Analysis of Outage Probability and Throughput for Half-Duplex Hybrid-ARQ Relay Channels

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 7, SEPTEMBER 2012 3061

Analysis of Outage Probability and Throughput forHalf-Duplex Hybrid-ARQ Relay Channels

Behrouz Maham, Member, IEEE, Aydin Behnad, Member, IEEE, and Mérouane Debbah, Senior Member, IEEE

Abstract—We consider a half-duplex wireless relay networkwith hybrid-automatic retransmission request (HARQ) andRayleigh fading channels. In this paper, we analyze the averagethroughput and outage probability of the multirelay delay-limited(DL) HARQ system with an opportunistic relaying scheme indecode-and-forward (DF) mode, in which the best relay is se-lected to transmit the source’s regenerated signal. A simple anddistributed relay selection strategy is considered for multirelayHARQ channels. Then, we utilize the nonorthogonal cooperativetransmission between the source and selected relay for retrans-mission of source data toward the destination, if needed, usingspace-time codes. We analyze the performance of the system. Wefirst derive the cumulative density function (cdf) and probabilitydensity function (pdf) of the selected relay HARQ channels. Then,the cdf and pdf are used to determine the exact outage probabilityin the lth round of HARQ. The outage probability is requiredto compute the throughput-delay performance of this half-duplexopportunistic relaying protocol. The packet delay constraint isrepresented by L, which is the maximum number of HARQrounds. An outage is declared if the packet is unsuccessful afterL HARQ rounds. Furthermore, simple closed-form upper boundson outage probability are derived. Based on the derived upperbound expressions, it is shown that the proposed schemes achievethe full spatial diversity order of N + 1, where N is the numberof potential relays. Our analytical results are confirmed by sim-ulation results. In addition, simulation shows that our proposedscheme can achieve higher average throughput, compared withdirect transmission and conventional two-phase relay networks.

Index Terms—Hybrid ARQ, outage probability, performanceanalysis, relay channels, throughput.

I. INTRODUCTION

COOPERATION among devices has been considered toprovide diversity in wireless networks where fading may

significantly affect single links [1]. Initial works have em-phasized on relaying, where a cooperator node amplifies (ordecodes) and forwards, possibly in a quantized fashion [2], theinformation from the source node to help decoding at the desti-nation node [3]–[5]. The achieved throughput can be increased

Manuscript received November 14, 2011; revised April 12, 2012; acceptedMay 31, 2012. Date of publication June 8, 2012; date of current versionSeptember 11, 2012. This work was presented in part at the IEEE VehicularTechnology Conference, September 6–9, 2010, Ottawa, ON, Canada. Thereview of this paper was coordinated by Dr. C. Ibars.

B. Maham and A. Behnad are with School of Electrical and ComputerEngineering, College of Engineering, University of Tehran, Tehran 14395-515,Iran (e-mail: [email protected], [email protected]).

M. Debbah is with the Alcatel-Lucent Chair on Flexible Radio, SUPÉLEC,91192 Gif-sur-Yvette, France (e-mail: [email protected]).

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

Digital Object Identifier 10.1109/TVT.2012.2203342

with the integration of cooperation and coding, i.e., by lettingthe cooperator send incremental redundancy to the destination[6]. In particular, it has been shown in [6] that coded coopera-tion achieves a diversity order of two, whereas DF reaches onlya diversity order one, when the transmissions of source and co-operator are orthogonal. The capacity of cooperative networksusing both the DF and coded cooperation has been extensivelystudied [6], [7] for simple networks with simple medium-access-control protocols. In [8], a system with two transmissionphases that makes use of convolutional codes is analyzed andcharacterized by means of partner choice and performanceregions. Resource allocation for space-time-coded cooperativenetworks has been studied in [9] and [10], where the analysisof bit error rate and outage probability are also derived. Un-fortunately, in cooperative relaying, diversity gain is increasedat the expense of throughput loss due to the half-duplex con-straint at relay nodes. Different methods have been proposed torecover this loss. In [11], successive relaying using repetitioncoding has been introduced for a two-relay wireless networkwith flat fading. In [12], relay selection methods have beenproposed for cooperative communication with DF relaying.

In cooperative relaying, communication between the sourceand destination nodes is performed with the aid of multiple re-lays acting as the retransmitters. This technique presents a gen-eralization of the classical Automatic Repeat reQuest (ARQ)mechanisms. It provides a substantial increase in the diversitygain, compared to conventional ARQ, particularly in the caseof the so-called long-term (LT) quasi-static ARQ channels,where the channel connecting the source and destination nodesvery slowly varies from round to round (see, e.g., [13]). Aprominent alternative to reducing the throughput loss in relay-aided transmission mechanisms is the combination of bothARQ and relaying. This approach would significantly reducethe half-duplex multiplexing loss by activating ARQ for rare er-roneously decoded data packets, when they occur. Approachestargeting the joint design of ARQ and relaying in one commonprotocol have recently received more interest (see, for instance,[14] and [15]). Boujemaa [16] evaluated the expected waitingtime and the sojourn time of packets in the network of queuesof cooperative truncated hybrid-automatic retransmission re-quest (HARQ). A cross-layer approach to design an adaptationscheme over the relay channel in the presence of HARQ isdeveloped in [17] to enhance the system performance for datapacket transmission over block-fading relay channels. In [18],two smart HARQ schemes with incremental redundancy weredeveloped for a dual-hop network of two relays implementingcooperative communication. Motivated by the aforementionedsuggestion, we investigate and analyze throughput efficient

