =

{1 xn even

0 xn odd.(37)

Fig. 8 shows E[T ] where the average is taken over the MTposition. This figure remarks that RS brings considerablesavings to total resource usage with respect to SC. In fact, forexample, for q = 0.5 and JD-HARQ-IR, RS requires 64% lesspacket transmissions via fiber optic links, on average, whilethe saving increases up to 64% for JD-HARQ-CC at the samevalue of q.3) Payload Throughput: The normalized (with respect to

the packet duration) decoding delay is the time required for

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

E[T

]

q

RS−JD−HARQSC−JD−HARQES−JD−HARQ

Fig. 8. Total average of number backhaul transmissions for JD-HARQ as afunction of q. Solid line: HARQ-IR; dashed line: HARQ-CC.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.36

0.365

0.37

0.375

0.38

0.385

0.39

E[π

] [bi

t/s/H

z]

q

RS−HARQ−IRSC−HARQ−IRES−HARQ−IR

Fig. 9. Average payload throughput as a function of q for HARQ-IR. Solidline: JD; dashed line: LD.

decoding a packet when transmitted in uplink from a MT, i.e.

D =∑

n

pn(x,Γ), (38)

which depends on the MT positions. This delay is to beintended as the number of transmissions that are needed toachieve correct decoding and not as the effective time, whichmay be affected by delays in the retransmission requestsfrom the BS. Indeed, the backhaul must be designed inorder to support both the traffic of the soft information andthe signaling traffic about channel conditions that allows BSselection, in order not to impact on the overall delay.As a more comprehensive performance metric we consider

the normalized payload throughput π = RD , i.e. the ratio

between the required spectral efficiency and the time requiredto positively acknowledge the transmitted packets.Fig. 9 shows E[π] (with the average taken with respect to

the MT position) as a function of the interference probabilityfor a system using HARQ-IR. It can be observed that SCslightly outperforms RS. We explain this by observing that


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.36

0.365

0.37

0.375

0.38

0.385

0.39

E[π

] [bi

t/s/H

z]

q

RS−HARQ−CCSC−HARQ−CCES−HARQ−CC

Fig. 10. Average payload throughput as a function of q for HARQ-CC. Solidline: JD; dashed line: LD.

the minimization of the total active BSs reduces the numberof active BSs during the earliest frames while increases it inthe last frames. The consequence is that in general a reductionin the average total number of active BSs is observed with anincreased decoding delay associated to a higher probability ofa late decoding. We also note that both ES and RS approacheshave a very close performance.Fig. 10 shows E[π] as a function of the interference proba-

bility for a system using HARQ-CC. It can be observed thatSC outperforms RS also in this case. By comparing Fig.s 5and 6 with Fig.s 9 and 10 we observe that for a very closepayload throughput the various techniques require a differentnumber of active BSs, i.e. different backhaul occupancies.Hence, 1/f(x) from Fig.s 5 and 6 can be read as the energyefficiency of the transmission.

VI. CONCLUSIONS

We analyzed an uplink scenario in a cooperative multi-cellnetwork with error control based on (H)ARQ, with the purposeof minimizing the total average number of cooperating BSsunder a constraint on the outage probability. We proposed anefficient and effective recursive algorithm for both HARQ andARQ cases by exploiting properties of the general solution ofthe minimization problem and compared its performance to thecase in which the number of cooperating BSs is fixed (SC).Results from simulations confirm our claim in both HARQand ARQ cases, thus making our algorithm a computationallycheap solution to the minimization of the backhaul usage.

APPENDIX AGAUSSIAN APPROXIMATION FOR JD-HARQ-IR

We notice that Wj �∑M

m=1 ym(j)|hm(j)|2 is gammadistributed with scale 1 and shape xj and

E[log2(1 + ΓWj)] =1

(xj − 1)!×∫ +∞

0

log2(1 + Γw)wxj−1e−wdw.

