Maximum Throughput of a Cooperative Energy Harvesting Cognitive Radio User
aa r X i v : . [ c s . I T ] J u l Maximum Throughput of a Cooperative EnergyHarvesting Cognitive Radio User
Ahmed El Shafie,
Member, IEEE,
Tamer Khattab,
Member, IEEE, and H. Vincent Poor,
Fellow, IEEE
Abstract —In this paper, we investigate the maximum through-put of a saturated rechargeable secondary user (SU) sharingthe spectrum with a primary user (PU). The SU harvestsenergy packets (tokens) from the environment with a certainharvesting rate. All transmitters are assumed to have databuffers to store the incoming data packets. In addition to itsown traffic buffer, the SU has a buffer for storing the admittedprimary packets for relaying; and a buffer for storing the energytokens harvested from the environment. We propose a newcooperative cognitive relaying protocol that allows the SU to relaya fraction of the undelivered primary packets. We consider aninterference channel model (or a multipacket reception (MPR)channel model), where concurrent transmissions can survivefrom interference with certain probability characterized by thecomplement of channel outages. The proposed protocol exploitsthe primary queue burstiness and receivers’ MPR capability.In addition, it efficiently expends the secondary energy tokensunder the objective of secondary throughput maximization. Ournumerical results show the benefits of cooperation, receivers’MPR capability, and secondary energy queue arrival rate on thesystem performance from a network layer standpoint.
Index Terms —Cognitive radio, relaying, protocol, cooperation,throughput analysis, queue stability.
I. I
NTRODUCTION S econdary utilization of a licensed primary band can ef-ficiently enhance the spectrum usage and improve itsscarcity. Secondary users (SUs) can use the spectrum undercertain quality of service requirements for the primary users(PUs). High performance wireless communication networksrelies, among other technologies, on cooperative communica-tions, where nodes cooperate to mitigate fading.In many practical situations and applications in wirelesssensor networks, the SU is a battery operated device. Thesecondary operation, which involves spectrum sensing andaccess, is accompanied by energy consumption. Consequently,an energy-aware (energy-efficient) SU must optimize its sens-ing and access decisions to efficiently invest the availableenergy. When the SU is capable of relaying, it should alsooptimize its decision on accepting other nodes packets forrelaying. This is because accepting a packet for relaying willrequire its retransmission and therefore consumes energy. Part of this paper has been accepted in Personal, Indoor and Mobile RadioCommunications (PIMRC), 2014 [1].A. El Shafie is with Wireless Intelligent Networks Center (WINC), NileUniversity, Giza, Egypt. He is also with Electrical Engineering, Qatar Uni-versity, Doha, Qatar (e-mail: ahmed.salahelshafi[email protected]).T. Khattab is with Electrical Engineering, Qatar University, Doha, Qatar(email: [email protected]).H. V. Poor is with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544 USA (email: [email protected]).This research work is supported by Qatar National Research Fund (QNRF)under grant number NPRP 09-1168-2-455.
Energy harvesting technology is an emerging technologyfor energy-constrained terminals which allows the transmit-ter to collect (harvest) energy from its environment. For acomprehensive overview of the different energy harvestingtechnologies, the reader is referred to [2] and the referencestherein.Data transmission by an energy harvester with a recharge-able battery has got a lot of attention recently [3]–[11]. In[3], the optimal online policy for controlling admissions intothe data buffer is derived using a dynamic programmingframework. In [4], energy management policies which stabilizethe data queue are proposed for single-user communicationand some delay-optimal properties are derived. In [5], theoptimality of a variant of the back-pressure algorithm usingenergy queues is shown.The authors of [6] considered a cognitive scenario wheretwo different priority nodes share a common channel. Thehigher priority user (PU) has a rechargeable battery, whereasthe lower priority user (SU) is plugged to a reliable power sup-ply and therefore has energy each time slot without limitations.In [7], the authors investigated a cognitive setting with onePU and one rechargeable SU. The SU randomly accesses andsenses the primary channel and can possibly leverage primaryfeedback. Receivers are capable of decoding under interferenceas they have multipacket reception (MPR) capabilities. Theauthors investigated the maximum secondary throughput understability and delay constraints on the primary queue. In [8],the SU randomly accesses the channel at the beginning of thetime slot to exploit the MPR capability of receivers. The SUaims at maximizing its throughput under stability and queueingdelay constraints on the primary queue. In [9], El Shafie etal. investigated the maximum stable throughput of an energyharvesting SU under stability of an energy harvesting primarytransmitter. The SU selects a sensing duration each timeslot from a predefined set such that its stable throughput ismaximized under the stability of the primary queue.Cooperative cognitive relaying has got extensive attentionrecently [10]–[14]. In [12], Sadek et al. proposed cognitiveprotocols for a multiple access system with a single relay thataids the transmitting nodes. The proposed cooperative proto-cols enable the relaying node to help a set of buffered transmit-ters operating in a time-division multiple access network whentheir queues are empty due to source burstiness. The secondarythroughput of the proposed protocol as well as the delayof symmetric nodes were investigated. The authors of [13]investigated a network composing of one primary transmitter-receiver pair and one secondary transmitter-receiver pair. Thecognitive radio transmitter aims at maximizing its throughputvia optimizing its transmit power such that the primary and the relaying queues are maintained stable.Integrating cooperative communications and energy harvest-ing technologies has been considered in several works such as[10] and [11]. In [10], the authors investigate the effects of net-work layer cooperation in a wireless three-node network withenergy harvesting nodes and bursty data traffic. The authorsderived the maximum stable throughput of the source as wellas the required transmitted power for both a non-cooperativeand an orthogonal decode-and-forward cooperative schemes.In [11], the authors study the impact of the energy queue on themaximum stable throughput of a cooperative energy harvestingSU that utilizes the spectrum whenever the PU’s queue isempty. The authors assume an energy packet consumptionin either data decoding or data transmission. Inner and outerbounds are derived for the secondary throughputIn this work, we investigate the maximum throughput foran energy harvesting SU in presence of a PU. In contrast to[11] and [10], we consider a generalized MPR channel modeland propose a new cooperative protocol which exploits theMPR capability of the receivers. In the proposed cooperativecognitive relaying protocol, the SU cooperatively relays a frac-tion of undelivered primary packets. The flow of the primarypackets through the SU’s relaying queue is controlled usingsome tunable parameters which depend on the channels qualityand other queues states. The proposed cooperative cognitiverelaying protocol allows the SU to transmit simultaneouslywith the PU at a fraction of the