1

Delay and Throughput Analysis of CognitiveGo-Back-N HARQ in the Face of Imperfect

SensingAteeq Ur Rehman, Lie-Liang Yang, Fellow, IEEE and Lajos Hanzo, Fellow, IEEE

Abstract—In order to mitigate spectrum scarcity,the cognitive radio (CR) paradigm has been invokedfor improving the overall exploitation of the licensedspectrum by identifying and filling the free spectrumholes without degrading the transmission of primaryusers (PUs). Hence, we conceive a CR communicationscheme, which enables a cognitive user (CU) to sensethe activity of the PUs over a primary radio (PR)channel, which is exploited to transmit data using themodified Go-Back-N hybrid automatic repeat request(GBN-HARQ) protocol, when PR channel is free fromthe PUs. This arrangement is termed as the cognitiveGBN-HARQ (CGBN-HARQ), where the activity of PUson the PR channel is modelled as a two-state Markovchain having ‘ON’ and ‘OFF’ states. However, the CUmay wrongly detect the ‘ON’/‘OFF’ activity of the PUsin the channel, hence resulting in false-alarm or mis-detection. Therefore, the two-state Markov chain isextended to four states by explicitly considering all thewrong sensing decisions. In this paper, we analyticallymodelled the CGBN-HARQ scheme with the aid of aDiscrete Time Markov Chain (DTMC). Explicitly, analgorithm is developed for deriving all the legitimatestates and for eliminating the illegitimate states, whichassists us in reducing both the dimensionality ofthe state transition matrix and the associated com-putational complexity. Furthermore, based on DTMCmodelling, we derive closed-form expressions for eval-uating the throughput, the average packet delay andthe end-to-end packet delay of CGBN-HARQ in realisticimperfect sensing environment. The results are alsovalidated by our simulations. Our performance resultsdemonstrate that both the achievable throughput and

The authors are with the School of Electronics and ComputerScience, University of Southampton, Southampton, SO17 1BJ,UK (email: {aur1g12, lly, lh}@ecs.soton.ac.uk).

A. U. Rehman is with the department of Computer Science,Abdul Wali Khan University, Mardan, Pakistan.

The financial support of the EPSRC projects EP/Noo4558/1and EP/L018659/1, as well as of the European Research Coun-cil’s Advanced Fellow Grant under the Beam-Me-Up project andof the Royal Society’s Wolfson Research Merit Award is gratefullyacknowledged.

A part of this work has been published in the IEEE VehicularTechnology Conference, Nanjing, China, May 2016.

The research data for this paper is available athttp://doi.org/10.5258/SOTON/404271

the delay are significantly affected by the activity ofthe PUs, as well as by the reliability of the PR channeland by the number of packets transmitted per TS. Toattain the maximum throughput and/or the minimumtransmission delay, the number of packets transmittedwithin a TS should be carefully adapted based on theactivity level of the PUs and on the quality of the PRchannel.

Index Terms—ARQ, GBN-ARQ, primary radio, PRchannel, cognitive radio, spectrum sensing, number ofpackets, cognitive GBN-HARQ, Discrete time Markovchain, throughput, delay.

LIST OF ACRONYMS

ACK Positive AcknowledgementARQ Automatic Repeat ReQuestCGBN Cognitive Go-Back-NCR Cognitive RadioCSW Cognitive Stop and WaitCU Cognitive UserDTMC Discrete Time Markov ChainDSA Dynamic Spectrum AccessFEC Forward Error CorrectionGF Galois FieldHARQ Hybrid Automatic Repeat ReQuestNACK Negative AcknowledgementOFF Channel free from PUON Channel occupied by PUPe Packet Error ProbabilityPMF Probability Mass FunctionPR Primary RadioPU Primary UserRS Reed-SolomonRTT Round-Trip TimeSW Stop and WaitTS Time-slot

LIST OF SYMBOLS

α Probability of traversing from ‘ON’ to‘OFF’ state

2

β Probability of traversing from ‘OFF’ to‘ON’ state

III Identity matrixk time duration used for sensing a TSKd Information Bits of the codewordLi Total number of new packets transmitted

in state SiMc Total number of packetsMT Maximum delayN Number of packets in a TSNd Coded symbolsnp(Si) Function for storing number of new pack-

ets associated with state SiPPP Transition matrixppp Stores probabilities of all state in a TSPPP d Distribution of end-to-end delay packet

delay by simulationPi,j {i, j}th element of the transition matrixPMF Distribution of end-to-end packet delay by

analysisPoff Channel free probabilityPon Channel busy probabilityΦΦΦ Steady-state vectorΦi ith element of the Steady-state vectorπππ steady-state vectorπi ith element of steady-state vectorRT Throughput by analysisR′T Normalized throughput by analysisRS Throughput by simulationS Sample setSi Represents state iSN a set of states containing new packetsSi Subset of SN , which contains states asso-

ciated with new packet[ ]T Represents matrix transposeT Total duration of the time-slotTd duration for data transmission in a TSTD Theoretically derived Total average packet

delayT ′DS Total average packet delay obtained by

simulationTp Time duration for packet deliveryTs during for sensing a TSτ Theoretically derived Average end-to-end

packet delayτs Average end-to-end packet delay obtained

by simulation

I. INTRODUCTION

Since the germination of the cognitive radio con-cept [1], it has been considered to be one of

the favourable regimes for mitigating the spectrumshortage that resulted from the inefficient spectrumutilization. The studies in [2]–[5] show that theinefficient utilization stems from the classic staticspectrum allocation policy. Based on this policy,a certain bandwidth is exclusively allocated to anetwork, which is only available for the primaryusers (PUs). By contrast, CR relies on dynamic spec-trum access (DSA), which allows both the PUs andCUs to share the licensed spectrum. For example,the CUs can sense and exploit the earmarked, butmomentarily unoccupied licensed spectrum, whichis referred to as a spectrum hole, and utilize itfor their own transmissions [6]–[10]. However, theCUs have to evacuate the spectrum as soon as thePUs are reclaiming it. These capabilities of CR haveencouraged the regulatory bodies to officially allowthe deployment of CR systems in order to maximizethe overall spectrum exploitation without havingto make any changes to the existing PR systemsoperated under the classic fixed spectrum alloca-tion approach [2], [10], [11]. As a result, variousIEEE standards have incorporated the CR concept,including IEEE 1900, 802.11y, 802.16h, 802.22, etc.,as shown in [12] and the references in it.

0

500

1000

1500

2000

2500

Nu

mb

ero

fP

aper

s

2000 2002 2004 2006 2008 2010 2012 2014

YearFigure 1. The number of contributions performed in the contextof CR per year as per IEEE Xplore database.

In recent years, CR has attracted substantial re-search attention and has been systematically stud-ied from diverse perspectives, as evidenced by thenumber of contributions dedicated to it, as shownin Fig. 1 [6]–[9], [13], [14]. Specifically, Haykin [8]has outlined the fundamental operating principles ofCR systems in the context of spectrum sensing, chan-nel estimation, cooperative communications, powercontrol and dynamic spectrum management. In [6],[7], [9], the authors have addressed the spectrumsensing and resource allocation problems of CR. Thearchitecture of the CR network and its impact on

3

Synchronization

Fading and

Shadowing

Hand−off

Spectrum

Spectrum

Sharing

Spectrum

Decision OFDMA

TDMA

Spectrum sensing

Non−cooperative

Cooperative

Matched−Filter

Energy−detection

Cycle−stationary

Imperfect

SensingInterference

Management

SW

Flow−control

Error−correction

Data Transmission

FEC

SR

Transmitter

Design

activity

PUs Channel

Noise

Channel

Coding

End−to−EndTransmission

reliability

Transmission

Opportunistic

SchedulingEstimation

ChannelSDR Design

Spectrum

QoS

CR access

Mode

control

Error

ARQ

GBN

Cognitive GBN-HARQ

Throughput

Delay HARQ

Optimal value of N

Figure 2. An overview of factors affecting the data transmission in CR systems.

dynamic resource allocation have been studied in[13]. By contrast, Tragos et al. [14] have inves-tigated the resource allocation techniques of bothcentralized and distributed CR networks, while con-sidering the factors of interference, power, delay,etc. In CR, in addition to the challenges imposedby both spectrum sensing and DSA, CR systems alsoface the extra challenges, when aiming for a lowBER. Therefore, powerful error-control approachesare required for achieving reliable communications,similarly to the traditional wireless communicationsystems [15], [16]. Some of the challenges associ-ated with achieving reliable communication in CRsystems are presented in the Fig. 2.

Similar to previous studied [17]–[21] and [22]–[26], the activity of the PUs is modelled using aMarkov chain having the ‘ON’ and ‘OFF’ states.Specifically, in the ‘ON’ state, the PR channel isconsidered to be occupied by the PUs, whereas inthe ‘OFF’ state, it is free from PUs. In order todetermine the status of the PR channel, the CRsystem invokes spectrum sensing and then accessesthe channel, but only when it is deemed to be inthe ‘OFF’ state. In practice, the sensing operationcan never be perfect, therefore, the CR system mayerroneously identify the PR channel’s state, hencegenerating either a false-alarm or a mis-detectionevent. In this case, the channel perceived by theCR system may be described by a four state Markovchain by considering both the actual states of the

PR channel and the states obtained after the sensingoperation. Moreover, similar to [27]–[29], the PR’schannel is organized in time-slots (TSs), where eachTS has a duration of T seconds, which is dividedinto two sections of Ts and Td seconds [27], [29].The first Ts seconds of a TS are used for sensingthe channel, while the remaining Td seconds areutilized for data transmission, which follows theGo-Back-N hybrid automatic repeat request (GBN-HARQ) principles [15], [19], [20], [30], providedthat the PR channel is deemed to be in the ‘OFF’state.