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3062 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 7, SEPTEMBER 2012

cooperative transmission techniques where both ARQ and re-laying are jointly designed. To this end, a diversity effect canbe introduced to relay networks by simply allowing the nodes tomaintain previously received information concerning each ac-tive message. Each time that a message is retransmitted, eitherfrom a new node or from the same node, every node in the relaynetwork will increase the amount of resolution information ithas about the message. Once a node has accumulated sufficientinformation, it will be able to decode the message and can actas a relay and forward the message (as in DF [3], [19]). Thisdiversity effect can be viewed as a space-time generalization ofthe time-diversity effect of hybrid ARQ (HARQ), as describedin [20]. Thus, the HARQ scheme, which is used in this paper, isa practical approach to designing wireless ad hoc networks thatexploit the spatial diversity, which is achievable with relaying.The retransmitted packets could originate from any node thathas overheard and successfully decoded the message. Currentand future wireless networks based on packet switching useHARQ protocols at the link layer. Hence, the performanceof HARQ protocols in relay channels has attracted recentresearch interest [14], [21], [22]. In [21], diversity–multiplexingtradeoffs for half-duplex single-relay protocols with HARQ arecomputed as the signal-to-noise ratio (SNR) tends to infinity.The tradeoffs are computed from the high-SNR asymptoticbehavior of the outage probability, which is defined as theprobability of packet failure after L HARQ rounds. However,in this work, we derive the exact expressions for the outageprobability and diversity–multiplexing tradeoffs in multiple-relay protocols with HARQ.

In this paper, we propose an efficient HARQ multirelayprotocol, which leads to full spatial diversity. We assume aDL network with the maximum number of HARQ roundsL, which represents the delay constraint. The protocol usesa form of incremental redundancy HARQ transmission withassistance from the selected relay via nonorthogonal transmis-sion in the second transmission phase if the relay decodes themessage before the destination. Note that, by nonorthogonaltransmission, we mean that the source node and the selectedrelay simultaneously retransmit the source data using space-time codes or beamforming techniques.

Our main contributions can be summarized here.

1) We introduce a distributed relay selection scheme forHARQ multirelay networks by using acknowledgment(ACK) or nonacknowledgment (NACK) signals transmit-ted by destination.

2) The exact and closed-form expressions are derived forthe outage probability of the system, which is defined asthe probability of packet failure after L HARQ rounds, inhalf-duplex.

3) For sufficiently high SNR, we derive a simple closed-form average outage probability expression for a HARQsystem with multiple cooperating branches, and it isshown that full diversity is achievable in the proposedHARQ relay networks.

4) The simulations show that the throughput of the HARQ-based relay channel is significantly larger than that ofdirect transmission as well as conventional two-phase

Fig. 1. Wireless relay network consisting of a source, a destination, and Nrelays.

relay networks for a wide range of SNRs, target outageprobabilities, delay constraints, and relay numbers.

The remainder of this paper is organized as follows: InSection II, the system model and protocol description are given.The performance analysis and the closed-form expressions forthe outage probability and asymptotic analysis of the systemare presented in Section III, which are utilized for optimizingthe system. In Section IV, the overall system performance ispresented for different numbers of relays and channel condi-tions, and the correctness of the analytical formulas is con-firmed by simulation results. Conclusions are presented inSection V.

Notations: Superscripts (·)t, (·)H , and (·)∗ stand for trans-position, conjugate transposition, and elementwise conjugation,respectively. The expectation operation is denoted by E{·}. Theunion and intersection of a collection of sets are denoted by⋃

and⋂

, respectively. Symbol |x| is the absolute value ofscalar x, whereas [x]+ denotes max{x, 0}. Logarithms log2and log are the base-two logarithm and the natural logarithm,respectively.

II. SYSTEM MODEL AND PROTOCOL DESCRIPTION

Consider a network consisting of a source, one or more relaysdenoted as i = 1, 2, . . . , N , and one destination. The wirelessrelay network model is shown in Fig. 1. It is assumed that eachnode is equipped with a single antenna. We denote the source-to-destination, source-to-ith-relay, and ith-relay-to-destinationlinks by f0, fi, and gi, respectively. Suppose that each link hasRayleigh fading, independent of the others. Therefore, f0, fi,and gi are i.i.d. complex Gaussian random variables (RVs) withzero mean and variances σ2

0 , σ2fi

, and σ2gi

, respectively. For anytwo nodes, σ2

x = k0/dνx is the path-loss coefficient, where dx is

the distance of link x, k0 depends on the operating frequency,and ν is the path-loss exponent, which typically lies in therange of 2 ≤ ν ≤ 6. As in [14], all links are assumed to beLT quasi-static, wherein all HARQ rounds of a single packetexperience a single channel realization. Subsequent packetsexperience independent channel realizations. Note that suchan assumption, which is applicable in low-mobility environ-ments such as indoor wireless local area networks, clearlyreveals the gains due to HARQ since temporal diversity is notpresent.

MAHAM et al.: ANALYSIS OF OUTAGE PROBABILITY AND THROUGHPUT FOR HALF-DUPLEX HARQ RELAY CHANNEL 3063

A. Relay Selection Strategy

In this paper, we use selection relaying, a.k.a. opportunisticrelaying [23], which selects the best relay among N availablerelays. Inspired by the distributed algorithm proposed in [23],which uses request-to-send and clear-to-send signals to selectthe best relay, we propose a selection procedure for HARQsystems using ACK/NACK signals.

1) In the first step, the source node broadcast its packettoward the relays and the destination. Thus, relays canestimate their source-to-relay channels.

2) If the destination correctly decodes the packet, the relayswould not cooperate. Otherwise, relays exploit the NACKsignal, which is transmitted by the destination to estimatetheir corresponding relay-to-destination channels.

3) The ith relay i = 1, . . . , N has a timer Ti, whose value isproportional to the inverse of min{|fi|2, |gi|2}.

4) The relay with maximum amount of min{|fi|2, |gi|2} hasthe smallest Ti. Whenever the first relay finished its timer,it broadcasts a flag packet toward the other relays to makethem silent and announce itself as the selected relay.

