Diversity-Multiplexing-Delay Tradeoffs in MIMO Multihop Networks with ARQ

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by  Yao Xie

Diversity-Multiplexing-Delay Tradeoffs in MIMO
Multihop Networks with ARQ
∗ ∗
Yao Xie , Andrea Goldsmith
∗
Department of Electrical Engineering, Stanford University, Stanford, CA.
Email: [email protected], [email protected]
Abstract—Tradeoff in diversity, multiplexing, and delay in TheDMDTanalysisassumesasymptoticallyinfiniteSNR.
0 multihopMIMOrelaynetworkswithARQisstudied,wherethe However, in the more realistic scenario of finite SNR, re-
1 random delay is caused by queueing and ARQ retransmission. transmissionisnotanegligibleeventandhencethequeueing
0 This leads to an optimal ARQ allocation problem with per- delay has to be brought into the picture (see discussions in
2 hop delay or end-to-end delay constraint. The optimal ARQ
allocation has to trade off between the ARQ error that the [5]).WithfiniteSNRandqueueingdelay,theDMDTwillbe

n receiver failstodecodeintheallocated maximumARQrounds different from that under the infinite SNR assumption. The
a andthepacketlossduetoqueueingdelay.Thesetwoprobability DMDT with queueingdelay is studied in [5] and an optimal
J of errors are characterized using the diversity-multiplexing-
ARQ adapted to the instantaneous queue state for the point-
5 delay tradeoff (DMDT) (without queueing) and the tail proba-
to-point MIMO system is presented therein.
bilityofrandomdelayderivedusinglargedeviationtechniques,
] respectively. Then the optimal ARQ allocation problem can be In this work, we extend the study [5] of optimal ARQ
T formulatedasaconvexoptimizationproblem.Weshowthatthe assuming high but finite SNR and queueing delay in point-
I optimal ARQ allocation should balance each link performance to-point MIMO systems to multihop MIMO networks. This
.
s aswellavoidsignificantqueuedelay,whichisalsodemonstrated work is also an extention our previous results in [4] to
c by numerical examples.
[ incorporate queueing delay. We use the same metric as that

4 I. INTRODUCTION usedin[5],whichcapturestheprobabilityoferrorcausedby
both ARQ error, and the packet loss due to queueing delay.
v 1In a multihop relaying system, each terminal receives
3 The ARQ erroris characterizedby informationoutageprob-
the signal only from the previous terminal in the route
5 ability,whichcanbefoundthroughadiversity-multiplexing-
and, hence, the relays are used for coverage extension.
3 delay tradeoff analysis [2], [4]. The packet loss is given
5 Multiple input-multipleoutput(MIMO) systems can provide by the limiting probability of the event that packet delay
. increaseddataratesbycreatingmultipleparallelchannelsand
2 exceeds a deadline. Unlike the standard queuing models for
1 increasingdiversitybyrobustnessagainstchannelvariations. networks (e.g., [6], [7]) where only the number of messages
9 Anotherdegreeoffreedomcanbeintroducedbyanautomatic
awaiting transmission is studied, here we also need to study
0 repeat request (ARQ) protocol for retransmissions. With the
the amount of time a message has to wait in the queue
:
v multihop ARQ protocol, the receiver at each hop feeds back of each node. Our approach is slightly different from [5],
i tothe transmittera one-bitindicatoronwhetherthemessage
X where the optimal ARQ decision is adapted per packet; we
can be decoded or not. In case of a failure the transmitter
study the queues after they enter the stable condition, and
r
sends additional parity bits until either successful reception
a hence we use the stationary probability of a packet missing
ormessageexpiration.TheARQprotocolprovidesimproved
a deadline. An immediate tradeoff in the choice of ARQ
reliabilitybutalsocausestransmissiondelayofpackets.Here
round is: the larger the number of ARQ attempts we used
we study a multihop MIMO relay system using the ARQ
for a link, the higher the diversity and multiplexing gain we
protocol. Our goal is to characterize the tradeoff in speed
canachieve,meaninga lowerARQ error.However,thisisat
versus reliability for this system.
a price of more packet missing deadline. Our goal is to find
The rate and reliability tradeoff for the point-to-point
anoptimalARQallocationthatbalancesthesetwoconflicting
MIMO system, captured by the diversity-multiplexingtrade-
goals and equalizes performance of each hop to minimizes
off (DMT), was introduced in [1]. Considering delay as the
the probability of error.
thirddimensioninthisasymptoticanalysiswithinfiniteSNR,
The remainder of this paper is organized as follows.
the diversity-multiplexing-delay tradeoff (DMDT) analysis
Section II introduces system models and the ARQ protocol.
for a point-to-point MIMO system with ARQ is studied in
Section III presents our formulation and main results. Nu-
[2], and the DMDT curve is shown to be the scaled version
merical examples are shown in Section IV. Finally Section
of the correspondingDMT curve withoutARQ. The DMDT
V concludes the paper.
in relaynetworkshasreceiveda lotof attentionas well(see,
e.g., [3].) In our recent work [4], we extended the point-to- II. MODELS AND BACKGROUND
pointDMDTanalysistomultihopMIMOsystemswithARQ A. Channel and ARQ Protocol Models
and proposed an ARQ protocol that achieves the optimal
Consider a multihop MIMO network consisting of N
DMDT.
nodes:withthesourcecorrespondingtoi=1,thedestination
correspondingtoi=N,andi=2,··· ,N−1corresponding
101/04/2010. SubmittedtoTheIEEEInternational SymposiumonInfor-
mationTheory2010. to the intermediate relays, as shown in Fig. 1. Each node
is equipped with M antennas. The packets enter the net- (iv) We consider both the long-term static channel, where
i
work from the source node, and exit from the destination H = H for all l, i.e. the channel state remains
i,l i
node, forming an open queue. The network uses a multihop constantduringalltheARQrounds,andindependentfor
automatic repeat request (ARQ) protocol for retransmission. differenti.Ourresultscanbe extendedto the theshort-
With the multihop ARQ protocol, in each hop, the receiver term static channel using the DMDT analysis given in
feedsbacktothetransmitteraone-bitindicatoraboutwhether [4].
the message can be decoded or not. In case of a failure
the transmitter retransmits. Each channel block for the same Message n(cid:13) Message n-1(cid:13) Message n-2(cid:13) Message n(cid:13) Message n-1(cid:13)
time t(cid:13) time t(cid:13)
message is called an ARQ round. We consider the fixed
Source(cid:13) Destination(cid:13) Source(cid:13) Destination(cid:13)
ARQ allocation, where each link i has a maximum of ARQ Message n+1(cid:13) Message n(cid:13) Message n-1(cid:13) Message n(cid:13)
time t+1(cid:13) time t+1(cid:13)
rounds L , i = 1,···N −1. The packet is discarded once
i Source(cid:13) Destination(cid:13) Source(cid:13) Destination(cid:13)
the maximum round has been reached. The total number of Message n+2(cid:13) Message n+1(cid:13) Message n(cid:13) Message n+1(cid:13) Message n(cid:13)
ARQroundsislimitedtoL: N−1L ≤L.ThisfixedARQ time t+1(cid:13) time t+1(cid:13)
i=1 i Source(cid:13) Destination(cid:13) Source(cid:13) Destination(cid:13)
protocol has been studied in our recent paper [4].
