Queue management for two-user cognitive radio with delay-constrained primary user

ContentslistsavailableatScienceDirect

Computer

Networks

journalhomepage:www.elsevier.com/locate/comnet

Queue

management

for

two-user

cognitive

radio

with

delay-constrained

primary

user

Kamal

Adli

Mehr

a,1

_,

_Javad

_Musevi

_Niya

a

_,

_Nail

_Akar

b,2,∗

a WiLab, Department of Electrical and Computer Engineering, University of Tabriz, Iran b Electrical and Electronics Engineering Department, Bilkent University, Turkey

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 27 September 2017 Revised 11 April 2018 Accepted 30 May 2018 Available online 31 May 2018 Keywords:

Queue management

Multi-regime Markov fluid queue Overlay cognitive radio Physical layer security

a

b

s

t

r

a

c

t

Inthispaper,twonovelQueueManagementPolicies(QMP)areproposedforQualityofService(QoS) en-hancementofatwo-userCognitiveRadioNetwork(CRN)comprisingaPrimaryUser(PU)andSecondary User(SU),thelatterhavingnon-causalinformationonPU’smessages(orpackets).Specifically,weaim tomaximizethethroughputoftheSUwhilesatisfyingthedelaycriterionofthePrimaryUser(PU).The firstproposedQMPisahybridinterweave/overlayschemewherealltheSU’sresourcesaredevotedtothe transmissionofPU’spackets.ThesecondproposedQMPadaptivelyusesallorsomeoftheSU’sresources towardsthetransmissionofthePU’spacket,thisdecisionbeingbasedonthepacket’sdelayexperienced inthePUqueue.ForthisadaptiveQMP,anovelmulti-regimeMarkovfluidqueuemodelisproposedvia whichclosed-formexpressionsarederivedandvalidatedfortheexactdelaydistributionforPoissonPU trafficandexponentiallydistributedpacketlengths.Usingthisanalyticaltool,weoptimallytunethe pa-rametersoftheadaptiveQMPandweshowthroughnumericalexamplesthatitconsistentlyoutperforms thehybridinterweave/overlaymodelaswellastwootherconventionalschemesintermsofSU through-put.WealsoshowthattheperformanceimprovementattainablebytheproposedQMPdependsonthe intensityofPUtrafficaswellasthechannelconditions.Aheuristicsuboptimalparametertuningscheme isalsoproposedwithlessercomputationalcomplexity.

© 2018ElsevierB.V.Allrightsreserved.

1. Introduction

Demand for access to wireless spectrum has been growing dramatically. However, conservative spectrum allocation policies havegiven riseto underutilizedlicensedchannels[1].CRNshave emerged as a promising solution to cope with spectrum ineffi-ciency inthese channels[2].Key toCRNs isthe incorporation of SecondaryUsers(SU)oflowerprioritywhichshouldtransmittheir messages in a way that the Primary Users (PU) of the licensed channel wouldnot be adversely affected.There arethree distinct deploymentparadigmsforcognitiveradio,namelyunderlay, inter-weave,andoverlayparadigms[3].Theseparadigmsare described briefly inTable 1.In the underlayparadigm, thecognitive trans-mission is allowed if theinterference generated by the cognitive useron theprimary receiverisbelowapredefined threshold.On

∗ _{Corresponding author.}

E-mail addresses: k.adlimehr@tabrizu.ac.ir (K.A. Mehr), niya@tabrizu.ac.ir (J.M. Niya), akar@ee.bilkent.edu.tr (N. Akar).

1 The research of K. A. Mehr was carried out when he was visiting Bilkent Uni- versity, Turkey.

2 His research is supported in part by Tübitak project 115E360.

theotherhand,intheinterweaveparadigm,theSUutilizes tempo-raryspectrumholesinspace,time,and/orfrequency opportunisti-callytotransmititsmessages,i.e.,SUtransmitsitsmessages when-everthereisnoongoing primary transmissiononthechannel. In contrast,intheoverlayparadigm,thesecondary transmitter com-pletelycooperateswiththeprimarytransmittertodeliverthePU’s messages to the intended receivers. In exchange, the PU permits thecooperativeSUtoutilizethelicensedchannelofPU[3].Inthis paradigm,the primarymessage isknownatthesecondary trans-mittercausally ornon-causally. The overlay paradigm is suitable formodeling primary networks that sublease their spectrum for monetaryorsecuritypurposesforwhichalevelofcooperation be-tweentheprimaryandsecondarynetworksisnotunusual[4].For instance, a cognitive radio system where the cognitive transmit-terhasacquiredthemessageofthe primaryuserintheprevious phases,or,acognitiveradiosystem, whereboth primaryand sec-ondarytransmitters belong to the sameauthority (such ascloud empoweredcellularnetwork[5])areexamplesofoverlaycognitive radiosystems.

In this paper, a novel framework is introduced for managing thequeuesinatwo-useroverlaycognitiveradioscenariowiththe SUhavingnon-causalinformationaboutPU’smessagesandthePU https://doi.org/10.1016/j.comnet.2018.05.028

Ta b le 1 Comparison of under la y, int e rw ea v e , and ove rl ay cogniti v e ra dio par a digms. N e tw or k-side inf o rmation kno w n at the SU (Simult a neous) tr ansmission condition SU’s tr ansmit po w e r is limit e d by Un d e rl a y Int e rf er ence ge n e ra te d to the primar y re ce ive r When the int e rf er ence ge n e ra te d to the primar y re ce ive r is be lo w a pr e d efine d thr e shold Limit e d by the int e rf er ence ca use d to the primar y tr ansmitt e r Int e rw ea v e Spectrum holes in space, time, and/or fr eq uency When ther e is de te ction of loss of PU acti vity Limit e d by the ra n g e of PU acti vity that the SU can de te ct Ov er la y Channel ga in s, encoding t e chniq u es, and possibl y PU’s messag es When the int e rf er ence to the PU can be of fs e t by using part of SU’s po w e r to re la y PU’s messag es SU can tr ansmit at an y po w e r le v e l

is subjectto statistical delayconstraints. Furthermore, two novel QMPs are proposed for the packet queue held at the PU. Two-useroverlayscenarios(alsoknownasCognitiveInterFerence Chan-nel(CIFC))havebeenstudiedbyinformation-theoreticmethodsin [3,6,7]. A recent survey on the extension of CIFC to the case of multipleusers ispresented in[8]. Inthispaper,we focus onthe two-user CIFC scenario similar to some recent studies on cogni-tive radio with QoS constraints [9–11]. Building upon the exist-ing information-theoreticresults, wepropose queuemanagement policies so as to provide as many transmission opportunities as possibleforthe SU(providing maximumbenefitforthe coopera-tivecognitiveuser)whilesatisfyingthePU’sdelayconstraintgiven interms ofthe probability thata packet’s queuingdelay exceed-ingagiventhresholdbeinglessthanacertainvalue.Nonetheless, the proposed modelcan also directly be usedto model thecase of multiple PU-SU pairs where each pair receives a static time-frequencyresource allocation. Moreover,the proposed model can beviewedasapreliminarysteptowardsgaininginsightin delay-constrainedoverlay cognitiveradio andalso towardsmodeling of dynamicalsharingofresourcesacrossPU-SUpairswithdistributed MAC(MediumAccessControl)protocolswhichisleftforfuture re-search.