The transmission of NACK from the destination allows forthe estimation of the instantaneous wireless channel betweenrelay and destination at each relay according to the reciprocitytheorem [24]. We assume that the forward and backward chan-nels between the relay and destination are the same due tothe reciprocity theorem. Note that these transmissions occuron the same frequency band and the same coherence interval.Therefore, the relay with the minimum of its backward andforward channels is selected as the best relay. In other words,this policy selects the “bottleneck” of the two paths. In [25], therequest to an ARQ is served by the relay closest to the desti-nation, among those that have decoded the message. However,distance-dependent relay selection does not consider the fadingeffect of wireless networks and leads to a maximum diversity oftwo. Therefore, in this work, the request to an ARQ is served bythe relay under the best instantaneous channel conditions. Simi-lar to [23], we choose the relay with the maximum of min{γfi ,γgi}, i = 1, . . . , N , as the best relay, where γfi = |fi|2, andγgi = |gi|2. Thus, the index of the best relay is written by

r = arg maxi=1,...,N

{min{γfi , γgi}} (1)

and we define

γmax � min{γfr , γgr} = maxi=1,...,N

{min{γfi , γgi}} . (2)

As it can be seen from (1), index r depends on all relaychannels γfi and γgi , i = 1, 2, . . . , N . Hence, although γfi andγgi , i = 1, 2, . . . , N are independent RVs, γfr and γgr couldnot be independent.

B. Transmission Strategy

Let s and s denote the transmitted signals from the sourceand the selected relay, respectively. As shown in Fig. 2, duringthe first HARQ round, the relays and destination listen tothe source transmit block s. At the end of the transmission,

Fig. 2. Example of HARQ protocol for relay selection system. The selectedrelay decodes the message after HARQ round k. The source and selected relaysimultaneously transmit sl and sl, respectively, for all HARQ rounds l > k. Inthis figure, the destination decodes the message after HARQ round M , wherek < M ≤ L.

the destination transmits both the source and relays a one-bit ACK or NACK indicating the success or failure of thetransmission, respectively. The NACK/ACK is assumed to bereceived error-free and with negligible delay. Then, with theprocedure previously given, the best relay is selected. As longas NACK is received after each HARQ round and the max-imum number of HARQ rounds is not reached, the sourcesuccessively transmits subsequent HARQ blocks of the samepacket. As shown in Fig. 2, suppose the selected relay decodesthe message after HARQ round k, whereas the destination hasnot yet correctly decoded the message. For all HARQ roundsl > k, the source and the selected relay simultaneously transmits and s, respectively. For this nonorthogonal transmission, thedestination can be benefited from a spatial diversity technique,such as distributed space-time codes.

The Alamouti code can be used to transmit the coded pack-ets; hence, no interference occurs due to the simultaneous trans-missions of the source and relay. The effective coding rate afterl HARQ rounds is R/l b/s/Hz, where R is the spectral efficiency(in bits per second per hertz) of the first HARQ round. Let x andx denote the Alamouti code transmitted signals from the sourceand the selected relay, respectively. The received signal y at thedestination can be written as follows:

y =

{f0x+ grx+ n, if l > kf0x+ n, if l ≤ k

(3)

where index r refers to the index of the selected relay, and n isa complex white Gaussian noise sample with variance N0.

III. PERFORMANCE ANALYSIS

A. Average Throughput

Two definitions of throughput are considered. A frequentlyused metric for throughput analysis is the LT average through-put given by [13]

GLT =R

E{l} =R∑L−1

l=0 Pout(l)(4)

where E{l} is the average number of HARQ rounds spent trans-mitting an arbitrary message, and Pout(l) denotes the proba-bility that the packet is incorrectly decoded at the destinationafter l HARQ rounds. In the next section, we calculate closed-form solutions for the outage probability terms Pout(l) used in(4). The definition in (4) relies on the steady-state behavior ofseveral message transmissions. During this time, probabilitiesPout(l) are assumed to be constant. This assumption is removed


by considering the DL throughput, which is the throughput of asingle packet, defined by [14]

GDL =

L∑l=1

R

l[Pout(l − 1)− Pout(l)] . (5)

An advantage of definition (5), which does not resort to LTbehavior, is the ability to track slow time variations in thechannels.

In this section, we calculate the outage probability of theHARQ relay selection system proposed in the previous section.In addition to achieving a performance metric, outage proba-bility expression is needed in both throughput definitions in (4)and (5).

B. Exact Outage Probability

The outage probability in the first HARQ transmission round,i.e., Pout(1) is similar to the outage probability of the directtransmission. For direct transmission, the mutual informationbetween source and destination at each HARQ round is

If0 = Is,d = log2

(1 +

P

N0γf0

)(6)

where P is the average transmit power from the source. Sincethe total mutual information after the first HARQ round for LTquasi-static channels is If0 , the outage probability after the firstHARQ round is

Pout(1) = Pr[If0 < R] = 1 − exp

(−N0(2R − 1)

P σ20

). (7)

Next, we calculate Pout(l), l = 2, 3, . . . , L. Let χ denote theearliest HARQ round after which the relay stops listening to thecurrent message. The outage probability for the relay channelafter l HARQ rounds, where l > 1, is given by [21]

Pout(l) =

l−1∑k=1

Pout(l |l > k) Pr[χ = k]

+

L∑k=l

Pout(l |l ≤ k)Pr[χ = k]. (8)

To compute Pr[χ = k], the mutual information between sourceand relay for each HARQ round is given by

Ifr = log2

(1 +

P

N0γfr

)(9)

where γfr is an exponentially distributed RV with mean σ2fr

.The event χ = 1 means that the relay is able to decode themessage in the first transmission from the source, and thus, wehave

