Cognitive Access Policies under a Primary ARQ process via Forward-Backward Interference Cancellation

0.58 MB·English

by  Nicolò Michelusi

1
Cognitive Access Policies under a Primary ARQ process via
Forward-Backward Interference Cancellation
Nicolò Michelusi, Petar Popovski, Osvaldo Simeone,
Marco Levorato, Michele Zorzi
Abstract—This paper introduces a novel technique for access the PU (underlay cognitive radio paradigm [8]). Within this
byacognitiveSecondaryUser(SU)usingbest-efforttransmission framework,we propose to exploitthe intrinsic redundancy, in
to a spectrum with an incumbent Primary User (PU), which
theformofcopiesofPUpackets,introducedbytheType-IHy-
3 uses Type-I Hybrid ARQ. The technique leverages the primary
brid Automatic Retransmission reQuest (Type-I HARQ [10])
1 ARQ protocol to perform Interference Cancellation (IC) at
0 the SU receiver (SUrx). Two IC mechanisms that work in protocol implemented by the PU by enabling Interference
2 concert are introduced: Forward IC, where SUrx, after decod- Cancellation (IC) at the SU receiver (SUrx). We introduce
ing the PU message, cancels its interference in the (possible) two IC schemes that work in concert, both enabled by the
n following PU retransmissions of the same message, to improve
underlyingretransmissionprocessofthePU.WithForwardIC
a the SU throughput; Backward IC, where SUrx performs IC
J (FIC), SUrx, after decoding the PU message, performs IC in
on previous SU transmissions, whose decoding failed due to

0 severe PU interference. Secondary access policies are designed thenextPUretransmissionattempts,iftheseoccur.WhileFIC
3 that determine the secondary access probability in each state provides IC on SU transmissions performed in future time-
of the network so as to maximize the average long-term SU slots, Backward IC (BIC) provides IC on SU transmissions
] throughput by opportunistically leveraging IC, while causing
T performed in previous time-slots within the same primary
bounded average long-term PU throughput degradation and SU
I powerexpenditure.Itisprovedthattheoptimalpolicyprescribes ARQ retransmission window, whose decoding failed due to
s. thattheSUprioritizesitsaccess inthestateswhereSUrxknows severeinterferencefromthePU.BICreliesonbufferingofthe
c the PU message, thus enabling IC. An algorithm is provided to received signals. Based on these IC schemes, we model the
[
optimally allocate additional secondary access opportunities in state evolution of the PU-SU network as a Markov Decision
the states where the PU message is unknown. Numerical results
1 Process [11], [12], induced by the specific access policy used
areshowntoassessthethroughputgainprovidedbytheproposed
v by the SU, which determines its access probability in each
techniques.
9
stateofthenetwork.Followingtheapproachputforthby[13],
7 Index Terms—Cognitive radios, resource allocation, Markov
we study the problem of designing optimal secondary access
0 decision processes, ARQ, interference cancellation
1 policies that maximize the average long-term SU throughput
. by opportunistically leveraging FIC and BIC, while causing
2
I. INTRODUCTION a bounded average long-term throughput loss to the PU and
0
3 Cognitive Radios (CRs) [3] offer a novel paradigm for im- a bounded average long-term SU power expenditure. We
1 provingtheefficiencyofspectrumusageinwirelessnetworks. show that the optimal strategy dictates that the SU prioritizes
v: Smartusers,referredtoas SecondaryUsers(SUs),adapttheir its channel access in the states where SUrx knows the PU
i operation in order to opportunistically leverage the channel message,thusenablingIC;moreover,weprovideanalgorithm
X
resourcewhilegeneratingboundedinterferencetothePrimary tooptimallyallocateadditionalsecondaryaccessopportunities
r
a Users(PUs)[4]–[6].Fora surveyoncognitiveradio,dynamic in the states where the PU message is unknown.
spectrum access and the related research challenges, we refer The idea of exploiting PU retransmissions to perform IC
the interested reader to [6]–[9]. on future packets (similar to our FIC mechanism) was put
In a standard model for cognitive radio, the PU is a legacy forth by [14], which devises several cognitive radio protocols
system oblivious to the presence of the SU, which needs to exploitingthehybridARQretransmissionsofthePU.Therein,
satisfy given constraints on the performance loss caused to thePUemployshybridARQwithincrementalredundancyand
the ARQ mechanismis limited to at most one retransmission.
N. Michelusi and M. Zorzi are with the Department TheSUreceiverattemptstodecodethePUmessageinthefirst
of Information Engineering, University of Padova, Italy
time-slot.Ifsuccessful,theSUtransmittersendsitspacketand
({michelusi,zorzi}@dei.unipd.it); P. Popovski is with
the Department of Electronic Systems, Aalborg University, Denmark the SU receiverdecodesit byusing ICon the receivedsignal.
([email protected]);O.Simeoneis withtheCenter forWireless Com- In contrast, in this work, we address the more general case
municationandSignalProcessingResearch(CWCSPR),NewJerseyInstitute
of an arbitrary number of primary ARQ retransmissions, and
of Technology, New Jersey, USA ([email protected]); M.
Levorato is with the Ming Hsieh Department of Electrical Engineering, we allow a more general access pattern for the SU pair over
University of Southern California, Los Angeles, USA, and with Stanford the entire primaryARQ window.We also modelthe interplay
University, USA([email protected]).
between the primary ARQ protocol and the activity of the
This work is a generalization of [1], presented at the Information Theory
andApplicationsWorkshopinFebruary2011,andof[2],presentedatthe49th SU, by allowing for BIC. It should be noted that IC-related
Annual Allerton Conference on Communication, Control, and Computing in schemes are also used in other context, e.g., decoding for
September2011.
graphical codes [15] and multiple access protocols [16].
Manuscript received date:Apr.15,2012.
Manuscript revised date:Sep.1,2012. Other related works include [17], which devises an oppor-
2
ACK/NACK as i.i.d. over time-slots and independent over the different
links, and we denote the average SNR as γ¯ =E[γ ].
x x
PUtx PUrx We assume that no Channel State Information (CSI) is
γ
p
available at the transmitters, so that the latter cannot allocate
their rate based on the instantaneous link quality, to ensure
γ
sp correct delivery of the packets to their respective receivers.
Transmissions may thus undergo outage, when the selected
γ
ps rate is not supported by the current channel quality.