P Fig.2:Left:fullduplexmultihoprelaynetwork.Right:halfduplex
relay multihop MIMO relay network.
ARQ(cid:13) ARQ(cid:13)
M1(cid:13)(cid:13) M2(cid:13)(cid:13) MN(cid:13)-1(cid:13) MN(cid:13)(cid:13)
Source(cid:13) H1(cid:13)(cid:13) Relay(cid:13) H2(cid:13)(cid:13) HN(cid:13)-2(cid:13) Relay(cid:13) HN(cid:13)-1(cid:13) Destination(cid:13)
Fig. 1: Upper: relay network with direct link from source to Function 0.05
destination. Lower: multihop MIMO relay network without direct Cost −0.5
link. Log of −13−.015
20 5 6
Assume the packets are delay sensitive: the end-to-end 10 3 4
transmission delay cannot exceed k. One strategy to achieve k1 0 2 L1
this goal is to set a deadline ki for each link i with Fig. 3: The logarithm of the cost function (7) for the (4, 1, 2)
Ni=−11ki ≤ k. Once a packet delays more than ki it multihop MIMO relay networks. SNR is 20 dB.
is removed from the queue. This per-hop delay constraint
P
correspondstothefinitebufferateachnode.Anotherstrategy
istoallowlargeper-hopdelaywhileimposinganend-to-end
delay constraint. Other assumptions we have made for the
4 6
channel models are 3 5
4
(i) Thechannelbetweentheithand(i+1)thnodesisgiven *L12 *L223
1
1
by: 40 40
3 150 3 150
2 100 2 100
SNR 1 50 1 50
Y = H X +W , 1≤l ≤L . (1) r 0 0 k r 0 0 k
i,l i,l i,l i,l i
M
r i
The message is encoded by a space-time encoder into 8 80
7
a1,·s·e·qu,eLn}c,ewohferLe Tmaistritchees b{lXocik,lle∈ngtCh,Mai×ndT,Y: il,l =∈ ** + LL213456 *k1246000
2
CMi+1×T, i = 1,··· ,N −1, is the received signal at 41 40
3 150 3 150
the (i + 1)th node, in the lth ARQ round. The rate 2 100 2 100
1 50 1 50
of the space-time code is R. Channels are assumed r 0 0 k r 0 0 k
to be frequency non-selective, block Rayleigh fading Fig.4:AllocationsofoptimalARQ:L∗1,L∗2,L∗1+L∗2,k1∗,ina(4,1,
and independent of each other, i.e., the entries of the 2)MIMOrelaynetwork.SNRis20dB.(Theoptimalk2∗ =k−k1∗.)
channel matrices Hi,l ∈ CMi+1×Mi are independent
andidenticallydistributed(i.i.d.)complexGaussianwith
B. Queueing Network Model
zero mean and unit variance. The additive noise terms
W are also i.i.d. complex Gaussian with zero mean We use an M/M/1 queue tandem to model the multihop
i,l
andunitvariance.TheforwardlinksandARQfeedback relaynetworks.Thepacketsarriveatthe sourceasa Poisson
links only exist between neighboring nodes. processwithmeaninterarrivaltimeµ,(i.e.,thetimebetween
(ii) We considerboththe full-duplexandhalf-duplexrelays the arrival of the nth packet and (n − 1)th packet.) The
(see, e.g., [4]) where the relays can or cannot transmit random service time depends on the channel state and is
and receive at the same time, respectively, as shown upper bounded by the maximum ARQ rounds allocated L .
i
in Fig. 2. Assume the relays use a decode-and-forward As an approximation we assume the random service time at
protocol (see, e.g., [4]). Nodeiforeachmessageisi.i.d.withexponentialdistribution
(iii) We assume a short-term power constraint at each node andmeanL .Withthisassumptionwecantreateachnodeas
i
for each block code. Hence we do not consider power an M/M/1 queue. This approximation makes the problem
control. tractableandcharacterizesthequalitativebehaviorofMIMO
2
multihop relay network. Node i has a finite buffer size. The WecannotassumeinfiniteSNRbecauseotherwisethequeue-
packets enter into the buffer and are first-come-first-served ing delay will be zero, as pointed out in [5]. However we
(FCFS). Assume µ ≥ L so that the queues are stable, i.e., assumehighSNRtousetheDMDTresultsinoursubsequent
i
the waiting time at a node does not go to infinity as time analysis.
goeson.Burke’stheorem(see,e.g.,[7])saysthatthepackets
depart from the source and arrive at each relay as a Poisson III. DIVERSITY, MULTIPLEXING,AND DELAYTRADEOFF
process with rate pi/µ, where pi is the probability that a VIA OPTIMALARQ ROUND ALLOCATION
packet can reach the ith node. With high SNR, the packet
A. Full-Duplex Relay in Multihop Relay Network
reaches the subsequent relays with high probability: p ≈ 1
i
(theprobabilityofa packetdroppingissmallbecauseituses 1) Per-Hop Delay Constraint: The probability of error
upthemaximumARQ round.)Henceallnodeshavepackets depends on the ARQ window length allocation L , deadline
i
arrive as a Poisson process with mean inter-arrival time µ. constraint k , multiplexing rate r, and SNR ρ. For a given r
i
and ρ, we have
C. Throughput
P ({L },{k }|ρ,r)=
e i i
Denote by b the size of the information messages in bits,
N−1
B[t] the number of bits removed from transmission buffer P (ρ,{L })+ P (D >k ). (4)
ARQ i Queue i i
at the source at time slot t. Define a renewal event as the
i=1
event that the transmitted message leaves the source and X
Here D denotes the random delay at the ith link when the
eventually is received by the destination node possibly after i
queue is stationary. This P expression is similar to that
one or more ARQ retransmissions. We assume that under e
givenbyEquation(33)of[5].Ourgoalistoallocateper-hop
full-duplexrelaysthetransmittercannotsendanewmessage
ARQ round {L } and delay constraint {k } to minimize the
until the previous message has been decoded by the relay at i i
probability of error P .