time slots to exploit theMPR capability of the receiving nodes. The proposed protocolis simple and doesn’t require continuous estimation of thechannel state information (CSI) at the transmitting terminals.The contributions of this paper can be summarized asfollows: • We propose a new cooperative cognitive relaying protocolwhich exploits primary queue burstiness and receiversMPR capability while efficiently managing the SU’senergy expenditure. • We derive the channel outages probabilities in case ofenergy harvesting nodes and the presence of delays intransmission with and without concurrent transmissions.Moreover, we shed light on some important issues relatedto access delays on channel outages with and withoutconcurrent transmissions. • We derive service and arrival mean service rates expres-sions. • We investigate the maximum throughput of the SU underthe stability of all other queues in the system. • We study the impact of the MPR capability of the re-ceivers and the secondary energy queue on the secondarythroughput. • To make the characterization of the secondary throughputfeasible, we consider three approximated systems: two ofthem are shown to be inner bounds on the performance ofthe original system, whereas the third is shown to be anouter bound on the performance of the original system.This paper is structured as follows: Next we describe thesystem model adopted in this paper. We explain the proposedcooperative cognitive relaying protocol and provide the analy-sis of the queues rates and the problem formulation in Section III. In Section IV, we provide some numerical results. Theconclusions are drawn in Section V.II. S
YSTEM M ODEL
We consider a simple configuration comprised of onerechargeable battery operated secondary transmitter ‘ s ’, onesecondary destination ‘ d s ’, one primary transmitter ‘ p ’ andone primary destination ‘ d p ’. The network model is shownin Fig. 1. The primary transmitter-receiver pair operates overslotted channels. Time is slotted and a slot is T secondsin length. Each transmitter has an infinite-length data buffer(queue) to store its own incoming fixed-length data packets,denoted by Q ℓ , ℓ ∈ { p , s } . In addition to its own traffic queue,the cognitive user has an infinite capacity buffer to store theenergy packets harvested from the environment; and an infinitecapacity relaying queue to store the accepted primary packetsfor relaying. Let Q r denote the secondary relaying queue and Q e denote the secondary energy queue with mean arrival rates ≤ λ r ≤ packets/slot and ≤ λ e ≤ energy packets/slot,respectively. The secondary data queue is assumed to besaturated (always backlogged). Arrivals at queues Q p and Q e are assumed to be Bernoulli random variables [15], [16]. Thearrivals at each queue are assumed to be independent andidentically distributed (i.i.d.). The Bernoulli model is simple,but it captures the random availability of ambient energysources. In the analysis of discrete-time queues, Bernoulliarrivals see time averages (BASTA) is an important feature,which is equivalent to the Poisson arrivals see time averages(PASTA) property in continuous-time systems [8]. Arrivals arealso independent from queue to queue. The mean arrival rateto the primary queue, Q p , is λ p ∈ [0 , packets/slot. All datapackets are of size B bits. We assume that one energy packetis needed for the transmission of one data packet. The energyqueue has energy packets each of e energy units. For similarassumptions of infinite size of data buffers and modeling thearrivals of data and energy queues as Bernoulli arrivals, thereader is referred to [6]–[10] and the references therein.The proposed cooperative cognitive relaying protocol andthe theoretical development in this work can be readily gen-eralized to networks with more than one PU and more thanone SU, where several PUs may choose one or more SUs orthe best SU for cooperation. All wireless links exhibit a stationary non-selective Rayleighblock fading. The instantaneous channel fading coefficient oflink j → k (link connecting nodes j and k ) remains constantduring a time slot T ∈ { , , , . . . } , but changes indepen-dently and identically from one slot to another according toa circularly symmetric complex Gaussian distribution withzero mean and variance σ jk . Received signals at node k arecorrupted by complex additive white Gaussian noise (AWGN)with zero mean and variance N k Watts. This is a reasonable approximation if the energy contained inside oneenergy packet is much less than the total capacity of the energy buffer (or thebattery storage capacity) [8], [10]. The considered network can be seen as part of a larger network withmultiple primary nodes assigned to orthogonal frequency bands or differenttime slots via employing frequency division multiple-access or time divisionmultiple-access, respectively. p Q r Q p r e e Q s Q ps p d s d Fig. 1. Primary and secondary queues and links. In the figure, the solid linesare the communication channels while the dashed lines are the interferencechannels.
Let ζ jk denote the fading gain of link j → k . We do notassume the availability of CSI at the transmitters. Since the PUtransmits from the beginning of the time slot over the wholeslot duration if its queue is nonempty, the spectral efficiencyof the primary terminal is R p = B / ( T W ) bits/sec/Hz, where W is the channel bandwidth. The cognitive radio user maytransmit either at the beginning of the time slot or after τ seconds from the beginning of the time slot. Hence, thesecondary transmission time is T ( i )s = T − iτ , where i = 0 ifthe SU transmits at t = 0 , and i = 1 if the SU transmits at t = τ . The spectral efficiency of the secondary transmission iseither R (0)s = B / ( T W ) bits/sec/Hz or R (1)s = B / (( T − τ ) W ) bits/sec/Hz for i = 0 and i = 1 , respectively. Note that the decision duration , τ , should be long enough to justify theperfect detection of the primary queue state. The PU transmitsdata with a fixed power P p Watts, whereas the SU transmitswith power P ( i )s = e /T i Watts, i ∈ { , } . The secondarytransmit power is a function of the time instant in which theSU starts data transmission within the time slot. Outage ofa link occurs when the instantaneous capacity of that link islower than the transmitted spectral efficiency rate [7], [8], [10].Assume that node j transmits a packet to node k and at thesame time node v transmits to its respective receiver. Due tothe broadcast nature of the wireless communication channel,the signal transmitted by node v arrives at node k and causesinterference with the signal transmitted by node j . Let usassume that node j starts transmission at t = iτ , whereas node v starts transmission at t = nτ , where i, n ∈ { , } . Underthis setting, the probability that a transmitted packet by node j being successfully received at node k is P ( v ) jk,in = 1 − P ( v ) jk,in (see Appendix A for the exact expression). If transmitter j sends its packet alone (without interference) to node k , andstarts transmission at t = iτ , the probability of that packetbeing successfully decoded at k is P jk,i . The physical layer We assume here that the PU and the SU dedicate a special channel withsmall bandwidth for sharing state information of the PU. Specifically, duringthe first τ seconds of the time slot, the PU cooperatively sends its own queuestate, i.e., empty or nonempty, to the SU each time slot over the dedicatedbandwidth. This can be done through one-bit signal sent from the PU to theSU. is explained with details in Appendix A.A fundamental performance measure of a communicationnetwork is the stability of its queues. Stability can be definedrigorously as follows. Denote by Q ( t ) the length of queue Q at the beginning of time slot T . Queue Q with meanarrival rate λ and mean service rate µ is said to be stableif lim κ →∞ lim T →∞ Pr { Q ( T ) < κ } = 1 [12], where Pr { . } denotes the probability of the event in the argument. Forstrictly stationary arrival and service processes, queue Q isstable if µ ≥ λ . In a multiqueue system, the system is stablewhen all queues are stable.III. P ROPOSED COOPERATIVE COGNITIVE RELAYINGPROTOCOL