In this paper, we specifically extend the classicGBN-HARQ protocol to the CR system, when imper-fect sensing is considered. Here the GBN protocolhas been favoured over other ARQ schemes, becauseits performance is better than that of the classic stop-and-wait HARQ (SW-HARQ), while its complexityis lower than that of the classic selective-repeatHARQ (SR-HARQ). Hence, we are interested in thecharacteristics and performance of the GBN-HARQoperated in CR communication environments, whichis hence referred to as CGBN-HARQ. As mentionedabove, in our CGBN-HARQ scheme, first the CRtransmitter senses the status of the PR channel.If it is sensed to be free from the PUs, it trans-mits a number of packets based on the GBN-HARQprotocol. The CR receiver detects the packets oneafter another. Then the corresponding feedback isgenerated for each packet and fed back to the trans-

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mitter. When the transmitter receives a feedback, it(re)transmit the packets based on the feedback sig-nal. Particularly, after each ACK flag, the transmittersends a new packet, whereas for each NACK flag,the erroneous packet and all the subsequent packetsare retransmitted in the next free TS.

This paper further develops our research on cog-nitive HARQ [17]–[21]. In a nutshell, in [17], [20],[21] we have studied both the throughput and delayof cognitive SW-HARQ (CSW-HARQ), cognitive Go-Back-N HARQ (CGBN-HARQ) and that of cognitiveSR-HARQ (CSR-HARQ), all under the assumption ofperfect spectrum sensing. By contrast, in [18], [19],we have characterized the throughput and delay ofthe CSW-HARQ and CGBN-HARQ schemes both forperfect and imperfect sensing. In more detail, in[18], we have conceived the throughput and delay ofthe CSW-HARQ scheme. On the other hand, in [19],we have investigated the performance of a CGBN-HARQ scheme in terms of its achievable throughputand delay. We assumed that the feedback flags arealways received after the sensing duration of a TS,hence resulting in a simplified operational modelfor analysing the CR transmitter, but at the cost ofa longer round-trip delay and increased number ofpacket retransmissions. By contrast, in this paper,we relinquished the above constraint by enablingthe transmitter to receive feedback flags any timeduring a TS (namely both during sensing as wellas in the transmission epoch). Since the feedbackinformation is well-protected and typically relies ona single-bit information [31], [32], the feedbackoverhead does not affect the sensing process andthe PU’s communication. By eliminating the abovesimplifying assumptions, the analytical modelling ofthe CGBN-HARQ scheme associated with imperfectsensing became challenging, due to: 1) the receptionof the feedback flags both with in the sensing dura-tion and data transmission duration; 2) the imper-fect sensing decisions seen in our four-state Markovchain. By doing so, the transmitter circumvents theretransmission of packets for which ACK flags arereceived in the sensing period. As a result, both theoverall throughput and the delay of the system areenhanced. Moreover, in contrast to [19], [20], in thistreaties, the CGBN-HARQ is modelled by a DTMC,based on which closed-form expressions are derivedboth for the throughput as well as for the averagepacket delay and for the end-to-end packet delay.Our performance studies show that for achievingan increased throughput and a reduced delay, thenumber of packet transmission per TS requires a

careful consideration for the sake of maximizingboth the utility of PR channel and of the CR aideddata transmission.

A. Related Work

The performance of the classical ARQ protocolshas been extensively characterized [15], [30], [58].Specifically, the conventional GBN-ARQ has beenmodelled with the aid of DTMC and its throughputand delay was investigated in [59]–[66]. In moredetail, Turin as well as Ausavapattanakun and Nos-ratinia [61], [64] invoked hidden Markov modellingand studied the throughput of GBN-ARQ, whenconsidering both perfectly reliable and practicallyunreliable feedback in fading environments. In [62],Chakraborty and Liinaharja introduced an adaptiveGBN-ARQ scheme operating in a time-varying fad-ing channel and characterized its throughput. Inthis adaptive GBN-ARQ, the transmitter switchesits transmission mode based on the reception ofcontiguous ACK’s and NACK’s. Moreover, Hayashidaand Komatsu [59] as well as Zorzi and Rao [60]have studied the delay of the classic GBN-ARQregime. As further advances, Zorzi [63] has designedblock-coded error-control scheme operating underdelay constraints and studied its performance. Qinand Yang [67]–[70] invoked ARQ techniques for abutterfly network topology and analyzed both itsthroughput as well as the probability mass functionof its delay and its average packet delay. Similarly,Chen et al. proposed an algorithm based on Markovmodelling for characterizing both the probabilitydistribution and the average packet delay [65]. Liet al. proposed a cooperative GBN-HARQ schemerelying on a limited number of retransmissions [71].The HARQ technique constitutes a promising designfor achieving reliable communication, therefore ithas been widely employed in wireless systems, asseen in the Timeline 1.

In the context of ARQ aided CR systems, thereare only a limited number of references [79], [80],[93]–[103], which are illustrated in Timeline 2.However, none these references have provided thedetailed theoretical throughput and delay analysisof ARQ-assisted CR systems. Specifically, in [93]–[95], it is assumed that the CUs infer the activity ofthe PUs via observing their feedbacks flags. Touatiet al. [96] have performed the seminal performanceanalysis of ARQ-aided and relay-assisted CR systems.Yue et al. [97], [98] have proposed an improvedHARQ scheme for the family of CR systems by

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1975 The family of ARQ techniques was studied in satellite communications [33].

1984 ARQ and HARQ techniques were studied by Lin et al. [34].

1988 De et al. studied the performance of Type-II HARQ techniques [35].

1990 Kallel introduced the code combining technique [36].

1994 Decker investigated the family of HARQ techniques in GSM [37].

1997 Zorzi et al. characterized the ARQ regimes in fading mobile radio channels [38].

1999 Hamorsky and Hanzo studied the performance of turbo coded Type-II HARQ [39].

2000 Sozer et al. characterized various ARQ paradigms in underwater acoustic networks [40].

2001 Caire and Tuninetti investigated the performance HARQ in Gaussian fading channels [41].

2003 Kim et al. quantified the performance of HARQ protocols in mobile communication [42].

2005 Zhao and Valenti studied ad hoc networks, when HARQ was employed [43].

2007 Beh et. al investigated the HARQ techniques of LTE OFDMA systems [44].

2009 Vuran and Akyildiz provided the cross-layer analysis of error control techniques [45].

2010 Zhang and Hanzo proposed HARQ aided superposition coding [46], [47].

2011 Chen et al. introduced multi-component turbo coded HARQ techniques [48]–[50].

2013 Chen et al. discussed the challenges of concatenating HARQ and turbo codes [51], [52].

2014 Ngo and Hanzo presented the state-of-the-art of HARQ techniques in cooperative communications [53].

2015 Chen et al. introduced an adaptive HARQ scheme by amalgamating Raptor codes and HARQ [54].

2015 Xu et al. proposed Type-III HARQ based on network coding and distributed turbo coding [55].

2016 Zhu et al introduced truncated HARQ technique for reducing the video distortion [56].

2016 Makki et al. characterized the performance of HARQ-aided and relay assisted networks [57].

TIMELINE 1: Milestones of the conventional HARQ-aided wireless systems.

invoking anti-jamming coding for achieving reliablecommunication. Furthermore, Liang et al. and Hu etal. [99], [100] studied the combination of spectruminterweave and underlay sharing access modes forthe sake of maximizing the performance of the CUs.Specifically, Hu et al. [100] proposed ARQ tech-niques for maximizing the transmission reliability.On the other hand, in [101]–[103] comprehensivesurveys have been presented for various communi-cation techniques employed in CR systems.

However, to the best of the our knowledge, nopublished treaties has theoretically analyzed thedelay and throughput of ARQ protocols in contextof CR systems. The dynamic activities of PUs andthe unreliable nature of spectrum sensing make themodelling and analysis of the ARQ protocols in CRsystems challenging. Therefore, our objective is tomodel the family of HARQ-aided CR systems by

deriving closed-form expressions for the throughputand delay of CGBN-HARQ schemes when imperfectspectrum sensing is assumed.

B. Contributions and Paper Structure

Against the above background, the contributionsof this paper are summarized as follows,

(a) The cognitive protocol proposed in [20] isextended by enabling the transmitter to simul-taneously sense and receive the feedback flag ofthe packet transmitted in the previous free TSin a realistic imperfect sensing environment. Todo so, the CR transmitter requires significantimprovements for reformulating the transmis-sion principles of the classic GBN-HARQ, whenincorporating it in CR systems.

(b) In contrast to [19], in this paper, the CGBN-HARQ has been theoretically modelled and anal-

6

2001 Ramos and Madani studied software defined radio systems, especially their protocol layer [72].

2005 Cabric and Brodersen introduced a transmission scheme based on wide-band power control andspectrum shaping [73].

2006 Fujii et al. focused on space-time-block-coding assisted distributed ARQ in ad hoc CR networks [74].

2006 Devroye et. al investigated the communication limits of CR systems [75].

2007 Weingart et. al investigated the parameters of the physical-network-and-application-layer for the sakeof improving both the spectrum efficiency and reliability [76].