The first proposed QMP is inspired by a combination of con-ventionalinterweaveandoverlay paradigmsandinthisQMP,the PUleveragesall ofSU’sresources unlessthePU’squeueisempty. TheSUisonlyallowed totransmitits ownpacketswheneverthe PU’s queue is empty.The second proposed QMP adaptively uses allorsome oftheSU’sresources towardsthetransmissionofthe PU’s packetthrougha quantitycalledthepower allocationfactor. Thisdecisiondependsonthepacket’sdelayexperiencedinthePU queue exceeding a certain threshold, i.e., delay-aware QMP. Con-ventionalqueuingmodels fall shortofmodeling thisdelay-aware QMP. Themaincontribution ofthispaperliesinthenovel queu-ing model ofthe delay-aware QMP assuming that the packet ar-rival process atthe PU queue isPoisson whilethe SU’s queueis assumedtobesaturated,i.e., SUalwayshasa packettotransmit. Moreover,PUpacketlengthsareassumedtobeexponentially dis-tributed.Keytotheproposedqueuingmodelisthetoolof Multi-RegimeMarkovFluidQueues(MRMFQ)whichisusedtoobtainthe exactsteady-statedelaydistributionfortheproposed delay-aware QMP. We refer to [12] andthe references therein fordescription andsteady-statesolutionsofMRMFQs.

In the numerical experimentation of the proposed QMPs, we focus on a fairly general secure overlay cognitive radio scenario duetothefact thatphysicallayersecurity isgainingmore atten-tioninthecontextofcognitiveradio[13].TheSUperformance en-counteredwithbothQMPsarethoroughlyinvestigatedinthis sce-narioandcomparedwiththeconventionalinterweaveandoverlay paradigms.For thispurpose,we first derive an achievablesecure rateregion forthe AdditiveWhite Gaussian Noise(AWGN) chan-nel model. As the next step, based on theanalytical expressions derived using MRMFQs, the optimal power allocation factor and delaythresholdvalues are numericallyobtainedto maximize the throughputoftheSUwhilemeetingthePU’sdelayconstraint.

The remainder of the paper is organized as follows: Section 2 provides a briefliterature overview.Section 3 presents the system model. Analytical models of the QMPs are presented inSection4.InSection5,parameteroptimizationoftheproposed delay-aware QMP isaddressed. Section 6isdevoted tonumerical (bothsimulationandanalysis)results.Finally,weconclude. 2. Relatedwork

The interaction of cognitive radio with the higher layers of the network hierarchy has attracted significant attention of re-searchers. Forthe overlay paradigm, the SUdevotes a portion of

its power to keep the PU’s rate unchanged, whilededicating the rest of its power to deliver its own messages, without consider-ing QoS constraints [3]. However, most of the wireless services are QoS-constrained andtheconsidered policy in[3]maynot be an appropriate choice for serving delay-sensitive PU applications that havebeen gainingpopularity.Inthisscenario, thePU carry-ing delaysensitiveinformation(likeTV broadcastingforcognitive radiosinTVwhitespace)maybewilling topermit theSUto uti-lizeitslicensedspectrumaslongasitsQoSrequirements,e.g., de-layrequirements,arenotcompromised.Furthermore,thecognition capabilities of SUs can be deployedto improvethe user experi-enceofPUs.QoS-constrainedSUinanunderlaycognitiveradio re-lay channel is studied in[4] andthe so called effective capacity [14]isderived.Thisscenarioisextendedtoanenergy-constrained SU in [9] where a proper band from a pool of available spec-trumbandsis tobe chosen.The reference[10]considers optimal power allocationforadelay-QoS awaresecureunderlaycognitive network which consists of one PU-SU pair andan eavesdropper. In [11], queuing analysis ofa cooperative cognitive radio with a QoS-constrainedPU andacooperativeSUisconsidered.Optimum admission controlparameters oftheprimary packetsattheSUis derived tomaximizetheSU’sthroughputwhilemeetingPU’s QoS conditions. Despitethe rich literature in regardswiththe under-laycognitiveradioparadigmwithQoS-constrainedSU,theoverlay cognitive radioandQoSconstraintsofthePUdidnot getenough attention,whichmotivatesustoconductthisresearch.

To become mainstream, another major challenge of cognitive radios issecurity. Recently, physicallayer approachesare gaining moreattentiontoprovidesecuritymechanismsforcognitiveradio [13]. Due to node limitations, cooperative methods are deployed toimprovesecrecyperformance. ThecooperationbetweenthePU andSUincognitiveradioistypicallystudiedinthecontextofthe overlay paradigm. On the other hand, performance of QoS con-strained wireless communicationsystems withPhysical Layer Se-curity (PLS) isinvestigatedin afew studies.In [15],across-layer resource allocation problem is investigated in a wireless single-hopuplinkscenariobytakingtheinformation-theoreticsecrecyas a QoSmetric. Asimilar problemisstudied in [16]tocontrol the uplink ofacellularnetworkby deployinghybridARQ.They max-imizea network utility function whilekeepingthe queues stable andmeeting secrecyconstraintina block-fadingchannel. In[17], a secret keyqueue isdeployed tosecure theprivate information ina single-hopwireless communicationsystemover ablock fad-ingchannelwhilemeetingthedelayconstraints.Similarly,fairrate allocationis studiedin[18]fora securebroadcast channelwhere unintendedreceiversareconsideredasinternaleavesdroppers. In-steadofdevotingall availableresources totheuserwiththebest channel condition, securetransmissionoflessfavorableusers are facilitatedsimultaneouslybyusingsecretkeys,forthesakeof fair-ness. Furthermore,a dynamicnetworkcontrolmechanismis pro-posed in[19]for amulti-hop wireless network whiletaking pri-vacyanddelayintoaccount.Theyinvestigatetheoptimalrate allo-cationtopreservetheconfidentialityofthetransmitteddatawhile meetingdelayconstraints.

3. Systemmodel

In this paper, we focus on the two-user cognitive radio sce-nario depicted in Fig. 1 where gij refers to channel coefficients. In order to apply the proposed QMPs to the overlay cognitive radio system, the achievable rate region of the setup should be determined. The achievable rate region for the overlay cognitive interference channel without secrecy is derived in [6] and [7]. As an illustrative example,the achievable rate region for a sam-ple scenariowithg11=1,g12=0.5,g21=0.6,andg22=1, unity

noisevariances,andP1=P2=1mWisdepictedinFig.2,whichis

derived based on the cognitive interference channel model pre-sentedin[7].ThesetofachievableratesisdenotedwithA,which isessentially the set ofpoints locatedbelow the curve inFig. 2. Moreover,thesetofallpointsonthecurveinFig.2isdenotedby C. Inthispaper, we envision asystem withone PU-SUpair only (optionallyalongwithaneavesdropper).However, themodelalso coversthecasewithmultiplePU-SUpairswhereeachpairreceives a static time-frequency resource allocation. The modeling of dy-namicalsharing ofresources across PU-SUpairs with distributed MAC(MediumAccessControl)isleftforfutureresearch.

ThePUpermitstheSUtoutilizeitslicensedbandaslongasthe delayconstraints arenotviolated. AQMP controlstheinteraction betweentheprimaryandthesecondarytransmittersbymeansof decidingwhichachievableratepairtouse(andalsowhen)forthe transmissionofPUandSUpackets.Clearly,inordertoachieve op-timumperformance, aQMP willtry toselecttheratepairson C. Any arbitrary ratepair on C is denoted by (rp,c, rs,c). The max-imum achievable rate of the PU (SU) when all of the resources oftheSUaredevoted toPU’s (SU’s)messagesisdenoted byrp,m (rs,m),whichislocatedattheintersectionofCandther1(r2)axis.