In this section, we describe in details the proposed cooper-ative cognitive relaying protocol, denoted by S . The time slotstructure is shown in Fig. 2. At the beginning of the time slot,if the secondary energy queue is nonempty, the SU may decideto receive the primary packet with probability f or decide toaccess the channel using one of its queues with probability f . Accessing the channel at the beginning of the time slot ismotivated by the following facts: • First, it may be the case that using the whole time in datatransmission provides higher throughput than wasting τ seconds for channel sensing, specially at low primaryarrival rate as the PU will be inactive during most ofthe time slots. Moreover, the probability of being inoutage of a link decreases with the total time used indata transmission over that link. This fact is discussedand its formula is proved in Appendix A. • Second, the presence of MPR capability at the receivingnodes allows packets decoding under interference withnonzero probability, which can be exploited by the SU toboost its throughput. • Third, as will be explained in details later, due to the fixedenergy transmission property of the energy harvestingSU, secondary delays of channel access may increasethe interference at the primary destination due to theincreases of the secondary transmit power, which in turnreduces the probability of successful decoding of theprimary packets at the primary destination.Based on these observations, channel accessing at the begin-ning of the time slot may be useful for certain scenariosand under specific system and channel parameters. On thecontrary, if the SU decides to receive the primary packet in atime slot, it will take another action/decision after τ secondsfrom the beginning of the time slot. The decision duration τ is designed such that the information signal sent from thePU to the SU about the primary queue current state, emptyor nonempty, is received correctly with probability one atthe SU. This is important for designing an efficient accessprotocol on the basis of the actual state of the time slot, i.e.,busy/free. As mentioned earlier, nodes dedicate a small bandto cooperatively exchange information regarding the actualstate of the PU. The transmission of the state occurs over the Throughout this paper, φ =1 − φ . time interval [0 , τ ] , where τ is assumed to be the transmissiontime of the information and is chosen to result in a negligibledecoding errors of these information at the SU.We summarize the medium access control (MAC) as fol-lows: • The PU transmits the packet at the head of its queue. • During the time interval [0 , τ ] , the PU sends its queuestate (empty or nonempty) to the SU over the dedicatedbandwidth for information exchange. • If the SU has energy packets and decides to access thechannel at the beginning of the time slot, it ignores the in-formation sent from the PU and resumes its transmissiontill the end of the time slot. This happens with probability − f . • If at the beginning of the time slot the SU decides to re-ceive the primary packet, which happens with probability f , it adjusts its receiving end to the receiving mode andstarts to collect data from the primary transmission. • Based on the received state signal from the PU, the SUperfectly discerns the state of the PU. • If the PU’s queue is nonempty and the secondary energyqueue is nonempty, the SU decides whether to resume pri-mary packet reception, which occurs with probability ω ;or to access the channel concurrently with the PU usingone of its data queues, which occurs with probability ω .In the latter case, accessing the channel simultaneouslywith the PU is motivated by the presence of the MPRcapability at receivers. • If the PU’s queue is empty and the secondary energyqueue is nonempty, the SU accesses the channel withprobability using one of its data queues. • If at the beginning of the time slot the SU has no energypackets in its energy queue, it decides whether to receivethe primary packet, which occurs with probability α , ornot. Note that since there is no energy in the secondaryenergy queue, there is no need to take another decisionat t = τ seconds. This is because the SU is incapableof establishing any data transmission due to the lack ofenergy. In such cases, the probability of receiving theprimary packet is α , whereas the probability of remainingsilent till the end of the current time slot is α . • At the far end of the time slot, the SU decides, on thebasis of its ability to decode the primary packet andthe status of primary packet decoding at the primarydestination, whether to accept or reject the admission ofthe primary packet to the relaying queue. The acceptanceprobability of a primary packet is β , whereas the rejectionprobability is β = 1 − β .If the relaying queue is nonempty, the SU selects one ofits packets for transmission with probability Γ = 1 − Γ ; orselects one of the relaying packets with probability Γ . If therelaying queue is empty, the SU accesses the channel usingits own packets with probability . The selection probability Γ represents the relative importance of the primary relaying We assume very small energy needed for packets decoding, which isreasonable due to its small value relative to the energy per packet (or energyneeded for data transmission). (cid:2028) (cid:3)(cid:1872) (cid:3404) (cid:3) (cid:2028) (cid:3)(cid:1872) (cid:3404) (cid:3) (cid:1846) (cid:3)(cid:1872) (cid:3404) (cid:882) (cid:3) (cid:1846)
Fig. 2. Time slot structure. packets and is used for controlling the throughput of therelaying queue. Choosing
Γ = 1 gives full priority to therelaying packets over the secondary packets, while
Γ = 0 favors the secondary packets (i.e., no selection for the relayingpackets). By varying Γ between and , we can maximize thesecondary throughput under stability of the other queues.We would like to emphasize here the importance of havingdifferent parameters associated with the different state ofthe queues in the system. Having such parameters enhancethe system performance and help in achieving the optimalperformance of the network under investigation.It should be noted that the probability of outage of a certainlink depends on the time available for data transmission.Hence, the probability of outage when the SU transmits atthe beginning of the time slot is less than the probability ofoutage when it starts data transmission at t = τ . Althoughusing lower transmission time raises the secondary transmittedpower, e / ( T − τ ) , the channel outage raises as well [7], [9] (seeAppendix A for proof). We should note that the interferencecaused by the SU on the PU’s transmission increases with thedelay in secondary data transmission. This happens becausethe secondary transmit power raises as mentioned earlier. Thereader is referred to Appendix A for more details.At the far end of each time slot, a feedbackacknowledgement/negative-acknowledgement (ACK/NACK)signal is sent from the receiver to inform the respectivetransmitter about the decodability status of its packet. Thefeedback message is overheard by all nodes in the networkdue to the wireless channel broadcast nature. Decoding errorsof the feedback messages at the transmitters are negligible,which is reasonable for short length packets as low rate andstrong codes can be employed in the feedback channel [12],[15]. If a packet is received correctly at its destination, it isthen removed from the system.For the primary packets, if the primary destination candecode the transmitted packet, it sends back an ACK and thepacket leaves the system. If the SU can decode the packetand the packet is admitted (accepted) for relaying while theprimary destination cannot, the SU sends back an ACK andthe PU drops that packet. If the SU cannot decode the primarypacket; or if it can correctly decode the packet but decidesto reject it and the primary destination fails in decoding thepacket, the PU retransmits that packet at the following timeslot. We note that the feedback signals sent by the SU and theprimary destination are separated either in time or frequency. A. Queues Service and Arrival Processes