In order to improve reliability, the PU employs Type-I
γ
s
SUtx SUrx HARQ[10]withdeadlineD ≥1,i.e.,atmostDtransmissions
of the same PU message can be performed, after which the
packet is discarded and a new transmission is performed (the
Buffering/PU message knowledge
PUisassumedtobebacklogged).WedefinetheprimaryARQ
state t ∈ N(1,D)1 as the number of ARQ transmission at-
Fig.1. Systemmodel
temptsalreadyperformedonthecurrentPUmessage,plusthe
current one. Namely, t =1 indicates a new PU transmission,
and the counter t is increased at each ARQ retransmission,
tunistic sharing scheme with channel probing based on the
until the deadline D is reached. We assume that the ARQ
ARQfeedbackfromthePUreceiver.Aninformationtheoretic
feedback is received at the PU transmitter by the end of
framework for cognitive radio is investigated in [18], where
the time-slot, so that, if requested, a retransmission can be
the SU transmitter has non-casual knowledge of the PU’s
performed in the next time-slot.
codeword. In [19], the data transmitted by the PU is obtained
causally at the SU receiver. However, this model requires a Ontheotherhand,theSU,ineachtime-slot,eitheraccesses
joint design of the PU and SU signaling and channel state thechannelbytransmittingitsownmessage,orstaysidle.This
information at the transmitters. In contrast, in our work we decision is based on the access policy µ, defined in Sec. III.
explicitly model the dynamic acquisition of the PU message The activity of the SU, which is governed by µ, affects the
at the SU receiver, which enables IC. Moreover, the PU is outage performanceof the PU, by creating interferenceto the
oblivious to the presence of the SU. PU overthe link SUtx→PUrx. We denote the primary outage
The paper is organized as follows. Sec. II presents the probability when the SU is idle and accesses the channel,
systemmodel.Sec.IIIintroducesthesecondaryaccesspolicy, respectively, as2
the performancemetrics and the optimizationproblem,which
q(I)(R ),Pr R >C(γ ) ,
is addressed in Sec. IV. Sec. VI presents and discusses the pp p p p
numerical results. Finally, Sec. VII concludes the paper. The (cid:18) (cid:19)
γ
proofs of the lemmas and theorems are provided in the q(A)(R ),Pr R >C p , (1)
pp p p 1+γ
appendix. (cid:18) (cid:18) sp(cid:19)(cid:19)
where R denotes the PU transmission rate, measured in
p
bits/s/Hz, C(x) , log (1+x) is the (normalized) capacity
2
II. SYSTEMMODEL of the Gaussian channel with SNR x at the receiver [20].
Thisoutagedefinition,aswellastheonesintroducedlateron,
We consider a two-user interference network, as depicted
assume the use of Gaussian signaling and capacity-achieving
in Fig. 1, where a primary transmitter and a secondary
coding with sufficiently long codewords.However,ouranaly-
transmitter, denoted by PUtx and SUtx, respectively, transmit
sis can be extended to include practical codes by computing
to their respective receivers, PUrx and SUrx, over the di-
theoutageprobabilitiesforthespecificcodeconsidered.In(1),
rect links PUtx→PUrx and SUtx→SUrx. Their transmissions
it is assumed that SU transmissionsare treated as background
generate mutual interference over the links PUtx→SUrx and
Gaussian noise by the PU. This is a reasonable assumptionin
SUtx→PUrx.
CRs in which the PU is oblivious to the presence of SUs. In
Timeisdividedintotime-slotsoffixedduration.Eachtime-
general,wehaveq(A)(R )≥q(I)(R ),whereequalityholdsif
slot matches the length of the PU and SU packets, and the pp p pp p
and only if γ ≡0 deterministically.We denotethe expected
transmissions of the PU and SU are assumed to be perfectly sp
PU throughputaccruedin each time-slot, when the SU is idle
synchronized. We adopt the block-fading channel model, i.e.,
andaccessesthechannel,asT(I)(R )=R [1−q(I)(R )] and
the channel gains are constant within the time-slot duration, p p p pp p
and change from time-slot to time-slot. Assuming that the Tp(A)(Rp)=Rp[1−qp(Ap)(Rp)], respectively.
SU and the PU transmit with constant power P and P ,
s p
respectively, and that noise at the receivers is zero mean 1WedefineN(n0,n1)={t∈N,n0≤t≤n1}forn0≤n1∈N
Gaussianwithvarianceσw2,wedefinetheinstantaneousSignal 2 Herein,wedenotetheoutageprobabilityasqx(Zy),wherexandyarethe
toNoiseRatios(SNR)ofthelinksSUtx→SUrx,PUtx→PUrx, source and the recipient of the message, respectively (PU if x,y = p, SU
SUtx→PUrx and PUtx→SUrx, during the nth time-slot, as if x,y =s), and Z ∈{A,I} denotes the action of the SU (A if the SU is
active and it accesses the channel, I if the SU remains idle). For example,
γs(n), γp(n), γsp(n) and γps(n), respectively. We model the qp(As) istheprobabilitythatthePUmessageisinoutageatSUrx,whenSUtx
SNR process {γx(n),n=0,1,...}, where x∈{s,p,sp,ps}, transmits.
3
A. Operation of the SU
-PUmessagedecoded,
meUchnalinkiesmth,eit iPsUasstuhmatedustehsatathseimSUpleusTesyp"eb-eIstHeyffborritd"tAraRnsQ- StrUeaitnetderafsernenocisee C(γ)ps PunUdeacnoddeSdU:cmaepsascaigtyesof
-SUmessageundecoded= interferencefreechannels
mission. Moreover, the SU is provided with side-information Rp exceeded
about the PU, e.g., ARQ deadline D, PU codebook and
feedback information from PUrx (ACK/NACK messages). RsU RsU=C(γs)
PUandSUmessagesundecoded:rx
e
This is consistent with the common characterization of the at signalisbufferedforBICrecovery
R
PU as a legacysystem, and of the SU as an opportunisticand RsU=C(γs/(1+γps))
cognitive system, which exploits the primary ARQ feedback PUandSUmessages γ))s -SPUUminetsesrfaegreendceecoded,
ttohecrfleoawtecaobnetrsot-lemffoecrthalinniksmwsitahremlaexfitmtoizethdethurpopuegrhlpauyte,rws.hBilye jointlydecoded γ/(1+ps RsU+Rp=-PtrUeatmedesasasgneouisnedec.
overhearingthe feedback information from PUrx, the SU can =C( C(γs+γ
thus track the primary ARQ state t. Moreover, by leveraging Rp ps)
the PU codebook, SUrx attempts, in any time-slot, to decode 0 Rate,Rp
the PU message, which enables the following IC techniques
at SUrx: Fig. 2. Decodability regions for PU message (rate Rp) and SU message
• Forward IC (FIC): by decoding the PU message, SUrx (rate RsU)atSUrx,forafixedSNRpair (γs,γps)
can performIC in the currentas well as in the following
ARQ retransmissions, if these occur, to achieve a larger
PUtx→SUrx, with SNR γ . We denote the corresponding
SU throughput; ps
• Backward IC (BIC): SUrx buffers the received signals outage probability as qp(Is)(Rp)=Pr(Rp >C(γps)).
correspondingtoSUtransmissionswhichundergooutage If the SU accesses the channel, SU transmissions are per-
due to severe interference from the PU. These transmis- formedwithrateRsU (bits/s/Hz)andareinterferedbythePU.
sions can later be recovered using IC on the buffered SUrx thus attempts to decode both the SU and PU messages;
received signals, if the interfering PU message is suc- moreover, if the decoding of the SU message fails due to
cessfullydecodedbySUrxinasubsequentprimaryARQ severeinterferencefromthePU,thereceivedsignalisbuffered
retransmission attempt. for future BIC recovery.Using standard information-theoretic
We define the SU buffer state b ∈ N(0,B) as the number results [20], with the help of Fig. 2, we define the following
SNR regions associated with the decodability of the SU and
of received signals currently buffered at SUrx, where B ∈
N(0,D−1)3 denotesthe buffersize. Moreover,we define the PU messages at SUrx, where Ac denotes the complementary
set of A:4
PU message knowledge state Φ∈{K,U}, which denotes the
knowledgeatSUrxaboutthePUmessagecurrentlyhandledby Γ (R ,R ), (γ ,γ ):R ≤C(γ ),R ≤C(γ ),
p sU p s ps sU s p ps
thePU.Namely,ifΦ=K,thenSUrxknowsthePUmessage,
n
thus enabling FIC/BIC; conversely (Φ=U), the PU message R +R ≤C(γ +γ ) (2)
sU p s ps
is unknown to SUrx.