2008 Cheng and Zhen combined power controlled modulation with truncated ARQ for improving theoverall throughput of CR systems [77].

2009 Yue and Wang introduced an anti-jamming coding technique in the context of CR systems for improv-ing the reliability [78].

2010 Ao and Chen introduced HARQ assisted amplify-and-forward cooperative relaying into CR sys-tems [79].

2010 Yang et al. invoked network coding for CR systems for improving the efficiency of classic ARQ tech-niques [80].

2011 Cheng et al. focussed their attention on improving the data rate of CR system by observing the ARQfeedback flags of PUs [81].

2012 Makki et al. characterized the throughput and outage probability of CR aided HARQ systems [82].

2013 Andreotti et al. introduced a link adaptation paradigm for increasing the goodput, whilst reducing thecomplexity [83].

2014 Ozcan et al. characterized the error rate performance of CR systems [84].

2014 Chen et al. focused their efforts on the end-to-end transmission improvements of CR-aided ad hocnetworks [85].

2015 Zou et al proposed an opportunistic relay assisted scheme for improving the reliability of CR systems[86].

2015 Makki et al. invoked finite-length codewords in spectrum sharing CR networks and studied theirthroughput [87], [88].

2016 Khan et al. reviewed the networking techniques, applications and CR communication protocolsemployed in smart grids [89].

2016 Zhang et. al introduced energy-efficient DSA protocols for improving the overall channel utilizationwithout degrading the PUs performance [90].

2016 Patel et al. focused on maximizing the achievable rates of CR systems [91], [92].

TIMELINE 2: Milestones in CR-aided HARQ systems.

ysed by a DTMC-based approach in imperfectsensing environments. Based on this model,closed form expressions are derived for thethroughput, average packet delay and end-to-end packet delay. For the case of end-to-endpacket delay, both its probability distributionand average end-to-end packet delay are anal-ysed

(c) Finally, the theoretical analysis has been verifiedby the simulations.

The rest of this paper is organized as follows.In Section II, we model the primary radio systems

as well as the cognitive systems. In Section III,we outlined the transmission principles of CGBN-HARQ, whereas the operation of the CR transmitterand receiver is presented in Subsections III-A andIII-B, respectively. Then, we detailed our DTMC-based analytical model in Section IV. Then, closed-form expressions are derived for the throughput anddelay in the Subsections IV-D and IV-E, which arethen verified by our simulations presented in SectionV. Finally, the conclusions of this contribution areprovided in Section VI.

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II. SYSTEM MODEL

In this section, we will model both the PR and CRsystems. Furthermore, the main assumptions usedare described.

A. Modelling of Primary Radio Systems

0 (OFF)

1(ON)

β

α

1 - β 1 - α

Figure 3. Two-state discrete-time Markov chain model for PRsystems [23]–[25].

Similar to our previous studies [19], [20], a two-state Markov chain having ‘ON’ and ‘OFF’ states isinvoked for modelling the activities of PUs in the PRchannel, as shown in Fig. 3. In the ‘ON’ state, a PUis assumed to be utilizing the PR channel, while inthe ‘OFF’ state, the PR channel is free from the PUs.The transition probability from the ‘OFF’ to the ‘ON’and that from ‘ON’ to ‘OFF’ state are represented byβ and α, respectively. Let Pon and Poff = 1 − Pon

represent the probabilities of the PR channel beingin the ‘ON’ and ‘OFF’ states. Then, when the Markovchain reaches its steady state, we have [30]

Ponα = Poffβ, (1)

which results in

Pon =β

α+ β, Poff =

α

α+ β. (2)

Similar to our previous contributions [18]–[20]and as shown in Fig. 4(a), the PUs utilize the channelon the basis of TSs, whereas the activation of eachTS is independent and has the same probability.Similarly, we also assume that when a TS is found inthe ‘OFF’ or ‘ON’ state, the PUs activity remains thesame until the end of that TS. For instance, whena TS is deemed to be in the ‘OFF’ state, the CUis allowed to access the TS and send its own data.Otherwise, the CU waits until the next TS [22].

B. Modelling of Cognitive Systems

In CR systems, each TS is further partitionedinto the sensing time duration of Ts seconds anddata transmission in the remaining of Td = T − Tsseconds, as presented in Fig. 4(b). As mentionedabove, in the sensing duration, the CUs perform

ChannelOFF OFFON

T TT

t

(a) Channel utilization by PU

Transmission Channel Sensing Transmission Sensing Transmission Sensing

T

Ts

T

Td = T − TsTs

T

Td = T − Ts No-t

Ts

(b) Channel Utilization by CU

Figure 4. Time-slot structure of PR and CR systems, where aCR TS consists of a sensing duration of Ts and a transmissionduration of Td = T − Ts, when given the total duration T of atime-slot.

P1,0

P0,1

P2,3

P3,2

P0,3

P0,2 P1,2P1,3

P2,0 P3,1P3,0P2,1

P2,2

P0,0

P3,3

P1,1S0 S1

S3S2

Figure 5. Modelling the CR system in the face of realisticimperfect sensing with the help of a four-state Markov chain,where S0 = 00 represents the case in which the TS is free inreality and the CU correctly sensed it as being free. S1 = 01illustrates that the channel is free in reality, but the CU detectedit to be busy. Likewise S2 = 10 and S3 = 11.

sensing for the sake of determining the ‘ON’ and’OFF’ activity of the PUs in the channel considered.If the PR channel is deemed to be free, then inthe remaining Td seconds the CU embarks on datatransmission using the classic GBN-HARQ protocol,which will be detailed below. Otherwise, it waitsuntil the next TS, during which the above procedureis repeated.

It is widely recognized that the sensing opera-tion cannot be perfectly reliable in practice, hencethere is always a chance of incorrect sensing, dueto numerous problems, such as for example thechannel-induced shadowing and fading, aggravatedby the background noise. Similarly, in the contextof CR systems, it may also wrongly detect the pres-ence/absence of PUs in the PR channel, hence re-sulting either in false-alarm (Pfa) or in mis-detection(Pmd). To elaborate a little further, false-alarm rep-resents the scenario, in which the PU is inactive butthe CR system deems it to be active. On the otherhand, mis-detection represents the case, in which thePU is active, yet the CR system deemed the channelto be free. In summary, the states of this realistic CR

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system are shown in Fig. 5, where the index of eachstate is the combination of a two-digit sequence,where the first digit illustrates the real status ofthe PR channel, whereas the second digit representsthe sensed status of the channel. Additionally, thetransition probability between states can be readilyarranged in a matrix form as

PPP =

P0,0 P0,1 P0,2 P0,3

P1,0 P1,1 P1,2 P1,3

P2,0 P2,1 P2,2 P2,3

P3,0 P3,1 P3,2 P3,3

.

Then, with the aid of (2) and Fig. 5, we obtain

P0,0 = P1,0 = (1− β)(1− Pfa)P0,1 = P1,1 = (1− β)Pfa

P0,2 = P1,2 = βPmd

P0,3 = P1,3 = β(1− Pmd)P2,0 = P3,0 = α(1− Pfa)P2,1 = P3,1 = α(1− Pfa)P2,2 = P3,2 = (1− α)(Pmd)P2,3 = P3,3 = (1− α)(1− Pmd).

(3)

Let us express the steady-state probabilities of theMarkov chain as ΦΦΦ = [Φ0,Φ1,Φ2,Φ3]

T . Then, wehave

ΦΦΦ = PPP TΦΦΦ, (4)

which demonstrates that ΦΦΦ is the right eigenvectorof PPPT having an eigenvalue of one. Therefore, basedon (4), we have

ΦΦΦ = [Φ0 Φ1 Φ2 Φ3]T

= λ×[α(1− PFa)

β(1− Pmd)

α(PFa)

β(1− Pmd)

(Pmd)

(1− Pmd)1

]T,

(5)

where λ ∈ R is applied to satisfy3∑

i=0

Φi = 1, (6)

giving

λ =β(1− Pmd)

α+ β. (7)

Upon substituting (7) into (5), the steady-state prob-abilities of the CR system at states S0, S1, S2 and S3are given by

Φ0 =α(1− Pfa)

α+ β, Φ1 =

α(Pfa)

α+ β(8)

Φ2 =β(Pmd)

α+ β, Φ3 =

β(1− Pmd)

α+ β. (9)

III. PRINCIPLES OF COGNITIVE GO-BACK-N HYBRID

AUTOMATIC REPEAT REQUEST

CR Transmitter

sensing

PR channel

Wireless channel

NY

block of N packets

Transmit/Retransmit

free?

Is PR channel N

Y

Is the packet

correct?Transmit ACK Transmit NACK

Discard other

received packets

CR Receiver

Figure 6. Flow chart showing the operations of the proposedCGBN-HARQ scheme.