Let us first consider the packets of the primary queue, Q p . Apacket departs the primary queue in either one of the followingevents. If the link p → d p is not in outage; or if the link p → d p is in outage, the link p → s is not in outage, andthe SU decides to admit the packet to the relaying queue. Asuccessfully received packet by either the primary destinationor the SU will be dropped from the primary queue. The meanservice rate of the primary queue is then given by µ p = P pd p , Pr { Q e = 0 } + f Pr { Q e = 0 } ω +Pr { Q e = 0 } ( δ pd p , f + δ pd p , f ω ) ! + P pd p , P ps , ( α Pr { Q e = 0 } + f Pr { Q e = 0 } ω ) β (1)where δ pd p , and δ pd p , denote the reduction in P pd p , dueto concurrent transmission when the SU accesses the channelat t = 0 and t = τ , respectively. The definition and derivationof P jk,i and δ jk,in are provided in Appendix A. It should bepointed out here that without cooperation the maximum meanservice rate for the primary queue is P pd p , , whereas withcooperation the maximum achievable primary mean servicerate is P pd p , + P pd p , P ps , , which is attained when theSU sets β = α = f = ω = 1 . Thus, the maximum achievablethroughput of the PU is increased by P pd p , P ps , packets pertime slot.A packet from Q s is served if the secondary energy queueis nonempty, the SU decides to access the channel using Q s ,and the link s → d s is not in outage. The mean service rateof Q s is given by µ s = P sd s , (cid:16) f (cid:16) Pr { Q p = 0 , Q e = 0 } δ sd s , +Pr { Q p = 0 , Q e = 0 } (cid:17) + ˆ δ sd s f (cid:16) ω Pr { Q p = 0 , Q e = 0 } δ sd s , +Pr { Q p = 0 , Q e = 0 } (cid:17)(cid:17) × (cid:16) ΓPr { Q r = 0 } + Pr { Q r = 0 } (cid:17) , (2)where ˆ δ jk = P jk, P jk, is defined in Appendix A.Similarly, the mean service rate of Q r is given by µ r = P sd p , Γ (cid:16) f (cid:16) Pr { Q p = 0 ,Q e = 0 } δ sd p , +Pr { Q p = 0 ,Q e = 0 } (cid:17) + ˆ δ sd p f (cid:16) ω Pr { Q p = 0 , Q e = 0 } δ sd p , +Pr { Q p = 0 , Q e = 0 } (cid:17)(cid:17) . (3)The mean arrival rate of the relaying queue is obtaineddirectly from (1). That is, λ r = P pd p , P ps , α Pr { Q e = 0 } + f Pr { Q e = 0 } ω ! β Pr { Q p = 0 } , (4)where Pr { Q p = 0 } in (4) means that the arrival of a primary packet at Q r occurs when the primary queue is nonempty.An energy packet is consumed from the secondary energyqueue in a time slot if the SU decides to transmit a data packetfrom one of its data queues. The mean service rate of Q e isthen given by µ e = f +Pr { Q p = 0 } f ω + f Pr { Q p = 0 } = 1 − Pr { Q p = 0 } f ω. (5)In (5), f means that the SU accesses the channel at t = 0 ; Pr { Q p = 0 } f ω means that the SU decides to access thechannel at t = τ seconds, which occurs with probability ω when { Q p = 0 } ; and f Pr { Q p = 0 } means that the SU decidesto access the channel after τ seconds with probability onewhen { Q p = 0 } .Relaying the primary packets by the SU may seem to wastethe time slots that could be used otherwise for its own packets.However, it turns out that the SU is indeed gaining sinceopportunistic relaying of primary packets results in emptying(servicing) the primary queue faster as the service process ofthe primary queue increases; in return, more network resourcescan be utilized for delivering the secondary packets. As aresult, all users simultaneously achieve performance gains. B. Approximated Systems
The service processes of the primary data queue and thesecondary energy queue are coupled, i.e., interacting queues.This means that the departure of a packet at any of themdepends on the state of the other. Hence, we cannot analyze thesystem performance or compute the service process of eachqueue directly. For this reason, we study three approximatedsystems. Two of them provide inner bounds and the thirdprovides an outer bound on the actual performance.In the first approximated system, we assume that the PUtransmits dummy packets when its queue is empty. Thesepackets may interfere with the SU in case of concurrenttransmissions, but do not contribute on the throughput of thePU. The essence of such assumption is to cause a constantinterference with the SU to decouple the queue interactionand to render the computation of nodes’ service rates possible.Under such assumption, the probability of the primary queuebeing empty is set to zero; that is, Pr { Q p = 0 } = 0 and Pr { Q p = 0 } = 1 . Since the PU is always backlogged (has atleast one packet at its queue in each time slot), the probabilityof the SU finds a free time slot is zero. Thus, all time slots thatthe SU decides to access in are occupied by the PU. Hence, theservice rates of the secondary queues, Q s and Q r , are reducedrelative to the original system in which the PU’s queue may beempty in some time slots and the SU can access the channelalone. Accordingly, this system is an inner bound for theoriginal system.In the second approximated system, we assume an energypacket dissipation in each time slot, which implies that µ e = This is actually the stochastic dominance approach extensively investigatedin the literature, see for example [7], [8], [10], [12], [17], [18]. Accessing the channel alone (without interference) provides a successfulpacket decoding at the relevant receiver higher than the case of concurrenttransmission as is obvious. The reader is referred to Appendix A for proofsand further details. energy packet per time slot. Under such assumption, theprobability of the energy queue being empty is significantlyincreased relative to the original system. Consequently, thesecondary packets get service less frequently. Furthermore, therelaying packets get service in a lower rate, hence the event ofprimary queue being empty decreases as the SU may decreasethe acceptance ratio of the relaying packets to maintain itsrelaying queue stability. Thus, the possibility of having a freetime slot or an interference-free time slot for the SU is reducedas well. Accordingly, this system is an inner bound on theoriginal system.In the third approximated system, we assume that thedeparture of the energy queue is almost zero, or equivalently,the probability of having an energy packet stored in thesecondary energy queue in any time slot is one. This systemis an outer bound on the original system as the SU willalways be able to access the channel for transmitting its ownpackets or retransmitting the relayed primary packets eachtime slot, if there is a chance for the SU to access the channel.Hence, all service rates of the data queues will be increasedsimultaneously.