which point the relay can begin transmission over the next e
Forthelong-termstaticchannel,usingtheDMDTanalysis
hop(Fig.2a.)Underhalfduplexrelaysweassumetransmitter
results [4] we have:
cannot send a new message until the relay to the next hop
completes its transmission (Fig. 2b.) N−1
The number of bits B¯ transmitted in each renewal event, PARQ(ρ,{Li})= ρ−fi(cid:16)Lri(cid:17). (5)
for full-duplexing B¯ = (N − 1)b, and for half-duplexing Xi=1
B¯ =(N−1)b/2whenN isodd,andB¯ =Nb/2whenN is Here fi(r) is the diversity-multiplexing tradeoff (DMT) for
even.Thelong-termaveragethroughputoftheARQprotocol a point-to-pointMIMO system formedby nodesi and i+1.
is defined as the transmitted bits per channel use (PCU) [2], Assuming sufficient long block lengths, f (r) is given by
i
which can be found using renewal theory [8]: Theorem 2 in [1] quoted in the following:
Theorem 1: [1] For sufficiently long block lengths, the
1 s B¯ . B¯
η = liminf B[t]= = diversity-multiplexing tradeoff (DMT) f(r) for a MIMO
s→∞ Ts t=1 E(τ) (N −1)T system with Mt transmit and Mr receive antennas is given
X
R, Full duplex; by the piece-wise linear function connecting the points
= R, Half duplex, N is odd; (2) (r,(Mt−r)(Mr −r)), for r =0,··· ,min(Mt,Mr).
 R2 1 + 1 , Half duplex, N is even. Denote the amount of time spent in the ith node by
 2 2N
the nth message as Di. The probability of packet loss
wathtehreeτsoisurtcheeauvnet(cid:0)rilagiet dreuarca(cid:1)htieosnfthroemdtehsetintiamtieonapnaocdkee,taanrrdive=.s PQueue(Di > ki) can be fnound as the limiting distribution of
lim P(Di >k ) (adapted from Theorem7.4.1 of [8]):
denotes asymptotic equality. A similar argument as in [2] n→∞ n i
. Lemma 2: The limiting distribution of the event that the
shows that E(τ)=(N −1)T for high SNR.
delay at node i exceeds its deadline k , for M/M/1 queue
i
models, is given by:
D. Diversity-Multiplexing-Delay Tradeoff
The probability of error Pe in the transmission has two PQueue(Di >ki)=nl→im∞P(Dni >ki)= Lµie−ki(cid:16)L1i−µ1(cid:17). (6)
sources: from the ARQ error: the packet is droppedbecause
the receiver fails to decode the message within the allocated Here the differencein the service rate andpacketarrivalrate
numberofARQ rounds,denotedasP ,andtheprobability 1 − 1 ≥ 0 and utility factor Li both indicate how “busy”
ARQ Li µ µ
that a message misses its deadline at any node due to large node m is. Using the above results, (4) can be written as
AquReQueirnegladyelnaeyt,wdoernkost.edFoalsloPwQuienueg.WtheewfrilalmgeivweoPrke foofr[v1a]r,iowues Pe({Li},{ki}|ρ,r)=N−1 ρ−fi(cid:16)Lri(cid:17)+ Lie−ki(cid:16)L1i−µ1(cid:17) . (7)
µ
assumethesizeofinformationmessagesb(ρ)dependsonthe i=1 (cid:20) (cid:21)
X
operatingsignal-to-noiseratio(SNR)ρ,andafamilyofspace
Notethatthequeueingdelaymessagelosserrorprobabilityis
time codes {C } with block rate R(ρ) = b(ρ)/T , rlogρ.
ρ decreasinginL ,andtheARQerrorprobabilityisincreasing
i
We use the effective ARQ multiplexing gain and the ARQ
inL .HenceanoptimalARQroundsallocationateachnode
i
diversity gain [2]
L should trade off these two terms. Also, the optimal ARQ
i
η(ρ) logP (ρ) allocationshouldalso equalizethe performanceofeachlink,
r , lim , d,− lim e . (3)
e ρ→∞logρ ρ→∞ logρ as the weakest link determines the system performance [4].
3
Hence the optimal ARQ allocation can be formulated as (ii) is always true for moderate k values. When (i) and (ii)
the following optimization problem: are violated, which means one error dominates the other,
then the optimal solution lies at the boundary of A. With
{Li}m,{ikni}∈APe({Li},{ki}|ρ,r) (8) the totalARQ roundsconstraintin (8), using the Lagrangian
multiplier an argument similar to above still holds.
where
2) End-to-End Delay constraint: When the buffer per
N−1L ≤L,
i=1 i nodeislargeenoughaperhopdelayconstraintisnotneeded,
A= 1≤L ≤µ, i=1,··· ,N −1 (9)
 PNi=−11iki ≤k.  aTnhdewexeacctanexipnrsetessaidonimfoprosteheantaielnpdr-otob-aebnidlitdyeloafythceonesntrda-itnot-.
The followingPlemma (proof omitted due to thespace limit) end delay is intractable. However a large deviation result is
shows that the total transmission distortion function (15) is available. The following theorem can be derived using the
convex in the interior of A. main theorem in [9]:
Lemma 3: The transmission distortion function (15) is Theorem 4: ForastationaryM/M/1queuetandem(with
convex jointly in L and k in the convex set full-duplex relays):
i i
N−1
L 1
({Li},{ki}:ki > 2(Lµi −i 1), i=1,···N −1.), kl→im∞nl→im∞k logPQueue Xi=1 Dni ≥k!=−θ∗,
Lemma 3 says that except for the “corners” of A the cost where θ∗ =minN−1 1 − 1 .
i=1 Li µ
function is convex. However these “corners” have higher
n o
probabilityoferror:k andL takeextremevaluesandhence This theorem says that the bottleneck of the queueing net-
i i
one link may have a longer queueing delay then the others. work is the link with longest mean service time Li. Hence
So we only need to search the interior of A where the cost theoptimalARQroundallocationproblemcanbeformulated
function is convex. as:
To gain some insights into where the optimal solution min P ({L },{k }|ρ,r) (14)
e i i
resides in the feasible domain for the above problem, we {Li}∈B
present a marginal cost interpretation. Note that the proba- where
bility of error can be decomposed as a sum of probability
P ({L },{k }|ρ,r)
of error on the ith link. The optimal ARQ rounds allocated e i i
on this link should equalize the “marginalcost” of the ARQ N−1
=P (ρ,{L })+P Di ≥k ,
error and the packet loss due to queueing delay. For node i, ARQ i Queue n
!
