Le et al. EURASIP Journal on Wireless Communications andNetworking(2015) 2015:49 DOI 10.1186/s13638-015-0289-2RESEARCH Open AccessDistributed back-pressure scheduling withopportunistic routing in cognitive radionetworksYuan Le1*, Xiuzhen Cheng1, Dechang Chen2, Nan Zhang1, Taieb Znati3, Mznah A Al-Rodhaan4and Abdullah Al-Dhelaan4AbstractA challenging problem in cognitive radio networks (CRNs) is to design a throughput efficient routing and schedulingalgorithm for end-to-end communications between secondary users (SUs) in a distributed manner. Due to theopportunistic nature of CRN routing, we consider a framework with randomized path selection in this paper.Motivated by the back-pressure scheduling policy, we adopt the differential queue backlog as the routing metric toachieve throughput efficiency. Moreover, we demonstrate the sufficient condition for our framework to achievethroughput optimality by analyzing the Lyapunov drift. We further propose a distributed medium access controlalgorithm that can approximately satisfy the required condition when the transmission attempt probability is low. Theperformance of our proposed scheme is evaluated in a simulation platform, and the evaluation results verify theeffectiveness of our distributed scheme.Keywords: Cognitive radio networks; Opportunistic routing; Back-pressure scheduling; Throughput optimal routing1 IntroductionCognitive radio networking has been extensively studiedinrecent years [1-6]. Thecognitiveradio(CR) tech-nique allows unlicensed users (secondary users (SUs)) toopportunistically access the licensed spectrum withoutinterfering with licensed users (primary users (PUs)) toexploit the under-utilized portion of the spectrum [7-9].The coexistence of PUs and SUs makes it infeasible todirectly adopt the routing algorithms designed for generalmulti-hop wireless networks, as the links can be highlydynamic due to the existence of PUs. In the worst case,it may not be possible to deliver a packet along a cer-tain path without interfering with any PU. With such adynamic environment, designing a reliable routing pro-tocol foraCRnetwork(CRN)becomesachallengingproblem.There exist many routing algorithms that simultane-ously consider spectrum sensing and management. For*Correspondence: ppzt@gwmail.gwu.edu1Department of Computer Science, The George Washington University,Washington, DC, USAFull list of author information is available at the end of the articleinstance, a spectrum-aware routing algorithm was devel-oped in [10] to integrate spectrum discovery with routeestablishment and then assign channels to links to min-imize interference. But because the network topology ina CRN varies over time due to the dynamic nature ofPU activities, many existing routing algorithms are ren-deredinapplicableastheircomputedpathsusuallydonot adapt to the network dynamism. Thus whenever thenetwork topology changes, routes may need to be recal-culated, leading to a high computational cost. To addressthis challenge, opportunistic routing (OR) is introducedto CRN, in which SUs select only the next hop relay fortheir transmissions based on the routing metric computedfrom the current link states. For instance, a delay efficientopportunistic routing algorithm is proposed in [11].On the other hand, the back-pressure (BP) frameworkproposedbyTassiulas et al. [12] introduces through-put optimal routingandschedulingtosupport multi-pleflowsinwirelessadhoc networks. Observing thatOR provides an opportunity to overcome the challengescaused by network dynamics (OR selects the next relayfor routing based on the current link quality) and thatBP scheduling has the potential to guarantee throughput 2015 Le et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly credited.Le et al. EURASIP Journal on Wireless Communications and Networking(2015) 2015:49Page 2 of 14optimality(BPselectstheflowwiththehighest backpressure), wedecidetocombinetheconceptsofbackpressure and opportunistic routing, aiming at achievingthroughput optimal routing in dynamic multi-hop cogni-tive radio networks. Furthermore, to avoid the centralizedlink controller in BP, we adopt CSMA/CA for collisionavoidance among all selected flows. To the best of