γo
Remark 1 (Feedback Information). Note that PUrx needs (γs,γps):RsU >C(γs),Rp ≤C ps (3)
1+γ
to report one feedback bit to inform PUtx (and the SU, (cid:26) (cid:18) s(cid:19)(cid:27)
[
which overhears the feedback) on the transmission outcome
Γ (R ,R ), (γ ,γ ):R ≤C(γ ),R ≤C(γ ),
(ACK/NACK). On the other hand, two feedback bits need to s sU p s ps sU s p ps
be reported by SUrx to SUtx: one bit to inform SUtx as to n
R +R ≤C(γ +γ ) (4)
whether the PU message has been successfully decoded, so sU p s ps
that SUtx can track the PU message knowledge state Φ; and (γ ,γ ):R >C(γ ),R ≤C oγps (5)
one bit to inform SUtx as to whether the received signal has s ps p ps sU 1+γ
(cid:26) (cid:18) s(cid:19)(cid:27)
been buffered, so that SUtx can track the SU buffer state b. [ c
Γ (R ,R ), Γ (R ,R )∪Γ (R ,R ) (6)
Herein, we assume ideal (error-free) feedback channels, so buf sU p p sU p s sU p
that the SU can track (t,b,Φ), and the PU can track the n o
(γ ,γ ):R ≤C(γ ) .
s ps sU s
ARQstatet.However,optimizationispossiblewithimperfect
observations as well [21]. The SN\Rnregions (2) and (4) guaranotee that the two rates R
p
We now further detail the operation of the SU for andRsU arewithinthemultipleaccesschannelregionformed
Φ ∈ {K,U}. bythetwotransmitters(PUtxandSUtx)andSUrx[20],sothat
1) PU message unknown to SUrx (Φ=U): When Φ=U both the SU and PU messages are correctly decoded via joint
and the SU is idle, SUrx attempts to decode the PU message, decodingtechniques.Ontheotherhand,intheSNRregion(5)
so asto enableFIC/BIC. A decodingfailure occursif the rate (respectively, (3)), only the SU (PU) message is successfully
of the PU message, R , exceeds the capacity of the channel
p
4Herein, weassume optimal joint decoding techniques ofthe SUand PU
3 Note that B ≤ D−1, since the same PU message is transmitted at messages.Usingothertechniques, e.g.,successive IC,theSNRregions may
most D times by PUtx. Once the ARQ deadline D is reached, a new PU change accordingly, without providing any further insights in the following
transmissionoccurs,andthebufferisemptied. analysis.
4
decodedatSUrxbytreatingtheinterferencefromthePU(SU) we obtain
as background noise. If the SNR pair falls outside the two
T (R )≥T (R )=T (R ,R )
regions(4)and(5)(respectively,(2)and(3)),thenSUrxincurs sK sK sK sU sU sU p
a failure in decoding the SU (PU) message. Therefore, when +ps,buf(RsU,Rp)RsU >TsU(RsU,Rp). (9)
(γ ,γ ) ∈ Γ (R ,R ), SUrx successfully decodes the SU
s ps s sU p Conversely, the choice of the rate R is not as straightfor-
sU
message. The corresponding expected SU throughput is thus
ward,sinceitsvaluereflectsatrade-offbetweenthepotentially
given by
larger throughput accrued with a larger rate R and the
sU
T (R ,R ),R Pr((γ ,γ )∈Γ (R ,R )). (7) corresponding diminished capabilities for IC caused by the
sU sU p sU s ps s sU p
more difficult decoding of the PU message by SUrx.
Similarly, when (γs,γps) ∈ Γp(RsU,Rp), In the following treatment, the rates RsK, RsU and Rp
SUrx successfully decodes the PU message. We
are assumed to be fixed parameters of the system, and they
denote the corresponding outage probability as
are not considered part of the optimization (see Sec. VI for
qp(As)(RsU,Rp) , Pr((γs,γps)∈/ Γp(RsU,Rp)). Note further elaboration in this regard). For the sake of notational
that qp(As)(RsU,Rp) > qp(Is)(Rp), since SU transmissions convenience,weomitthedependenceofthequantitiesdefined
interfere with the decoding of the PU message. above on them. Moreover, for clarity, we consider the case
Finally, in (6), the decoding of both the SU and PU B = D−1 in which SUrx can buffer up to D−1 received
messagesfails, since the SNR pair(γ ,γ ) falls outside both signals. However,the following analysis can be extended to a
s ps
regionsΓ (R ,R )andΓ (R ,R ).However,therateR generic value of B.
p sU p s sU p sU
is within the capacity region of the interference free channel
(RsU ≤C(γs)), so thatthe SU message can be recoveredvia III. POLICY DEFINITION ANDOPTIMIZATIONPROBLEM
BIC,shouldthePUmessagebecomeavailableinafutureARQ
We model the evolution of the network as a Markov
retransmissionattempt.The receivedsignalis thusbufferedat
Decision Process [11], [12]. Namely, we denote the state of
SUrx. We denote the buffering probability as
the PU-SU system by the tuple (t,b,Φ), where t ∈ N(1,D)
p (R ,R ),Pr((γ ,γ )∈Γ (R ,R )) is the primary ARQ state, b ∈ N(0,B) is the SU buffer
s,buf sU p s ps buf sU p
state and Φ ∈ {U,K} is the PU message knowledge state.
=Pr((γ ,γ )∈Γ (R ,0)) (8)
s ps s sU
(t,b,Φ) takes values in the state space S ≡ S ∪S , where
U K
−Pr((γs,γps)∈Γs(RsU,Rp))>0, S ≡ {(t,0,K) : t ∈ N(2,D)} and S ≡ {(t,b,U) : t ∈
K U
where the second equality follows from inspection of Fig. 2. N(1,D),b∈N(0,t−1)} are the sets of states where the PU
message is known and unknown to SUrx, respectively.
2) PU message known to SUrx (Φ = K): When Φ = K,
The SU followsa stationary randomizedaccess policy µ∈
SUrx performs FIC on the received signal, thus enabling
U ≡{µ:S 7→[0,1]}, which determinesthe secondaryaccess
interferencefreeSUtransmissions.TheSUtransmitswithrate
probability for each state s ∈ S. Note that, from [22], this
R , and the accrued throughput is given by T (R ) =
sK sK sK choice is without loss of optimality for the specific problem
R Pr(R <C(γ )).
sK sK s athand.Namely,instate(t,b,Φ)∈S, theSUis"active",i.e.,
WenowprovideanexampletoillustratetheuseofFIC/BIC
it accesses the channel, with probability µ(t,b,Φ) and stays
at SUrx.