Similar to our previous studies [19], [20], in thistreatise, the CR system also transmits its data in theform of packets, where each packet is assumed tobe encoded by Reed-Solomon (RS) codes definedover the Galois Field of GF(q)=GF(2m). Each RScoded packet is comprised of Nd and Kd symbols.Specifically, Nd represents the information symbols,whereas Kd denotes the coded symbols. Moreover,we assumed that each RS coded packet requires aduration of Tp seconds for its transmission. As aresult, the transmitter is capable of sending N =Td/Tp packets in each free TS. Furthermore, we alsoassumed that the RS code is capable of perfectlydetecting and rectifying t errors. However, whenmore than t symbol errors are received in a packet,it discards the packet.

Following the above assumptions and as illus-trated in Fig. 6, the data packets are transmittedbetween a pair of CR users, namely the CR transmit-ter and CR receiver, while relying on the principlesof the CGBN-HARQ scheme. Similar to the classicGBN-HARQ [15], [30], [58], the CR transmitterseamlessly transmits N packets one after another ina free TS, without waiting for their feedback flags,where the feedback for each packet is programmedto be received after the round-trip-time (RTT) of Tdseconds. The RTT can be defined as the time du-ration spanning from the commencement of packettransmission until the reception of its feedback flag[15], [30], [58]. Note that during the RTT of apacket, the subsequent (N − 1) packets are also

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transmitted.Below we provide more details concerning the

operational principles of the CGBN-HARQ based CRtransmitter and CR receiver, which was described byAlgorithm 1 of [20] in a perfect sensing scenario.By contrast, the additional consequences imposed byimperfect sensing are discussed as follows.• When a TS is mis-classified, the transmitter

sends N packets to the receiver following thenormal procedure described in Algorithm 1 of[20].

• In the case of false-alarm, even the though thechannel is free, the CR transmitter does nottransmit packets.

• At the receiver side, in the mis-detection sce-nario, the receiver discards the packet with aprobability of one, since the probability of thepacket being in error is too high both due toits collision with the PUs transmission and thechannel noise.

A. Operation of the CR transmitter

In the conventional GBN-HARQ protocol, thepackets are continuously transmitted by the trans-mitter until it receives a negative feedback flag(NACK). In this case, the retransmission of the cor-rupted and of all subsequent packets has to takeplace. In the CGBN-HARQ scheme of this contribu-tion, however, the PR channel is first sensed by theCR transmitter for Ts seconds. If the TS is detectedto be free from PUs, the corresponding packets aretransmitted, otherwise the transmitter waits for thenext TS. Additionally, in realistic imperfect sens-ing environments the transmission of packets takesplace, when the PR channel is correctly detected tobe in the ‘OFF’ state, or when it is deemed to be inthe ‘OFF’ state but it is actually ‘ON’. On the otherhand, if the PR channel is in the ‘ON’ state or falselydeemed to be in the ‘ON’ state, the transmitter waitsfor the next TS and repeats the process of sensing. Insummary, in each free TS, the transmitter sends Npackets. These packets may be either new or thosethat require retransmission.

Moreover, as in the classical GBN-ARQ, it assumedthat the CR transmitter has a buffer size of N packetsand it follows the first-in-first-out principle [15],[30], [58]. In the transmitter buffer the copies ofthe transmitted packets are stored until it receives apositive feedback (ACK) flag. For example, if an ACKis received for a packet, the CR transmitter removesits copy from the buffer and replaces it with a new

packet. On the other hand, if a NACK flag is received,the transmitter buffer remains unchanged. Further-more, it is assumed that the transmission of feedbackflags is always error-free. This may be justified, sincea single bit of feedback information is sufficient forthe CGBN-HARQ, which can be strongly protectedfrom errors [31], [32]. This justifies our assumptionof allowing the CR transmitter to receive feedbackflags at any time in a busy or free TS, without ham-pering the sensing process or the PU transmission,as shown in Fig. 7 and 8.

It is worth noting that when feedback flags arereceived during sensing time, the transmitter onlyupdates its buffer and waits until the decision aboutthe TS takes place. For instance, as shown in Fig. 7,the ACK flag for packet 1 is received during sensingtime, hence the transmitter replaces the copy ofpacket 1 by packet 5 and waits for the sensing result.Soon after the sensing decision took place and ACKflags were received in the transmission period, thetransmitter sends the new packets 5, 6, 7, 8 in thefree TS, as illustrated in TS T2 of Fig. 7. The processcontinues until a NACK flag is received, as shownin Fig. 8. The transmitter immediately stop sendingnew packets and prepares for retransmitting boththe corrupted and the subsequent packets. Moreover,in the case of a busy TS, the transmitter prepares thesequence of packets in the buffer for transmittingthem in the next free TS, as shown in TS T3 and T4of Figs. 7 and 8.

It is worth noting that in Fig. 7, TS T2 is mis-detected, which implies that the TS is actually oc-cupied by the PU, but due to the wrong sensingdecision, it is declared to be free. As a result, newpackets, i.e., packets 5, 6, 7 and 8, of Fig. 7 aretransmitted. However, due to their collisions with thePU transmissions, these packets are corrupted andhence all N packets are received in error. Moreover,in the case of false-alarm, the free TS is sensed tobe busy, hence no transmission takes place.

B. Operation of the CR receiver

In this contribution, the CR receiver works in asimilar way as that proposed in our previous studies[19], [20]. This is because, the receiver has noinformation about the sensing or the PU’s activity inthe channel. However, it only performs RS decodingof the received packets and generates the respectivefeedback flags, which are then sent to the transmit-ter. For example, as shown in Fig. 7 and 8, if a packetis declared to be error-free after error-correction, the

10

Sen

sing

Sen

sing

Sen

sing

Sen

sing

DDDTransmission

kk

ACK

ACK

ACK

k

ACK

NACK

k

CR-receiver

Transmitter1

1

3 42

2 3 4

5

6 7 8

No

RTT=Td=NTp

T1 T2 T3 T4

Miss Free576

Busy

5

8 8765

5 6 7 8

Tp

transmission

6 7 8

5 6 7 8

Figure 7. Packet transmission using the CGBN-HARQ scheme, when k = 1 and N = 4 in the case of imperfect sensing.

DDD

Sen

sing

Sen

sing

Sen

sing

Sen

sing

TransmissionACK

ACK

ACK

NACK

Receiver

TransmitterTp

1

1

3 42

2

35

3

No

Free Free4 5

RTT=Td=NTp

4 5 6 7

4 5 6 73

T1 T2 T3 T4

transmission

4 5 6 7Busy

NACK

3 4e3

e4

Figure 8. Packet transmission using the CGBN-HARQ scheme, when k = 1 and N = 4, illustrating the effect of the NACK flag inboth free and busy TSs.

receiver transmits an ACK flag to inform the trans-mitter that the packet is correctly received, hence anew packet is required. On the other hand, if theNACK flag is transmitted due to erroneous receptionof a packet, the transmitter sends both the erroneousand the subsequent packets again. Moreover, in thecase of mis-detection, the CR receiver considers thepacket to be erroneous and feeds back a NACK flagto the transmitter’s control channel receiver for therespective packet.

IV. DISCRETE TIME MARKOV CHAIN BASED

ANALYSIS

In contrast to the classic GBN-HARQ protocol, themodelling of the CGBN-HARQ transmission schemein imperfect sensing environments hinges on thedynamic activity of the PU and on the potentiallyincorrect sensing of the PR channel. In this pa-per, the CGBN-HARQ transmission scheme has beenmodelled by DTMC in a realistic imperfect sensingenvironment. Based on the DTMC modelling, wethen evaluate the steady-state throughput, averagepacket delay and the end-to-end delay consideringboth its probability distribution and the average end-to-end packet delay.

To commence the analysis, the states of the CGBN-HARQ system can be jointly determined by thefollowing factors:

(a) The actual status of the PR channel.(b) The sensed status of the PR channel.(c) The status of the N packets stored in the

transmitter buffer, (which may be ‘new’, ‘old’and ‘packets transmitted twice in a TS’).

Let us formulate the list of states as

S = {S0, S1, . . . , Si, . . . ST }, (10)

where ST represents the total number of states, i.ewe have ST = |S|, and Si is the ith state in S andhas an (N + 2)-length base-3 digit formulated as

Si = {Si0Si1Si2 · · ·Si(N+1)}, i = 0, 1, · · · (11)

For convenience, Si represents the ith state se-quence. Specifically, in the state sequence of Eq (11),the first digit Si0 illustrates the real status of the PRchannel,

Si0 =

{0, if the PR channel is free in real,1, if the PR channel is busy in real.

(12)

The second digit Si1 represents the sensed status of

11

the PR channel, which is given by

Si1 =

{0, if the PR channel is sensed free ;1, if the PR channel is sensed busy.

(13)

Finally, the digit Sij for j = 2, . . . , N + 1, is deter-mined as

Sij =

0, if the jth packet is a new one;1, if the jth packet is a retransmitted one,

when the TS is free; or the one to beretransmitted, when the TS is busy;

2, if the jth packet is a repeated replicaof a previous packet in the same TS.

(14)Having determined Sij according to (12), (13) and(14), we can find a one-to-one mapping between iand Si as

i =

N+1∑j=0

Sij3N−j+1, (15)

which gives the subscript i of Si, representing theith state of the DTMC.

For the sake of deep understanding, a pair ofexamples are provided below, which explains theDTMC-based modelling in the face of imperfect sens-ing in terms of the associated state transitions.