TheachievablerateofPUin theabsenceoftheSUisdenoted by

rp,w.Moreover,thecorresponding SUrateofrp,w onCisdenoted byrs,w.Thatis,PUandSUcanachieverp,w andrs,wrates, respec-tively,iftheyoperatesimultaneously.

Now,theconventional interweave andoverlay models are de-scribed:

• Conventional Interweave Model (CIM): The cognitive user does notcooperatewiththePU.Ittransmitsitsmessagesduringthe inactivitytimesofPUwiththemaximumraters,mwhiletrying to keep the primary transmissions unharmed [3]. Hence, the system alternates between the rate pairs (0, rs,m) and (rp,w, 0), depending on whether the PU’s queue is empty or not, respectively.

• ConventionalOverlayModel(COM):Thecognitiveuser,whohas non-causalknowledgeofthePU’smessage,cooperateswiththe PU.Bydevotingaportionofitsresources totheprimary mes-sage, the cognitive transmitter keeps the rate of the PU un-changedatrp,w.Simultaneously,theSUtransmitsitsown mes-sages by the remaining of its resources, which results in the rate ofrs,w forthe SU [3]. Nevertheless,SU devotes all ofits availableresourcestotransmititsownmessages,wheneverthe PU’s queueis empty.Thus,the systemalternatesbetweenthe ratepairs(0, rs,m) and(rp,w,rs,w),dependingonwhetherthe PU’squeueisemptyornot,respectively.

In this paper, we propose two novel QMPs for the overlay CRN with the delay-constrained PU which are described in the following.

• HybridOverlay-InterweaveScheme(HOIS):Thisschemeisa com-bination ofCIM andCOM. Whenever the PU has a packet to transmit, allSU’sresources are devotedtothe transmissionof the packet. When the PU’s queueis empty,the SU transmits its ownpackets atthe maximum rate. Thus, the systemwith HOIS alternatesbetween (0, rs,m) and (rp,m,0) depending on whetherthePU’squeueisemptyornot,respectively.

• Delay-AwareAdaptiveRateControl(DAARC):Thequeuedelay ex-perienced by each packetin theprimary transmitter is moni-tored.When apacketgetstobetransmitted,its queuingdelay is checked.Ifthisdelayisbelow acertain threshold(denoted by T(1)_{), i.e., in regime 1, the SU simultaneously transmits its}

own messages and the PU’s packet at rates rs,c and rp,c, re-spectively.IfthequeuedelayisabovethethresholdT(1)_,i.e.,in

regime2,allofSU’sresourcesaredevotedtotransmitonlyPU’s packets atraterp,mwhile theSU’srateiszero.Wheneverthe primary transmitter’s queue becomes empty,the SU achieves

Fig. 1. Overlay cognitive radio system model.

Fig. 2. Achievable rate region of an overlay cognitive radio system for channel coefficients set as g 11 = 1 , g 12 = 0 . 5 , g 21 = 0 . 6 , and g 22 = 1 , unit noise variances, and power budget of the PU and SU set to P 1 = P 2 = 1 mW .

theraters,m bydeployingallitsresources. Withthe appropri-atechoiceofthepowerallocationparameter

ζ

(orequivalently the ratepair (rp,c,rs,c))along withthe choice of the thresh-oldT(1)_{,theperformanceofDAARCcanbecontrolled.Notethat}

whenT(1)_{=0,allPUpacketsareservedataraterp,m,i.e.,the}

systemreducestotheHOISscheme.

ThePU packetsareassumedtoarriveattheprimary transmit-ter accordingto a Poisson process withrate

λ

p packets per mil-liseconds(ms),and,thelengthofeachpacketisexponentially dis-tributedwithmeanL.Itisassumedthatthechannelbandwidthis

BkHz.Thus, theservicetimeofapacket tobeserved atrater_i,j

fori∈{p,s}andj∈{c,m,w}throughoutitstransmissionwillbe ex-ponentiallydistributedwithmean _R1

i, j = L

Br_{i, j} ms.Furthermore,itis assumedthattheSU’strafficissaturated,i.e.,italwayshaspackets totransmit.Themotivationbehindthesaturatedtrafficassumption inthe SUis two-fold. First, themaximum achievable throughput forthe SUoccurswhenSU hasalwaystraffic to sendandinthis waythe throughputcapacity ofthe SU can be obtained. Second, the saturated traffic decouples the primary queue from the sec-ondaryqueue,which facilitatestheanalytical solutionofthe sys-temand makes it possible to obtain closed-formexpressions for thesystemperformancemetrics.

4. StochasticmodelfortheQMPs

InCIM,COM,andHOIS, thepacketsattheprimarytransmitter areservedatafixedrate,whichisequaltorp,wforCIMandCOM, andrp,m forHOIS. Assuming a Poissonpacket arrival process for the PU queue with rate

λ

p, the queue of the PU is modeled by thewell-knownM/M/1 queuemodel[20].Since theSUtransmits

only when the primary queueis empty forCIM (with rate rs,w) andHOIS (with raters,m), thethroughput ofthe SU, denoted by

S2,canbeexpressedasfollowsinbitspersecond(bps): SCIM

2 =

(

1−

λ

p/Rp,w

)

Brs,w, SHOIS2 =

(

1−

λ

p/Rp,m

)

Brs,m. (1) Ontheotherhand,theSUtransmitswithraters,m(rs,w)when theprimary queueisempty (notempty)forCOM.Thus, theSU’s throughputcanbeexpressedasfollows:

SCOM

2 =

(

1−

λ

p/Rp,w

)

Brs,m+

(

λ

p/Rp,w

)

Brs,w. (2) DAARC deploys a delay-aware service mechanism to manage the primary queue, i.e., the transmission rate of both users de-pends on the delay experienced by the Head of the Line (HoL) packet in the primary queue. Delay-awaremechanisms including DAARC cannot be modeled withconventionalqueuemodels like

M/M/1. Hence, we deploy multi-regime Markov fluid queues for modelingDAARC andderivingthesteady-state distributionofthe queuing delay. The following thresholds are defined: 0₌T(0)_<

T(1)_<T(2)_{→∞.In DAARC,a serviceratedecisionisto bemade}

basedonthedelayalreadyexperienced bythepacket(queue de-lay denoted by DQ(t)). Whenever T(0)≤ DQ(t)< T(1), the system is inregime 1andthe packet isto be served withrate R(p1)=Rp,c, andwheneverT(1)_{≤ D}

Q(t)<T(2),thesystemisinregime2andthe packetistobeserved withrateR_p(2)=Rp,m bydevotingallofthe resourcesoftheSUtothePU’smessage;seealsoFig.1.Intuitively, itshouldholdthatR(_p2)_>R(_p1)formeetingthedelayconstraints.