Pr[χ = 1] = Pr[Ifr > R] = Pr[γfr > μ1] (10)

where μ1 = N0/P (2R − 1). For k = 2, . . . , l − 1, χ = k if themessage is successfully decoded by the relay at the kth HARQ

round, and we have

Pr[χ = k] = Pr [(k − 1)Ifr < R, k Ifr > R]

= Pr [(k − 1)Ifr < R]− Pr[k Ifr < R]

= Pr[γfr < μk−1]− Pr[γfr < μk] (11)

where

μk =N0

P(2R/k − 1). (12)

For k = l, . . . , L, χ = k if the relay did not decode the messagesuccessfully after (l − 1) HARQ rounds, and thus, we have

Pr[χ = k] = Pr [(l − 1)Ifr < R] = Pr[γfr < μl−1]. (13)

From (11) and (13), Pr[χ = k], i.e., the probability that thenumber of retransmission needed for the relay to decode isequal to k, can be calculated as

Pr[χ = k]

=

⎧⎨⎩

Pr[γfr > μ1], if k = 1Pr[γfr < μk−1]− Pr[γfr < μk], if 1 < k < lPr[γfr < μl−1], if k ≥ l.

(14)

Since the index r given in (1) is dependent on channels,γfr and γgr are not independent for N > 1. Thus, obtain-ing a closed form for probability density function (pdf) isnot straightforward. As it is seen from (14), for computingPr[χ = k], the cumulative density function (cdf) of RV γfr isrequired. In the following, the cdf of the RV γfr is derived.

Proposition 1: Let γfi and γgi , i = 1, . . . , N , be indepen-dent exponential RVs with means σ2

fiand σ2

gi, respectively. The

cdf and pdf of γfr , where r is defined as (1), are given by

Fγfr(γ) =

N∏i=1

⎛⎜⎝1 − e

−

(1

σ2fi

+ 1

σ2gi

)γ

⎞⎟⎠−

N∑j=1

1σ2gj

e− γ

σ2fj

×γ∫

0

e− β

σ2gj

N∏i=1i�=j

(1 − e

−( 1

σ2fi

+ 1

σ2gi

)β)

dβ,

γ � 0 (15)

fγfr(γ) =

N∑j=1

⎛⎜⎝ 1σ2fj

e−

(1

σ2fj

+ 1

σ2gj

)γ

×N∏i=1i�=j

⎛⎜⎝1 − e

−

(1

σ2fi

+ 1

σ2gi

)γ

⎞⎟⎠+

1σ2fjσ2gj

e− γ

σ2fj

×γ∫

0

e− β

σ2gj

N∏i=1i�=j

⎛⎜⎝1 − e

−

(1

σ2fi

+ 1

σ2gi

)β

⎞⎟⎠dβ

⎞⎟⎠(16)

respectively.


Proof: The proof is given in Appendix A. �It is noteworthy that the integrals in (15) and (16) can be

easily calculated, as all terms in the expansion of integrands areof the exponential form.

Corollary 1: When σ2fi

= σ2f and σ2

gi= σ2

g for all i =1, 2, . . . , N , the cdf and pdf of γfr are simplified as follows:

Fγfr(γ) =

⎛⎝1 − e

−(

1

σ2f

+ 1

σ2g

)γ

⎞⎠

N

−Nσ2

f

σ2f + σ2

g

e− γ

σ2f

× B

⎛⎝1 − e

−(

1

σ2f

+ 1

σ2g

)γ

;N,σ2f

σ2f + σ2

g

⎞⎠

fγfr(γ) =

N

σ2f

e−(

1

σ2f

+ 1

σ2g

)γ

⎛⎝1 − e

−(

1

σ2f

+ 1

σ2g

)γ

⎞⎠

N−1

+N

σ2f + σ2

g

e− γ

σ2f

× B

⎛⎝1 − e

−(

1

σ2f

+ 1

σ2g

)γ

;N,σ2f

σ2f + σ2

g

⎞⎠ (18)

respectively, where B(x; a, b) =∫ x

0 ta−1(1 − t)b−1dt is the in-complete beta function [26].

Corollary 2: For high values of SNR, i.e., when γ=μk→0,the closed-form solution for (15) can be obtained as

Fγfr(γ) ∼= γN

(N∏i=1

(1σ2fi

+1σ2gi

))(1N

N∑i=1

σ2gi

σ2fi+ σ2

gi

).

(19)

From (15), Pr[χ = k] in (14) can be written as

Pr[χ = k] =

⎧⎨⎩

1 − Fγfr(μ1), if k = 1

Fγfr(μk−1)− Fγfr

(μk), if 1 < k < lFγfr

(μl−1), if k ≥ l.(20)

Next, the conditional probabilities Pout(l|l > k) andPout(l|l ≤ k) in (11) will be calculated. After correct decodingof the source packet at the relay, the relay helps the sourceby simultaneous transmission according to the Alamouti code.Hence, assuming that the relay transmits the same power P asthe source, the mutual information of the effective channel isgiven by

Is,r,d = log2

(1 +

P

N0γf0 +

P

N0γgr

). (21)

Let Itot,k,l denote the total mutual information accumulatedat the destination after l HARQ rounds and when χ = k. Fork < l, the relay listens for k HARQ rounds and simultaneouslytransmits the message with the source using the Alamouti codefor the remaining (l − k) HARQ rounds. For k ≥ l, the relaydoes not help the source during the l HARQ rounds. Hence

Itot,k,l =

{kIf0 + (l − k) Is,r,d, if k = 1, . . . , l − 1lIf0 , if k = l, . . . , L

(22)

where If0 is the mutual information between the source anddestination at each HARQ round and can be written as If0 =log2(1 + P/N0γf0).