A. Example 1

Let us examine the CGBN-HARQ scheme using asingle Tp for the sensing process, which is only ca-pable of transmitting a single packet (i.e. k = 1 andN = 1). In other words, the transmitter buffer storesthe copy of a single packet. In this scenario, a packetwill never be repeated in the same TS and thereforedigit 2 of Eq. (14) will never appear in the statesequence. According to the above definitions, we canshow that the DTMC of this system has ST = 8states, which are explicitly shown in Table I. Corre-

State (Si0 Si1 Si2) State (Si0 Si1 Si2)S0 (0 0 0) S9 (1 0 0)S1 (0 0 1) S10 (1 0 1)S3 (0 1 0) S12 (1 1 0)S4 (0 1 1) S13 (1 1 1)

Table IPOSSIBLE STATES OF THE CGBN-HARQ, WITH N = 1 AND k = 1.

spondingly, the state transitions are shown in Fig. 10,where the transitions and their transition proba-bilities are detailed below. Let us assume that the

S0

S9

S4

S1

S3

S12 S13

S10

Figure 10. State diagram illustrating the modelling of theCGBN-HARQ scheme for N = 1 and k = 1. The dashedlines denote the transitions to the busy states only, due to thedetection of a busy TS. On the other hand, solid lines representthe transitions to the free states during the detection of free TSs.The solid circles illustrate the free states due correct detectionand mis-detection, whereas dashed circles denote the busy statesdue to correct detection and false-alarm.

transmitter is initially in state S0 = 000. This state isdefined by the events that the TS is actually free (i.e.Si0 = 0) and it is also sensed to be free by the CRsystem (i.e. Si1 = 0). Therefore, the transmission ofa new packet takes place (Si2 = 0). Moreover, whenthe transmitter receives an ACK flag related to thetransmitted packet and the next TS is again correctlydetected to be free, the transmitter remains in thesame state S0. This gives the transition probabilityof P0,0 = Poff (1− Pfa)× (1− Pe), where Pe is thepacket’s error probability. By contrast, if the trans-mitter receives a NACK flag and the TS is found tobe free from PUs, the system traverses to state S1 =001. Correspondingly, the transition probability isP0,1 = Poff (1− Pfa)× Pe. In a similar manner, if thenext TS is correctly detected to be busy and an ACK(or a NACK) flag is received, then the transmitterchanges its state from S0 to S12 (or S13), with a tran-sition probability of P0,12 = Pon(1− Pmd)× (1− Pe)(or P0,13 = Pon(1− Pmd)× Pe). Likewise, if the nextbusy TS is mis-detected and an ACK (or NACK)flag is received, then the transmitter traverses fromS0 to S9 (or S10), with a transition probability ofP0,9 = PonPmd × (1− Pe) (or P0,10 = PonPmd × Pe).

When the transmitter is in state S9 or S10, in thenext TS the transmitter can only make a transition toone of the following states S1, S4, S10 and S13. This isbecause the transmissions associated S9 and S10 arealways assumed to be incorrect due to mis-detection.

12

DD D D

Sen

sing

Sen

sing

Sen

sing

Sen

sing

Transmission

Sen

sing

ACK

ACK

NACK

S0=000000transmitter

Receiver

TransmitterTp

1

1

3 42

2

35 4 5 3 6

T1 T2 T3 T4

transmission

3 4 5 6

NACK

3e3 3

e3

No

T5

33transmission

33 3 3 3 3 3

54

3

3 5 64

3 4 5 6State of the S363=111110 S39=001110S257=100112 S120=011110

No

Free Miss False Busy Free

Figure 9. Packet transmission using CGBN-HARQ, where, Miss, and, False, represents mis-detection and false-alarm, respectively.At the end of the TS, the transmitter state is observed.

If the next TS yields false-alarm or it is correctly de-tected to be busy, then the transmitter moves to stateS4 or S13, associated with the transition probabilityof P9,4 = PoffPfa or P9,13 = Pon(1− Pmd). Similarly,based on Fig. 10, the remaining state transitions canbe readily analyzed, where in Table II, we presentedthe corresponding state to state transition probabil-ities.

B. Example 2

In the second example, we considered the CGBN-HARQ scheme having the parameters of k = 1 andN = 4. For clarity, we consider the scenario shownin Fig. 9, which illustrates that the transitions takeplace across five TSs, with the following justification.

(a) Let us consider in TS T1 that the PR channel iscorrectly detected to be free. Hence, we haveSi0 = 0 and Si1 = 0. In this case, we assumethat N = 4 new packets are transmitted, whichgive Sij = 0 for j = 2, 3, 4, 5. Therefore, atthe end of TS T1, the state observed by thetransmitter is S0 = {0, 0, 0, 0, 0, 0}. As shownin the figure, the first two packets are receivedwithout errors, therefore ACK flags are trans-mitted. By contrast, packet 3 is erroneouslyreceived, hence the receiver feeds back a NACKflag to the transmitter.

(b) Due to mis-detection, the TS T2 is deemed tobe free, although it is actually busy. Therefore,the transmitter receives an ACK after the sens-ing duration and it transmits the new packet5. However, after the reception of a NACK flagat the end of the first Tp of the transmissionduration of T2, both the corrupted as wellas the subsequent packets are retransmitted,where the indices of the 4 packets transmittedduring T2 are 5, 3, 4, 5. It is worth noting thatpacket 5 is transmitted twice in the sameTS, while packets 3 and 4 are old packets

transmitted in the previous TS, as depictedin Fig. 9. All the above-mentioned events re-sult in the state associated with the index ofS257 = 100112 in TS T2. Correspondingly, thetransition probability from state S0 to S257is P0,257 = PonPmd × (1 − Pe)

2Pe. Moreover,since the TS T2 is mis-detected, the packetstransmitted in TS T2 are all received in error.Hence, the receiver feeds back a NACK flag,while packets 4 and 5 are discarded, regardlesswhether they are received correctly or incor-rectly.

(c) During TS T3, the channel is falsely detectedto be busy. Since all packets transmitted dur-ing the TS T2 are erroneous, they requireretransmission. Hence, the transmitter changesits state from S257 to S120 = 011110 with aprobability of P257,120 = Pon × Pmd.

(d) Following the same pattern, the state associ-ated with TS T4 is S363 and the correspondingtransition probability from state S120 to S363 isP257,363 = Pon × (1− Pmd).

(e) Finally, the TS T5 is correctly detected to befree. Therefore, all the N packets waiting inthe buffer are transmitted, which results in atransition from state S363 to the state S39 witha transition probability of P363,39 = Poff ×(1−Pfa).

According to the principles of our CGBN-HARQ,the characteristics of the DMTC derived for theCGBN-HARQ relying on realistic imperfect sensingare given below.

• In a busy TS, there will no transmission orretransmission of a packet, therefore the respec-tive state sequence does not contain digit 2. Thisbusy state can be encountered either due tocorrect sensing of an occupied PR channel, orowing to the false-alarm encountered in a freePR channel.

13

• When a transition from a busy state to a freestate occurs, only the first two digits of thecurrent state sequence Si0 and Si1 may changein the new state sequence. This is explicitlyshown in TS T5 of Fig. 9. Hence, digit 2 doesnot appear in the new state sequence.

• The first two digits of a state sequence, i.e. Si0and Si1, never assume a value of 2.

With the aid of these properties, large fractionsof the states can be eliminated from consideration.However, there is still a large number of states.In this context, the mathematical formulation re-quired for deriving total number of states and forgenerating state sequences with the aid of generalexpression remains an open challenge. However, thetransition probability for any pair of state can bereadily derived. Hence, we propose the algorithmshown in Fig. 11 for finding the number of statesand the state sequences. The algorithm is conceivedas follows.

Fundamentally, this algorithm uses a search tech-nique for finding the legitimate states. Initially, thetransmitter is set to the state S0 = 000000. Letthe legitimate states form the set S, which initiallycontains only S0. In the process for generating newstates, we arrange for an erroneous transmission tooccur at each possible position of the packets sent inthe TS for the current state S0. Let us simultaneouslyassume that the next TS is either free or busy, whichis correctly sensed, mis-detected or sensed subject toa false-alarm.

For example, during the TS T1 of Fig. 7, onlynew packets are transmitted, which are correctlyreceived. Hence ACK flags are fed back. In the algo-rithm, this case is represented by e = 0. Continuingthis example, when the next TS is busy, but it issensed to be free, 4 new packets are transmittedin TS T2. Consequently, the state is changed toS243 = 100000. Moreover, in the same case of e = 0,it is possible that TS T2 may be sensed to be busydue to false detection of a free PR channel or owingthe correct detection of a busy PR channel. If this isthe case, the current state changes to S81 = 010000for the false-alarm or to S324 = 110000 for thecorrect detection, respectively. When a new stateis generated, for instance S324, it is then checkedagainst the state-set S. If state S324 is already in S,it is ignored. Otherwise, S324 is added to the set S.Similarly, by considering the cases of e = 1 to e = N ,the algorithm will generate all legitimate states thatthe transmitter may encounter upon emerging fromstate S0. Once all the error-free transmission to

ST=|S|STOP

Assume errorat eth position

Yes

Ise ≤ N

Yesset Si0 = 0

freeTS sensed

Is

Yesset Si1 = 0

Replaceit by 2

in realTS Free

Is

set Si1 = 1

S∩Si==∅ e = e + 1

AddSiinto

Se=e+1.