(39)

We then obtain [22]

E[log2(1 + ΓWj)] =1

(xj − 1)! log 2×[(

1Γ

)xj π

xj sin(xjπ) 1F1

(xj ; xj + 1;

1Γ

)− 1

(xj − 1)!p−xj×{[

log1Γ− Ψ(xj)

]− 1

Γ(1 − xj)2F2

(1, 1; 2, 2− xj ;

1Γ

)}],

where

pFq (r1, . . . , rp; s1, . . . , sq; z) �∞∑

k=0

∏pi=1(ri)k∏qi=1(si)k

zk

k!(40)

Ψ(z) � 1(z − 1)!

∂(z − 1)!∂z

(41)

and (r)k = r(r + 1) . . . (r + k − 1) is the rising factorial ofr. We finally have

μφ =n∑

j=1

E[log2(1 + ΓWj)]. (42)

On the other hand, a closed-form expression for E[log22(1 +

ΓWj)] does not exist, therefore σ2φ can not be determined

analytically.

APPENDIX BCLOSED-FORM EXPRESSION OF PROBABILITY (13) AND

ARQ

Probability (13) can be rewritten thanks to [23] as

P

⎡⎣ n∑

j=1

xj∑m=1

Γm|hm(j)|2 < 2R − 1

⎤⎦ = 1−

(M∏

m=1

ΓYm(n)m

)M∑

m=1

Ym(n)∑l=1

φm,l(−Γm)

ΓYm(n)−l+1m

×

Ψ(Γm(2R − 1), Ym(n) − l

),

(43)

where Ψ(a, b) is the CDF of a Poisson random variable ofparameter a,

φm,l(b) = (−1)l−1∑

Ω(M,m,l)

∏j

(ij + Yj(n) − 1

ij

)τj(b) ,

the set Ω(M, m, l) defines partitions of l − 1 through thepositive integer indices ij , such that

∑Mj=1,j �=m ij = l − 1

and τj(b) = (Γj + b)−(Yj(n)+ij).

APPENDIX CPROOF OF THEOREM 1.1

We prove theorem 1.1 in the case N = 2; the proofcan be easily extended to the general case. Let x+ ≤ Mand x− ≤ x+ be two integers. First observe that bothconfigurations of vector x, [x−, x+] and [x−, x+], satisfy theoutage probability constraint since for LD-ARQ the outageprobability is only determined by the set of active BSs withinthe whole transmission irrespective of the frames in whichthey are active [see (21)]. Then both above configurations arefeasible and we aim at showing that

f(x−, x+) ≤ f(x+, x−) , (44)


which corresponds to the thesis. From (1) and (21) we have

f(x+, x−) − f(x−, x+) =x+ + x−x+∏

m=1

am+

−[x− + x+

x−∏m=1

am

].

(45)

Using (23) we have

f(x+, x−) − f(x−, x+) ≥(x+ − x−)+

+x−∏

m=1

am(x−ax+−x−x− − x+) .

(46)

Then we observe that for fixed∏x−

m=1 am, ax− is minimizedwhen a1 = . . . = ax− = a � x−

√∏x−m=1 am, therefore

defining

g(a) � x+ + x−ax+ − (x− + x+ax−) (47)

we havef(x+, x−) − f(x−, x+) ≥ g(a) . (48)

Now, we note that g(0) = x+ − x− ≥ 0 and g(1) = 0 andwe also have ∂g

∂a = x−x+a (ax+ − ax−) ≤ 0 for 0 ≤ a ≤ 1.

Since g(a) is a continuous function of a, the minimum ofg(a) is assumed for a = 1 and it is equal to g(1) = 0, henceg(a) ≥ 0, which concludes the proof.

APPENDIX DCLOSED-FORM SOLUTION OF (2) FOR Γm = Γ.