1) First approximated system, Inner bound:
In this case,denoted by S , the PU is always backlogged. If the SU decidesnot to access the channel at the beginning of the time slot, itwill not access later at t = τ . This is because the PU is alwaysactive and wasting τ seconds for knowing the activity state ofthe PU will not lead to any gains in terms of secondary queuesthroughput. Therefore, the optimal ω is ω ∗ = 1 . Moreover,the decision on accessing the channel or receiving of theprimary packet is taken at the early beginning of the timeslot, specifically at t = 0 . If the secondary energy queue isnonempty, the SU decides to access the channel by one of itsqueue with probability f or decides to receive the possibleprimary transmission with probability f . If the secondaryenergy queue is empty, the SU cannot transmit data and itsdecision becomes whether to receive of the possible primarytransmission with probability α or remain idle with probability − α . At the end of the time slot, the SU decides whether toadmit the primary packet or to reject it, as explained earlier.Under the first approximated system, the mean service rate ofthe energy queue is given by µ e = 1 − f. (6)Using the results provided in Appendix B (setting µ = µ e =1 − f ), the probability of the energy queue being empty is givenby ν ◦ = 1 − λ e − f . (7)Based on this, the relaying queue departure and arrival meanrates are, respectively, given by µ r = P sd p , Γ f ν ◦ δ sd p , , (8) λ r = P pd p , P ps , αν ◦ + f ν ◦ ! β. (9) This is actually the highest probability for a queue to be empty becausethe service rate is packets/slot. The probability of the relaying queue being nonempty is givenby Pr { Q r = 0 } = π r = λ r µ r . (10)The mean service rate of Q s becomes µ s = P sd s , f ν ◦ δ sd s , (cid:16) Γ π r + π r (cid:17) . (11)The primary queue mean service rate is given by µ p = P pd p , (cid:16) (1 − ν ◦ f )+ δ pd p , f ν ◦ (cid:17) + P pd p , P ps , ( αν ◦ + f ν ◦ ) β. (12)We note that the queues are not interacting anymore. Hence,we can apply Loynes theorem to check the stability of thequeues and obtain the maximum stable throughput basedon the first approximated system via solving the followingconstrained optimization problem. max . β,f,α, Γ µ s , s . t . λ r ≤ µ r , λ p ≤ µ p , (13)where µ r , λ r , µ s and µ p are in (8), (9), (11) and (12),respectively.For a given f and β , we can get a closed-form expressionsfor Γ and α , then we solve a family of convex optimizationproblems parameterized by β and f . Specifically, the optimalsolutions of Γ and α are a set of points which satisfies thestability constraint of the primary and relaying queues stability,respectively. Using (12), the optimal α for a fixed f and β isgiven by α ∗ ≥ λ p − P pdp , (cid:16) (1 − ν ◦ f )+ δ pdp , fν ◦ (cid:17) P pdp , P ps , β − f ν ◦ ν ◦ . (14)Using the constraint on the stability of the relaying queue,the optimal Γ is given by Γ ∗ ≥ P pd p , P ps , ( α ∗ ν ◦ + f ν ◦ ) βP sd p , f ν ◦ δ sd p , , (15)where α ∗ is given in (14). The optimal β and f are obtainedvia grid search and are selected as the pair of parametersthat yields the highest objective function in (13). From (15),we note that the optimal selection probability of the relayingqueue for transmission, Γ ∗ , increases with increasing the ac-ceptance probability of the primary undelivered packets, β , andthe flow rate to the relaying queue P pd p , P ps , ( α ∗ ν ◦ + f ν ◦ ) β .This is because the SU should increase the selection of Q r for transmission to maintain the relaying queue stability. Inaddition, Γ ∗ increases with decreasing of P sd p , f ν ◦ δ sd p , .This is because P sd p , f ν ◦ δ sd p , determines the probabilityof certain transmitted packet from the relaying queue beingcorrectly received at the primary destination and therefore ifthis term is high, the SU will not need several transmission forthe same packet each time slot. Hence, the SU can reduce the The expression in (21) is obtained via solving the Markov chain modelingthe relaying queue when its arrival and service processes are decouple of theother queue and become computable. probability of choosing the relaying queue for transmissionat a time slot and rather it could use that time slot for thetransmission of its own packets.
2) Second approximated system, Inner bound:
In this ap-proximated system, denoted by S , we assume that an energypacket is consumed per time slot. That is, µ e = 1 energypackets per time slot. The probability of the energy queuebeing empty is given by ν ◦ = 1 − λ e µ e = 1 − λ e . (16)We can interpret the probability λ e as the fraction of time slotsthat can be used by the SU for data transmission. It should bepointed out here that under this approximation the buffer sizedoes not change the state probabilities. Hence, does not haveany impact on the queues’ rates. The Markov chain modelingthe energy queue in this case is composing of two states only:state where the energy queue has no packets, and state where the energy queue has only one packet. The probabilityof the energy queue having more than one packet, ν k , k ≥ ,is zero.The mean service rate of Q p is given by µ p = P pd p , (cid:16) ( λ e + f λ e ω )+ λ e ( δ pd p , f + δ pd p , f ω ) (cid:17) + P pd p , P ps , ( αλ e + f λ e ω ) β. (17)The probability of the primary queue being nonempty is givenby Pr { Q p = 0 } = π p = λ p µ p . (18)The relaying queue mean service and arrival rates are, respec-tively, given by µ r = λ e P sd p , Γ (cid:16) f (cid:16) π p δ sd p , + π p (cid:17) + f ˆ δ sd p (cid:16) δ sd p , ωπ p + π p (cid:17)(cid:17) , (19) λ r = P pd p , P ps , αλ e + f λ e ω ! βπ p . (20)The probability of the relaying queue being nonempty isgiven by Pr { Q r = 0 } = π r = λ r µ r . (21)The mean service rate of Q s is then given by µ s = P sd s , λ e (cid:16) f (cid:16) π p δ sd s , + π p (cid:17) + ˆ δ sd s f (cid:16) ωπ p δ sd s , + π p (cid:17)(cid:17) × (cid:16) Γ π r + π r (cid:17) . (22)Since the queues are decoupled in the second approximatedsystem, the maximum secondary throughput is given by solv- The expression in (21) is obtained via solving the Markov chain modelingthe relaying queue when its arrival and service processes are decouple of theother queue and become computable. ing the following problem. max . β,f,α,ω, Γ µ s , s . t . λ r ≤ µ r , λ p ≤ µ p , (23)where µ p , µ r , λ r and µ s are in (17), (19), (20), and (22),respectively.We conjecture that the throughput region of the secondapproximated system contains that of the first approximatedsystem. This is because, in contrast to the second approxi-mated system where there can be free time slots, in the firstapproximated system, the PU is always backlogged; hence,it always interferes with the SU and the probability of afree time slot for the SU is zero. Since the probability ofsuccess transmission under interference is low, the servicerates of the SU’s queues are decreased significantly underthe first approximated system. Based on that, the secondapproximated system always provides better performance thanthe first approximated system.