ourknowledge,we are the first to combine BP and OR totake advantage of the benefits brought by OR and BP ina distributed fashion. Because the general back-pressureframework consists of a max-weight optimization prob-lem,which is NP complete,we have to seek a greedyalgorithm to implement our framework of distributed BPscheduling with OR routing in CRNs for a suboptimalsolution.Inthispaper, weadopt thequeuemodel employedby the back-pressure algorithm in [12] to combine ORwith BP and employ the differential queue backlog as ourrouting metric.We further analyze the Lyapunov driftof our framework and derive the sufficient condition toachieve throughput optimality. Leveraging this condition,we propose a distributed medium access control (MAC)algorithm that stabilizes the network when the traffic loadis light.The core idea of our algorithm is to tune thecontention window size (CW) corresponding to the max-imum queue backlog to achieve the required transmissionprobability. We evaluate our protocol in a simulation plat-form, and our results indicate that the proposed protocolcanachieveboththroughputoptimalityandreliabilityunder a light traffic load.The rest of the paper is organized as follows. Section 2discusses the most related work. The distributed back-pressureschedulingwithopportunisticroutingframe-work, together with its throughput optimality analysis, arepresented in Section 3. We propose a distributed MACalgorithm that can stabilize the network when the trans-mission probability is low in Section 4. The evaluationsettings and simulation results are reported in Section 5.We conclude this paper with a discussion on our futureresearch in Section 6.2 Related workThere exist many studies on the routing protocoldesign for cognitive radio networks [10,13-17]. A high-throughput spectrum-aware routing protocol was inves-tigatedin[10]. In[13], theauthorsdevelopedatree-based routing algorithm that can achieve low end-to-enddelay.These protocols all use a predefined single pathfor all transmissions. Sincetheconnectivityof alinkmay be highly dynamic in a cognitive radio network, theperformance can be significantly decreased if a cruciallink is frequently unavailable. To address this problem,a multi-path-based routing algorithm was proposed in[14]. Toestablishamorereliableend-to-endpath, anopportunistic routing framework was adopted in [15]. Toenhance throughput and obtain a global-path stable rout-ing path, a network formation game was introduced tomodel the process of multi-hop relay selection in a coop-erative CRN [16]. Mobility-assisted routing to deal withthe intermittently connected characteristic of mobile adhoc CRNs was investigated in [17].Opportunisticroutinghas alsobeeninvestigatedinwireless mesh networks (WMNs) [11,18-23]. The widelystudied ExOR routing protocol was proposed by Biswaset al. in [18]. ExOR selects the next hop relay node afterthe transmission is done. It adopts sequential acknowl-edgements to exchange priority information among allthe receivers when a transmission occurs and selects thereceiver with the highest priority as the next hop relay toforward the packet towards its destination. In contrast,Chachulski et al. developed a MAC-independent oppor-tunisticroutingschemenamedMOREin[19], whichcombinesroutingwithnetworkcodingtoavoidnodecoordination. A contention-aware delay analysis of oppor-tunistic routing was presented in [20], which provides amathematical model about the contention to compute theexpected delay.The inherent properties of informationdissemination in opportunistic networks was studied in[21], and [22] proposes two algorithms for intermittentlyconnected mobile P2P networks, which exploit the spa-tial locality, spatial regularity, and activity heterogeneity ofhuman mobility to select relays.The original back-pressure algorithmproposed byTassiulas et al. in [12] has become an appealing researcharea due to its guarantee of throughput optimality. How-ever, it involves a max-weight optimization problem, mak-ing the algorithm computationally inefficient. To addressthis challenge, many suboptimal solutions have beendeveloped, which can be generally categorized into twoclasses. The first class [24,25] requires a centralized con-troller tomaketheschedulingdecisions andfocuseson developing scheduling algorithms whose stability canbe guaranteed in a fraction of the capacity region. Theimpact of an imperfect scheduling policy on the capac-ity region was demonstrated by Lin et al. in [24]. Dimakiset al. [25] indicated that the full capacity region can beachieved by a greedy scheduling policy for certain net-works. Thesecondclassalgorithms[26-28]studydis-tributed scheduling policies. In [26], the authors consid-eredadatagramnetworkanddevelopedadistributedscheduling policy using only local information. Neely et al.