In orderto modelthe queuingsystemof thePU as aMarkov fluid queue, three auxiliary random processes are defined. First, we introduce the sojourn time process U(t) which is the overall time spentinthe systemincludingservicetime forthe packetto be served by the server. Ifthere are no packets being served at

Fig. 3. Sample paths of the following processes: (a) U ( t ), (b) A ( t ), and (c) X ( t ).

time t, thenU

(

t

)

₌0. Moreover, let the unfinished work process

A(t)denotetheamountoftimeneededtoserveallwaitingpackets includingtheoneintheserviceattimet.Clearly,apacketthathad arrivedattimetwithT(0)_{≤ A(t)<T}(1)_(T(1)_{≤ A(t)<T}(2)_)will

eventu-allybeservedatrateR(_p1)(R(_p2)).Thesamplepathsforthetwo pro-cessesU(t)andA(t)aregiveninFig.3(a)andFig.3(b),respectively, foran examplescenario. Thepacket arrivalinstantsare indicated bythesmallarrows.Duetoabruptjumps,neitherthesojourntime process, northeunfinishedwork processare suitableto be mod-eled asa fluid queue [21].Therefore, the random process X(t) is introduced,byreplacingtheabruptjumpsinthesojourntime pro-cesswithlineardecrementscorrespondingtoadriftofminusone (see Fig. 3(c)). Moreover, itis clear fromsamplepath arguments that thesteady-statedistributionoftheprocess U(t)(A(t))canbe derivedfromthatofX(t)bycensoringoutthestatescorresponding tonegative(positive)drifts.

We first focus on the MRMFQ model for X(t) for which we define two servicestates 1 and 2 during which the packets are servedwithrateR(p2)andR(p1),respectively,andX(t)increaseswith driftequalto1.Whentheserviceofthecurrentjobcompletesin states1and2,thesystemtransitsintostate3.Duringstate3,X(t) experiences a decrease with slope one for an exponentially dis-tributedamountoftimewithmean1/

λ

p,sothedelayofthenew HoL packet is reduced by an amount corresponding to its inter-arrival time. IfT(0)_{≤ X(t)<T}(1) _(T(1)_{≤ X(t)<T}(2)_{), the system}

tran-sitsfrom state 3to state 2(state 1).Also, it is possiblethat X(t) may hit zerowhile instate 3. Inthis case, upon the arrival ofa new packetto the system, thispacket isgoing to be served ata rateofR(p1).Therefore,onlytransitionfromtheboundaryX

(

t

)

=0 occurs out of state 3 to state 2 with rate

λ

p. In summary, the backgroundmodulatingMarkovsystemdenoted byZ(t) hasthree different states, namely, 1, 2, and 3. Therefore, the sample path followed by X(t) can well be modeled asthe modulated process of an MRMFQ, namely the process (X(t), Z(t)), withtwo regimes and three states.This MRMFQ is characterized by two infinitesi-mal generator matrices for regimes1 and 2, denoted by Q(i) _for i=1,2,twoinfinitesimalgeneratormatricesforthefinite bound-aries,denotedbyQ˜(i)_{fori=0,1,andthecorrespondingdrift} ma-triceswhicharedenotedbyR(i)_{fori=1,2,andR}˜(i)_fori=0,1for theregimesandboundaries,respectively.See[12]foranelaborate description of MRMFQs. The state transitiondiagrams of Z(t) are depictedinFig.4forbothregimesandboundaryX

(

t

)

=0.

Onthebasis oftheabovedescription,theinfinitesimal genera-tormatricesforthetworegimescanbeobtainedasfollows:

Q(1)₌



_{0 0 0} 0 −R(1) p R(p1) 0

λ

p −

λ

p



, Q(2)₌

⎡

⎣

−R (2) p 0 R(p2) 0 −R(1) p R(p1)

λ

p 0 −

λ

p

⎤

⎦

. (3)

Fig. 4. State transition diagrams of Z ( t ) for (a) boundary X(t) = 0 , (b) first regime, and (c) second regime.

NotethatsinceX(t) increasesintheservicestate2,theremay be transitions from state 2 to state 3 in regime 2. Furthermore, thetransition matrixat thresholdT(1) _{isequal to Q}(2)_{.Also, Q}˜(0)

issimilartoQ(1)_{exceptthatthereisnotransitionfromstate2to}

state3attheboundaryX

(

t

)

=0.Therefore,itholdsthat ˜ Q(0)₌



_{0 0 0} 0 0 0 0

λ

p −

λ

p



. (4)

Furthermore, the drift matrices for each regime and bound-ary are written as R(1)₌R˜(1)_=R(2)_{=diag[1,1,−1] and} R˜(0)₌

diag[1,1,0] where diag denotes a diagonal matrix. The follow-ingtheoremprovidesaclosed-formexpressionforthesteady-state distributionofqueuingdelayfortheDAARCsystemofinterest. Theorem1. Thesteady-statecumulative distributionfunction (CDF) oftheDAARCqueuingdelay,denotedbyF(x),isexpressedasfollows:

F(x)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ c₃(0)  , x=0, a(1) 1 R(p1)−λp  1−e− R(_p1)−λpx  , 0<x≤ T(1), ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ a(2) 1 R(_p1)  e−R(p1)x_{− e}−Rp(1)T(1)₋ a (1) 1 R(_p1)−λp  1− e− R(_p1)−λpT(1) + a (2) 2 R(_p2)−λp  e− R(_p2)−λpx − e− R(_p2)−λpT(1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦  , x>T(1), (5) where c₃(0)=

⎡

⎢

⎢

⎢

⎢

⎢

⎣

R(p1)+λp R(p1)−λp + 2λp  R(p2)− R(p1)  R(p1)  R(p2)− R(p1)−λp eλpT(1) − 2λp  R(p2)− R(p1)  R(p2)−λp   R(p1)−λp  R(p2)−λp  R(p2)− R(p1)−λp e−  R(_p1)−λp  T(1)

⎤

⎥

⎥

⎥

⎥

⎥

⎦

−1 , (6) a(₁1)=c₃(0)

λ

p, a(₁2)= R (2) p − R(p1)−

λ

p R(p2)− R(p1) c(₃0)

λ

peλpT (1) , a(₂2)= −

λ

p R(p2)− R(p1)−

λ

p c(₃0)

λ

pe  R(p2)−R( 1) p  T(1) , (7) and



=c(₃0)+ a( 1) 1 R(p1)−

λ

p



1− e−  R(1) p −λp  T(1)



+a( 2) 1 R(p1) e−R(p1)T(1) + a (2) 2 R(p2)−

λp

e−  R(2) p −λp  T(1) .

Proof.ThedetailedproofisprovidedinAppendixA.

Eq.(5)provides a closed-formexpression forthe queuedelay CDFofPU’spackets fortheDAARC QMPandthisexpression con-stitutesthemaincontributionofthispaper.Basedonthis expres-sion,theSU’sthroughputiscalculatedasfollows.IfthePU’squeue

isempty,theSUtransmitsatrateBrs,m.Ontheotherhand,ifthe PU’squeueisnotempty,thesecondarytransmits atrateBrs,w or zero,wheneverthedelayexperienced bytheHoLpacketofPU is beloworaboveT(1)_{,respectively.Therefore,}

SDAARC

2 =Brs,mF

(

0

)

+Brs,w

F

(

T(1)

)

− F

(

0

)



. (8)

5. SystemoptimizationforDAARC

Upon deriving the closed-form analytical expressions for the performancemetricsofDAARC,we focusourattentionto the op-timum selection of the two DAARC parameters, namely the pa-rameter

ζ

which controls the primary and secondary rate pairs asillustratedinFig.2,andthedelaythresholdT(1)_.Ourproposed

QMPsare applicable toanyoverlay cognitive radio system. How-ever,motivated by the increasing application of PLS in cognitive radio,amoregeneralsecureoverlaycognitiveradiosystemis con-sideredinthispaperasan exampletoapplytheproposed QMPs. Thismodelisinspiredbythesecureoverlaycognitiveinterference channel model presented in [7]with an additional eavesdropper totakeintoaccountthesecurityofthePU’smessagesaswell.This secureoverlay cognitiveradio modelisformed by introducingan externaleavesdroppertothesystemmodeldepictedinFig.1.