3) Third approximated system, Outer bound:
In this case,denoted by S , we consider a backlogged energy queue.This means that there exists at least one energy packet eachtime slot in Q e . This case can happen when λ e = 1 energypackets/slot regardless of the value of µ e . In this case, theprobability of the energy queue being nonempty approachesthe unity. The service and arrival rates are obtained directlyfrom (1), (2), (3), (4) and (5) with Pr { Q e = 0 } = 1 . Themean service and arrival rates of the queues are then given by µ p = P pd p , (cid:16) f ω +( δ pd p , f + δ pd p , f ω ) (cid:17) + P pd p , P ps , f ωβ, (24) µ r = P sd p , Γ (cid:16) f (cid:16) π p δ sd p , + π p (cid:17) + f ˆ δ sd p (cid:16) ωπ p δ sd p , + π p (cid:17)(cid:17) , (25) λ r = P pd p , P ps , f ωβπ p , (26)where π p in (25) and (26) follows (18) with µ p in (24), and µ s = P sd s , (cid:16) f (cid:16) π p δ sd s , + π p (cid:17) + f ˆ δ sd s (cid:16) ωπ p δ sd s , + π p (cid:17)(cid:17) × (cid:16) Γ π r + π r (cid:17) , (27)where π r follows (21) with µ r and λ r in (25) and (26),respectively.The outer bound can be computed by solving the followingproblem: max . β,f,ω, Γ µ s , s . t . λ r ≤ µ r , λ p ≤ µ p , (28)where µ p , µ r , λ r and µ s are in (24), (25), (26), and (27),respectively.The optimization problems are solved numerically at the SUfor a given channels and system parameters. Specifically, fora given parameters, the SU solves the optimization problemand use the optimal parameters for the system’s operation. C. Some Important Remarks
Following are some important remarks.
1) First Remark:
Using the results in Appendix A, thecomplement of outage probability of link p → d p when theSU starts transmission at the beginning of the time slot isgiven by P (c)pd p , = 11+ (cid:16) B WT − (cid:17) γ sdp , σ sdp γ jk σ jk exp (cid:16) − B WT − γ pd p , σ pd p (cid:17) , (29)while the probability of that link being not in outage when theSU starts transmission at t = τ is given by P (c)pd p , = 11+ (cid:16) B WT − (cid:17) γ sdp , σ sdp γ jk σ jk exp (cid:16) − B WT − γ pd p , σ pd p (cid:17) . (30)The ratio of (29) to (30) is given by ρ = P (c)pd p , P (c)pd p , = 1 + (cid:16) B WT − (cid:17) γ sdp , σ sdp γ pdp , σ pdp (cid:16) B WT − (cid:17) γ sdp , σ sdp γ pdp , σ pdp = 1 + a a − τ/T . We note that γ sd p , = γ sd p , / (1 − τ /T ) and a = (cid:16) B WT − (cid:17) γ sdp , σ sdp γ pdp , σ pdp . If a ≫ , the reduction of the primarypacket correct reception probability due to secondary accessdelay (when the SU accesses the channel at t = τ ) is ρ ≈ − τ /T . Therefore, if the secondary decides to access after τ seconds of primary packet reception based on the primaryactivity, the probability of the primary packet decoding reducesby a factor − τ /T relative to the case when the SU accessesthe channel at the beginning of the slot. The reduction of theprimary packet correct reception is a linear function of τ . Ifthe decision time, τ , is high, the primary packet decoding willbe reduced significantly.Assume that the primary transmits with a very low power.This makes a much greater than . Thus, we can approximatethe reduction, due to secondary access delay, of the probabilityof the primary channel not being in outage by ρ ≈ − τ /T .At the same time, since the primary transmit power is low,the required τ for perfect primary detection is high. Thismeans that the reduction of the primary packet decoding at theprimary destination due to concurrent transmissions is signif-icantly high. In this case, the secondary access probability at t = 0 is definitely higher than the access probability at t = τ when the PU is detected to be active and the SU decides toaccesses the channel. Moreover, it may be better for the SU toaccess the channel at t = 0 to use the whole slot time in datatransmission; and at t = τ if the PU is declared to be inactive,if the PU is declared to be active, it may be better to resumereceiving the primary packet because concurrent transmissionwould be harmful for the PU as explained earlier.
2) Second Remark:
Assume that the current primary arrivalrate is λ p = λ ⋆ p . Increasing the primary arrival rate to λ ⋆ p +∆ λ p , ∆ λ p ≥ , increases the probability of the primary queuebeing nonempty. This is because the probability of having anarrival at a certain time slot is increased. Consequently, thenumber of empty time slots that the SU can detect or accessalone decreases as well. In addition, the probability of relayingqueue selection, Γ , must be increased to maintain the stabilityof the relaying queue as the arrival rate of the relaying queue isincreased due to the increasing of λ p . These two observations lead to the fact that the achievable secondary rate is increasedrelative to the case of λ p = λ ⋆ p . This means that the secondaryservice rate, µ s , is a non-increasing function of the primaryarrival rate λ p .