[27] proposed a distributed approach using randomizedtransmission selection. In [28], a distributed greedy proto-col was presented to achieve stability in a reduced capac-ityregion. Alearning-baseddistributedback-pressurescheduling algorithm requiring no information exchangewasproposedin[29]; butitrequiresarelativelylongconvergence time.Le et al. EURASIP Journal on Wireless Communications and Networking(2015) 2015:49Page 3 of 14Inadditiontofindingsuboptimal solutions for theoriginal max-weight optimization problem,utility opti-mization based protocol design using the back-pressureframeworkhasalsobeingextensivelystudied. In[30],Georgiadis et al. claimed that the back-pressure algorithmperforms within O(1/V) of optimalthroughput utility,with O(V) average delay, where Vis a control parame-ter that can be chosen as large as desired. This gives anexplicit [ O(1/V), O(V)] utility-delay tradeoff. With differ-ent definitions of the utility functions, [31-33] proposeddifferent back-pressure policies that may jointly con-sider power control, fairness, end-to-end delay, and flowcontrol.Inthis paper, we propose a distributedframeworkthat combines opportunistic routing with back pressureschedulingandderivethesufficient conditionforourframework to achieve throughput optimality.Based onthis condition, we design a MAC protocol that stabilizesthe network when the transmission probabilities are low.Toourbestknowledge, thispaperisthefirsttotakeadvantage of the benefits provided by BP and OR to tacklethechallengesfacedbycognitiveradionetworkswithdynamic link statuses.3 A framework of distributed back-pressurescheduling with opportunistic routing3.1 Our distributed scheduling frameworkIn this section, we detail our framework of distributedback-pressure scheduling with opportunistic routing fora CRN with multiple simultaneous flows. Back-pressurescheduling is introduced to select the flow with maximumback pressure at each node while opportunistic routingis introduced to address the high dynamism of CRNs,such that the next hop relay for a flow can be selectedopportunisticallyfromall thefeasiblepaths. Sincetheoriginal back-pressure algorithm assumes the existenceof aglobal controller that prevents thesimultaneousactivation of potentially conflicting links/flows for colli-sion avoidance, we adopt CSMA/CA in our distributedframeworkformediumaccesscontrol (MAC) asit isnotfeasibletoimplementacentralizedlinkcontrollerin a CRN. In summary, our distributed framework takesback-pressure scheduling for flowselection, opportunisticrouting for path determination, and CSMA/CA for MACregulation.We consider a multi-hop CR network with N nodes andL links. Assume that there exist F flows in the network. Forsimplicity, we further assume that there are F flow queuesat each node,with the cth queue serving the cth flow.The exogenous packet arrival rate at node i for an arbi-trary flowc is denoted by aci, and the corresponding arrivalrate vector is denoted by Ac. Denote by Xci (t) the queuelength of flow c in node i. Let sc(l) and dc(l) be the send-ing and receiving nodes of link l for flow c, respectively.The connectivity and routing matrix Rc = _rcil_for flow cis defined by a N L matrix, with rcil representing the lthelement in the ith row of Rc:rcil =___1, if i = dc(l),1, if i = sc(l),0, otherwise.(1)Note that the link connectivity in a CR network mayvary over time and thus cause the Rc matrix to change.Let pcl be the probability that link l is used to transmitpackets for flow c. Thus, the queue length in time slot(t +1) can be represented as follows:Xci (t +1) = Xci (t) +L

