On Optimal Spectrum Access of Cognitive Relay With Finite Packet Buffer

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by  Kedar Kulkarni

On Optimal Spectrum Access of Cognitive Relay With
Finite Packet Buffer
Kedar Kulkarni, Student Member, IEEE, and Adrish Banerjee, Senior Member, IEEE
Abstract—We investigate a cognitive radio system where sec- using time sharing, with some access probability. Furthermore,
ondaryuser(SU)relaysprimaryuser(PU)packetsusingtwo-phase we consider that both PU and SU have finite capacity queues.
relaying.SUtransmitsitsownpacketswithsomeaccessprobability
SU’s finite queue size affects cooperation offered to PU. Thus,
in relaying phase using time sharing. PU and SU have queues of
queue sizes at both PU and SU impact PU’s packet loss. Our
finitecapacitywhichresultsinpacketlosswhenthequeuesarefull.
Utilizing knowledge of relay queue state, SU aims to maximize its aim is to find optimal access policy of SU that maximizes SU
packetthroughputwhilekeepingpacketlossprobabilityofPUbelow packet throughput while satisfying PU’s packet loss constraint.
7athreshold.Byexploitingstructureoftheproblem,weformulateit Specifically, our contribution is as follows.
1as a linear program and find optimal access policy of SU. We also
0proposelowcomplexitysub-optimalaccesspolicies,namelyconstant • We model PU and relay queues as discrete time Markov
2probability transmission and step transmission. Numerical results chains (DTMC). Using DTMC analysis, we characterize
are presented to compare performance of proposed methods and packet loss probability of PU and SU packet throughput.
n
astudy effect of queue sizes on packet throughput. • We formulate the problem of maximizing SU throughput
IndexTerms—Blockingprobability,cognitiveradio,finitecapacity
J under PU packet loss constraint, which is non-convex. By
queue, optimal access, relaying
2 exploiting structure of the problem, we transform it into a
2 linearprogramming(LP)problemoverthefeasiblerangeof
I. INTRODUCTION

PUpacketthroughput.Wealsoproposetwolow-complexity
] Incognitiveradio(CR)networks,secondaryusers(SUs)access
T suboptimal access methods that transform original multi-
spectrum allocated to primary users (PUs) in such a way that
I dimensional problem into one dimensional problem.
.given quality-of-service (QoS) requirement of PUs is satisfied.
s • Finally,wepresentnumericalresultstostudyeffectofqueue
Users store packets arriving from higher layers in queues before
c sizes, path loss and time sharing on SU packet throughput.
[transmission over wireless link. Various works have studied SU
We also compare the performance with infinite capacity
packet throughput for non-cooperation scenarios where SU’s
1 queue system under queue stability constraint.
vaccess probability is optimized under queue stability constraint
4of PU [1]–[3]. Cooperation between SU and PU improves II. SYSTEMMODEL
0throughput of both users as shown in [4] and [5]. These works
As shown in Fig. 1, a PU source P transmits packets to PU
2consider queues of infinite storing capacity. In practice, queues
destination D with assistance of a SU node S using two-phase
6
are of finite size. If a queue is full, new packets cannot be
0 relayingasdonein[8].NodesP andS areequippedwithpacket
admitted to the queue and are lost. Queueing performance of
. queue Q of capacity N and relay queue Q of capacity N
1 P P S S
finite sized SU queue was studied in [6]. In [7]–[11], authors
0 respectively. In a slot of duration T, P transmits its packet with
studied relay selection problem for finite buffer-aided relaying
7 powerP fortimeβT, β ∈[0, 1].IfDfailstoreceivethepacket,
P
1systems. These works considered dedicated relay nodes that do it is admitted to the relay queue at S, provided that the packet is
:not have their own data to transmit. In [12] and [13], authors
v correctlyreceivedatS andtherelayqueueisnotfull.Inrelaying
consideredcooperativeCRnetworkswithfinitesizedrelayqueue
i phase of duration (1−β)T, S relays the PU packet to D with
Xandproposedpacketadmissioncontrolassumingthatrelayqueue
power P . With some access probability, SU also transmits its
S
rinformation is available at SU. However, underlying assumption
a own packets to SU destination R using time sharing, that is, SU
in these works is that PU queue has infinite buffer length.