All 4

consideredcases of TS

?

Anyrepetition in

the TS

Nostates

considered

All

?

Transmit N new packetsset S0 = 0, 0, . . . , 0,S={S0}, l = 1 and e = 0.

Yes

set Scur to lth value in S

l=l+1

set Si0 = 1

No

No

No

No

No

Yes

No

Yes

Yes

Generate a state Siwhere Sij ∈ {0, 1, 2}for 2 ≤ j ≤ N .

Figure 11. Illustration of the process of deriving states relayingon (N+2)-length base-3 digits. For each value of e, the followingfour cases are considered: 1) correct detection of a free PRchannel 2) correct detection of a busy PR channel 3) mis-detection; and 4) false-alarm . A special case of a completelyerror-free packet is represented by e = 0, while in a TS, edenotes the position of the corrupted packet.

erroneous transmission scenarios of each packet ina TS have been considered, the algorithm is thenactivated for the next state in S, which is S243 in theexample considered in Fig. 7. The algorithm repeatsall the above steps for the sake of deriving newlegitimate states. Moreover, due to the properties ofirreducibility and the aperiodic nature of the DTMC,the state generating algorithm stops, once each state

14

in S has been examined as current state and nonew state is generated. Finally, at the end of thealgorithm, the cardinality of the set S gives the totalnumber of states ST , i.e. we have ST = |S|.

C. State Transition Probability Matrix

Let the state transition probability matrix be ex-pressed as PPP and the state at the ith TS be repre-sented by S(i). Then, for the example of Fig. 9, wehave S(1) = S0, S(2) = S257, S(3) = S120 S(4) =S363 and S(5) = S39. Given the state Si in TS m,we have S(m) = Si. Then the probability Pi,j oftransition to the next state is Sj , i.e. S(m+ 1) = Sjcan be expressed [30], [104] as:

Pi,j = P {S(m+ 1) = Sj | S(m) = Si, . . . , S(1) = S0}= P{S(m+ 1) = Sj |S(m) = Si}, (16)

which is the (j, i)th element of PPP. According to theproperties of the DTMC, we have

0 ≤ Pi,j ≤ 1∑Sj∈S

Pi,j = 1, ∀ Si ∈ S. (17)

It is worth noting that the transition probabilityPi,j is set to zero, if the transmitter cannot makea transition from Si to Sj in a single step (i.e. inthe next TS). To continue our modelling, let theprobability Pi(m) represent the transmitter state inTS m. Then we have

∑i∈S Pi(m) = 1. Again, let

ppp(m) = [P0(m), P1(m), . . . , PST(m)]T store the ST

probabilities in TS m. Furthermore, we assume thatthe transmission commences from state S(1) = S0associated with the probabilities of

ppp(1) = [1, 0, . . . , 0]T . (18)

Then, with the aid of the law of total probability[30], [105], the specific probability that the Markovchain transitions into state j during the TS (m+ 1)may readily be expressed as

Pj(m+ 1) =∑∀Si∈S

Pi(m)Pi,j , ∀ Sj ∈ S and m > 1.

(19)

By considering all the states in S, we readily arriveat the recursive equation of

ppp(m+ 1) = PPPTppp(m), m = 1, 2, 3, 4, . . . (20)

given the above equation, we know that

ppp(m+ 1) = (PPPT )mppp(1). (21)

Eq (17) shows that the state transition probabilitymatrix PPPT is a left stochastic matrix [105], becausethe sum of each column is equal to one. Conse-quently, based on the Perron-Frobenius theorem, thelimit of limm→∞(PPPT )m exists [105], and when wehave m → ∞, the DTMC reaches its steady state[30], implying that

ppp(m+ 1) = ppp(m). (22)

Let us represent the steady-state probabilities by πππ,i.e. πππ = limm→∞ppp(m). Then, based on (20) and (22),we have [30], [104],

πππ = PPPTπππ, (23)

where πππ is the right eigenvector of PPPT , having aneigenvalue of one. As a result, the steady-state vectorπππ can be found by determining the eigenvector of PPPT

[30], [104], [105].

D. Throughput of CGBN-HARQThe throughput of the CGBN-HARQ scheme is

quantified in terms of the average number of pack-ets successfully transmitted per TS. The successfultransmission of a packet in a TS depends on twofactors: 1) a free PR TS is correctly sensed by the CRsystem, and 2) the packet is delivered error-freely. Asshown in (23), when the DTMC reaches its steadystate, the throughput is determined by those states,which have one or more new packets successfullytransmitted. Let us analyze the throughput in detailas follows.

As mentioned above, when the DTMC reaches itssteady state, the additional throughput generatedduring the TS considered depends on two factors: 1)the PR channel is sensed free, corresponding to thescenario, when the first and second digits in the statesequence are either 00 or 10; and 2) the transmissionof new packets in a respective state, provided thatthe channel is sensed to be free. Let us assume thatnp(Si) is the number of new packets transmittedin association with state Si ∈ SN . Explicitly, np(Si)equals the number of zeros in the state sequence ofSi minus two, when the first two digits are 00 orminus one, if the first two digits are 10. Consideringthe first event, let us collect the states with thefirst two digits being 00 or 10 into a set denotedas SN . Then, when we obtained the steady stateprobabilities πππ, the throughput of the CGBN-HARQscheme can be evaluated as

RT =∑

Si∈SN

πi × np(Si), (packets / TS). (24)

15

Current State Si State Sj probability State Sj probability State Sj probability State Sj probabilityPi,j=i = 0, 1 & Pi,0=Poff (1-Pfa)(1-Pe) Pi,1=Poff (1-Pfa)Pe Pi,3=PoffPfa(1-Pe) Pi,4=PoffPfaPe

j = 0, 1, 3, . . . 13 Pi,9=PonPmd(1-Pe) Pi,10=PonPmdPe Pi,12=Pon(1-Pmd)(1-Pe) Pi,13=Pon(1-Pmd)Pe

Pi,j=i = 3, 12 & Pi,0=Poff (1-Pfa) Pi,1 = 0 Pi,3=PoffPfa Pi,4 = 0j = 0, 1, 3 . . . 13 Pi,9=PonPmd Pi,10 = 0 Pi,12=Pon(1-Pmd) Pi,13 = 0

Pi,j=i = 4, 8, 9, 13 Pi,0 = 0 Pi,1=Poff (1-Pfa) Pi,3 = 0 Pi,4=PoffPfa

& j = 0, 1 . . . 7 Pi,9 = 0 Pi,10=PonPmd Pi,12 = 0 Pi,13=Pon(1-Pmd)

Table IISTATE TO STATE TRANSITION PROBABILITIES WITH RESPECT TO TS FOR N = 1 AND k = 1 CASE.

Furthermore, if we express the throughput in termsof the number of packets per Tp (packet duration),we have

R′T =TpT×RT =

1

k +N×RT (packets / Tp).

(25)

Let us now analyze the delay imposed by our CGBN-HARQ scheme.

E. Delay of CGBN-HARQ

In the traditional GBN-HARQ, the transmission de-lay of a packet is contributed by the time required forits first transmission as well as the time associatedwith its retransmission. By contrast, in the proposedCGBN-HARQ, in addition to the above delay, extradelay may be imposed by the unavailability of thePR channel, which may either be due to the correctsensing of the busy PR channel or owing to thefalse sensing of the free PR channel. Therefore, inour CGBN-HARQ scheme [20], it is desirable toconsider two types of delays, namely, the averagepacket delay and end-to-end packet delay. Specifi-cally, the average packet delay may be defined asthe total time required from the instant of sensinga TS until the error-free transmission of all packetsdivided by the total number of packets that had tobe transmitted during this time. On the other hand,the end-to-end delay is the delay of a packet fromits first transmission attempt until the instant that itis finally correctly received. Below we consider bothtypes of delays.

1) Average Packet Delay: In the previous section,we have obtained the throughput quantified in termsof the average number of the packets per TS (or Tp).Hence, given the throughput as expressed in (24) or(25), we can readily infer that the average packetdelay can be evaluated as

TD =1

Rs(TS per packet) (26)

=k +N

RT(Tp per packet). (27)

2) End-To-End Packet Delay: In this subsection,we will first commence by studying the probabilitymass function (PMF) of the end-to-end packet delayand then investigate the average end-to-end packetdelay.

Let us define SN for storing all the states in whichone or more new packets are transmitted. In otherwords, SN ⊂ S. Given a state Si ∈ SN , we defineanother set S(m)

i for accumulating the specific states{Sj} in which li,j ≥ 1 new packets transmitted fromstate Si are correctly received with a delay of mTp,i.e. we have:

S(m)i ={Sj | li,j ≥ 1 new packets transmitted at

state Si are correctly received at state

state Sj with exactly the delay of mTp}.(28)

Let Li represents the number of new packets trans-mitted from Si. Then, the PMF of the end-to-endpacket delay can be evaluated as:

P (m) =1

c

∑Si∈SN

∑Sj∈S(m)

i

πi × li,j × P (m)i,j

Li, m = 1, 2, . . . .

(29)

where c =∑

Si∈SNπi and P

(m)i,j denotes the proba-

bility of transition from state Si to state Sj with thedelay of mTp.