with fixed ki, the marginalcosts (partial differentials) of the Xi=1
N−1
ARQ errorprobability,andthepacketlossprobabilitydueto =. ρ−fi(cid:16)Lri(cid:17)+e−θ∗k. (15)
queueing delay, with respect to L are given by
i
i=1
X
∂ρ−∂fLi(cid:16)Lri(cid:17) = Lr2fi′ Lr ρ−fi(cid:16)Lri(cid:17)lnρ<0, (10) B = Ni=−11Li ≤L, (16)
i i (cid:18) i(cid:19) 1≤L ≤µ, i=1,··· ,N −1
and (cid:26) P i (cid:27)
ForhighSNR,thiscanbeshowntobeaconvexoptimization
∂PQueue(Di >ki) = 1 1+ k e−ki(cid:16)L1i−µ1(cid:17) >0. (11) problem. A simple argument can show that the packet loss
∂Li µ(cid:18) Li(cid:19) probability with the per-hop delay constraint is larger than
Note that f′ < 0. The optimal solution equalizes these that using the more flexible end-to-end constraint.
i
two marginal costs by choosing L ∈ [1,µ]. Note that
i
these marginalcost functionsare monotonein L , hence the
i
equalizing L∗ exists and 1 < L∗ < µ if the following two B. Half-duplex Relay in Multihop Network
i i
conditions are true for L =1 and L =µ:
i i Half-duplex relay is not a standard queue tandem model.
(i): ∂PQueue(Di >ki) <− ∂ρ−fi(cid:16)Lri(cid:17) , (12) Hthoewteavilerprwobeabcialnityaflsoor tdheeriveneda-tol-aerngde ddeelvaiyatoiofnarmesuullttihfoopr
∂L ∂L (cid:12)
i (cid:12)L=1 i (cid:12) network with half-duplex relays (proof in the Appendix):
(cid:12) (cid:12)L=1
(ii): ∂PQueue(Di >ki)(cid:12)(cid:12) >− ∂ρ−fi(cid:16)Lri(cid:17)(cid:12)(cid:12)(cid:12) , (13) haTlfh-deuoprelemx 5re:laFyosr),awshtaetniotnhaernyuMmb/Mer/o1f qnuoedueeNtanisdelamrg(we:ith
∂L ∂L (cid:12)
i (cid:12)(cid:12)L=µ i (cid:12)(cid:12)L=µ 1 N−2
(cid:12) (cid:12) lim lim logP Di ≥k =−θ∗. (17)
These conditions involv(cid:12)e nonlinear inequaliti(cid:12)(cid:12)es involving k→∞n→∞k Queue n !
µ, ρ, r, Mi and Mi+1, which defines the case when the Xi=1
optimal solution is in the interior of A. Analyzing these FromthistheoremweconcludethattheoptimalARQalloca-
conditions reveals that these conditions tend to satisfy at tion problem with the end-to-end constraint and half-duplex
lower multiplexing gain r, small Mi or Mi+1, small ki, and relays can be formulated the same as that with full-duplex
larger µ (light traffic). Note that with high SNR condition relays (14).
4
IV. NUMERICAL EXAMPLES Lindley’s recursion has the solution:
Wi = max(σi −τi ), i=2,···N −2,
Consider a MIMO relay network consists of a source, a n ji≤n ji,n−1 ji+1,n
relay, and a destination node. The relay is full-duplex. The W1 = max(σ1 +σ2 −τ1 ). (22)
numberofantennasoneachnodeis(M1,M2,M3),M1 =4, n j1≤j2 j1,j2−1 j1,j2−1 j1+1,j2
M2 =1, and M3 =2, where the relay has a single antenna. where the partial sum τi = p Ai and σ = p Si.
Other parameters are: ρ = 20dB, k = 30, L = 8, and the l,p k=l k l,p k=l k
Hence
multiplexing gain is r = 2. The base 10 logarithm of the P P
cost function (7) is shown in Fig. 3. We have optimized the Di =max(σi +Si −τi ), i=2,···N −2. (23)
cost function with respect to L2 and k2 so we can display it n ji≤n ji,n−1 n ji+1,n
in three dimensions. Note that the surface is convex in the From(20)wehaveτi =τi−1+Di−1−Di−1 forl ≤p+1,
l,p l,p p l−1
interior of the feasible region. The optimal L∗, L∗, k∗ are and 0 otherwise. Plug this into (23) we have
1 2 1
showninFig.4.Alsonotethatasrincreasestothemaximum
Di =max(σi+1 +Si −τi−1 −Di−1+Di−1). (24)
possible r = 4, the total number of ARQ rounds allocated n ji≤n ji,n−1 n ji+1,n n ji
L∗+L∗ gradually increases to the upper bound L=8 as k
1 2 Hence the recursive relation if we move Di−1 to the left-
increases. n
hand-side:
V. CONCLUSIONS AND FUTURE WORK Dni +Dni−1 =mji≤anx(σji+i,n1−1+Sni −τjii−+11,n+Dji−i 1). (25)
We have studied the diversity-multiplexing-delaytradeoff Now from (23) we have Dji−i 1 = maxji−1≤ji(σji(i−1),ji−1+
in multihop MIMO networks by considering an optimal Si−1−τi−1 ). Plug this in the above (25) we have
ARQallocationproblemtominimizetheprobabilityoferror, ji j(m−1)+1,jm
which consists of the ARQ error and the packet loss due to Dni +Dni−1
queueingdelay.Ourcontributionistwo-fold:wecombinethe = max (σi+1 +Si +σi +Si−1−τi−1 )
DMDT analysis with queueing network theory, and we use j(i−1)≤ji≤n ji,n−1 n j(i−1),ji−1 ji j(i−1)+1,n
the tail probability of random delay to find the probability Do this inductively, we have
of packet loss due to queueing delay. Numerical results
N−2 N−2
show that optimal ARQ should equalize the performance Di = max (σi+1 +Si )−τ1 .
of each link and avoid long service times that cause large i=2 n j2≤···≤jN−1=n"m=2 ji,j(i+1)−1 ji+1 j2+1,n#
queueing delay. Future work will investigate joint source- X X
If we also add D1 =W1+S1 to the above equation, after
channelcodinginmultihopMIMOrelaynetworks,extending n n n
rearranging terms we have:
the results of [5].