relays PU packet for duration α(1−β)T and transmits its own
Also, whole slot is used by the relay either for transmission or
packet for duration (1−α)(1−β)T, α ∈ [0, 1]. The access
for reception. A relaying protocol where packet reception and
probability is p when there are n, 0≤n≤N packets in Q .
n S S
transmissiontakesplaceinthesameslotusingtimesharing,may
If Q is empty, whole relaying duration (1−β)T is used to
S
enable the relay to improve its throughput by transmitting own
transmit SU packet, with probability p =1.
0
packets more frequently.
We assume that all channels are independent block-fading in
In this paper, we investigate SU throughput in a cooperative
nature, that is, channel gains remain constant during a slot and
CR system where SU relays failed packets of PU using two-
varyindependentlyfromslottoslot.Channelpowergainbetween
phaserelaying.SUtransmitsitsownpacketsintherelayingphase
source s and destination d is denoted as g and is exponentially
sd
distributed with mean σ2 , s,d ∈ {P, S, D, R}. The distance
Copyright(c)2015IEEE.Personaluseofthismaterialispermitted.However, sd
permissiontousethismaterialforanyotherpurposesmustbeobtainedfromthe between s and d is denoted by rsd and path-loss exponent
[email protected]. is denoted by κ. Additive white Gaussian noise (AWGN) at
The authors are with Department of Electrical Engineering, Indian Institute receivers has power σ2 . PU and SU packets have fixed length
of Technology Kanpur, Kanpur 208016, INDIA (e-mail: [email protected]; N
[email protected]). of B bits. A packet is assumed to be delivered successfully to
Receive Relay Tx
(a)
Fig.1. SystemmodelofSUrelayingPUpacketsandtransmittingownpackets
usingtwo-phaserelayingandtimesharing
intendedreceiverifinstantaneouschannelcapacityisgreaterthan
required transmission rate. Then probability of successful packet (b)
transmission is given by [3]
Fig.2. DiscretetimeMarkovchain(DTMC)modelof(a)PUqueueQP and
θsd=Pr(cid:20)log2(cid:18)1+ gsdσPN2srs−dκ(cid:19)≥WBTs(cid:21)=exp−σN2Ps(cid:16)r2s−WdκBTσss2−d(11(cid:17)), (abr)eTrehglaeivyneqnpureoubebyaQbνSi1lit=ies1o−f QwP0 abnedingνNnPon=-emwpNtyPa=nd(Q1−P1µPbe)iγnNgPfwul0l
respectively. To keep PU packet loss below a limit, probability
where P is transmit power, T is transmission duration and W
s s of Q being full should remain below a threshold (cid:15), i.e. ν ≤
ischannelbandwidth.Wedenoteprobabilityofsuccessfulpacket P NP
(cid:15). From Fig. 2(a), we observe that, any increase in µ would
transmission on s−d link without time sharing by θ , s,d ∈ P
sd decreaseprobabilityofQ beingfull,thatis,ν monotonically
{P, S, D, R}. In case of time sharing, time available for relay- P NP
decreases in µ . Thus, for given value of λ , we can find µ ∈
ing/transmissionofPUandSUpacketsislessthan(1−β)T.We P P P
[0, 1] such that ν (µ ) = (cid:15), using Bisection method. Packet
use θSD and θSR to denote successful transmission probabilities loss constraint ν NP≤ P(cid:15) can now be written as µ ≥ µ . SU
in case of time sharing1. As required transmission rate is higher, NP P P
should choose its access probability in such a way that packet
probabilities of successful packet transmission decrease. Thus,
loss constraint of PU is satisfied.
we have θ < θ and θ < θ . Successful transmission
SD SD SR SR
probabilities on all links are known to the SU [2]–[4].