Given πππ,PPP and S, the PMF of end-to-end packetdelay can be obtained as follows. Let

PPPMF = [P (1), P (2), . . . , P (MT )]T , (30)

where MT represents the maximum delay having avery high value such as, 10−8. Using this approx-imation, the probability of events having a delayhigher than MT can be ignored. Let us first initializeppp(0) = IIIi, where IIIi is the ith column of the identitymatrix IIIST

. Then, based on the properties of theDTMC, we have

ppppppppp(j) = PPPT .ppp(j−1) = (PPPT )jIIIi, j = 1, 2, . . . (31)

16

From (31) we can obtain the following insights:a) If a packet originally transmitted in state Si is

successfully received in state Sj , the effectiveend-to-end delay is jTps;

b) The probabilities of transition from state Si toany other states in S are given by ppp(j) in (31);

Therefore, with the aid of the above information, wecan update PPPMF using the following formula:

P (m)← P (m− 1) +πi × li,j × P (m)

i,j

Li, (32)

for m = 1, 2 . . . ,MT , Sj ∈ S(m)i , Si ∈ SN . Having

formulated the PMF of the end-to-end packet delay,we now evaluate the average end-to-end packetdelay as follows

τ =

MT∑i=1

iTp × P (i). (33)

Below we proceed by providing our performanceresults and validate our theoretical analysis by com-paring the analytical results to those obtained fromour simulations.

V. PERFORMANCE RESULTS

In this section, the performance of the CGBN-HARQ scheme is characterized, when both perfectand imperfect sensing are assumed. We quantifythe throughput, the average packet delay and theaverage end-to-end packet delay. More specifically,we quantify the impact of false-alarm probability(Pfa), mis-detection probability (Pmd), channel busyprobability (Pon), packet error probability (Pe), aswell as the number of packets N transmitted perTS. Note that in the perfect sensing environment of[20], both Pfa and Pmd are zero.

For our simulations, we configure the CGBN-HARQ transmission scheme in Matlab in a realisticimperfect sensing environment, where the transmit-ter seamlessly transmits N packets in each free TSand the receiver generates feedback flags for thereceived packet. The PU’s activity is based on auniformly distributed random process. To analysethe performance of the CGBN-HARQ transmissionscheme, we performed 50, 000 Monte Carlo simu-lations for all the parameters used in this study.Specifically, the parameters used in our simulationsand in our theoretical modelling as well as theirimpact on the performance are summarized in TableIII. Furthermore, the observation period commencesfrom the first TS and continues until the CR receiver

successfully receives all packets. In our simulations,the period of observation commences from sensingthe first TS and continues until the successful recep-tion of all the packets considered. Correspondingly,the throughput is evaluated as

R′S =Ns

Nt × (k +N)(packets per Tp), (34)

where Nt denotes the overall number of TSs re-quired for the error-free transmission of Ns packets.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

No

rmal

ized

Th

rou

gh

pu

t(P

acket

/T

p)

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Packet Error Probability (Pe)

Pmd = 0.1, Pfa = 0.1, N = 4, k = 1

Theory

Simulation

Perfect Sensing

Theory

Simulation

Imperfect Sensing

Pon = 0 : 0.1 : 0.4

Figure 12. Throughput of the CGBN-HARQ versus Pe forvarious values of Pon, when k = 1 and N = 4. The theoreticalresults were calculated from (25) and the simulation resultswere obtained from (34).

Fig 12 shows that for a given probability of Pon,the achievable throughput of CGBN-HARQ is at itsmaximum in both idealistic perfect and realisticimperfect sensing scenarios, when the channel usedby the CU is perfectly reliable, yielding Pe = 0.However, when the channel becomes less reliable,i.e. when Pe increases, the throughput reduces withPe due to the increase in the number of packetretransmissions. As shown in Fig 12, at a givenPe, the highest throughput is attained, when thePR channel is always free to use, correspondingto Pon = 0. By contrast, when Pon increases, thethroughput of the CGBN-HARQ system decreases,since less time is available for the CU to transmitsits data. Moreover, Fig 12 demonstrates the effectof unreliable sensing on the throughput, exhibitinga substantial drop due to the inaccurate sensing ata given Pe and/or Pon. Finally, as demonstrated inFig 12, the results obtained from our analysis arevalidated by the simulation results, which confirmsthe accuracy of our analytical modelling.

17

0.0

0.1

0.2

0.3

0.4

0.5

0.6N

orm

aliz

edT

hro

ug

hpu

t(P

acket

/T

p)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Number of Packets in TS (N)

Pfa = 0.2, Pmd = 0.2, Pon = 0.2, k = 1

Pe = 0

Pe = 0.05

Pe = 0.1

Pe = 0.2

Pe = 0.3

Pe = 0.4

Lines = Analysis, Marker = Simulation

Figure 13. Investigating the optimal throughput of the CGBN-HARQ scheme based on the number of packet transmission (N)per TS, for various values of Pe in a perfect sensing environment.

Continuing with the throughput analysis now theoptimal value of N is investigated in Fig. 13 versusPe, Pfa, Pmd and Pon. It can be seen that the through-put always increases upon increasing the value of N ,provided that the channel is perfectly reliable, i.e.we have Pe = 0. When Pe increases, an increasedvalue of N may result in a reduced throughput. Thisis because for a large value of N the number ofretransmissions increases, once there is even a singlepacket in error. Furthermore, as seen in Fig. 13, for agiven packet error probability Pe ≥ 0, there exists anoptimum value for N , which results in the highestthroughput. The optimum value of N reduces, as thepacket error probability Pe increases.

Having characterized the attainable throughput,we now continue by quantifying the delay imposed.In our simulations, the average packet delay is com-puted as the total time duration required for theerror-free transmission of Ns packets, divided by Ns,which is formulated as

T ′DS =Nt(k +N)

Ns× Tp (seconds). (35)

The results presented in Figs. 14 and 15 are normal-ized by Tp.

Specifically, Fig. 14 shows the average packetdelay versus the packet error probability Pe,parametrized b the channel busy probability Pon. Asshown in Fig. 14, the minimum delay is observed,when the channel is ideal, i.e. when Pe = 0 and/orwhen the channel is always free for the CU to use,corresponding to Pon = 0. The average packet delayincreases upon increasing Pe and/or of Pon, as a

0

1

2

3

4

5

6

7

8

9

Aver

age

Pac

ket

Del

ayN

orm

aliz

edb

yT

p

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Packet Error Probability (Pe)

Pmd = 0.1, Pfa = 0.1, N = 4, k = 1

Theory

Simulation-based

Perfect Sensing

Theory

Simulation

Imperfect Sensing

Pon = 0 : 0.1 : 0.4

Figure 14. Average packet delay of the CGBN-HARQ systemversus packet error probability for various channel busy proba-bilities, when both perfect and imperfect sensing are considered.The theoretical results were calculated from (27) and the sim-ulation results were obtained by taking into account the totalnumber of TS used for transmission (35).

0

2

4

6

8

10

12

14

16

18

20

22

Aver

age

Pac

ket

Del

ayN

orm

aliz

edb

yT

p

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Packet Error Probability (Pe)

Pfa = 0.2, Pmd = 0.2, Pon = 0.2, k = 1

N = 1, 3, 5, 7, ... 15

Lines = Analysis, Marker = Simulation

Figure 15. Investigating the optimal average packet delay ofthe CGBN-HARQ system based on the number of packets (N)per TS versus Pe in a realistic imperfect sensing environment,when Pon = 0.2 and Ts = Tp.

result of the increased number of retransmissionsand/or the reduced transmission opportunities forthe CU. Furthermore, similar to the above discus-sions, the unreliable sensing also increases the aver-age packet delay by deeming a free TS to be busy. Bycontrast, activating packet transmissions in a busyTS also increases the average packet delay due tothe increase in the number of erroneous receptionsand retransmissions. Additionally, as shown in Fig.14 and also in Fig. 15, the results of our analytical

18

approach are validated by the simulation results.In Fig. 15, the average packet delay versus Pe re-

lationship is characterized in terms of the number ofpackets transmitted in a single TS, given Pe, Pfa, Pmd

and Pon. As shown in the figure, at Pe = 0, a highervalue of N gives a lower delay. However, when Pe

is sufficiently large, a higher value of N results ina higher delay. For a given value of N , the averagepacket delay increases, as the channel becomes lessreliable, i.e. when Pe increases. Again, as shown inFig. 15, the simulation results validate our analyticalresults.

In Fig. 16, the PMF of the end-to-end packet delayis quantified for N = 4, 7 and 9, when realisticunreliable sensing is considered. In our simulation-based study the end-to-end delay of a packet isquantified in terms of the time duration spanningfrom its first transmission to the instant, when it iscorrectly received. Specifically, let the vector ddd oflength Ns store the end-to-end delay in terms ofTp of the Ns packets, where Ns is a large numberand d(j) denotes the end-to-end delay of the jthpacket. Hence, the PMF of the end-to-end packetdelay presented in Fig. 16 is evaluated as

PPP d(i) =

∑Ns

j=1 δ [d(j)− i]Ns

, 1 ≤ i ≤ max(ddd), (36)

where max(ddd) denotes the maximum delay of theNs packets. Note that in Fig. 16 the PMF associatedwith a delay beyond 120T ′ps is not shown, since thevalues are all close to zero.