"idle" with probability 1−µ(t,b,Φ). We denote the "active"
and "idle" actions as A and I, respectively.
Example 1. Consider a sequence of 3 primary retransmis-
With these definitions at hand, we define the following
sion attempts in which the SU always accesses the channel.
averagelong-termmetrics under µ: the SU throughputT¯ (µ),
Initially, the PU message is unknown to SUrx, hence the PU s
the SU power expenditure P¯ (µ) and the PU throughput
messageknowledgestateissettoΦ=Uinthefirsttime-slot, s
T¯ (µ), given by
andtheSUtransmitswithrateR .AssumethattheSNRpair p
sU
(γs(1),γps(1)) falls in Γbuf(RsU,Rp). Then, neither the SU 1 N−1
northePUmessagesaresuccessfullydecodedbySUrx,butthe T¯ (µ)= lim E R 1 {Q =A}∩Oc s
roencdeitvimede-ssiglonta,l(iγsb(2u)ff,eγred(2fo))rf∈utΓure(RBIC,Rrec)ov∩eΓry.(IRnthe,Rsec)-, s N→+∞N "nX=N0−1sΦn (cid:0) n s,n(cid:1)(cid:12)(cid:12)(cid:12) 0#
s ps s sU p p sU p 1 (cid:12)
hence both the SU and PU messages are correctly decoded +N→lim+∞NE" RsUBn1(Opcs,n)(cid:12)s0#, (cid:12)(10)
by SUrx, and the PU message knowledge state switches to nX=0 (cid:12)
Φ = K. At this point, SUrx performs BIC on the previously 1 N−1 (cid:12)(cid:12)
P¯ (µ)=P lim E 1({Q =A}) s(cid:12) , (11)
buffered received signal to recover the corresponding SU s s n 0
N→+∞N " (cid:12) #
message. Inthe thirdtime-slot, SUtx transmitswith rate RsK, N−nX1=0 (cid:12)(cid:12)
and decoding at SUrx takes place after cancellation of the 1 (cid:12)
T¯ (µ)= lim E R 1 Oc s ,(cid:12) (12)
interference from the PU via FIC. p N→+∞N " p p,n (cid:12) 0#
We now briefly elaborate on the choice of the transmission where n is the time-slontX=in0dex, s(cid:0)0 ∈ S(cid:1)(cid:12)(cid:12)(cid:12)is the initial state in
(cid:12)
rate R . Since its value does not affect the outage perfor- time-slot 0; Φ ∈{K,U} is the PU message knowledge state
sK n
manceatPUrx(1)andtheevolutionoftheARQprocess,R and B is the SU buffer state in time-slot n; Q ∈ {A,I}
sK n n
is chosen so as to maximize T (R ). Therefore, from (8) is the action of the SU, drawn according to the access policy
sK sK
5
µ; O and O denote the outage events at SUrx for the as
s,n ps,n
decodingofthe SU andPU messages, so thatOsc,n andOpcs,n W¯s(µ), s∈Sπµ(s)µ(s),
denote successful decoding of the SU and PU messages by F¯ (µ), D π (t,0,K)µ(t,0,K)(T −T ),
SUrx, respectively;Op,n denotesthe outageeventat PUrx,so B¯s(µ), PDt=2 µt−1π (t,b,U)bR sK sU (15)
that Oc denotes successful decoding of the PU message by s Pt=1 b=0 µ sU
PUrx;pa,nnd 1(E)istheindicatorfunctionoftheeventE. Note P× 1P−qp(Is)−µ(t,b,U) qp(As)−qp(Is) .
that all the quantities defined above are independent of the h (cid:16) (cid:17)i
initial state s . In fact, starting from any s ∈ S, the system In (14), TsUW¯s(µ) is the SU throughput attained without
0 0 FIC/BIC, while the terms F¯ (µ) and B¯ (µ) account for the
reacheswithprobability1thepositiverecurrentstate (1,0,U) s s
throughputgainsofFICandBIC,respectively.Conversely,the
(new PU transmission) within a finite number of time-slots,
PU accrues the throughput T(I) if the SU is idle and T(A) if
due to the ARQ deadline. Due to the Markov property, from p p
the SU accesses the channel, so that (12) is given by
this state on, the evolution of the process is independent of
the initial transient behavior, which has no effect on the time T¯ (µ)=T(I)−(T(I)−T(A))W¯ (µ). (16)
p p p p s
averages defined in (10), (11) and (12).
The quantity (T(I) − T(A))W¯ (µ) is referred to as the PU
In this work, we study the problem of maximizing the p p s
throughputlossinducedbythesecondaryaccesspolicyµ[13].
average long-term SU throughput subject to constraints on
Thefollowingresultfollowsdirectlyfrom(13),(14)and(16).
the average long-term PU throughput loss and SU power.
Specifically, Lemma 1. The problem (13) is equivalent to
µ∗ =argmµaxT¯s(µ) s.t. T¯p(µ)≥Tp(I)(1−ǫPU), µ∗ =argmaxµ∈UT¯s(µ) (17)
P¯s(µ)≤Ps(th), (13) s.t. W¯ (µ)≤min (1−qp(Ip))ǫPU,Ps(th) ,ǫ .
where ǫ ∈ [0,1] and P(th) ∈ [0,P ] represent the (nor- s ( qp(Ap)−qp(Ip) Ps ) W
PU s s
malized) maximum tolerated PU throughputloss with respect
to the case in which the SU is idle and the SU power
Inthenextsection,wecharacterizethesolutionof(17).We
constraint,respectively.Thisproblementailsatrade-offin the
will need the following definition.
operation of the SU. On the one hand, the SU is incentivized
to transmitin orderto increase its throughputand to optimize Definition 1. Let µ be the policy such that secondary access
the buffer occupancy at SUrx (i.e., failed SU transmissions takes place if and only if the PU message is known to SUrx,
which are potentially recovered via BIC). On the other hand, i.e., µ(s) = 1, ∀s ∈ SK, µ(s) = 0, ∀s ∈ SU. We denote
SUtransmissionsmightjeopardizethecorrectdecodingofthe the SU access rate achieved by such policy as ǫth = W¯(µ).
PU message at SUrx, thus impairing the use of FIC/BIC, and The system is in the low SU access rate regime if ǫW ≤ ǫth
might violate the constraints in (13). in (17). Otherwise, the system is in the high SU access rate
regime.