In this model, the delay-constrained PU intends to transmit its packets securely to both primary and secondary receivers in a timely manner in cooperation with the secondary transmitter, while keeping it secret from the external eavesdropper. On the otherhand,theSUintendstodeliveritsmessagestothesecondary receiverwhilekeepingthemsecretfromtheprimaryreceiverand theeavesdropper.Itisassumedthatthesecondarytransmitterhas non-causalknowledgeaboutthePU’smessage.

InordertoapplytheproposedQMPs,thesecureachievablerate regionofthesystemisrequired,whichisderivedasfollows.LetP1

(P2)betransmissionpowerofthePU(SU)inmilliwatts(mW).The

SUdevotesaportionofitspower,i.e.,

ζ

P2,tohelpthePUdeliver

itsmessageswhereastheremainingpower,i.e.,

(

1−

ζ

)

P2,isused

fortransmitting SU’s packets. Let the achievable rates of the PU andSU, bedenotedby r1 andr2,respectively,inunitsofbitsper

secondperHertz(bps/Hz).Alsolet

r1j=C



|

g1j

|

2 P1+

|

g2j

|

2 ζP2

|

g2j

|

2 (1−ζ )P2+σ2j



, r2j=C



|

g2j

|

2 (1−ζ )P2 σ2 j



, for j=1,2,3, (9) where_C

(

x

)

₌1

2log

(

1+x

)

,g13andg23arerespectivelythechannel

coefficientsbetweentheeavesdropperandprimaryandsecondary transmitters,and

σ

2

j denotesthevariance ofthereceivernoiseat receiverj.Based onBloch etal.[22],thesetofachievable ratesis derivedintermsofthepowerallocationfactor

ζ

:

r1≤ min

{

r11,r12

}

− r13, r2≤ r22− max

{

r21,r23

}

. (10)

Note that inthis scenario, the eavesdropper does not acquire neitherPU’s norSU’smessages.However,inordertoconsiderthe worstcasescenario,itisassumedthattheeavesdropperknowsthe PU’smessageandcancancelitsinterferenceout, whileittriesto decodetheSU’smessage.Notethat theratesprovided in(10)are thesecureratesofthecognitiveinterferencechannel.The achiev-ablerateforthesamechannelwithoutsecurityconstraintscanbe derivedby ignoringthepresence oftheillegitimatereceivers,i.e., therate of thePU isthe achievable rate ofthe worst legitimate receiverandtherateofSUistheachievablerateofthesecondary receiver.Asanillustrativeexample,theachievablesecrecyrate re-gionwhichisderivedbyvaryingthevalueof

ζ

isdepictedinFig.5 forasamplescenariowithg11=1,g12=0.5,g13=0.25,g21=0.6, g22=1, and g23=0.25, unity noise variances, P1=1mW, and P2=2mW.

Fig. 5. Achievable secrecy rate region of secure overlay cognitive radio system for channel coefficients set as g 11 = 1 , g 12 = 0 . 5 , g 13 = 0 . 25 , g 21 = 0 . 6 , g 22 = 1 , and g 23 = 0 . 25 , unit noise variances, power budget of the PU set to P 1 = 1 mW , and power budget of the SU set to P 2 = 2 mW .

In the cognitive radio network, the SU desiresto achieve the highestthroughputpossiblewhilemeetingthePUdelayconstraint. Inthissection,twodifferentversionsofDAARCareconsidered de-pendingontheparameterselectionmethod.

In the firstversion named DAARC†_{,simultaneous optimization}

ofthe powerallocation factor

ζ

andthe thresholdT(1) _is

consid-ered. Thus, the optimization problemis the maximization of the SU’s throughput while satisfying the PU’s delay constraint with appropriate choicesof

ζ

andT(1)_{. Theclosed-formexpressions of}

the SU’sthroughput andPU packets’ delaydistribution facilitates numerical calculation of the optimum parameters by deploying standardNLP(Non-LinearProgramming)methods.First,the inter-section point of r11 withr12,and r21 withr23 is determined for

a given channel condition. Hence, the min and max functionsin the rate region can be eliminated by dividing the region into at most3sub-regions.Then,theoptimumpointiscalculatedforeach sub-region byusingNLP. Finally,theglobaloptimum pointis ob-tainedby comparingthe optimumpointsineach sub-region;see Algorithm1 forthepseudo-codeforDAARC†_.

Algorithm1:Pseudo-codeofDAARC†_.

Input:Channelgainsandtransmissionpowers Output:T(1)†_,

_ζ

†

1 Find

ζ

(1)suchthatr₁₁=r₁₂andfind

ζ

(2)suchthatr₂₁=r₂₃, 2 Dividethesupportsetof

ζ

into3sub-regions:

a)S1=



0_,min



ζ

(1)_,

_ζ

(2)



_, b)S2=



min



ζ

(1)_,

_ζ

(2)



_,max



_ζ

(1)_,

_ζ

(2)



_, c)S3=



max



ζ

(1)_,

_ζ

(2)



_,1



_,

3 PerformNLPineachsub-regioninordertosolvethe

followingproblemfori∈

{

1,2,3

}

:



T_i(1),

ζi



=argmax T(1)_,ζS DAARC 2 s.t. Pr

{

Dq>DT

}

<

ε

,

ζ

∈Si Choosethe



T_i(1),

ζ

i



pairamongthethreecandidatesthat providethemaximumSUthroughputas



T(1)†_,

_ζ

†



_.

Two-dimensional NLP used for DAARC† _{is computationally}

costly. Hence,a heuristicsub-optimalDAARC versionis proposed, which is named asDAARC‡_{, to reduce the complexity ofthe}

op-timization problem. Thisheuristic method stems fromthe obser-vation that theoptimalvalue of

ζ

doesnot varymuch beyonda certain value of T(1)_{. In DAARC}‡_{, the PU’s delay constraint is}

ne-glected in the first step, andthe optimal

ζ

is calculated by de-ployingaone-dimensionalNLP,usingasufficientlylargevaluefor

T(1)_{. Then, given the optimal}

_ζ

_{and corresponding r}(1)

denoted by r(_p1)‡_, the optimal value of T(1) _{is calculated. For this}

purpose, we first derive the maximum value of T(1)_{, denoted by}

¯

T(1)‡_{, that satisfies the delay constraint Pr}

_{

_D

q>DT

}

=

ε

. Since the SU’s throughput is an increasing function of T(1) _{except at}

the origin, the optimum choice for T(1) _{denoted by T}(1)‡ _will

ei-ther be T¯(1)‡ _{or zero, depending on whicheveryields higher SU}

throughput; the pseudo-code of this second version is given in Algorithm2.

Algorithm2:Pseudo-codeofDAARC‡_.

Input:Channelgainsandtransmissionpowers Output:T(1)‡_,

_ζ

‡

1Sameasthesteps1-3ofAlg.1, 2SetT(1)toasufficientlylargevalue,

3Performone-dimensionalNLPjustfor

ζ

foreachsub-region

inordertosolvethefollowingoptimizationproblemfor

i∈

{

1,2,3

}

:

ζi

=argmax

ζ S

DAARC

2

s.t.