Therefore, for k ≥ l, we have

Pout(l |l ≤ k) = Pr[lIf0 < R] = 1 − exp

(−μl

σ20

). (23)

From (22), the conditional probability Pout(l|l > k) can becalculated as

Pout(l |l > k)

= Pr[Itot,k,l < R]

= Pr

{log2

[(1 +

P

N0γf0

)k

×(

1 +P

N0γf0 +

P

N0γgr

)l−k]< R

}

= Pr

⎧⎪⎨⎪⎩γgr <

2R/(l−k)

PN0

(1 + P

N0γf0

)k/(l−k)− γf0 −

N0

P

⎫⎪⎬⎪⎭

=

μl∫γf0

=0

β(γf0)∫

γgr=0

e−

γf0σ20

σ20

fγgr(γgr ) dγf0 dγgr � Υ(l, k) (24)

where β(γf0) = (2R/(l−k)N0/P (1 + (P/N0)γf0)k/(l−k))−

γf0 −N0/P . Due to symmetry, the pdf of RV γgr , i.e., fγgr(γ),

is the same as the pdf of γfr , with perhaps different mean.Thus, the pdf of γgr can be found from (16) as

fγgr(γ) =

N∑j=1

⎛⎜⎝ 1σ2gj

e−

(1

σ2fj

+ 1

σ2gj

)γ

×N∏i=1i�=j

⎛⎜⎝1 − e

−

(1

σ2fi

+ 1

σ2gi

)γ

⎞⎟⎠+

1σ2fjσ2gj

e− γ

σ2gj

×γ∫

0

e− β

σ2fj

N∏i=1i�=j

⎛⎜⎝1 − e

−

(1

σ2fi

+ 1

σ2gi

)β

⎞⎟⎠dβ

⎞⎟⎠ .

(25)

By substituting fγgr(γ) from (16) into (24), Pout(l|l > k) is

obtained. Therefore, using (14), (23), and (24), the outage prob-ability in the lth stage of the HARQ process can be achieved as

Pout(l) =(1 − Fγfr

(μ1))Υ(l, 1)

+

l−1∑k=2

(Fγfr

(μk−1)− Fγfr(μk)

)Υ(l, k)

+

L∑k=l

Fγfr(μl−1)

(1 − e

−μlσ20

)(26)

where Υ(l, k) is defined in (24).


IV. ASYMPTOTIC ANALYSIS

In this section, we will study the achievable diversity gainsin a HARQ-based parallel relay network containing N relays.In particular, we consider the relay selection network with theselection strategy as stated in (1) and HARQ channels with themaximum number of HARQ rounds of L.

A tractable definition of the diversity gain is [27, eq. (1.19)]

Gd = − limρ→∞

log(Pout)

log(ρ)(27)

where ρ = P/N0. Thus, in the following, we investigate theasymptotic behavior and diversity order of Pout(l) in (8).

For calculating the minimum diversity gain of HARQ wire-less relay networks when the selection strategy in (1) is used,it is enough to derive an upper bound on the outage probabilityPout(l).

The RV γfr , which corresponds with the source–relay chan-nel of the selected relay, can be bounded as

γmax ≤ γfr ≤ γsmax (28)

where γmax is given in (2), and γsmax is defined as

γsmax = max

i=1,...,N{γfi}. (29)

The cdf of γsmax can be written as

Pr {γsmax < γ} = Pr{γf1 < γ, γf2 < γ, . . . , γfN < γ}

=

N∏i=1

(1 − e

− γ

σ2fi

). (30)

The cdf of γmax can also be written as

Pr{γmax < γ} = Pr{γ1 < γ, γ2 < γ, . . . , γN < γ} (31)

where γi = min{γfi , γgi} is again an exponential RV with theparameter equal to the sum of parameters of exponential RV γfiand γgi , i.e., 1/σ2

fiand 1/σ2

gi, respectively.

Thus, assuming that all channel coefficients are independentof each other, we can rewrite (31) as

Pr{γmax < γ} =

N∏i=1

⎛⎜⎝1 − e

−γ

(1

σ2fi

+ 1

σ2gi

)⎞⎟⎠ . (32)

Thus, it is easy to show that the cdf of γfr can be bounded as

Pr {γsmax < γ} ≤ Pr{γfr < γ} ≤ Pr{γmax < γ}. (33)

Therefore, combining (14) and (33), an upper bound onPr[χ = k] will be obtained as follows:

Pr[χ = 1] ≤ 1 − Pr [γsmax < μ1] (34)

Pr[χ = k] ≤ Pr[γmax < μk−1]− Pr [γsmax < μk] (35)

for 1 < k < l. From (32), (30), and (35), we have

Pr[χ = 1] ≤ 1 −N∏i=1

(1 − e

− μ1σ2fi

)� Λ1(1) (36)

and Pr[χ = k] for 1 < k < l can be calculated as

Pr[χ = k] ≤N∏i=1

⎛⎜⎝1 − e

−μk−1

(1

σ2fi

+ 1

σ2gi

)⎞⎟⎠

−N∏i=1

(1 − e

− μkσ2fi

)� Λ1(k). (37)

For k ≥ l, by combining (14), (32), and (33), we have

Pr[χ = k] ≤ Pr[γmax < μl−1]

=

N∏i=1

⎛⎜⎝1 − e

−μl−1

(1

σ2fi

+ 1

σ2gi

)⎞⎟⎠ � Λ2(l). (38)

Next, Pout(l|l > k) in (24) can be upper bounded as

Pout(l |l > k)

≤ Pr

⎧⎪⎨⎪⎩γmax <

2R/(l−k)

PN0

(1 + P

N0γf0

)k/(l−k)− γf0 −

N0

P

⎫⎪⎬⎪⎭

=

μl∫γf0

=0

β(γf0)∫

γmax=0

e−

γf0σ20

σ20

fγmax(γmax) dγf0 dγmax. (39)

The pdf of RV γmax, i.e., fγmax(γ) can be found by the

derivative of Pr{γmax < γ} in (32). Thus, we have

fγmax(γ) =

N∑i=1

(1σ2fi

+1σ2gi

)e−γ

(1

σ2fi

+ 1

σ2gi

)

×N∏j=1j �=i

⎛⎜⎝1 − e

−γ

(1

σ2fj

+ 1

σ2gj

)⎞⎟⎠ . (40)

Therefore, by substituting Pr[χ = k] from (37) and (38) andPout(l|l ≤ k) and Pout(l|l > k) from (23) and (39), respec-tively, in (8), an upper bound on outage probability of the lthstage of HARQ process, i.e., Pout(l), can be achieved.