N−2 N−2
Di = (σi +Si )−τ1
APPENDIX n j(i−1),ji−1 j(i+1) j2+1,n
i=1 i=2
X X
Proof of Theorem 5 +Sj12 +σj11,j2−1+Sj22 +σjNN−−12,j(N−1)−1. (26)
Fornodei,i=1···N,lettherandomvariableSni denotes Note that σi is independent of Si . For long
theservicetimerequiredbythenthcustomerattheithnode j(i−1),ji−1 j(i+1)
queue we can ignored the last four terms caused by edge
(thenumberofARQsusedforthenthpacket),andAi bethe
n effect (the source and end queue of the multihop relay
interarrivaltimeofthenthpackets(i.e.,thetimebetweenthe network).Bystationarityoftheserviceprocessσi +
arrival of the nth and (n−1)th packages to this node). The j(i−1),ji−1
Si has the same distribution as σi . Then (26)
waiting time of the nth packet at the ith node Wi satisfies j(i+1) 0,ji−j(i−1)
n reduces to the case studied in [9] and we can borrow the
Lindley’s recursion (see [9]):
large deviation argument therein to derive the exponent θ∗.
Wi =(Wi +Si+1 −Ai)+, 2≤i≤N −2, (18)
n n−1 n−1 n
where (x)+ =max(x,0). The total time a message spent in
a node is its waiting time plus its own service time, hence REFERENCES
Di =Wi +Si. (19)
n n n [1] L.ZhengandD.N.C.Tse,“Diversityandmultiplexing:Afundamental
tradeoff in multiple-antenna channels,” IEEE Trans. Inform. Theory,
The arrival process to the (i + 1)th node is the departure vol.49,pp.1073–1096,2003.
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Diversity-multiplexing-delay tradeoff,” IEEE Trans. Inform. Theory,
Ai =Ai−1+Di−1−Di , 2≤i≤N −2. (20) vol.52,pp.3601–3621,2006.
n n n n−1
[3] T.Tabet,S.Dusad,andR.Knopp,“Diversity-multiplexing-delay trade-
with Ai a Poisson process with rate 1/µ. Also the waiting off in half-duplex ARQ relay channels,” IEEE Transactions on Infor-
n
mationTheory,vol.53,pp.3797–3805, October2007.
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[4] Y.Xie,D.Gunduz,andA.Goldsmith,“MultihopMIMOrelaynetworks
W1 =(W1 +S1 +S2 −A1)+. (21) withARQ,”IEEEGlobecom2009CommunicationTheorySymposium,
n n−1 n−1 n−1 n Dec.2009.
[5] T.Holliday,A.J.Goldsmith,andH.V.Poor,“Jointsourceandchannel
A well-known result is that (see, e.g. [9]), if the arrival
coding for MIMO systems: Is it better to be robust or quick?,” IEEE
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5
[6] N.BisnikandA.Abouzeid,“Queueingnetworkmodelsfordelayanaly-
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AnnalsofOperations Research,vol.79,pp.3–26,Jan.1998.
6
Diversity-Multiplexing-Delay Tradeoffs in MIMO
Multihop Networks with ARQ
∗ ∗
Yao Xie , Andrea Goldsmith
∗
Department of Electrical Engineering, Stanford University, Stanford, CA.
Email: [email protected],[email protected]
Abstract—Tradeoff in diversity, multiplexing, and delay in delay has to be brought into the picture (see discussions in
0 multihopMIMOrelaynetworkswithARQisstudied,wherethe [?]).WithfiniteSNRandqueueingdelay,theDMDTwillbe
1 random delay is caused by queueing and ARQ retransmission. different from that under the infinite SNR assumption. The
0 This leads to an optimal ARQ allocation problem with per- DMDT with queueingdelay is studied in [?] and an optimal
2 hop delay or end-to-end delay constraint. The optimal ARQ
allocation has to trade off between the ARQ error that the ARQ adapted to the instantaneous queue state for the point-

n receiver failstodecodeintheallocated maximumARQrounds to-point MIMO system is presented therein.
a andthepacketlossduetoqueueingdelay.Thesetwoprobability In this work, we extend the study [?] of optimal ARQ
J of errors are characterized using the diversity-multiplexing-
assuming high but finite SNR and queueing delay in point-
5 delay tradeoff (DMDT) (without queueing) and the tail proba-
to-point MIMO systems to multihop MIMO networks. This
bilityofrandomdelayderivedusinglargedeviationtechniques,
] respectively. Then the optimal ARQ allocation problem can be work is also an extention our previous results in [?] to
T formulatedasaconvexoptimizationproblem.Weshowthatthe incorporate queueing delay. We use the same metric as that
I optimal ARQ allocation should balance each link performance usedin[?],whichcapturestheprobabilityoferrorcausedby
.
s aswellavoidsignificantqueuedelay,whichisalsodemonstrated both ARQ error, and the packet loss due to queueing delay.
c by numerical examples.
[ The ARQ erroris characterizedby informationoutageprob-
I. INTRODUCTION ability,whichcanbefoundthroughadiversity-multiplexing-
4
In a multihop relaying system, each terminal receives delay tradeoff analysis [?], [?]. The packet loss is given
v
3 the signal only from the previous terminal in the route by the limiting probability of the event that packet delay
5 and, hence, the relays are used for coverage extension. exceeds a deadline. Unlike the standard queuing models for
3 Multiple input-multipleoutput(MIMO) systems can provide networks (e.g., [?], [?]) where only the number of messages
5 increaseddataratesbycreatingmultipleparallelchannelsand awaiting transmission is studied, here we also need to study
.
2 increasingdiversitybyrobustnessagainstchannelvariations. the amount of time a message has to wait in the queue
1 Anotherdegreeoffreedomcanbeintroducedbyanautomatic of each node. Our approach is slightly different from [?],
9
repeat request (ARQ) protocol for retransmissions. With the where the optimal ARQ decision is adapted per packet; we
0
multihop ARQ protocol, the receiver at each hop feeds back study the queues after they enter the stable condition, and
:
v tothe transmittera one-bitindicatoronwhetherthemessage hence we use the stationary probability of a packet missing
i
X can be decoded or not. In case of a failure the transmitter a deadline. An immediate tradeoff in the choice of ARQ
sends additional parity bits until either successful reception round is: the larger the number of ARQ attempts we used
r
a ormessageexpiration.TheARQprotocolprovidesimproved for a link, the higher the diversity and multiplexing gain we
reliabilitybutalsocausestransmissiondelayofpackets.Here canachieve,meaninga lowerARQ error.However,thisisat
we study a multihop MIMO relay system using the ARQ a price of more packet missing deadline. Our goal is to find
protocol. Our goal is to characterize the tradeoff in speed anoptimalARQallocationthatbalancesthesetwoconflicting
versus reliability for this system. goals and equalizes performance of each hop to minimizes
The rate and reliability tradeoff for the point-to-point the probability of error.