Underµ∈U,thestateprocessisastationaryMarkovchain,
with steady state distribution π [12], [23]. π (s),s ∈ S, is
µ µ
the long-term fraction of the time-slots spent in state s, i.e., IV. OPTIMALPOLICY
πµ(s) = N→lim+∞N1 Nn=−01Pr(µn)(s|s0), where Pr(µn)(s|s0) is In this section, we characterize in closed form the optimal
the n-step transitionPprobability of the chain from state s0.5 policy in the low SU access rate regime, and we present an
In state (t,b,U), the SU accesses the channel with proba- algorithm to derive the optimal policy in the high SU access
bility µ(t,b,U), thus accruing the throughput µ(t,b,U)T . rate regime.
sU
Moreover,ifSUrxsuccessfullydecodesthePUmessage(with
probability 1−q(I) −µ(t,b,U)(q(A) −q(I))), bR bits are
ps ps ps sU A. Low SU Access Rate Regime
recoveredbyperformingBIConthebufferedreceivedsignals,
The next lemma shows that, in the low SU access rate
yielding an additional BIC throughput. Similarly, in state
regime, an optimal policy prescribes that secondary access
(t,0,K),the SU accruesthe throughputµ(t,0,K)T . Then,
sK
onlytakesplacein the states wherethePU messageis known
we can rewrite (10) and (11) in terms of the steady state
toSUrx,withanequalprobabilityinallsuchstates.Itfollows
distribution and of the cost/reward in each state as
that only FIC, and not BIC, is needed in this regime to attain
T¯s(µ)=TsUW¯s(µ)+F¯s(µ)+B¯s(µ), P¯s(µ)=PsW¯s(µ), (14) optimal performance.
where the SU access rate W¯s(µ), i.e., the average long-term Lemma 2. In the low SU access rate regime ǫW ≤ ǫth, an
number of secondary channel accesses per time-slot, the FIC optimal policy is given by6
throughput F¯s(µ) and the BIC throughput B¯s(µ) are defined µ∗(s)= ǫW, ∀s∈S , µ∗(s)=0, ∀s∈S . (18)
K U
ǫ
th
6Theoptimal policy in the low SUaccess rate is not unique. In fact, any
5Similarly to(10),(11)and(12),πµ(s)isindependent oftheinitial state policy µ such that µ(s) = 0, ∀s ∈ SU and W¯s(µ) = ǫth is optimal,
s0,duetotherecurrence ofstate (1,0,U). attaining thesamethroughput T¯s(µ)=TsKǫth as(18).
6
Moreover, T¯ (µ∗)=T ǫ , P¯ (µ∗)=P ǫ , and π (s)=0 and dT¯s(µ) = dW¯s(µ) =0. One exampleis the idle
s sK W s s W µ dµ(s) dµ(s)
T¯ (µ∗)=T(I)−(T(I)−T(A))ǫ . policy µ(s) = 0, ∀s: since the SU is always idle, the buffer
p p p p W
at SUrx is always empty,hence states (t,b,U) with b>0 are
Proof: For any policy µ ∈ U obeying the SU access
neveraccessed.Toovercomethisproblem,aformaldefinition
rate constraint W¯ (µ) ≤ ǫ , we have T¯ (µ) ≤ W¯ (µ)T ≤
s W s s sK isgiveninApp.B,bytreatingtheMarkovchainofthePU-SU
ǫ T .ThefirstinequalityholdssinceW¯ (µ)T isthelong-
W sK s sK system as the limit of an irreducible Markov chain. η (s) is
term throughputachievablewhen the PU message is known a µ
explicitly derived in Lemma 6 in App. B.
priori at SUrx, which is an upper bound to the performance;
the second from the SU access rate constraint. The upper We denote the indicator function of state s as δs : S 7→
bound ǫ T is achieved by policy (18), as can be directly {0,1}, with δs(s) = 1, δs(σ) = 0, ∀σ 6= s. Moreover, we
W sK
seen by substituting (18) in (14), (15). denote the policy at the ith iteration of the algorithm as µ(i).
We are now ready to describe the algorithm that obtains an
Remark 2. Note that secondary accesses in states S , where
U optimalpolicyin thehighSU accessrate regime.Anintuitive
the PU message is unknown to SUrx, would obtain a smaller
explanation of the algorithm can be found below.
throughput, namely at most T +p R ≤ T , where
sU s,buf sU sK
T is the "instantaneous" throughput and p R is the Algorithm 1 (Derivation of the optimal policy).
sU s,buf sU
BIC throughput, possibly recovered via BIC in a future ARQ
retransmission. Therefore, SU accesses in states SK are more 1) Initialization:
"cost effective". • Let µ(0) be the policy µ(0)(s)=0, ∀ s∈SU,
µ(0)(s)=1, ∀ s∈S , and i=0.
K
B. High SU Access Rate Regime • Let Si(d0l)e ≡ {s∈S : µ(0)(s)=0}≡ SU be the set
of states where the SU is idle.
In this section, we study the high SU access rate regime
in which ǫ > ǫ , thus complementing the analysis above 2) Stage i:
W th
for the regime where ǫW ≤ǫth. It will be seen that, if ǫW > a) Compute ηµ(i)(s), ∀ s ∈ Si(di)le and let s(i) ,
ǫth, unlike in the low SU access rate regime, the SU should argmaxs∈S(i) ηµ(i)(s).
generally access the channel also in states SU where the PU b) If η (s(i)i)dle≤ 0, go to step 3). Otherwise, let
message is unknown to SUrx in order to achieve the optimal µ(i)
performance. Therefore, both BIC and FIC are necessary to µ(i+1) =µ(i)+δs(i), Si(di+le1)≡Si(di)le\ s(i) .
attain optimality.Inthis section,we derivetheoptimalpolicy. c) Seti:=i+1.IfSi(di)le ≡∅,gotostep3)(cid:8).Oth(cid:9)erwise,
repeat from step 2).
We first introduce some necessary definitions and notations.
3) LetN =i,thesequenceofstates(s(0),...,s(N−1))and
Definition 2 (Secondary access efficiency). We define the of policies (µ(0),...,µ(N−1)).
secondaryaccessefficiencyunderpolicy µ∈U in state s∈S
4) Optimal policy: given ǫ ,
W
as
a) If W¯ (µ(N−1))≤ǫ , then µ∗ =µ(N−1).
ηµ(s)= dddWT¯¯µss(((sµµ))). (19) b) mOtahxerswii:sWe¯,sµ∗µ=(i)λ≤µW(ǫjW)+(a1n−dλλ)µ∈(j+(01,)1,]wuhneirqeuje,ly
dµ(s) solves W¯ (λµ(j)+(1−λ)µ(j+1))=ǫ .
(cid:8) s (cid:0) (cid:1) (cid:9) W
Thealgorithm,startingfromthe optimalpolicyforthecase
The secondary access efficiency can be interpreted as fol- ǫ = ǫ (Lemma 2), ranks the states in the set S in
W th U
lows. If the secondary access probability is increased in state decreasingorderofsecondaryaccessefficiency,anditeratively
s ∈ S by a small amount δ, then the PU throughput loss allocates the secondary access to the state with the highest
is increased by an amount equal to δ(T(I) − T(A))dW¯s(µ) efficiency,amongthestateswheretheSUisidle.Therationale
p p dµ(s)
(from (16)), the SU power is increased by an amount equal of this step is that secondary access in the most efficient
to δP dW¯s(µ) (from (14)), and the SU throughput augments state yields the steepest increase of the SU throughput, per
s dµ(s)
or diminishes by an amount equal to δdT¯s(µ) (depending unit increase of the SU access rate or, equivalently, of the
dµ(s) PU throughput loss and of the SU power expenditure. The
on the sign of the derivative). Therefore, η (s) yields the
µ optimality of Algorithm 1 is established in the following
rate of increase (or decrease if η (s) < 0) of the SU
µ theorem.
throughputperunitincreaseof the SU accessrate, asinduced
by augmenting the secondary channel access probability in Theorem 1. Algorithm 1 returns an optimal policy for the
state s. Equivalently, it measures how efficiently the SU can optimization problem (17).
access the channel in state s, in terms of maximizing the SU
Proof: See App. C.
throughput gain while minimizing its negative impact on the
PU throughput and on the SU power expenditure.