III. OPTIMALSPECTRUMACCESS
A. Queue blocking and packet loss
Packet arrival process at PU queue QP is Bernoulli with DTMC of the relay queue QS is as shown in Fig. 2(b) where
average rate λ ∈[0, 1] packets/slot. A packet is removed from state n denotes number of packets in relay queue at the end
P
QP only when it is received at D or S. A PU packet is admitted of receiving phase. Probability of a PU packet arriving at QS
to the relay queue QS when all of the following events are true– is q. When QS is in state n, SU transmits its own packets
1) Packet transmission on P − D link fails, 2) PU packet is with probability pn using time sharing. Thus, with probability
successfully received at S, and 3) QS is not full. Thus, packet (1−pn), PU packet is relayed for duration (1−β)T and with
departurerateatQP,denotedasµP,dependsonchannelbetween probability pn, PU packet is relayed for duration α(1−β)T.
P−S and state of QS. When QP is full, new packets cannot be Then probability of a PU packet departing QS in state n>0 is
admitted to the queue and are dropped.
(cid:0) (cid:1)
r =(1−p )θ +p θ =θ −p θ −θ . (4)
PU queue can be modeled as a discrete time Markov chain n n SD n SD SD n SD SD
(DTMC) as shown in Fig 2(a) where states denote number of
When PUis present,a packet isreceived at S with probability
packets in PU queue. Let w , n=0, 1,..., N be steady state
n P
probabilityofPUqueuebeinginstaten.Alsoletγ = λP(1−µP). (1−θPD)θPS. Thus, we have q = ν1θPS(1−θPD). For
Then we can write local balance equations for DTMC(1o−fλQP)µPas 1 ≤ n < NS, state transition from n to (n+1) occurs when
P
packet transmission of a packet in Q fails and a new PU
γ S
w = w , (2) packet is received, which happens with probability q(1−r ).
1 (1−µ ) 0 n
P State transition from n to (n−1) occurs when a packet is
wn+1 =γwn, n=1, 2,..., NP −1. (3) successfully relayed and no new packet arrives, which happens
Noting that w = γn−1w , n > 1 and (cid:80)NP w = 1, we get with probability (1−q)rn. Let πn, n=0, 1,..., NS be steady
n 1 n=0 n state probability of Q being in state n. Then we write local
probability of Q being empty as S
P
balance equations as
(cid:40) (1−µP)(1−γ) for γ (cid:54)=1
w0 = 1−1µ−Pµ(1P−γ)−γNP+1 for γ =1. π1 = (1−qq)r π0, (5)
NP+1−µP 1
q(1−r )
1Note that notation θsd only signifies success probability with time sharing πn+1 = (1−q)rn πn, n=1, 2,..., NS −1. (6)
andθsd(cid:54)=1−θsd. n+1
Forgivenvaluesofλ andµ ,steadystateprobabilitiesofrelay From (8) and (10), we see that the feasible values of µ are
P P P
queue can be calculated from (5), (6) using
(cid:8) (cid:9)
max µ , θ +θ θ (1−θ )
(cid:88)NS P PD PS SD PD
πn =1. (7) ≤µP ≤θPD+θPS(1−θPD). (16)
n=0
The linear program in (15) is solved over feasible values of
At the start of receiving phase, Q is full with probability
S
µ . Value of µ that corresponds to the maximum SU packet
π (1−r ). As a PU packet is admitted to Q only when P P
NS NS S throughputischosen.Fromoptimala andπ ,optimalSUaccess
Q is not full, we obtain packet departure rate of PU queue as n n
S µP =θPD+θPS(1−θPD)[1−πNS(1−rNS)]. (8) upsroedbaCbViliXtiepsaacrkeagfoeufnodrMasApTnL=ABπann[1,4]nto=s0o,lv1e,(.1.5.),iNnSp.olWyneohmaivael