MIMO system, captured by the diversity-multiplexingtrade- The remainder of this paper is organized as follows.
off (DMT), was introduced in [?]. Considering delay as the Section II introduces system models and the ARQ protocol.
thirddimensioninthisasymptoticanalysiswithinfiniteSNR, Section III presents our formulation and main results. Nu-
the diversity-multiplexing-delay tradeoff (DMDT) analysis merical examples are shown in Section IV. Finally Section
for a point-to-point MIMO system with ARQ is studied in V concludes the paper.
[?], and the DMDT curve is shown to be the scaled version
II. MODELS AND BACKGROUND
of the correspondingDMT curve withoutARQ. The DMDT
A. Channel and ARQ Protocol Models
in relaynetworkshasreceiveda lotof attentionas well(see,
e.g., [?].) In our recent work [?], we extended the point-to- Consider a multihop MIMO network consisting of N
pointDMDTanalysistomultihopMIMOsystemswithARQ nodes:withthesourcecorrespondingtoi=1,thedestination
and proposed an ARQ protocol that achieves the optimal correspondingtoi=N,andi=2,··· ,N−1corresponding
DMDT. to the intermediate relays, as shown in Fig. 1. Each node
TheDMDTanalysisassumesasymptoticallyinfiniteSNR. is equipped with M antennas. The packets enter the net-
i
However, in the more realistic scenario of finite SNR, re- work from the source node, and exits from the destination
transmissionisnotanegligibleeventandhencethequeueing node, forming an open queue. The network uses a multihop
automatic repeat request (ARQ) protocol for retransmission. constantduringalltheARQrounds,andindependentfor
With the multihop ARQ protocol, in each hop, the receiver differenti.Ourresultscanbe extendedto the theshort-
feedsbacktothetransmitteraone-bitindicatoraboutwhether term static channel using the DMDT analysis given in
the message can be decoded or not. In case of a failure [?].
the transmitter retransmits. Each channel block for the same
message is called an ARQ round. We consider the fixed Message n(cid:13) Message n-1(cid:13) Message n-2(cid:13) Message n(cid:13) Message n-1(cid:13)
time t(cid:13) time t(cid:13)
ARQ allocation, where each link i has a maximum of ARQ
Source(cid:13) Destination(cid:13) Source(cid:13) Destination(cid:13)
rounds L , i = 1,···N −1. The packet is discarded once Message n+1(cid:13) Message n(cid:13) Message n-1(cid:13) Message n(cid:13)
i
time t+1(cid:13) time t+1(cid:13)
the maximum round has been reached. The total number of
Source(cid:13) Destination(cid:13) Source(cid:13) Destination(cid:13)
ARQroundsislimitedtoL: N−1L ≤L.ThisfixedARQ Message n+2(cid:13) Message n+1(cid:13) Message n(cid:13) Message n+1(cid:13) Message n(cid:13)
i=1 i time t+1(cid:13) time t+1(cid:13)
protocol has been studied in our recent paper [?].
P Source(cid:13) Destination(cid:13) Source(cid:13) Destination(cid:13)
ARQ(cid:13) ARQ(cid:13) Fig.2:Left:fullduplexmultihoprelaynetwork.Right:halfduplex
relay multihop MIMO relay network.
M1(cid:13)(cid:13) M2(cid:13)(cid:13) MN(cid:13)-1(cid:13) MN(cid:13)(cid:13)
Source(cid:13) H1(cid:13)(cid:13) Relay(cid:13) H2(cid:13)(cid:13) HN(cid:13)-2(cid:13) Relay(cid:13) HN(cid:13)-1(cid:13) Destination(cid:13)
Fdeigst.in1a:tioUn.ppLeorw:erre:lamyulntiehtwoporMkIwMiOthrdeliareyctneltiwnkorkfrowmithsoouutrdceiretcot Function 0.05
link. Cost −0.5
Assume the packets are delay sensitive: the end-to-end Log of −13−.015 20 5 6
transmission delay cannot exceed k. One strategy to achieve 10 3 4
this goal is to set a deadline k for each link i with k1 0 2 L1
i
Ni=−11ki ≤ k. Once a packet delays more than ki it Fig. 3: The logarithm of the cost function (7) for the (4, 1, 2)
is removed from the queue. This per-hop delay constraint multihop MIMO relay networks. SNR is 20 dB.
P
correspondstothefinitebufferateachnode.Anotherstrategy
istoallowlargeper-hopdelaywhileimposinganend-to-end
delay constraint.
Other assumptions we have made for the channel models
4 6
are 3 5
4
(i) Thechannelbetweentheithand(i+1)thnodesisgiven *L12 *L223
1
1
by: 40 40
3 150 3 150
2 100 2 100
SNR 1 50 1 50
Y = H X +W , 1≤l ≤L . (1) r 0 0 k r 0 0 k
i,l i,l i,l i,l i
M
r i
The message is encoded by a space-time encoder into 8 80
7
a1,·s·e·qu,eLn}c,ewohferLe Tmaistritchees b{lXocik,lle∈ngtCh,Mai×ndT,Y: il,l =∈ ** + LL213456 *k1246000
2
CMi+1×T, i = 1,··· ,N −1, is the received signal at 41 40
3 150 3 150
the (i + 1)th node, in the lth ARQ round. The rate 2 100 2 100
1 50 1 50
of the space-time code is R. Channels are assumed r 0 0 k r 0 0 k
to be frequency non-selective, block Rayleigh fading Fig.4:AllocationsofoptimalARQ:L∗1,L∗2,L∗1+L∗2,k1∗,ina(4,1,
and independent of each other, i.e., the entries of the 2)MIMOrelaynetwork.SNRis20dB.(Theoptimalk2∗ =k−k1∗.)