Remark3. Itisworthnotingthatthedefinitionofη (s)given V. SPECIAL CASE:DEGENERATECOGNITIVE RADIO
µ
in Def. 2 is not completely rigorous. In fact, under a generic NETWORK SCENARIO
policy µ, the Markov chain of the PU-SU system may not be WepointoutthatAlgorithm1determinestheoptimalpolicy
irreducible [23], so that state s may not be accessible, hence fora genericset ofsystem parameters.However,the resulting
7
thesequenceofpolicies(µ(0),...,µ(N−1))returnedbyAlgo-
rithm 1 is such that, ∀i∈N(0,N −1),
e e
g g
n n
ra ra µ(i)(s)=1, ∀s∈S , (22)
X X K
T T
1 b<b(i)(t)
µ(i)(t,b,U)= , ∀(t,b,U)∈S , (23)
SUtx SUrx PUtx PUrx 0 b≥b(i)(t), U
(cid:26)
where b(i)(t) is non-increasing in t and non-decreasing in i,
with b(0)(t)=0 and b(N−1)(t)=¯b (t), i.e.,
max
¯b (t)=b(N−1)(t)≥···≥b(i)(t)≥b(i−1)(t) (24)
max
≥···≥b(0)(t)=0.
b(i)(1)≥b(i)(2)≥...≥b(i)(t−1)≥b(i)(t)≥...≥b(i)(D),
Fig.3. Degenerate cognitive radionetwork
(25)
where
ionptteirmpraeltepdo.liIcnytdhoisessencottioanlw, wayeschoanvseidaerstaruscptuecreiatlhcaatsiesoefastihlye RTssUU 1−qpp qp(As)−qp(Is) A0(t+1)
general model discussed so far, a degenerate cognitive radio  + h1−qp(As) p(cid:16) −∆ (cid:17)× i 
network, where the activity of the PU is unaffected by the  (cid:18)qp(As)−qp(Is) s,buf s(cid:19) 
StrUaCnstormnasnisisdsmieorinttstehreoafnsctdheenthaSeriUoP,Udie.repe.ic,cettiehvdeericnihsaFzneingreo.l.3g,aiwnhbereetwPeUenrxthies ¯bmax(t)= qp(As)×−qpqpp((cid:16)Is)qp(As)1−−qqp(pIsp)(cid:17)(1A−0(qtp(+Is))1A)0(t+1) −1
outside the transmission range of SUtx, whereas SUrx is (cid:16) (cid:17)(cid:16) (cid:17)
 
inside the transmission range of both SUtx and SUrx. In this  
 
scenario,theinterferenceproducedbySU toPU isnegligible.  
 
In contrast, the PU produces significant interference at the  (26)
 
SU receiver. The SU thus potentially benefits by employing  
and we have defined
the BIC and FIC mechanisms. We denote this scenario as
a Degenerate cognitive radio network, and we model it by 1−qD−τ+1q(I)(D−τ+1)
A (τ), pp ps , (27)
assuming that the SNR of the interfering link SUtx→PUrx is 0 1−q q(I)
pp ps
deterministically equal to zero, i.e., γ = 0. From (1), we
sp 1−qD−τ+1
then have qp(Ip) = qp(Ap) , qpp, i.e., the outage performance of A1(τ), pp . (28)
1−q
thePUisunaffectedbytheactivityoftheSU,andtheprimary pp
ARQ process is independent of the secondary access policy.
Proof: See App. D.
We define
Remark 4. Interestingly,this is the same result derived in our
T −T −p R
∆s , sK sU s,buf sU. (20) work [2] for D = 2. However, therein the result was shown
R
sU to hold for general q(A) ≥ q(I) (not necessarily a degenerate
pp pp
From(9),itfollowsthat∆s ≥0,withequalityifRsU =RsK. cognitiveradio network),whereasLemma3 holdsforgeneral
Therefore, RsU∆s is the marginalthroughputgain accrued in D butonlyfora degeneratecognitiveradio networkscenario.
the states where the PU message is known to SUrx, over the
throughput accrued in the states where the PU message is The lemma dictates that, in the degenerate cognitive radio
unknown (instantaneousthroughputT plus BIC throughput network scenario, the SU should restrict its channel accesses
sU
p R , possibly recovered in a future ARQ retransmis- to the states corresponding to a low primary ARQ index and
s,buf sU
sion). The following lemma proves that, if the marginal small buffer occupancy at the SU receiver. Alternatively, the
largerthe ARQ indexorthe bufferoccupancy,the smallerthe
throughput gain ∆ is "small", the secondary accesses in the
s
high SU access rate regime in a degenerate cognitive radio incentive to access the channel. By doing so, the SU maxi-
networkareallocated,inorder,tothestatesin S (Lemma2), mizesthebufferoccupancyintheearlyHARQ retransmission
K
then to the idle states (t,b,U) in S , giving priority to states attempts, and invests in the future BIC recovery. When the
U
with low b and t over states with high b and t, respectively. primary ARQ state t approaches the deadline D, the SU is
An illustrativeexample of the optimalpolicy forthis scenario incentivized to idle so as to help SUrx to decode the PU
is given in Fig. 4. message, thus enabling the recovery of the failed SU trans-
missions from the buffered received signals via BIC, before
Lemma 3. Inthedegeneratecognitiveradionetworkscenario the ARQ deadline D is reached and the buffer is depleted.
with qp(Ap) =qp(Ip) =qpp, if Moreover,whenthebufferstatebgrows,sinceqp(As) >qp(Is),the
1−q(A) instantaneousreward accruedby stayingidle ((1−qp(Is))bRsU)
∆ < ps p , (21) approachesand,atsomepoint,becomeslargerthanthereward
s qp(As)−qp(Is) s,buf accrued by transmitting (TsU +(1−qp(As))bRsU), hence the
8
PU
1,0,U 2,0,U 3,0,U 4,0,U 5,0,U
Rp ≃2.52 qp(Ip) ≃0.38 qp(Ap) ≃0.68
SU, R =argmax T (R ,R )
sU Rs sU s p
RsU =1.12 TsU ≃0.59
2,1,U 3,1,U 4,1,U 5,1,U qp(Is) ≃0.61 qp(As) ≃0.74 ps,buf =0.26
RsK≃1.91 TsK ≃1.10
SU, R =R
sU sK
RsU ≃1.91 TsU ≃0.40
3,2,U 4,2,U 5,2,U q(I) ≃0.61 q(A) ≃0.88 p =0.37
ps ps s,buf
RsK≃1.91 TsK ≃1.10
TABLEI
PARAMETERSOFTHESUANDPU,FORTHESNRSγ¯s=5,γ¯p=10,
4,3,U 5,3,U γ¯ps=5,γ¯sp=2.