A. SU throughput maximization complexity.
When there are n > 0 packets in Q , SU transmits its own
S B. Suboptimal methods
packet for duration (1−α)(1−β)T with probability p . If
n
Q is empty, whole duration (1−β)T is used to transmit SU From (4), (5), (6) and (7), we get steady state probabilities for
S
packet with probability p = 1. Given PU packet arrival rate Q as
0 S
λ , our objective is to maximize SU packet throughput while
P
ensuringthatpacketlossprobabilityofPUiskeptbelowspecified π =(cid:34)1+ 1 (cid:88)NS (cid:18) q (cid:19)n n(cid:89)−1(cid:18)1−rm(cid:19)(cid:35)−1, (17)
threshold. Thus, the optimization problem is written as 0 r 1−q r
1 m+1
n=1 m=1
max µS =θSRπ0+θSR(cid:88)NS πnpn (9) π =(cid:34)(cid:18) q (cid:19)n 1 n(cid:89)−1(cid:18)1−rm(cid:19)(cid:35)π , n>0. (18)
p,π,µP n=1 n 1−q r1 rm+1 0
m=1
s. t. µ ≥µ , (10)
P P
It can be proven that π is monotonically decreasing function
0≤p ,π ≤1, n=0, 1,..., N , 0
n n S of access probability p , n = 1, 2,..., N . Also we can prove
n S
p =1
0 that π , 0 < n ≤ N is a monotonically increasing function
n S
µP =θPD+θPS(1−θPD)[1−πNS(1−rNS)], of pm, m ≤ n and a monotonically decreasing function of
(5), (6), (7), pm, m > n. Intuitively, this can be explained from Fig. 2(b)
as follows. As access probability p increases, packet departure
where p = [p0,..., pNS]T and π = [π0,..., πNS]T. Opti- rate of QS decreases. Thus, more pmackets get queued up in QS.
mization problem in (9) is non-convex due to product terms Hence, the probability of Q having more than m packets in-
S
of optimization variables πn and pn. We transform it into a creases,whileprobabilityofQS havingpacketslessthanorequal
linear programming (LP) problem by exploiting structure of the to m decreases. Using this nature, we propose low complexity
problem. suboptimalmethodsthatsimplify(N +1)dimensionalproblem
S
Leta =π p .Thenwehavea =π and0≤a ≤π , n=
n n n 0 0 n n in (9) to a one-dimensional problem.
1, 2,..., N . From (7), we have
S 1) Constant probability transmission (CPT): In this method,
(cid:88)NS SU transmits its own packets with a fixed probability p when
0≤ an ≤1. (11) there are n>0 packets in relay queue. Thus, we have
n=0
(cid:40)
Using (4), we can transform balance equations (5) and (6) as 1 forn=0
p = (19)
given in (12) and (13) on next page. n p otherwise.
Similarly, constraint in (8) can be written as
(1−θSD)πNS+(cid:0)θSD−θSD(cid:1)aNS =1−θPµSP(1−−θPθPDD). (14) IθnSRtphi(cid:80)s Nnc=Sa1seπ,n.SUUsinpgac(cid:80)ketNn=St1hrπonug=hpu1t−isπ0µ,Swe=canθwSRritπe0th+e
problem of maximizing µ for fixed µ as
S P
Thus, constraints (5), (6), (8) are transformed into constraints
(11), (12), (13) and (14) which are affine in πn and an. The max θ p+π (cid:0)θ −θ p(cid:1)
SR 0 SR SR
optimization problem in (9) is still non-convex in µ . However, p∈[0,1]
P
for a given value of µ , the problem becomes a LP problem in s. t. (8), (5), (6), (7), (19).