ζ

∈Si

Choosethe

ζ

_iamongthethreecandidatesthatprovidesthe maximumSUthroughputas

ζ

‡_,

4Obtainr(_p1)‡ correspondingto

ζ

‡

i,

5FindthevalueofT(1)givenr(_p1)‡thatsatisfiesthedelay

constraintwithequality,i.e., ¯

T(1)‡_:Pr

_{

_D

q>DT

}

=

ε

ChooseT(1)‡_aseither0orT¯(1)‡_{dependingonwhichever}

yieldshigherSUthroughput:

T(1)‡_{=arg max}

T(1)_∈{0,T¯(1)‡}

SDAARC

2

6. Simulationsandnumericalresults

WeassumethatthechannelbandwidthisB=1MHz,andL=1 kbits. The time unit is set to ms. Therefore, the packet service times are exponentially distributed with mean 1/Ri, j=1/ri, j for

i∈{p, s} andj∈{c, m, w}. The numerical results are obtainedby MATLAB 2014a running on a machine with Intel core-i5 4300U CPUand8GBofRAM.

InordertovalidatetheanalyticalmethoddevelopedforDAARC, theanalyticalresultsarecomparedagainstsimulationsforthe par-ticular settingofR(_p1)₌r_p,w=0_.1327 andR(_p2)₌r_p,m=1_.2266. In particular, theprimary usertransmits atraterp,w whenit is op-eratinginregime 1andittransmits withthemaximumraterp,m when it switches to regime 2. In Fig.6, the CDF of the queuing delay inthe PU queueobtained by both the analytical model as well assimulationsareplottedforfourdifferentvaluesof

λ

p. An-alyticalresultsareintotalagreement withthesimulation results, whichvalidatesthe analyticmodel.Next,we considerthe perfor-manceofthefourQMPs inthefollowingfourseparate scenarios, associated withdifferentchannel conditions.Intwo ofthese sce-narios,thechannelcoefficientsg12andg13arevariedsuchthatthe

raterp,w becomespoorandgood,respectively:

• InthepoorPUscenario,g12=0.5,g13=0.45,g21=0.7.

• InthegoodPUscenario,g12=0.8,g13=0.25,g21=0.7.

Meanwhile,inthethirdandfourthscenarios,thechannel coef-ficientg21isalteredtochangetheimpactofsecondarytransmitter

onthePU’srate:

• InthepoorSUscenario,g12=0.5,g13=0.25,g21=0.3.

• InthegoodSUscenario,g12=0.5,g13=0.25,g21=0.7,

Notethattheremainingchannel coefficientsaresettog11=1, g22=1,andg23=0.25inallthefourscenarios.

Inthefirstnumericalexample,thefourQMPsCIM,COM,HOIS, and the particular policy DAARC† _{systems are studied for P}

i=1

mWfori=1,2.The delayconstraintparameters are set toDT= 80,320ms and

ε

=10−2.The NLPstep inDAARC†_{is conducted}

us-ingthe

fmincon

function inMATLAB.Thethroughputofthe SU,

whilethe primary’s delayconstraintis satisfied,is considered as the performance metric in all comparisons.This metric not only providesatooltocomparetheperformanceoftheSUindifferent QMPs,butalso,implicitlydeterminesthemaximumadmissible ar-rivalrateforthePU traffic,whichisthemaximumamountof ar-rivalratethatthePUcanservewhilesatisfyingthedelaycriterion. InFig.7,SUthroughputofDAARC† _{iscomparedtothatofCIM,}

COM and HOIS for the four channel scenarios. The policies CIM andCOMdonotcopewellwiththedelayrequirementsofthePU forrelativelyhighvaluesof

λ

p.Therefore,parts oftheir through-putcurves are not depictedin Fig. 7corresponding to situations whenthePU’s delayconstraintisviolated, i.e.,themaximum ad-missible rate for the PU traffic in these QMPs is relatively low. AlthoughCOMoutperformsHOIS forlow valuesof

λ

p,it cannot satisfythedelayrequirementofthe PUforlargevaluesof

λ

p.On theotherhand,DAARC† _{notonlyprovidessuperiorSUthroughput}

performance, butalso,it improves the maximumadmissible rate ofPUtraffic. TheSU achieveshigherthroughputby manipulating thedelaytolerance ofPU packets to its ownfavor inDAARC†_{. As}

showninFig.7,the achievablethroughputoftheSUincreasesin

DAARC†_{asthePUdelayconstraintisrelaxed.Furthermore,the}

per-formanceofDAARC†_{improveswithrespecttoHOISasthePU}

chan-nelconditionsimprovesincethesystemoperatesmorefrequently inregime 1andPU can handleits packetson its own.Notethat theperformance ofDAARC† _{reducestoHOISastheconditions,i.e.,}

channelgainsorarrival rate, turnout tobe unfavorable.Inthese situations,thesystembenefitsmorebytransmittingeachmessage oneatatimeinsteadofsimultaneoustransmissionofprimaryand secondarymessages.ThissituationhappenswhenT(1)_ischosenas

zero.This scenario is highlighted in the poorPU channel condi-tion(Fig.7(a))whereDAARC†_{performsidenticaltoHOIS}

indepen-dentofthedelaytoleranceofprimarypackets.Incontrast,DAARC†

performs very similarly to COM for some specific channel gains and/or arrival rates forexample when Algorithm 1 selects (rp,w,

rs,w) as the rate pair in the first regime and T(1)→∞. In these conditions,DAARC preferstotransmitthemessagesofPUandSU simultaneously.

In summary, the proposed DAARC† _{method provides a robust}

mechanismforthePUtohandleitsdelaysensitive trafficina re-sourcelimitedenvironment by leveraging SU’sresources. TheSU, whichdoesnotpossesslicensedbands,helpsthePUtodeliverits messages in a timely manner, in exchange for its own transmis-sionprivileges.ItisshownthattheSUachieveshigherthroughput comparedtoconventionalmethodsasCIMandCOM.Furthermore, asthePUdelayconstraintisrelaxed,theSUthroughputimproves. In the next step, we focus our attention to the performance analysisofDAARC‡_{,andcomparingtheresultswithDAARC}† _which

is the best performing method in the first set of numerical ex-amples. Again, the NLPsteps in DAARC‡ _{are conductedusing the}

fmincon

functioninMATLAB.

The two different DAARC versionsare compared in Fig. 8 for

DT=320 ms and P1=P2=1 mW. Furthermore, in the second

step of Algorithm 2, the sufficiently large number value is cho-senas600ms inthisexample.It isillustrated thattheproposed heuristicmethodDAARC‡_{performs veryclosetoDAARC}† _while

re-quiring less computational power. In particular, the average run timeofDAARC† _andDAARC‡_{,whicharetheaveragevaluesofover}

50runs, are 79.5743 and 6.5552 s, respectively, which translates intomore than90% reduction in computationtime withDAARC‡_.

Fig. 6. CDF of the delay obtained by both simulations and the proposed analytical method when R (1)

p = 0 . 1327 and R (p2 = 1 . 2266 . )

Fig. 7. SU throughput with COM, CIM, HOIS, and DAARC † for four different channel scenarios.