channel matrices Hi,l ∈ CMi+1×Mi are independent
andidenticallydistributed(i.i.d.)complexGaussianwith
B. Queueing Network Model
zero mean and unit variance. The additive noise terms
W are also i.i.d. complex Gaussian with zero mean We use an M/M/1 queue tandem to model the multihop
i,l
andunitvariance.TheforwardlinksandARQfeedback relaynetworks.Thepacketsarriveatthe sourceasa Poisson
links only exist between neighboring nodes. processwithmeaninterarrivaltimeµ,(i.e.,thetimebetween
(ii) We considerboththe full-duplexandhalf-duplexrelays the arrival of the nth packet and (n − 1)th packet.) We
(see, e.g., [?]) where the relays can or cannot transmit modelthe randomservice times for the nth packetat the ith
and receive at the same time, respectively, as shown nodeasindependentandidenticallyexponentiallydistributed
in Fig. 2. Assume the relays use a decode-and-forward with mean L that is also independent of the inter-arrival
i
protocol (see, e.g., [?]). time. This approximation leads to a tractable solution that
(iii) We assume a short-term power constraint at each node characterizesthequalitativebehaviorofthemultihopnetwork
for each block code. Hence we do not consider power queue.Nodemhasafinitebuffersize.Thepacketsenterinto
control. the buffer and are first-come-first-served (FCFS). Assume
(iv) We consider both the long-term static channel, where µ ≥ L so that the queues are stable, i.e., the waiting time
i
H = H for all l, i.e. the channel state remains at a node does not go to infinity as time goes on.
i,l i
2
Burke’stheorem(see,e.g.,[?])saysthatthepacketsdepart III. DIVERSITY, MULTIPLEXING,AND DELAYTRADEOFF
fromthesourceandarriveateachrelayasaPoissonprocess VIA OPTIMALARQ ROUND ALLOCATION
with rate p /µ, where p is the probability that a packet
m i
A. Full-Duplex Relay in Multihop Relay Network
can reach the mth node. With high SNR, the packet reaches
the subsequent relays with high probability: pi ≈ 1 (the 1) Per-Hop Delay Constraint: The probability of error
probability of a packet dropping is small because it uses up depends on the ARQ window length allocation L , deadline
i
the maximum ARQ round.) Hence all nodes have packets constraint k , multiplexing rate r, and SNR ρ. For a given r
i
arrive as a Poisson process with mean inter-arrival time µ. and ρ, we have
C. Throughput P ({L },{k }|ρ,r)=
e i i
Denote by b the size of the information messages in bits, N−1
B[t] the number of bits removed from transmission buffer PARQ(ρ,{Li})+ PQueue(Di >ki). (4)
at the source at time slot t. Define a renewal event as the i=1
X
event that the transmitted message leaves the source and Here D denotes the random delay at the ith link when the
i
eventually is received by the destination node possibly after queue is stationary. This P expression is similar to that
e
one or more ARQ retransmissions. We assume that under givenbyEquation(33)of[?].Ourgoalistoallocateper-hop
full-duplexrelaysthetransmittercannotsendanewmessage ARQ round {L } and delay constraint {k } to minimize the
i i
until the previous message has been decoded by the relay at probability of error P .
e
which point the relay can begin transmission over the next Forthelong-termstaticchannel,usingtheDMDTanalysis
hop(Fig.2a.)Underhalfduplexrelaysweassumetransmitter results [?] we have:
cannot send a new message until the relay to the next hop
N−1
comThpelenteusmitbsertraonfsbmitisssBi¯ontr(aFnisgm.i2ttbe.d) in each renewal event, PARQ(ρ,{Li})= ρ−fi(cid:16)Lri(cid:17). (5)
i=1
for full-duplexing B¯ = (N − 1)b, and for half-duplexing X
B¯ =(N−1)b/2whenN isodd,andB¯ =Nb/2whenN is Here fi(r) is the diversity-multiplexing tradeoff (DMT) for
a point-to-pointMIMO system formedby nodesi and i+1.
even.Thelong-termaveragethroughputoftheARQprotocol
Assuming sufficient long block lengths, f (r) is given by
is defined as the transmitted bits per channel use (PCU) [?], i
Theorem 2 in [?] quoted in the following:
which can be found using renewal theory [?]:
Theorem 1: [?] For sufficiently long block lengths, the
1 s B¯ . B¯ diversity-multiplexing tradeoff (DMT) f(r) for a MIMO
η = liminf B[t]= =
s→∞ Ts t=1 E(τ) (N −1)T system with Mt transmit and Mr receive antennas is given
X by the piece-wise linear function connecting the points
R, Full duplex;
(r,(M −r)(M −r)), for r =0,··· ,min(M ,M ).
= R, Half duplex, N is odd; (2) t r t r
 R2 1 + 1 , Half duplex, N is even. Denote the amount of time spent in the ith node by
 2 2N the nth message as Di. The probability of packet loss
n
whereτ istheave(cid:0)ragedura(cid:1)tionfromthetimeapacketarrive.s PQueue(Di > ki) can be found as the limiting distribution of
at the source until it reaches the destination node, and = lim P(Di >k ) (adapted from Theorem7.4.1 of [?]):
n→∞ n i
denotes asymptotic equality. A similar argument as in [?] Lemma 2: The limiting distribution of the event that the
.
shows that E(τ)=(N −1)T for high SNR. delay at node i exceeds its deadline k , for M/M/1 queue
i
models, is given by:
D. Diversity-Multiplexing-Delay Tradeoff
souTrhceesp:rforobmabitlhiteyAoRfQererorrrorP:ethienptahceketrtainssmdriospsipoendhbaescatuwsoe PQueue(Di >ki)=nl→im∞P(Dni >ki)= Lµie−ki(cid:16)L1i−µ1(cid:17). (6)
the receiver fails to decode the message within the allocated Here the differencein the service rate andpacketarrivalrate
numberofARQ rounds,denotedasPARQ,andtheprobability 1 − 1 ≥ 0 and utility factor Li both indicate how “busy”
that a message misses its deadline at any node due to large nLoide mµ is. µ
queueingdelay,denotedasP .WewillgiveP forvarious
Queue e Using the above results, (4) can be written as
ARQ relay networks.
N−1
infFoormlloawtioinngmthesesafgraemsebw(ρo)rkdeopfen[?d]s,ownethaessoupmeerattihnegssiizgenaol-f Pe({Li},{ki}|ρ,r)= ρ−fi(cid:16)Lri(cid:17)+ Lµie−ki(cid:16)L1i−µ1(cid:17) . (7)
i=1 (cid:20) (cid:21)
to-noise ratio (SNR) ρ, and a family of space time codes X
{C } with block rate R(ρ) = b(ρ)/T , rlogρ. We use the Notethatthequeueingdelaymessagelosserrorprobabilityis
ρ
effectiveARQ multiplexinggainandthe ARQ diversitygain decreasinginLi,andtheARQerrorprobabilityisincreasing
[?] inLi.HenceanoptimalARQroundsallocationateachnode
L should trade off these two terms. Also, the optimal ARQ
i
η(ρ) logP (ρ)
r , lim , d,− lim e . (3) allocationshouldalso equalizethe performanceofeachlink,
e
ρ→∞logρ ρ→∞ logρ as the weakest link determines the system performance [?].