P
variables π and a=[a , a ,..., a ]T and is written as
0 1 NS (cid:0) (cid:1)
The term π θ −θ p is monotonically decreasing in p
(cid:88)NS while term θ0 pSRis incrSeRasingin p. Thus, thereexists a unique p
max θ a +θ a (15) SR
SR 0 SR n
π,a that maximizes µ . Optimal solution can be found using Interval
n=1 S
halving method with complexity O(1).
s. t. 0≤π ≤1, n=0, 1,..., N ,
n S
2) Step transmission (ST): In this method, SU transmits its
a =π , 0≤a ≤π , n=1, 2,..., N ,
0 0 n n S ownpacketsusingtimesharingwithprobability1untillengthof
(7), (11), (12), (13), (14).
Q reaches a threshold N . Once it crosses N , the relaying
S th th
(cid:0) (cid:1)
θ (1−q)π −qπ = θ −θ (1−q)a , (12)
SD 1 0 SD SD 1
(cid:0) (cid:1) (cid:0) (cid:1)
θ (1−q)π −q(1−θ )π = θ −θ (1−q)a +q θ −θ a , n=1,..., N −1. (13)
SD n+1 SD n SD SD n+1 SD SD n S
0.9 0.7 0.7 σ2PD=-3 dB
Optimal, σ2 =-10 dB
0.8 NCPP=T∞ 0.6 0.65 σP2PDD=-13 dB
0.7 SCTRM ot) 0.5 0.6 σ2PD=-20 dB
(packets/slot)µS0000....3456 00.4.4675 0.445 0.45 0.455 (packets/slµS000...234 00..0045..5545
0.2 0.1 0.35
0.1
0 0.3
0 5 10 15 0 5 10 15
0 N N
0 0.2 0.4 0.6 0.8 1 P S
λ (packets/slot) (a) (b)
P
Fig.3. SUpacketthroughputµSversusPUpacketarrivalrateλP forNS =10 Fig.4. EffectofqueuesizesonSUpacketthroughputµS for(a)NS =10(b)
andNP →∞. NP =100andλP =0.5
phase duration of (1−β)T is used only to relay PU packets. 3) Throughput region: Fig. 3 plots throughput region of
Thus, we have proposed cooperation model. As λ increases, higher µ is
P P
(cid:40) required to satisfy PU packet loss constraint. To support high
1 forn≤N
p = th (20) µP, SU lowers its access probability. Thus, µS decreases with
n
0 otherwise. increase in λ . As λ increases further, constraint (10) becomes
P P
infeasible, at which point µ drops to zero. We also see that
S
In this case, the objective is
performance of constant probability transmission (CPT) method
and step transmission (ST) method is close to that of optimal
(cid:88)Nth
max θ π +θ π method.Hence,thesuboptimalmethodsaregoodlow-complexity
SR 0 SR n
Nth∈{0,1,...,NS} n=1 alternatives to the optimal method.
s. t. (5), (6), (7), (8), (20). Asabaselineforcomparison,wealsoplotthroughputregionof
cooperativerelayingmethod(CRM)in[13].InCRM,PUutilizes
With increasing Nth, π0 decreases while number of terms in whole undivided frame duration T for transmission/reception
summation increase. If value of θSR is very low compared to and optimizes SU access probability under PU queue stability
θSR, decrease in π0 is significant and µS initially decreases. constraint. In contrast, two-phase relaying model dedicates βT
But as π0 approaches zero, µS increases due to increasing value duration for reception in each slot. Thus, in CRM, probabilities
of θSR(cid:80)Nn=th1πn. For high value of θSR, µS increases with of successful transmission on P−D and S−R links are higher,
increasingNth.ThroughputdropstozerowhenπNS increasesto resulting in better performance of CRM at low and high values
such a value that constraint (4) cannot be satisfied. Value of Nth of λP. But in mid-range of λP, two-phase relaying benefits by
thatmaximizesµS canbefoundbylinearsearchwithcomplexity gaining time to transmit own packets as SU relays PU packets in
O(NS). the same slot.