10-7

2

5

10-6

2

5

10-5

2

5

10-4

2

5

10-3

2

5

10-2

2

5

10-1

2

5

Pro

bab

ilit

y

20 40 60 80 100 120Number of Tp’s

Pon = 0.2, Pe = 0.2, Pfa=0.2, Pmd=0.2, k = 1

N = 4

N = 7

N = 9

Lines = Analysis, Marker = Simulation

Figure 16. Probability distribution of end-to-end packet delayof the CGBN-HARQ systems in the case of imperfect sensing,when Pon = 0.2, Pe = 0.2, and N = 4, 7 and 9. The theoreticalresults were calculated from (32) and the simulation resultswere obtained by taking into account (36).

We observe from Fig. 16 that for the given param-eter values and N = 4, 47.7% of the packets are suc-cessfully received after their first transmission. Forthe packets that are successfully received after theirsecond transmissions, there are two cases. In thefirst case, if the TS following the first transmission isfree, on average 3.5%, 10% and 8.3% of the packetsare successfully received with the delays of 4Tp, 5Tpand 6Tp, respectively. In the second case, if the TSduring the first transmission is busy, then approxi-mately 1.5%, 2.3%, 4.3% and 4.2% of the packets aresuccessfully received with the delay of 8Tp, 9Tp, 10Tpand 11Tp, respectively. Similarly, we can find theprobabilities for the cases, when N = 7 and 9 areconsidered.

0

5

10

15

20

25

Aver

age

End-t

o-E

nd

Del

ayN

orm

aliz

edby

Tp

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Packet Error Probability (Pe)

Pmd = 0.1, Pfa = 0.1, N = 4, k = 1

Theory

Simulation

Perfect Sensing

Theory

Simulation

Imperfect Sensing

Pon = 0 : 0.1 : 0.4

Figure 17. Average end-to-end packet delay versus Pe forvarious channel busy probabilities Pon, when both perfect andimperfect sensing are considered. The theoretical results werecalculated from (33) and the simulation results were obtainedby taking into account (37)

Having characterized the PMF of the end-to-endpacket delay, we will now characterize the average-end-to-end packet delay of the CGBN-HARQ scheme.Here, the average end-to-end packet delay is formu-lated as:

τ =

max(ddd)∑i=1

PPP d(i)× i(Tp). (37)

Fig. 17 depicts the average end-to-end packet delayfor N = 4 versus Pe and Pon, when both perfectand imperfect sensing are considered. Similar to thetrends observed in the analysis of the average packetdelay, the average end-to-end packet delay increaseswith Pe and/or Pon. However, when comparing Fig.

19

14 and 17, there are two main noticeable character-istics associated with the minimum delays and withthe delay-gradient. Firstly, in terms of the minimumdelays, observe in Fig. 17 at Pe = 0 that theminimum end-to-end packet delay is one Tp in thecase of perfect sensing, regardless of the channel’sbusy probability of Pon. In the case of imperfectsensing, the minimum delay increases to an averageof 1.6 Tp, when Pon = 0.4. By contrast, we observedfrom Fig. 14 that the average packet delay has ahigher than a one Tp interval minimum delay, whichfurther increases upon increasing in Pon, regardlessof using perfect or imperfect sensing. The reasonfor this observation is that the average packet delayincludes the busy time before a packet’s transmis-sion. By contrast, the end-to-end packet delay iscalculated for each of packet individually, spanningfrom the instant of its first transmission attempt untilits successful reception. Secondly, upon comparingFig. 17 to Fig. 14, we observe that the average end-to-end packet delay gradient is higher than that ofthe average packet delay as a function of Pe and/orPon. This is because in the CGBN-HARQ scheme thedelay of every transmitted packet is dependent onthe success of (N −1) packets transmitted a head ofit. Therefore, if any of the (N − 1) previous packetis corrupted, then all the respective packets have tobe retransmitted, regardless whether they are error-free. This phenomenon results in longer end-to-endpacket delays.

0

20

40

60

80

100

120

Aver

age

Pac

ket

Del

ayN

orm

aliz

edb

yT

p

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Packet Error Probability (Pe)

Pfa = 0.3, Pmd = 0.3, Pon = 0.2, k = 1

N = 2

N = 3

N = 4

N = 5

N = 6

N = 7

N = 8

N = 9

Lines = Theory, Markers = Simulation

Figure 18. The optimal average end-to-end packet delay ofthe CGBN-HARQ system based on the number of packets (N)versus Pe in the scenario of imperfect sensing.

Moreover, in Fig. 18 we depict the effect of in-creasing the number packets per TS on the average

end-to-end packet delay. Explicitly, for a given valueof Pe, the average end-to-end packet delay increases,as the value of N increases. Furthermore, at a givenPe, the gradient of the average end-to-end delaycurve increases, as the value of N becomes higher.

VI. CONCLUSIONS

In this paper, we investigated a CGBN-HARQtransmission scheme conceived for CR systems,when both reliable and unreliable sensing are con-sidered. Both the throughput and delay of CGBN-HARQ has been investigated both by analysis andsimulations. The CGBN-HARQ system was mod-elled using DTMC-based approach, based on whichclosed-form expressions have been derived both forthe throughput and for the delay. Both the averagepacket delay and the average end-to-end packet de-lay have been studied. Finally, our analytical resultshave been validated by our simulation results. Theperformance results of the CGBN-HARQ recorded foridealistic and realistic sensing environments werecompared. The results presented in the above figuresare summarized in Table III, which reveals that thereliability of spectrum sensing, the activity of the PUsand the reliability of the CR communication have asubstantial impact on the throughput and delay ofthe CGBN-HARQ transmission scheme. When the CRcommunication become less reliable or when the PUhas a high probability of exploiting the channel, thethroughput reduces significantly, which ultimatelyresults in an increased transmission delay. Addition-ally, the optimal value of the throughput and delayof CGBN-HARQ have been quantified in terms of thenumber of packets transmitted per TS. This quan-titative analysis reveals that when the propagationenvironment is time-variant, the number of packetstransmitted within a TS should be appropriatelyadapted for achieving the highest possible through-put and the shortest average transmission delay.

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(a) For Pon = 0

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Ateeq Ur Rehman received his BEng de-gree in Computer Science and InformationTechnology from the Islamic University ofTechnology (OIC) Dhaka, Bangladesh, in2009. He obtained his Ph.D degree in wire-less communications from the Universityof Southampton in December 2016. Cur-rently, he is Lecturer at Abdul Wali KhanUniversity, Mardan, Pakistan, where he is

associated with the department of Computer Science. His mainresearch interests are next generation wireless communicationsand cognitive radio networks, Cooperative communication, Re-source allocation, particularly cross layer approach and HybridARQ.

Lie-Liang Yang received his BEng degree incommunications engineering from Shang-hai TieDao University, Shanghai, China in1988, and his MEng and PhD degreesin communications and electronics fromNorthern (Beijing) Jiaotong University, Bei-jing, China in 1991 and 1997, respectively.From June 1997 to December 1997 he wasa visiting scientist of the Institute of Radio

Engineering and Electronics, Academy of Sciences of the CzechRepublic. Since December 1997, he has been with the Universityof Southampton, United Kingdom, where he is the professorof wireless communications in the School of Electronics andComputer Science. He has research interest in a wide range oftopics in wireless communications, wireless networks and signalprocessing for wireless communications, as well as molecularcommunications. He has published over 330 research papersin journals and conference proceedings, authored/co-authoredthree books and also published several book chapters. The de-tails about his publications can be found at http://wwwmobile.ecs.soton.ac.uk/lly/. He is a fellow of both the IEEE and theIET, and a distinguished lecturer of the IEEE. He served asan associate editor to the IEEE Trans. on Vehicular Technologyand Journal of Communications and Networks (JCN), and iscurrently an associate editor to the IEEE Access and the Securityand Communication Networks (SCN) Journal.

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Lajos Hanzo FREng, FIEEE, FIET, Fellow ofEURASIP, DSc received his degree in elec-tronics in 1976 and his doctorate in 1983.In 2009 he was awarded the honorarydoctorate “Doctor Honoris Causa” by theTechnical University of Budapest. Duringhis 35-year career in telecommunicationshe has held various research and academicposts in Hungary, Germany and the UK.

Since 1986 he has been with the School of Electronics andComputer Science, University of Southampton, UK, where heholds the chair in telecommunications. He has successfullysupervised in excess of 100 PhD students, co-authored 20John Wiley/IEEE Press books on mobile radio communicationstotalling in excess of 10 000 pages, published 1600+ researchentries at IEEE Xplore, acted both as TPC and General Chairof IEEE conferences, presented keynote lectures and has beenawarded a number of distinctions. Currently he is directing a 60-strong academic research team, working on a range of researchprojects in the field of wireless multimedia communicationssponsored by industry, the Engineering and Physical SciencesResearch Council (EPSRC) UK, the European IST Programmeand the Mobile Virtual Centre of Excellence (VCE), UK. He is anenthusiastic supporter of industrial and academic liaison andhe offers a range of industrial courses. He is also a Governorof the IEEE VTS. During 2008 - 2012 he was the Editor-in-Chief of the IEEE Press and a Chaired Professor also at TsinghuaUniversity, Beijing. His research is funded by the EuropeanResearch Council’s Senior Research Fellow Grant. For furtherinformation on research in progress and associated publicationsplease refer to http://www-mobile.ecs.soton.ac.uk.