The average outage probability after n frames (21) can bewritten as

pLD−An+1 (x,Γ) = a

Pnj=1 xj , (49)

wherea � 1 − e−

2R−1Γ . (50)

By applying the Lagrange multipliers method we obtain theproblem

minx,δ

Λ(x, δ) = minx,δ

[f(x) − δ

(N∑

n=1

xn − loga θ

)]. (51)

Note that in defining the Lagrangian (51) we ignored thebounds (2c) on xn. Indeed, from Theorem 1.1 we know that ifthe solution of the unbounded problem (2a) with the constraint(2b) has some value of n for which xn > M , for sure wewill have xi > M, n ≤ i ≤ N . In such a case we setxi = M, n ≤ i ≤ N and solve the problem (2a) with theoutage constraint (2b) on the n − 1 unknowns x1, . . . , xn−1

(all the others being equal to M ).Setting to zero the derivatives of (51) we obtain

∂Λ∂xn

= aPn−1

j=1 xj

[1 +

(N−1∑l=n

xl+1aPl

j=n xj

)log a

]− δ = 0,

(52a)with n = 1, 2, . . . , N , and δ is provided by (31) and (32).Therefore

∂Λ∂δ

= −N∑

n=1

xn + loga θ = 0. (52b)

By manipulating the N + 1 equations (52) we obtain (30).Lastly, notice that (31) has a unique solution. In fact,

a product of two monotonically increasing functions is amonotone function if either of the two functions is positivevalued everywhere. Note also that both the feasible values ofδ in En(δ) and the fact that δ must be greater than zero, from(30), imply En(δ) > 0 for 1 ≤ n ≤ N − 1. Hence, EN (δ) is amonotone function and therefore (31) has a unique solution.

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Davide Zennaro (S’07) was born in Venice onAugust 26th, 1984. He received both the Laurea Tri-ennale (B.Sc degree) in Information Engineering andthe Laurea Specialistica (M.Sc degree) in Telecom-munications Engineering from the University ofPadova, Italy, in 2006 and 2008, respectively. Heis currently pursuing the Ph.D. degree in Informa-tion Engineering at the Department of InformationEngineering (DEI), University of Padova.Since January 2011 he is a visiting scholar at the

Department of Electrical and Computer Engineering,Texas A&M University, College Station, TX (USA), partially supported by an“A. Gini” fellowship. His research interests include clock synchronization inwireless networks, cooperation in cellular networks and distributed consensusalgorithms.

Stefano Tomasin (S’99, M’03) received the Laureadegree and the Ph.D. degree in TelecommunicationsEngineering from the University of Padova, Italy, in1999 and 2003, respectively. In the Academic year1999–2000 he was on leave at the IBM ResearchLaboratory, Zurich, Switzerland, doing research onsignal processing for magnetic recording systems.In the Academic year 2001–2002 he was on leaveat Philips Research, Eindhoven, the Netherlands,studying multicarrier transmission for mobile appli-cations. In the second half of 2004 he was visiting

at Qualcomm, San Diego (CA) doing research on receiver design for mobilecellular systems. Since 2005 he is Assistant professor at University of Padova,Italy. In 2007 he has been visiting faculty at Polytechnic University ofBrooklyn, NY, working on cooperative communications. His current researchinterests include signal processing for wireless communications, access tech-nologies for multiuser/multiantenna systems and smartgrid.

Lorenzo Vangelista (SM’02) was born in Bas-sano del Grappa, Italy, in 1967. He received theLaurea degree from University of Padova, Padova,Italy, in 1992, and the Ph.D. degree in Electricaland Telecommunication Engineering from Univer-sity of Padova, in 1995. He subsequently joined theTransmission and Optical Technology Departmentof CSELT, Torino, Italy. From December 1996 toJanuary 2002, he was with Telit Mobile Terminals,Sgonico (TS), Italy and then, up to May 2003, withMicrocell A/S, Copenaghen, Denmark. Until July

2006 he has been with the worldwide organization of Infineon Technologies,as program manager. Since October 2006 he is an Associate Professorof Telecommunication within the Department of Information Engineeringof Padova University, Italy. His research interests include signal theory,multicarrier modulation techniques, cellular networks, wireless sensors andactuators networks and smartgrid.

Base Station Selection in Uplink Macro Diversity Cellular Systems with Hybrid ARQ

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