To find the diversity order of Pout(l) from (40), an upperbound for fγmax

(γ) can be found as

fγmax(γ) ≤ NγN−1

N∏i=1

(1σ2fi

+1σ2gi

)(41)

which is a tight bound when γ → 0. Note that, in a high-SNRscenario, the behavior of the fading distribution around zero isimportant (see, e.g., [28]).

Using (41) and the fact that the exponential distribution is adecreasing function of γf0 , Pout(l|l > k) in (39) can be furtherupper bounded as

Pout(l |l > k)

≤μl∫

γf0=0

β(γf0)∫

γmax=0

N

σ20

γN−1max

N∏i=1

(1σ2fi

+1σ2gi

)dγf0 dγmax

≤ μl

σ20

μNl−k

N∏i=1

(1σ2fi

+1σ2gi

)� Ψ(l, k). (42)


Combining (8), (23), (37), (38), and (43), shown below, aclosed-form upper bound for the outage probability after lHARQ round can be obtained as

Pout(l) ≤l−1∑k=1

Λ1(k)Ψ(l, k) +L∑

k=l

Λ2(l)

(1 − e

−μlσ2f0

). (43)

Proposition 2: Assuming a HARQ system with N potentialrelays nodes, the relay selection strategy based on (1) canachieve the full diversity order of N + 1 for l > 1, and theoutage probability can be upper bounded as

Pout(l) ≤Δ(l)

ρN+1(44)

where

Δ(l) =(

2Rl − 1

)(2

Rl−1 − 1

)N L− l + 2σ20

N∏i=1

(1σ2fi

+1σ2gi

).

Proof: The proof is given in Appendix B. �Hence, similar to the DF selective relaying [3] or amplify-

and-forward-based opportunistic relaying [23], the full spatialdiversity is achievable when HARQ-based selection relayingis used.

Note that, at high SNR, a Taylor series expansion of Pout(1)in (7) around zero in the variable 1/ρ can be used. Thus, wehave Pout(1) = 2R − 1/ρσ2

0 +O(1/ρ2). Hence, the high-SNRdiversity order of the system with l = 1 is one, as expected.

V. NUMERICAL ANALYSIS

In this section, the performance of the proposed relay-selection HARQ system is studied through numerical results.We used the equal power allocation among the source and theselected relay. Assume that the relays and the destination havethe same value of noise power, and all the links have unit-variance Rayleigh flat fading, i.e., σ2

fi= σ2

gi= σ2

0 = 1. It isalso assumed that rate R is normalized to 1. We compare thetransmit SNR P/N0 versus outage probability performance.The block-fading model is used, in which channel coefficientsrandomly changed in time to isolate the benefits of spatialdiversity. The simulation result is averaged over 3 milliontransmitted symbols (channel realization trials).

Fig. 3 confirms that the analytical results attained inSection III for the outage probability have an accurate perfor-mance as the simulation results. We consider the maximumnumber of HARQ rounds to be L = 5. The outage probabilityat the second HARQ round, i.e., Pout(l = 2), is compared fortwo different number of relays N = 2, 4. One can see the exactoutage probability derived in (26) has a similar performanceas the simulated curved for all values of SNR. In addition,the closed-form outage probability expressions in (43) and(44) well approximate the simulated results, particularly undermedium- and high-SNR conditions. Furthermore, Fig. 3 showsthat the upper-bound expression in (43) is a tight upper bound.The asymptotic outage probability derived in (44) is also shownin Fig. 3, which confirms the full-diversity order of the proposedscheme.

Fig. 3. Outage probability Pout(l) curves of DL HARQ networks employingopportunistic relaying with two and four relays, when R = 1 b/s, L = 5 is themaximum number of rounds, and we consider HARQ round of l = 2.

Fig. 4. Outage probability performance of the proposed relay selection HARQsystem versus transmit SNR in a network with different numbers of relays,R = 1 b/s, L = l = 5 HARQ rounds, and σ2

fi= σ2

gi= σ2

0 = 1.

It is straightforward to show that the outage probability fordirect transmission after l HARQ round is [14, eq. (7)]

Pout,d(l) = 1 − exp

(−2R/l1

ρ σ20

). (45)

In Fig. 4, the outage probability at the l = L = 5th HARQround of the system with different numbers of relays are consid-ered. After selecting the best relay, Alamouti code is employedin the second transmission phase since the source and the bestselected relay are viewed as a two-antenna virtual MIMO.Compared to the single HARQ relaying system proposed in[14], the proposed HARQ opportunistic relaying system withN = 2, 3, and 4 relays considerably outperforms under all SNRconditions. For example, it can be seen that, in the outageprobability of 10−3, the system with two relays saves about8 dB in SNR, compared with the single-relay HARQ system.Furthermore, it can be checked that the system with N relayscan achieve the diversity order of N + 1.