5,4,U VI. NUMERICALRESULTS
We considera scenario with Rayleigh fading channels, i.e.,
the SNR γ , x ∈ {s,p,sp,ps}, is an exponential random
x
variable with mean E[γ ] = γ¯ . We consider the following
x x
parameters, unless otherwise stated. The average SNRs are
2,0,K 3,0,K 4,0,K 5,0,K
set to γ¯ =γ¯ =5, γ¯ =10, γ¯ =2. The ARQ deadline is
s ps p sp
D =5.R ischosenasR =argmax T (R ).ThePU
sK sK Rs sK s
rate R is chosen as the maximizer of the instantaneous PU
p
Fig.4. Illustrative exampleofthestructureoftheoptimal secondaryaccess throughput under an idle SU, i.e., Rp = argmaxRTp(I)(R).
policy for the degenerate cognitive radio network; the SU is active in the For the rate R , we evaluate the two cases R = R∗
black states, idle in the white ones, and randomly accesses the channel in and R = RsU, where R∗ = argmax TsU(R ,RsU).
thegraystate;thearrowsindicatethepossiblestatetransitions(transitionsto sU sK sU Rs sU s p
state(1,0,U)areomitted). The former maximizes the instantaneous throughput under
interference from the PU, thus neglecting the buffering ca-
pability at SUrx; therefore, the choice R = R∗ reflects a
sU sU
incentivetostayidlegrows.Ontheotherhand,if ∆ islarge, pessimisticexpectationoftheabilityofSUrxtodecodethePU
s
then the marginalthroughputgain accruedin the states where messageandtoenableBIC.Asto thelatter,from(9) wehave
thePUmessageisknowntoSUrx,overthethroughputaccrued R =R =argmax T (R ,R )+p (R ,R )R ,
sU sK Rs sU s p s,buf s p sK
in the states where the PU message is unknown,is large. The hence R = R maximizes the sum of the instantaneous
sU sK
SU is thusincentivizedto stay idle in the initialARQ rounds, throughput and the future throughput possibly recovered via
so as to help SUrx decode the PU message. Therefore, for BIC, thus reflecting an optimistic expectation of the ability
large ∆ , the optimal policy may not obey the structure of of SUrx to decode the PU message, which enables BIC. The
s
Lemma 3. PU throughput loss constraint is set to ǫ = 0.2, and the
PU
As a final remark, note that, in the degenerate cognitive constraint on the SU power is set to P(th) = P (inactive).
s s
radio network scenario, the only limitation to the activity The resulting values of the system parameters are listed in
of the SU is the secondary power expenditure P¯ (µ), since Table I.
s
the primary throughput is unaffected. In the special case We consider the following schemes: "FIC/BIC", which
P(th) = P in (13), neither the secondary power expenditure employs both FIC and BIC; the optimal "FIC/BIC" policy is
s s
nor the primary throughput degradation limit the activity of derived using Algorithm 1 and Lemma 2; "FIC only", which
the SU, hence the optimal policy solves the unconstrained does not employ the buffering mechanism [1]; "no FIC/BIC",
maximization problem µ∗ =argmax T¯ (µ), whose solution whichemploysneitherBICnorFIC.Inthiscase,theSUmes-
µ s
follows as a corollary of Lemma 3. sageisdecodedbyleveragingthePUcodebookstructure[24];
however,possibleknowledgeofthePUmessagegainedduring
Corollary 1. In the degenerate cognitive radio network sce-
the decoding operation is only used in the slot where the PU
nario, the solution of the unconstrained optimization problem
message is acquired, but is neglected in the past/future PU
µ∗ =argmax T¯ (µ) yields
µ s retransmissions.For"noFIC/BIC",theoptimalpolicyconsists
µ∗(s)=1, ∀s∈S , (29) inaccessingthechannelwithaconstantprobabilityinalltime-
K
slots, independently of the underlying state, so as to attain
µ∗(t,b,U)= 10 bb<≥¯¯bbmax((tt)), , ∀(t,b,U)∈SU, (30) the PU throughputloss constraint with equality. "PM known"
(cid:26) max refers to an ideal scenario where SUrx perfectly knows the
where ¯b (t) is defined in (26). currentPU message in advance, and removesits interference;
max
9

HighSecondary LowSecondary 1.1 BIC/FIC
0.7 AccessRate AccessRate
Regime Regime 1 FIConly
µ)0.6 0.9 noBIC/FIC
¯ut,T(s0.5 0.8 PMknown
UThroughp00..34 BIC/FIC,RsU=Rs∗U ¯T(µ)s000...567
S0.2 BIC/FIC,RsU=RsK
FIConly,RsU=Rs∗U 0.4
0.1 FIConly,RsU=RsK 0.3
noBIC/FIC,RsU=Rs∗U
0 0.2
0.9 1 1.1 1.2 1.3 1.4 1.5
PUThroughput,T¯p(µ) 0 0.1 0.2 0.3 0.4 0.5
γ¯sp/γ¯p
Fig.5. SUthroughputvsPUthroughput.γ¯s=γ¯ps=5,γ¯sp=2,γ¯p=10.
Theotherparameters aregiveninTableI. Fig.6. SUthroughputvsSNRratioγ¯sp/γ¯p.PUthroughputlossconstraint
ǫPU=0.2.γ¯s=γ¯ps=5,γ¯p=10.RsU=R∗sU.
specifically, SUtx transmits with rate RsK, thus accruing the
throughput T at each secondary access; "PM known" thus 0.45
sK
BIC/FIC
yields an upperbound to the performanceof any other policy
FIConly
considered. µ) 0.4 noBIC/FIC
In Fig. 5, we plot the SU throughput versus the PU ¯T(s PMknown
throughput,obtained by varyingthe SU access rate constraint ut,
p
ǫ in (17) from 0 to 1. As expected, the best performance gh0.35
W u
o
is attained by "FIC/BIC", since the joint use of BIC and FIC hr
T
enables IC at SUrx over the entire sequence of PU retrans- U 0.3
S
missions. "FIC only" incurs a throughput penalty (except in
the low SU access rate regime T¯ (µ) ≥ 1.37 where, from
p
Lemma 2, "FIC/BIC" does not employ BIC), since the SU 0.25
transmissionswhichundergooutageduetosevereinterference
0 0.5 1 1.5 2 2.5 3 3.5 4
from the PU are simply dropped. "no FIC/BIC" incurs a γ¯ps/γ¯s
further throughput loss, since possible knowledge about the
PchUoimceesosfagtheeistrnaontsmexispslioointedratteos,pewrfeornmoteICt.haCtotnhceersneilnegctitohne FǫPigU.7=. 0S.2U.γ¯thsro=ug5h,pγ¯ustpvs=S2N,Rγ¯pra=tio1γ¯0p.sR/γ¯sUs.=PURt∗shUro.ughputlossconstraint
R = R∗ outperforms R = R for the scenario
sU sU sU sK
considered. Note that, with RsU = Rs∗U, the SU accrues γ¯ps/γ¯s,whereγ¯s =5andRsU =Rs∗U,whichisafunctionof
a larger instantaneous throughput (T ), but FIC and BIC
sU γ¯ .Wenoticethat,whenγ¯ =0,theupperboundisachieved
ps ps
are impaired, since both the buffering probability (8), p ,
s,buf withequality,sincetheSUoperatesundernointerferencefrom
and the probability that SUrx does not successfully decode
the PU. The upper bound is approached also for γ¯ ≫ γ¯ ,
the PU message, qp(As), diminish. Hence, in this case the correspondingtoastronginterferenceregimewhere,wpsithhigsh
instantaneous throughputmaximization has a stronger impact probability, SUrx can successfully decode the PU message,
on the performance than enabling FIC/BIC at SUrx. remove its interference from the received signal, and then
In Fig. 6, we plot the SU throughputversus the SNR ratio attempt to decode the SU message. The worst performanceis
γ¯sp/γ¯p, where γ¯p = 5 and RsU = Rs∗U. Note that, for attained when γ¯ps ≃ γ¯s/2. In fact, the interference from the
γ¯sp/γ¯p ≤ 0.5, the SU throughput increases. In fact, in this PU is neither weak enough to be simply treated as noise, nor
regimetheactivityoftheSUcauseslittleharmtothePU,and strong enough to be successfully decoded and then removed.