WecannotassumeinfiniteSNRbecauseotherwisethequeue- Hence the optimal ARQ allocation can be formulated as
ing delay will be zero, as pointed out in [?]. However we the following optimization problem:
assumehighSNRtousetheDMDTresultsinoursubsequent
analysis. min Pe({Li},{ki}|ρ,r) (8)
{Li},{ki}∈A
3
where 2) End-to-End Delay constraint: When the buffer per
N−1L ≤L, nodeislargeenoughaperhopdelayconstraintisnotneeded,
i=1 i and we can instead impose an end-to-end delay constraint.
A= 1≤L ≤µ, i=1,··· ,N −1 (9)
i
 PNi=−11ki ≤k.  Tenhdedeexlaacytiesxipnrtersascitoanblfeo.rHtohweetvaeilrparolabragbeilditeyvioaftiothnereesnudl-ttois-
The followingPlemma (proof omitted due to thespace limit) available. The following theorem can be derived using the
shows that the total transmission distortion function (15) is main theorem in [?]:
convex in the interior of A. Theorem 4: ForastationaryM/M/1queuetandem(with
Lemma 3: The transmission distortion function (15) is full-duplex relays):
convex jointly in L and k in the convex set
i i N−1
1
lim lim logP Di ≥k =−θ∗,
{Li},{ki}:ki > 2( µL−i 1), i=1,···N −1. , k→∞n→∞k Queue Xi=1 n !
( Li ) where θ∗ =minN−1 1 − 1 .
Lemma 3 says that except for the “corners” of A the cost i=1 Li µ
function is convex. However these “corners” have higher This theorem says thnat the boottleneck of the queueing net-
probabilityoferror:ki andLi takeextremevaluesandhence work is the link with longest mean service time Li. Hence
one link may have a longer queueing delay then the others. theoptimalARQroundallocationproblemcanbeformulated
So we only need to search the interior of A where the cost as:
function is convex.
min P ({L },{k }|ρ,r) (14)
To gain some insights into where the optimal solution e i i
{Li}∈B
resides in the feasible domain for the above problem, we
where
present a marginal cost interpretation. Note that the proba-
bility of error can be decomposed as a sum of probability P ({L },{k }|ρ,r)
e i i
of error on the ith link. The optimal ARQ rounds allocated N−1
on this link should equalize the “marginalcost” of the ARQ =P (ρ,{L })+P Di ≥k ,
ARQ i Queue n
error and the packet loss due to queueing delay. For node i, i=1 !
X
AwRithQfiexrerodrkpir,otbhaebmiliatyrg,iannadltchoestpsa(cpkaerttilaolssdipffreorbeanbtiialiltsy)doufethtoe =. N−1ρ−fi(cid:16)Lri(cid:17)+e−θ∗k. (15)
queueing delay, with respect to Li are given by Xi=1
∂ρ−∂fLi(cid:16)iLri(cid:17) = Lr2ifi′(cid:18)Lri(cid:19)ρ−fi(cid:16)Lri(cid:17)lnρ<0, (10) B =(cid:26) P1≤Ni=−L11iL≤iµ≤, L,i=1,··· ,N −1 (cid:27) (16)
and ForhighSNR,thiscanbeshowntobeaconvexoptimization
∂PQueue(Di >ki) = 1 1+ k e−ki(cid:16)L1i−µ1(cid:17) >0. (11) proAblseimm.pleargumentcanshowthatthepacketlossprobabil-
∂L µ L
i (cid:18) i(cid:19) ity with the per-hopdelayconstraintis largerthanthatusing
Note that fi′ < 0. The optimal solution equalizes these the more flexible end-to-end constraint.
two marginal costs by choosing L ∈ [1,µ]. Note that
i
these marginalcost functionsare monotonein L , hence the B. Half-duplex Relay in Multihop Network
i
equalizing L∗ exists and 1 < L∗ < µ if the following two We can also find the tail probability for the end-to-end
i i
conditions are true for L =1 and L =µ: delay of a multihop network with half-duplex relays using
i i
largedeviationtechniques(see proofin the Appendix).Thus
(i): ∂PQueue(Di >ki) <− ∂ρ−fi(cid:16)Lri(cid:17) , (12) yields
∂Li (cid:12)L=1 ∂Li (cid:12)(cid:12) Theorem 5: ForastationaryM/M/1queuetandem(with
(cid:12) (cid:12)L=1 half-duplex relays), when the number of node N is large:
(cid:12) (cid:12)
(ii): ∂PQueue(Di >ki)(cid:12) >− ∂ρ−fi(cid:16)Lri(cid:17)(cid:12)(cid:12) , (13) 1 N−2
∂L ∂L (cid:12) lim lim logP Di ≥k =−θ∗. (17)
i (cid:12)(cid:12)L=µ i (cid:12)(cid:12)L=µ k→∞n→∞k Queue i=1 n !
(cid:12) (cid:12) X
These conditions involv(cid:12)e nonlinear inequaliti(cid:12)es involving FromthistheoremweconcludethattheoptimalARQalloca-
(cid:12)
µ, ρ, r, Mi and Mi+1, which defines the case when the tion problem with the end-to-end constraint and half-duplex
optimal solution is in the interior of A. Analyzing these relays can be formulated the same as that with full-duplex
conditions reveals that these conditions tend to satisfy at relays (8) and (14).
lower multiplexing gain r, small Mi or Mi+1, small ki, and
larger µ (light traffic). Note that with high SNR condition IV. NUMERICAL EXAMPLES
(ii) is always true for moderate k values. When (i) and (ii) Consider a MIMO relay network consists of a source, a
are violated, which means one error dominates the other, relay, and a destination node. The relay is full-duplex. The
then the optimal solution lies at the boundary of A. With numberofantennasoneachnodeis(M1,M2,M3),M1 =4,
the totalARQ roundsconstraintin (8), using the Lagrangian M2 =1, and M3 =2, where the relay has a single antenna.
multiplier an argument similar to above still holds. Other parameters are: ρ = 20dB, k = 30, L = 8, and the
4