In Fig. 5, we compare the opportunistic scheme based on (1),which is called best bottleneck link, with two other opportunis-tic relaying schemes. The first scheme is based on the selectionof a relay with the best source–relay link. This opportunistic


Fig. 5. Outage probability Pout(l) curves of multiple-relay HARQ networksemploying different opportunistic relaying schemes based on (1), (46), and(47) with two and four relays, when R = 1 b/s, when L = 5 is the maximumnumber of rounds, and when we consider HARQ round of l = 2.

relaying can be easily implemented by relays. Thus, unlike theselection based on (1), which needs the reciprocity of channels,this selection scheme can be implemented in frequency-divisionduplex (FDD) channels, as well as time-division duplex (TDD)channels. The index of the selected relay for this strategybecomes

α = arg maxi=1,...,N

{γfi}. (46)

Another scheme that we investigate is a fixed DF relaying withthe selection of the best relay–destination path. That is, the bestrelay can be selected by the destination, and the index of thebest relay is

ω = arg maxi=1,...,N

{γgi}. (47)

This opportunistic relaying can also be exploited in FDD chan-nels, as well as TDD channels (unlike the method based on (1),which requires the reciprocity of relay–destination channels).As can be observed from Fig. 5, our proposed HARQ-basedrelay selection outperforms two other schemes in which onlyone of the source–relay or relay–destination links is taken intoaccount in the decision of the selected relay. It can also be seenthat, under high-SNR conditions, both relay selections basedon (46) and (47) can only achieve the diversity of 2. However,using the selection strategy based on (1), full diversity of N + 1is obtainable, which confirms the validity of our analytic resultobtained in Proposition 2.

A DL throughput, which was defined in (5,) explicitlyaccounts for finite delay constraints and associated nonzeropacket outage probabilities. It can be shown that, for smalloutage probabilities, this DL throughput is greater than theconventional LT average throughput defined in (4). In additionto finite delay constraint, which is represented by the maxi-mum number L of HARQ rounds, higher layer applicationsusually require that Pout ≤ ρmax, where ρmax is a target outageprobability. The total LT and DL throughput are studied inFig. 6 subject to user quality-of-service constraints, whichare represented by outage probability target ρmax and delayconstraint L. In Fig. 6, the total LT and DL throughputs of

Fig. 6. DL and LT throughputs of direct transmission and relay selectionHARQ system versus transmit SNR for target outage probability 10−3, L =3 HARQ rounds, physical-layer rate R = 1 b/s, and σ2

fr= σ2

gr = σ20 = 1,

σ2fi

= σ2gi

= σ20 = 1.

opportunistic relaying HARQ system with N = 2 and 4 relaysare plotted as a function of SNR and compared with the direct-transmission HARQ system with L = 3, ρmax = 10−3, andthe following linear relay geometry: σ2

fr= σ2

gr= σ2

0 = 1. Asexpected, the presence of the relays significantly increases thethroughput. Furthermore, in agreement with [14, eq. (5)], it canbe seen that the DL throughput is greater than the LT averagethroughput. An interesting observation is that obtaining higherthroughputs is possible, as well as the diversity gain achieved bythe opportunistic relaying HARQ system, which was previouslyshown in Fig. 4. This behavior underscores the importance ofthe proposed system.

VI. CONCLUSION

In this paper, we have proposed a throughput-efficient relayselection HARQ system over Rayleigh fading. The throughput-delay performance of a half-duplex multibranch relay systemwith HARQ has been analyzed. A distributed relay selectionscheme has been introduced for HARQ multirelay networks byusing ACK/NACK signals transmitted by destination. We haveevaluated the average throughput and outage error probabilityperformance and have shown that the proposed technique sig-nificantly reduces the multiplexing loss due to the half-duplexconstraint while providing attractive outage error probabilityperformance. The closed-form expression outage probabilityhas been derived, which is defined as the probability of packetfailure after L HARQ rounds, in half-duplex. For sufficientlyhigh SNR, we have derived a simple closed-form averageoutage probability expression for a HARQ system with multiplecooperating branches. Based on the derived upper bound ex-pressions, it has been shown that the proposed scheme achievesthe full spatial diversity order of N + 1 in a nonorthogonalrelay network with N parallel relays. The analysis presentedhere allows quantitative evaluation of the throughput-delayperformance gain of the relay selection channel compared withdirect transmission. The numerical results confirmed that theproposed schemes can bring diversity and multiplexing gains inthe wireless relay networks.


APPENDIX APROOF OF PROPOSITION 1

First, we define the auxiliary RVs γi�= min{γfi , γgi} for

i = 1, 2, . . . , N . Since {γfi , γgi}Ni=1, are independent exponen-tial RVs, γi’s are also independent exponential RVs with thefollowing cdf:

Fγi(x) = 1 − e

−( 1

σ2fi

+ 1

σ2gi

)x

, x � 0. (48)

In addition, using partitioning theorem, we have

Pr{γfr � γ} =

N∑j=1

Pr {(γfr � γ) ∩ (r = j)} . (49)

For j = 1, 2, . . . , N , the summands of (49) can be obtained asfollows:

Pr {(γfr � γ) ∩ (r=j)}=Pr{(γj � γ) ∩ Γ ∩ (γfj � γ)

}=Pr {(γj � γ) ∩ Γ} − Pr

{(γj � γ) ∩ Γ ∩ (γfj > γ)

}(50)

where Γ =⋂N

i=1i�=j

(γi < γj). By substituting (50) in (49), we

obtain

Pr{γfr � γ} =

N∑j=1

Pr {(γj � γ) ∩ Γ}

−N∑j=1

Pr{(γj � γ) ∩ Γ ∩ (γfj > γ)

}= Pr{γmax � γ}

−N∑j=1

Pr{(γj � γ) ∩ Γ ∩ (γfj > γ)

}(51)

where γmax is defined in (2). Since γi’s are independent, thefirst term on the right side of (51) is given by

Pr{γmax � γ} = Pr

{N⋂i=1

(γi � γ)