the constraint on the PU throughput loss is inactive. The SU In Fig. 8, we plot the SU throughput versus the SU rate
thus maximizes its own throughput. As γ¯sp increases from 0 ratio RsU/RsK, where RsK ≃1.91 is keptfixed. Clearly, "no
to0.5γ¯p,theactivityoftheSUinducesmorefrequentprimary FIC/BIC" attainsthebestperformanceforRsU =Rs∗U,which
ARQ retransmissions, hence there are more IC opportunities maximizes the throughput T (R ,R ) achieved when nei-
sU sU p
available and the SU throughput augments. On the other therFICnorBICareused.Ontheotherhand,theperformance
hand, as γ¯sp grows beyond 0.5γ¯p, the constraint on the PU of "FIC/BIC" is maximized for a slightly larger value of
throughput loss becomes active, secondary accesses become R . In fact, this value reflects the optimal trade-off between
sU
more and more harmful to the PU and take place more and maximizing the throughput T (R ≃ 0.59R in Fig. 9),
sU sU sK
more sparingly, hence the SU throughput degrades. maximizing the buffering probability, p (R → 1), and
s,buf sU
In Fig. 7, we plot the SU throughputversus the SNR ratio minimizing the probability that SUrx does not successfully
10
0.45
RsU/RsK≃0.64 0.45
0.4
0.35 RsU/RsK≃0.52
µ) µ) 0.4
¯(Ts 0.3 ¯T(s
ut, RsU/RsK≃0.59 ut,
p0.25 p
gh gh0.35
u u
o 0.2 o
hr hr
T T
U0.15 U BIC/FIC
S BIC/FIC S 0.3 FIConly
0.1 FIConly noBIC/FIC
0.05 noBIC/FIC PMknown
PMknown 0.25
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10
RatioofSUrates,RsU/RsK ARQdeadline,D
Fig. 8. SU throughput vs SU rate ratio RsU/RsK. RsK ≃ 1.91 is kept Fig.10. SUthroughputvsARQdeadlineD.PUthroughputlossconstraint
fixed.PUthroughputlossconstraintǫPU=0.2.γ¯s=5,γ¯sp=2,γ¯p=10, ǫPU=0.2.γ¯s=γ¯ps=5,γ¯sp=2,γ¯p=10.RsU=R∗sU.
γ¯ps=5.
0.6 receiver (SUrx), after decoding the PU message, exploits this
RsU/RsK=0.59 knowledgetoperformForwardIC(FIC)inthefollowingARQ
0.5 retransmissions and Backward IC (BIC) in the previous ARQ
retransmissions, corresponding to SU transmissions whose
0.4 decoding failed due to severe interference from the PU. We
haveemployedastochasticoptimizationapproachtooptimize
0.3 theSUaccessstrategywhichmaximizestheaveragelong-term
SUthroughput,underconstraintsontheaveragelong-termPU
0.2 throughput degradation and SU power expenditure. We have
Bufferingprob.atSR,ps,buf provedthattheSUprioritizesitschannelaccessesinthestates
PMdecodingprob.atSR
0.1 withactiveSU,1-qp(As) where SUrx knows the PU message, thus enabling FIC, and
wehaveprovidedanalgorithmtooptimallyallocateadditional
norm.SUThroughput,TsU/TsK
0 secondary access opportunities in the states where the PU
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
RatioofSUrates,RsU/RsK message is unknown.Finally, we have shown numericallythe
throughputgain of the proposed schemes.
Fig.9. Probabilities ps,buf,1−qp(As) andnormalized SUthroughput TsU
vs theSU rate ratio RsU/RsK.RsK ≃1.91is kept fixed. γ¯s =γ¯ps =5, APPENDIXA
γ¯sp=2,γ¯p=10.
In this appendix, we compute T¯ (µ), W¯ (µ) and state
s s
properties of W¯ (µ).
s
decode the PU message,qp(As) (RsU →0). Finally, "FIC only" Definition 3. We define G (t,b,Φ), V (t,b,Φ) and
is optimized by RsU ≃ 0.52RsK < Rs∗U. Since "FIC only" D (t,b,Φ) as the average throuµghput, the avµerage number
µ
does not use BIC, this value reflects the optimal trade-off
of secondary channel accesses and the average number of
between maximizing T and minimizing q(A) (R →0).
sU ps sU time-slots, respectively, accrued starting from state (t,b,Φ)
In Fig. 10, we plot the SU throughput versus the ARQ
until the end of the primary ARQ cycle under policy µ (i.e.,
deadline D. We notice that, when D = 1, all the IC
until the recurrent state (1,0,U) is reached). Starting from
mechanisms considered attain the same performance as "no X (D +1,b,Φ) = 0, ∀b,∀Φ ∈ {U,K},7 where X stands
µ µ
FIC/BIC". In fact, this is a degeneratescenario where the PU for G , V or D (we write X ∈ {G,V,D}), these are
µ µ µ
does not employ ARQ, hence no redundancyis introduced in defined recursively as, for t∈N(1,D), b∈N(0,t−1),
the primary transmission process. Interestingly, by employing
FIC or BIC, the performance improves as D increases. In Xµ(t,b,U)=xµ(t,b,U)
fact, the larger D, the more the redundancyintroducedby the +Prµ(t+1,b,U|t,b,U)Xµ(t+1,b,U)
primary ARQ process, hence the more the opportunities for +Prµ(t+1,b+1,U|t,b,U)Xµ(t+1,b+1,U)
FIC/BIC at SUrx. +Prµ(t+1,0,K|t,b,U)Xµ(t+1,0,K), (31)
X (t,0,K)=x (t,0,K)
µ µ
VII. CONCLUSION + qp(Ip)+µ(t,0,K)(qp(Ap)−qp(Ip)) Xµ(t+1,0,K),
In this work, we have investigated the idea of leveraging h i
where x (t,b,Φ) is the cost/reward accrued in state (t,b,Φ)
µ
the redundancyintroducedbythe ARQ protocolimplemented
andPr (·|·)istheone-steptransitionprobability,whichcanbe
µ
by a Primary User (PU) to perform Interference Cancellation
(IC) at the receiver of a Secondary User (SU) pair: the SU 7Weintroducethefictitiousstate(D+1,b,Φ)fornotationalconvenience.