3) Third Remark:
From the expressions of the servicerates of the queues, the service processes are functions ofchannel outages probabilities. Based on the formulas of thechannel outage in Appendix A, the outage probability ofa certain link is a decreasing function of R p = B / ( T W ) .Therefore, increasing the targeted primary spectral efficiencyrate, R p , decreases all queues service rates. This leads to areduction in the maximum achievable secondary throughput, µ s . This means that the secondary service rate, µ s , is a non-increasing function of the primary targeted spectral efficiencyrate R p = B / ( T W ) .The following proposition summarizes the main observa-tions in the second and the third remarks. Proposition 1:
For a given channel and system parameters,let µ ∗ s ( λ p , R p ) be the maximum secondary throughput at thepair ( λ p , R p ) . The optimal secondary throughput satisfies thefollowing properties: • µ ∗ s ( λ p , R p ) ≥ µ ∗ s ( λ p + ∆ λ p , R p ) , ∆ λ p ≥ . • µ ∗ s ( λ p , R p ) ≥ µ ∗ s ( λ p , R p + ∆ R p ) , ∆ R p ≥ .IV. N UMERICAL R ESULTS AND C ONCLUSIONS
In this section, we provide some numerical results forthe optimization problems presented in this paper. We definehere the conventional scheme, denoted by S c , where the SUsenses the channel for τ seconds and if the primary dataqueue and the secondary energy queue are simultaneouslyempty and nonempty, respectively, the SU accesses the channelwith probability using one of its queues probabilisticallyif the relaying queue is nonempty. In addition, if the PU istransmitting a packet to its destination, the SU accepts withprobability one to relay and admit the transmitted packet ifthe primary destination fails in decoding that packet. Thesecondary throughput of the conventional system is obviouslya subset of the proposed cooperative system, S , and can beobtained from S via setting β = 1 , α = 1 , f = 1 and ω = 0 . Theother parameters are optimized over their domains to achievethe maximum secondary throughput.Fig. 4 represents the maximum secondary throughput of theapproximated systems of system S . The figures are gener-ated using the following common parameters: P sd p , = 0 . , δ sd p , = 0 . , P sd s , = 0 . , P ps , = 0 . , δ sd s , = 0 . , P pd p , = 0 , P (s)pd p , = P (s)pd p , = 0 , ˆ δ sd p = 0 . , ˆ δ sd s = 0 . , δ sd p , = 0 . , δ sd s , = 0 . . The outer bound which representsthe case of backlogged energy queue is close to the innerbound.Figs. 3 and 4 represent the maximum secondary throughputof the approximated systems of system S . The figures aregenerated using the following common parameters: P sd p , . , δ sd p , = 0 . , P sd s , = 0 . , P ps , = 0 . , δ sd s , = 0 . , K = 60 , P pd p , = 0 , P (s)pd p , = P (s)pd p , = 0 , ˆ δ sd p = 0 . , ˆ δ sd s = 0 . , δ sd p , = 0 . , δ sd s , = 0 . . In Fig. 3,the maximum secondary throughput under the approximated systems is plotted versus λ p . This figure is plotted with λ e = 0 . energy packets per time slot. The figure shows thatthe second approximated system provides throughput higherthan the first approximated system, hence the union, whichrepresents an inner bound on the actual performance of system S , is the second approximated system. The outer bound, whichrepresents the case of backlogged energy queue, is close to theinner bound. The gap between the two bounds shrinks as λ e increases.Fig. 4 reveals two important observations. First, the figurereveals the impact of the arrival rate of the secondary energyqueue on the system’s inner bound. Precisely, as the energyarrival rate increases, the inner and the outer bounds becomeclose to each other and they overlap for all λ p when λ e = 1 energy packets/slot. Second, the figure reveals that the innerbound of the proposed system can outperform the outer boundof the conventional cooperation protocol with reliable energysource plugged to the SU. Note that system S c is plotted with λ e = 1 energy packets per time slot (outer bound on S c ).We note that without cooperation the primary packets outageprobability is − P pd p , = 1 which implies that the probabilityof a primary packet being served in a given arbitrary time slotis zero. Hence, the primary queue is always backlogged andwill never be empty. On the other hand, with cooperation themaximum feasible primary arrival rate is . packets per timeslot.We note that for Figs. 3 and 4, without cooperation theprimary packets outage probability is − P pd p , = 1 whichimplies that the probability of a primary packet being servedat an arbitrary time slot is zero. Hence, the primary queue isalways backlogged and will never be emptied. On the otherhand, with cooperation the maximum feasible primary arrivalrate is . packets per time slot.The impact of MPR capability is shown in Fig. 5. The figurereveals the gains of the MPR capability on achieving higherthroughput for both users. The parameters are chosen to be: λ e = 0 . , P sd p , = 0 . , P sd s , = 0 . , P ps , = 0 . , P pd p , =0 . , ˆ δ sd p = 0 . , ˆ δ sd s = 0 . , and δ pd p , = δ sd s , = δ sd p , = δ sd s , = δ sd p , = X , which represents the MPR strength. Atstrong MPR, we can achieve orthogonal channels for terminalsover most λ p range. The plot also shows that the inner andthe outer bounds coincide for high λ p . This happens becausethe energy queue is backlogged under the used parameters.From the figures, it is noted that cooperation boosts bothprimary and secondary throughput. Furthermore, the energyarrival rate increases the probabilities of the secondary packetsand the relayed primary packets being served which, in turn,boost both primary and secondary throughput. The figuresalso show that the increasing of λ p decreases the maximumachievable secondary throughput.V. C ONCLUSION
In this paper, we have proposed a new cooperative cognitiverelaying protocol, where the SU relays some of the undeliveredprimary packets. We have considered a generalized MPR chan-nel model, and investigated the impact of the receivers’ MPRcapability on the users throughput. We also have investigated λ p [packets/slot] µ s [ p acke t s / s l o t] S S S Fig. 3. The maximum secondary throughput of the approximated systemsfor each λ p . λ p [packets/slot] µ s [ p acke t s / s l o t] S S S c (outer bound) λ e = 0 . λ e = 0 . S λ e = 0 . λ e = 0 . λ e = 0 . Fig. 4. The maximum secondary throughput of the conventional cooperativeprotocol and the second and third approximated systems for each λ p and fordifferent values of λ e . λ p [packets/slot] µ s [ p acke t s / s l o t] S S X = 0 . X = 1 X = 0,Collision wirelesschannel model Fig. 5. The maximum secondary throughput for each λ p and for differentvalues of MPR capability. the impact of the secondary energy queue on the systemperformance. We have provided two inner bounds and anouter bound on the secondary throughput, and showed thatthe bounds are coincide when the secondary energy queue isalways backlogged. The proposed protocol is designed suchthat the SU exploits the MPR capability and manages itsenergy packets to maximize its throughput under stability ofthe primary and the relaying queues.A possible extension of this work can be directed to spanthe case of having an SU equipped with multiple antennasand with the availability of CSI at the transmitting antennasto achieve the maximum rates for the queues.A PPENDIX