Suboptimal methods run over all feasible values of µ given
P 4) Effect of queue sizes: Fig. 4(a) plots SU packet throughput
in (16) and the value that corresponds to maximum SU packet
µ against PU queue capacity N for different values of P−D
S P
throughput is chosen.
channel gains. Low values of N cannot satisfy packet loss
P
constraint in (10). Packet throughput achieved in such infeasible
IV. NUMERICALRESULTSANDDISCUSSION
casesinzero.IncreasingN decreasesµ whichistheminimum
P P
Parameter values used to plot results are as follows. Transmit PU departure rate required to satisfy packet loss constraint. This
power is P = P = 0.1W. Frame duration is T = 100ms. allows SU to transmit its own packets with higher access proba-
P S
Time sharing factors are β = α = 0.5 unless stated otherwise. bilities.Thus,µS increaseswithincreaseinNP.AsNP increases
All channels have average channel gain σs2d = −10dB, s,d ∈ further,decreaseinµP isinsignificant.AccessprobabilitiesofSU
{P, S, D, R}. Noise power is σN2 =10−5W. We take B/W = becomeconstantandinturnµS becomesconstant.Forhighvalue
3 × 10−3bits/Hz. We consider rPS = rSD = rSR = 100m. of σP2D, PU packet arrival rate at QS is less which allows higher
Path loss exponent is κ=2. Packet loss probability threshold is SU access probabilities. Thus, µS increases as σP2D increases.
(cid:15)=0.01. Fig. 4(b) shows an interesting tradeoff involving relay queue
1 0.5
σ2 =-3 dB
0.9 PD
σ2 =-10 dB 0.4
0.8 σP2D=-13 dB 0.6
slot) 00..67 σPP2DD=-20 dB 0.4 0.3
kets/ 0.5 µS 0.2
c 0.2
(paS0.4 0.1
µ 0.3 0
0
0 0
0.2
0.2 0.2
0.1 0.4 0.4
0.6 0.6
020 40 60 80 100 120 140 160 180 β 0.8 0.8 α
rPS (m) 1 1
Fig. 5. Effect of distance between PU source and relay rPS on SU packet Fig.6. EffectoftimesharingfactorsβandαonSUpacketthroughputµS for
throughputµS forNP =100,NS =10,λP =0.5andrPD =200m. λP =0.5,NS =10,NP =50andσP2D =−20dB.
as objective in (9) is non-convex in β and α.
capacity N . Increase in N allows SU to transmit its own
S S
packets with higher probability. Also, from (17), we observe that V. CONCLUSION
increase in N decreases probability of relay queue being empty
S
π . For high values of σ2 , PU packet arrival rate at Q is WestudiedaCRsystemwhereSUemploystwo-phaserelaying
0 PD S
less. In this case, decrease in π (and subsequent decrease in torelayfailedPUpackets.Bothusershavepacketqueuesoffinite
0
θ π ) is significant compared to increase in µ due to higher capacity which results in packet loss. We proposed optimal as
SR 0 S
access probability. Thus, µ decreases with increase in N . For wellassuboptimalaccessmethodsforSUtomaximizeitspacket
S S
low values of σ2 , PU packet arrival rate at Q is more. In this throughput under packet loss constraint of PU. We observed that
PD S
case,increaseinSUthroughputduetohigheraccessprobabilityis two-phaserelayingmodelperformsbetterthancooperationmodel
significant. But as N increases further, π approaches zero and without time sharing for mid-range values of PU packet arrival
S 0
(cid:80)NS π p decreases. Thus, with increasing N , µ initially rate.Suboptimalmethodshavenegligiblelossintheperformance
n=1 n n S S
increases and then gradually decreases. and are good low complexity alternatives to the optimal method.
Furthermore, results revealed that as relay queue size increases,
5) Effect of distance: Fig. 5 plots SU throughput against
SU throughput improves initially but then decreases. PU queue
distancebetweenPUsourceandSUsourcer .Here,weassume
PS
size limits maximum supported PU packet arrival rate.
that D and R are in close vicinity and lie on the line connecting
P andS.Thenforgivenr ,wehaver =r =r −r .
PD SD SR PD PS REFERENCES
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