Thesecondarythroughputandtheirrelativegainswithrespectto HOIS (denoted by Gk _{for k∈{DAARC}†_{, DAARC}‡_{) for both versions}

of DAARC are given in Tables 2–4, for good primary, poor sec-ondary and good secondary scenarios, respectively. As expected,

DAARC† _{appearstobesuperior toalltheother methods.Thegain}

ofDAARC† _{dependsonthechannelconditionsandthearrivalrate}

ofthe PU traffic. We also observefrom Tables 2–4that the gain ofDAARC† _{growsconsistentlyasthearrivalrateincreasesandthis}

gainreachesover300%forrelativelyhighvaluesof

λ

pinthepoor secondaryscenario. We also observethat DAARC‡ _{performance is}

very close to DAARC† _{for most of the cases, which makes it an}

attractivechoice considering itssignificantly lower computational complexity.

NotethatforpoorPUchannelconditions,bothalgorithms per-formidenticallysincetheoptimizationisconductedforthe param-etersofthefirstregime whichispracticallynot usedduetovery

Table 2

The SU throughput (in units of kbps) obtained by HOIS and the two versions of DAARC with D T = 320 ms in the good primary scenario.

The percentage throughput gain of the two DAARC versions with respect to HOIS are also provided.

λp S HOIS2 S DAARC † 2 S DAARC ‡ 2 G DAARC † (%) G DAARC‡ (%) 0.09 0.1740 0.2123 0.2123 22.0115 22.0115 0.13 0.1642 0.2123 0.2123 29.2935 29.2935 0.19 0.1419 0.1968 0.1968 38.6892 38.6892 0.29 0.1049 0.1564 0.1564 49.0944 49.0944 0.31 0.0975 0.1478 0.1478 51.6053 51.6053 0.39 0.0678 0.1120 0.1120 65.0943 65.0943 0.49 0.0308 0.0638 0.0637 106.9178 106.7230

low performing PU.That is,in poor PU channel conditions, both algorithms prefer to devote all available resources to empty the

Fig. 8. Comparison of the two different versions of DAARC QMP for D T = 320 ms.

Table 3

The SU throughput (in units of kbps) obtained by HOIS and the two versions of DAARC with D T = 320 ms in the poor secondary sce-

nario. The percentage throughput gain of the two DAARC versions with respect to HOIS are also provided.

λp S HOIS2 S DAARC † 2 S DAARC ‡ 2 G DAARC † (%) G DAARC‡ (%) 0.01 0.4280 0.4378 0.4378 2.2897 2.2897 0.05 0.3888 0.4231 0.4191 8.8220 7.7932 0.09 0.3496 0.3885 0.3867 11.1270 10.6121 0.11 0.3300 0.3704 0.3704 12.2424 12.2424 0.13 0.3104 0.3521 0.3521 13.4343 13.4343 0.17 0.2712 0.3148 0.3148 16.0767 16.0767 0.21 0.2320 0.2771 0.2771 19.4397 19.4397 0.29 0.1536 0.2011 0.2011 30.9245 30.9245 0.39 0.0557 0.1050 0.1044 88.6793 87.6011 0.43 0.0165 0.0664 0.0582 303.3435 253.8602 Table 4

The SU throughput (in units of kbps) obtained by HOIS and the two versions of DAARC with D T = 320 ms in the good secondary sce-

nario. The percentage throughput gain of the two DAARC versions with respect to HOIS are also provided.

λp S HOIS2 S DAARC † 2 S DAARC ‡ 2 G DAARC † (%) G DAARC‡ (%) 0.01 0.2081 0.2123 0.2123 2.0183 2.0183 0.05 0.1911 0.2062 0.2043 7.9016 6.9073 0.09 0.1741 0.1916 0.1741 10.0517 0 0.11 0.1656 0.1838 0.1656 10.9903 0 0.13 0.1571 0.1758 0.1571 11.9032 0 0.17 0.1401 0.1592 0.1401 13.6331 0 0.27 0.0977 0.1148 0.0977 17.5266 0 0.37 0.0552 0.0669 0.0669 21.1918 21.1918 0.47 0.0127 0.0157 0.0156 23.1132 22.9560

PUqueuefirstandthenassigntheremainingresourcestotheSU. Therefore, the results associated with the poor channel scenario aredeliberatelynottabulated.

7. Conclusions

Two novel QMPs are proposed for an overlay cognitive radio network withdelay-constrained PU. The proposed QMPs attempt tomaximizetheSUthroughputwhilemeetingthePU’sdelay cri-terion.ThefirstQMP,namedHOIS, isahybridinterweave/overlay model, while the second QMP, called DAARC, deploys a delay-awareadaptivemechanism.ThroughanovelMRFMQbasedmodel, theclosed-formexpressionsfortheexactdelaydistributionofthe PUtrafficare derivedandvalidatedforDAARC.Moreover, analyti-calexpressions areemployedto optimallytunetheparameters of DAARC. Theproposed QMPsare applied to a secureoverlay CRN withdelay-constrainedPUwhichdeploysphysicallayersecurityto keep its messages confidential. The proposed methods are simu-latedandcomparedwiththeconventionalinterweaveandoverlay paradigmsusing thissecureoverlay CRN. Itis shownthat by in-telligentmanipulation ofthePU’s delayconstraint, DAARC consis-tently outperforms HOIS as well as CIM and COM withthe per-formancegapdependingonthechannelconditionsandthearrival ratetothePUqueue.Thecomputationalcomplexityoffinding off-linetheoptimumDAARCparametersandtherequirementtokeep trackofthequeuingdelayofeachpackettowardstheon-line im-plementationappeartobethedrawbacksofDAARC.Moregeneral trafficmodelsforPUandSUareleftforfutureresearch.

AppendixA. ProofofTheorem 1

The joint probability density function (pdf) vector of X(t) for regime k for k=1,2, when T(k−1)_{≤ D}

Q

(

t

)

<T(k) is defined as follows: f_i(k)

(

x

)

=lim t→∞ d dxPr

{

X

(

t

)

≤ x,Z

(

t

)

=i

}

, f(k)

₍

_x

₎

₌

_f(k) 1

(

x

)

,f (k) 2

(

x

)

,f (k) 3

(

x

)



. (A.1)

Similarly,thesteadystatemassaccumulationvectorfor bound-arypointsT(0)_andT(1)_{isdefinedasfollows:}

c_i(k)=lim t→∞Pr



X

(

t

)

=T(k)_,Z

₍

_t

₎

_=i



_{, c}(k)₌

_c(k) 1 ,c( k) 2 ,c( k) 3



. (A.2)

Followingthesameprocedure in[12],thefollowingsetof dif-ferentialequationsholdsforthejointpdfvectorofX(t):

d dxf(

i)

₍

_x

₎

_R(i)_=f(i)

₍

_x

₎

_Q(i)_{, fori=1,2. (A.3)}

Inordertoderivetheboundaryconditions,wefirstidentifythe setofstateswithpositive,negativeandzerodriftforeachregime andboundary.S(₊k),S₋(k),andS(₀k)representthestateswithpositive, negative,andzerodriftatregimek,respectively,andS˜(₊k),_S˜(k)

+ ,and

˜

S₊(k) representthestateswithpositive, negative, andzerodrift at boundaryk,respectively.Accordingtoaforementionedinfinitesimal generatormatricesanddrift matricesofthe systemmodel,these setsarepopulatedasfollows:

S₊(1)=

{

1,2

}

, S₋(1)=

{

3

}

, S(₀1)=∅, S₊(2)=

{

1,2

}

, S₋(2)=

{

3

}

, S(₀2)=∅, ˜ S₊(0)=

{

1,2

}

, S˜₋(0)=∅, S˜(₀0)=∅, ˜ S(₊1)=

{

1,2

}

, _S˜(1) − =

{

3

}

, S˜(01)=∅.