AWe derive here a generic expression for the outage proba-bility at the receiver of transmitter j (node k ) when there is aconcurrent transmission from the transmitter v . Assume thatnode j starts transmission at iτ and node v starts transmissionat nτ . Outage occurs when the spectral efficiency R ( i ) j = B W T ( i ) j exceeds the channel capacity P ( v ) jk,in = Pr (cid:26) R ( i ) j > log (cid:18) γ jk,i ζ jk γ vk,n ζ vk + 1 (cid:19) (cid:27) (31) where Pr {A} denotes the probability of the event A , γ jk,i = P ( i ) j / N k , P ( i ) j is the used transmit power by node j when itstarts transmission at t = iτ , γ vk,n = P ( n ) v / N k , and P ( n ) v is theused transmit power by node v when it starts transmission at t = nτ . The outage probability can be written as P ( v ) jk,in = Pr n γ jk,i ζ jk γ vk,n ζ vk + 1 < R ( i ) j − o (32) Since ζ jk and ζ vk are independent and exponentially dis-tributed (Rayleigh fading channel gains) with means σ jk and σ vk , respectively, we can use the probability density functionsof these two random variables to obtain the outage as P ( v ) jk,in = 1 −
11 + (cid:16) R ( i ) j − (cid:17) γ vk,n σ vk γ jk,i σ jk exp (cid:16) − R ( i ) j − γ jk,i σ jk (cid:17) (33)We note that from the outage probability (33), the numeratoris increasing function of R ( i ) j and the denominator is adecreasing function of R ( i ) j . Hence, the outage probability P ( v ) jk,in increases with R ( i ) j . The probability of correct packetreception P ( v ) jk,i = 1 − P ( v ) jk,i is thus given by P ( v ) jk,in = P jk,i (cid:16) B TW ( − iτT ) − (cid:17) γ vk,n σ vk γ jk,i σ jk = δ ( v ) jk,in P jk,i (34)where P jk,i = exp (cid:16) − R ( i ) j − γ jk,i σ jk (cid:17) is the probability of correctpacket reception when node j transmits alone (without inter-ference) and δ ( v ) jk,in ≤ is the reduction in the probabilityof correct packet reception P jk,i due to the presence ofinterference from node v . As is obvious, the probability ofcorrect packet reception is lowered in the case of interference. Based on (34), we note that P ( v ) jk,in P ( v ) jk,im = δ ( v ) jk,in δ ( v ) jk,im (35)Following are some important notes. First, note that if thePU’s queue is nonempty, the PU transmits its packet from thebeginning of the time slot (at t = 0 ) with a fixed transmitpower P p and data transmission time T p = T . Accordingly,the superscript ‘ i ’ in T ( i ) j which represents the instant that atransmitting node starts its data transmission in is removed incase of PU. In addition, the superscript ‘ ( v ) ’ is removed aswe have only one PU and one SU.Second, for the SU, the formula of probability of comple-ment outage of link s → k when the PU is active is givenby P (p)s k,i = exp (cid:16) − B TW ( − iτT ) − γ s k,i σ s k (cid:17) (cid:16) B TW ( − iτT ) − (cid:17) γ p k, σ p k γ s k,i σ s k (36)where n = 0 because the PU always transmits at t = 0 and γ s k,i = e / ( T (1 − iτ /T )) = γ s k, / (1 − iτ /T ) . The denom-inator of (36) is proportional to (cid:16) B TW ( − iτT ) − (cid:17) (1 − i τT ) ,which in turn monotonically decreasing with iτ . Using thefirst derivative with respect to iτ , the numerator of (36), P s k,i = exp (cid:0) − B TW ( − iτT ) − e T (1 − i τT ) σ s k (cid:1) , can be easily shown tobe decreasing with iτ as in [7], [9]. Since the numerator of(36) is monotonically decreasing with iτ and the denominatoris monotonically increasing with i , P (p)s k,i is monotonicallydecreasing with iτ . Therefore, the delay in the secondaryaccess causes reduction in the probabilities of the secondarypackets correct reception and the primary relayed packetscorrect reception at their destinations.Now, we compute the ratio P ( v ) jk, n P jk, . Using (34), we have ˆ δ ( v ) jk = P jk, P jk, = P ( v ) jk, n δ ( v ) jk, n P jk, (37)After some mathematical manipulations, the ratio P ( v ) jk, n P jk, isgiven by P ( v ) jk, n P jk, = δ ( v ) jk, n ˆ δ ( v ) jk (38)Note that throughout the paper, the superscript ‘ ( v ) ’ can beeliminated from symbols since we only have two nodes: onePU and one SU. That is, δ ( v ) jk,in = δ jk,in , ˆ δ ( v ) jk = ˆ δ jk and P ( v ) jk,in = P jk,in . A PPENDIX
BWhen the arrival and departure of the secondary energyqueue become decoupled from all other queues in the networkas in the approximated systems, we can construct and solve its e ! e ! e ! e ! e ! e ! e ! e Fig. 6. The Markov chain model for the secondary energy queue when itsservice rate is independent of the other queues and has a mean service rate ≤ µ ≤ . Markov chain. The Markov chain is shown in Fig. 6, where themean arrival rate is λ e and the mean service rate is µ . Solvingthe state balance equations of the Markov chain modeling thesecondary energy queue, it is straightforward to show that theprobability that the energy queue has ≤ ϑ ≤ ∞ packets, ν ϑ ,is ν ϑ = ν ◦ µ λ e µλ e µ ! ϑ = ν ◦ η ϑ µ , ϑ = 1 , , . . . , ∞ (39)where η = λ e µλ e µ . Since the sum of all states’ probabilities is theunity, P ∞ ϑ =0 ν ϑ = 1 . The probability of the secondary energyqueue being empty is obtained via solving the following linearequation: ν ◦ + ∞ X ϑ =1 ν ϑ = ν ◦ + ν ◦ ∞ X ϑ =1 µ η ϑ = 1 . (40)After some mathematical manipulations, ν ◦ is given by ν ◦ = 1 − λ e µ , (41)with λ e < µ . If λ e ≥ µ , the energy queue saturates, i.e.,becomes always backlogged. Thus, v ◦ = 0 , which boosts thesecondary rate. The probability of the primary energy queuebeing empty is − λ e /µ .If µ = 1 , η = 0 and the probability that the energy queuehaving more than one packet is zero. The states probabilitiesin such case are given by ν = 1 − λ e , ν = λ e , ν ϑ = 0 , ϑ = 2 , . . . , ∞ (42)R EFERENCES[1] A. El Shafie and T. Khattab, “Maximum throughput of a cooperativeenergy harvesting cognitive radio user,” Accepted in
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