}

=N∏i=1

Pr{γi � γ} =N∏i=1

Fγi(γ). (52)

In addition, the summand of the summation on the right sideof (51) is obtained as follows:

Pr{(γj � γ) ∩ Γ ∩ (γfj > γ)

}= Pr{γfj > γ}Pr

{(γj � γ) ∩ Γ

∣∣∣γfj > γ}

= Pr{γfj > γ}Pr{(γgj � γ) ∩ Γ

}= e

− γ

σ2fj

γ∫0

1σ2gj

e− β

σ2gj Pr{Γ} dβ

=1σ2gj

e− γ

σ2fj

γ∫0

e− β

σ2gj

N∏i=1i�=j

Fγi(β) dβ. (53)

Substituting from (48) into (52) and (53), one can obtain thecdf in (15) using (51)–(53). In addition, taking the derivativeof (15) with respect to γ results in the pdf of γfr , as givenby (16). �

APPENDIX BPROOF OF PROPOSITION 2

From a Taylor series expansion, it can be shown that Pout(l)in (43) can be rewritten as

Pout(l) ≤Ψ(l, 1)

(1 −

N∏i=1

μ1

σ2fi

)

+l−1∑k=2

[N∏i=1

μk−1

(1σ2fi

+1σ2gi

)−

N∏i=1

μk

σ2fi

]Ψ(l, k)

+L∑

k=l

μl

σ20

N∏i=1

μl−1

(1σ2fi

+1σ2gi

). (54)

By replacing Ψ(l, k) from (42) into (54), we have

Pout(l) ≤[μl

σ20

μNl−1

N∏i=1

(1σ2fi

+1σ2gi

)](1 −

N∏i=1

μ1

σ2fi

)

+l−1∑k=2

[N∏i=1

μk−1

(1σ2fi

+1σ2gi

)−

N∏i=1

μk

σ2fi

]

×[μl

σ20

μNl−k

N∏i=1

(1σ2fi

+1σ2gi

)]

+L∑

k=l

μl

σ20

N∏i=1

μl−1

(1σ2fi

+1σ2gi

). (55)

From (43) and by representing the factor μk in terms of theSNR ratio ρ, i.e., μk = 2R/k − 1/ρ, the outage probability inhigh SNR can be written as

Pout(l) ≤Δ(l)

ρN+1(56)

where

Δ(l) =(

2Rl − 1

)(2

Rl−1 − 1

)N L− l + 2σ20

N∏i=1

(1σ2fi

+1σ2gi

).

Hence, observing (56), the diversity order defined in (27) isequal to N + 1, which is the full spatial diversity for N + 1transmitting nodes.


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Behrouz Maham (S’07–M’10) received the B.Sc.and M.Sc. degrees in electrical engineering fromthe University of Tehran, Tehran, Iran, in 2005 and2007, respectively, and the Ph.D. degree from theUniversity of Oslo, Oslo, Norway, in 2010.

From 2006 to 2007, he was a System Engineerwith the Iran Telecommunication Research Center.From September 2008 to August 2009, he was withthe Department of Electrical Engineering, StanfordUniversity, Stanford, CA. He is currently an As-sistant Professor with the School of Electrical and

Computer Engineering, University of Tehran. He has held visiting appointmentswith Aalto University (formerly Helsinki University of Technology), Aalto,Finland, and the University of Oulu, Oulu, Finland, and has been the Alcatel-Lucent Chair at SUPÉLEC, Gif-sur-Yvette, France, and the University ofToronto, ON, Canada. He has more than 40 publications in major technicaljournals and conferences. Since April 2011, he has been serving as an Editorfor the Transactions on Emerging Telecommunications Technologies (formerlythe European Transactions on Telecommunications). His research interestsinclude scalable wireless communication and networking, with emphasis onrelay techniques, multiuser techniques, cognitive radio, interference alignment,and optimization theory.

Prof. Maham served as Technical Program Chair and Technical ProgramCommittee member of several major IEEE conferences.

Aydin Behnad (S’09–M’12) received the B.Sc. andM.Sc. degrees in electrical engineering from SharifUniversity of Technology, Tehran, Iran, in 2003 and2005, respectively, and the Ph.D. degree in electricalengineering from the University of Tehran, Tehran,in 2012.

He is currently with the School of Electricaland Computer Engineering, College of Engineering,University of Tehran. His current research interestsinclude the statistical analysis of interference, con-nectivity, and coverage in wireless ad hoc and sensor

networks, mobility modeling, and cooperative communications.

Mérouane Debbah (SM’08) received the M.Sc. andPh.D. degrees from the Ecole Normale Supérieure deCachan, France, in 1996.

He was with Motorola Labs, Saclay, France, from1999 to 2002, and the Vienna Research Center forTelecommunications, Vienna, Austria, from 2002 to2003. He then joined the Mobile CommunicationsDepartment, Institut Eurecom, Sophia Antipolis,France, as an Assistant Professor. Since 2007, hehas been a Full Professor with SUPÉLEC, Gif-sur-Yvette, France, and the holder of the Alcatel-Lucent

Chair on Flexible Radio. His research interests are information theory, signalprocessing, and wireless communications.

Dr. Debbah is a Wireless World Research Forum Fellow. He is an AssociateEditor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING. He receivedthe “Mario Boella” Prize Award in 2005; the 2007 General Symposium IEEEGLOBECOM Best Paper Award; the Wi-Opt 2009 Best Paper Award; the2010 Newcom++ Best Paper Award; the Valuetools 2007, Valuetools 2008, andCrownCom2009 Best Student Paper Awards; and the joint IEEE-SEE GlavieuxPrize Award in 2011.

Documents

Analysis of Outage Probability and Throughput for Half-Duplex Hybrid-ARQ Relay Channels