Hence,theintermediateboundarypointT(1)_{isanemittingstate}

[12] and the boundary conditions for the differential equations presentedin(A.3)areasfollows:

c₁(0)=c₂(0)=0, (A.4a) cm(1)=0, form∈

{

1,2,3

}

, (A.4b) f(1)

(

0+

)

R(1)=c(0)Q˜(0), (A.4c) f(2)

₍

_T(1)₊

₎

_R(2)_{− f}(1)

₍

_T(1)₋

₎

_R(1)_=c(1)_Q˜(1)_{, (A.4d)}



₂  k=1  T(k)− T(k−1)₊f (k)

₍

_x

₎

_dx+1 k=0 c(k)



(

1,1,1

)

T=1. (A.4e) As the boundary values for the differential equations are not knownyet,thisproblemshouldbeconsideredasaboundaryvalue problem.The spectral solution to MRMFQsare presentedin [12]. Forourmodel,wherethere isno statewithzero driftneither in theregimes,norintheboundaries,thegeneralsolutionto(A.3)is givenasfollows: f(k)

₍

_x

₎

₌ i a(_ik)eλ(ik)x

φ

(k) i ,forT( k−1)_<x<T(k)_{,1≤ k≤ 2 (A.5)} where

λ

(_ik),

φ

(k) i



istheitheigenvalue-lefteigenvectorpairofthe matrixQ(k)

₍

_R(k)

₎

−1_{[23]whicharederivedforthefirstandsecond}

regimeasfollows: det



λ

I− Q(1)

_R(1)



−1



_{=0, det}



λ

_{I− Q}(2)

_R(2)



−1



_=0,

which

resultin:

λ

(1) 1 ,

φ

( 1) 1



=

λ

p− R(p1),

(

0,1,1

)



,

λ

(1) 2 ,

φ

( 1) 2



=



0,

(

0,1,R( 1) p

λ

p

)



,

λ

(1) 3 ,

φ

( 1) 3



=

(

0,

(

1,0,0

)

)

, and

λ

(2) 1 ,

φ

( 2) 1



=



−R(1) p ,

(

λ

p R(p2)− R(p1) ,R( 2) p − R(p1)−

λ

p R(p2)− R(p1) ,1

)



,

λ

(2) 2 ,

φ

( 2) 2



=

λ

p− R(p2),

(

1,0,1

)



,

λ

(2) 3 ,

φ

( 2) 3



=



0,

(

λp

R(p2) ,0,1

)



.

Hence, thepdf ofthedelayforthe firstandsecond regime is derivedasfollows: f(1)

₍

_x

₎

_=a(1) 1 eλ (1) 1 x

φ

(1) 1 +a( 1) 2

φ

( 1) 2 +a( 1) 3

φ

( 1) 3 , f(2)

(

x

)

=a₁(2)eλ(12)x

φ

(2) 1 +a( 2) 2 eλ (2) 2 x

φ

(2) 2 +a( 2) 3

φ

( 2) 3 .

Inordertosolvethedifferentialequation,thereare6unknown

a coefficients,and6unknownc coefficientsthatshouldbe deter-mined using the boundary conditions. Since the queue model is ofinfinitesize, thestabilityconditionshouldalsobe satisfiedfor thesecondregimeas

π

(2)_R(2)_(1,1,1)T_<0where

_π

(2)_isthesteady

state vector of Q(2)_{. Furthermore, in order to acquire a bounded}

distribution forthesecond regime, thecoefficientscorresponding tothezeroeigenvaluesandtheeigenvaluesintherighthalfplane mustbeequaltozero.Tobeprecise,imposing(A.4a)and(A.4b) re-sultsin:

c₁(0)=c(₂0)=c₁(1)=c(₂1)=c₃(1)=a(₂1)=a₃(1)=a₃(2)=0.

Then,theEq.(A.4c)yieldsthefollowing:

a(₁1)

φ

₁(1)+a(₂1)

φ

₂(1)+a₃(1)

φ

₃(1)





_{1 0 0} 0 1 0 0 0 −1



=

0,0,c(₃0)





_{0 0 0} 0 0 0 0

λp

−

λp



whichimpliesthat

a₁(1)=c(₃0)

λ

p, a₂(1)=0, a(₃1)=0.

Inordertoderive aboundeddistribution,a₃(2) shouldbeequal tozero,sincea(₃2)isthecoefficientassociatedwiththeeigenvalue at the originin the last regime. Hence, employing(A.4d) results in: a(₁2)eλ(12)T( 1)

φ

(2) 1 +a( 2) 2 eλ (2) 2 T( 1)

φ

(2) 2 +a( 2) 3

φ

( 2) 3 − a( 1) 1 eλ (1) 1 T( 1)

φ

(1) 1 =

(

0,0,0

)

⎡

⎣

−R (2) p 0 R(p2) 0 −R(1) p R(p1)

λp

0 −

λp

⎤

⎦

=0

whichimpliesthat

a(₁2)=



R(p2)− R(p1)−

λ

p R(p2)− R(p1)



c(₃0)

λ

pe  λ(1) 1 −λ( 2) 1  T(1) , a(₂2)=



−

λp

R(p2)− R(p1)−

λp



c(₃0)

λp

e  λ(1) 1 −λ( 2) 2  T(1) , a(₃2)=0.

Furthermore, the stability condition is imposed as

λ

p<R(p2). ConsideringthestabilityconditionandRp(1)>0,wereplaceall un-knownvariablesinc(₃0)inEq.(A.4e)andderivec(₃0) asinEq.(6). Moreover,wederive theclosed-formexpressionforthea₁(1),a(₁2),

anda(₂2)coefficientsasin(7)andfortheremainingcoefficientsas follows:

whichgivethejointpdfvectorsasfollows: f(1)

₍

_x

₎

_=a(1) 1 e  λp−R(p1)  x

(

0,1,1

)

, f(2)

₍

_x

₎

_=a(2) 1 e−R (1) px



λ

p R(p2)− R(p1) ,R (2) p − R(p1)−

λ

p R(p2)− R(p1) ,1



+a(₂2)e  λp−R(p2)  x

(

1,0,1

)

.

Sincetheunfinishedworkprocess A(t)determinestheamount of delay that newly arriving jobs (which arrive to the system according to a Poisson process) will experience, the steady-state probabilitydistributionofstate3canbeusedtoobtainthe quan-tities of interest, by a direct consequence of the PASTA (Poisson ArrivalsSeeTimeAverages)property.Therefore,inordertoobtain thesteady-statedistribution ofA(t)fromthatofthe fluidprocess

X(t),wecensoroutthestates1and2,andsubsequentlynormalize thesteady-statedistributions.Inmathematicalterms,wecalculate the steady-stateCDFofA(t), whichisequalto thedistribution of thequeuedelayofanewlyarrivedpacket,fromthatof(X(t),Z(t)) asfollows:

F

(

x

)

=lim

t→∞Pr

{

A

(

t

)

≤ x

}

=tlim→∞

Pr

{

Z

(

t

)

=3,X

(

t

)

≤ x

}

Pr

{

Z

(

t

)

=3

}

(A.6) whichtogetherwiththeexpressionsobtainedforthejointpdf vec-torgivestheidentityin(5).

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