l=1rcil bl pcl +aci, (2)where aci is the arrival rate of flow c at node i and bl is thetransmission rate of link l.Let Pi be the probability that node i can successfullytransmit a packet. By adopting opportunistic routing, apacket can be successfully transmitted if at least one of itsfeasible relays can receive the packet correctly. Thus, Pican be represented asPi =

c

l:sc(l)=ipcl. (3)Let B be a diagonal matrix in which the lth diagonalelement is bl. Then, the equivalent matrix expression ofXc(t +1) becomes:Xc(t +1) = Xc(t) +RcBPc+Ac. (4)A link backlog Dl for each link is defined as follows. Theoriginal back-pressure routing policy [12] adopts the samedefinition.Dcl = bl _Xsc(l) Xdc(l)_, (5)Dl = maxcDcl(6)The best candidate flow and the corresponding backlogfor node i, denoted by ci and Di , respectively, are selectedaccording to the following policy:ci = argcmaxl:sc(l)=iDcl, (7)Di =maxl:sc(l)=iDcl, (8)That is, the maximum Dl is selected from all the out-boundlinks of nodei, andits correspondingflowisselected as the best candidate flow.In the next subsection, we derive the sufficient conditionfor our framework to achieve throughput optimality.Le et al. EURASIP Journal on Wireless Communications and Networking(2015) 2015:49Page 4 of 143.2 Throughput optimality of the proposed frameworkWe first define the stability of a network as follows.Definition A network is stable under a scheduling pol-icy , if the queue length Xci (t) satisfies:lim suptF

c=1N

i=1Xci (t) < D, (9)where D is a constant that is dependent on the arrival ratevectors Ac and transmission rate bl.Let C be the set of arrival rates for which the networkis stable under policy. The stability region C of a net-work is defined as C =

C.Definition A scheduling policy is throughput optimal ifthe network is stable for any arrival rate a C under thepolicy.In the following, we present a lemma to show that ourdistributed scheduling framework proposed in Section 3.1is throughput optimal under a certain success transmis-sion probability Pi.Lemma 1. In a CSMA/CA based CRN, a back-pressurescheduling with distributed routing framework is through-putoptimal if thefollowingequationholdsforall thenodes:Pi =_1 O_ 1Di__. (10)where O() stands for the big-O notation.The detailed proof of Lemma 1 can be found in theAppendix.Lemma 1 provides the sufficient condition for aCSMA/CA-based scheduling framework to achievethroughput optimality. In the next section, we proposeadistributedapproximatelythroughput optimal MACalgorithm based on Equation 10.4 A distributed approximate MAC algorithmIn this section, we first derive the contention window sizethat can satisfy Equation 10. Then, a detailed protocol ofour scheduling framework is proposed.4.1 Contention windowsize derivationIn our previous section, we show that a CSMA/CA-basedCRN is stable when its successful transmission probabil-ities {Pi} satisfy a certain constraint. In a network withacentralizedtransmissioncontroller, itisnothardtoachieve the required successful transmission probabilities.However, in a wireless network where stations competefor medium access in a distributed manner,it is diffi-cult to guarantee the transmission probabilities. In thissubsection, we propose a distributed medium access con-trol algorithm that approximately achieves the requiredtransmission probabilities for throughput optimality.In a CSMA/CA network, the transmission probabilityPi of station i is determined by the attempt probabilities{Pia, i = 1 . . . N}. We adopt the model in [34], in whichPi = Pia

j=i_1 Pja_, (11)and the attempt probability can be expressed as the fol-lowing function of the contention window size CWiPia =2CWi +1. (12)Notice that when the attempt probabilities approach to0, the product j=i_1 Pja_ converges to 1, leading toPi = Pia. In a typical CRN, transmissions occur sparselyand opportunistically due to the existence of PUs. There-fore, we can approximately represent Piby Pia. On theother hand, according to Lemma 1, the following equationshould hold for every Pia and Di :Pia = 1 O_ 1Di_. (13)This implies that whenthe maximumbacklog Dibecomes larger,the attempt probability Piais enlarged.Recall that Pi can be represented by Pia only when theattemptprobabilitiesapproachto0. Hence, themaxi-mum attempt probability should be bounded when themaximum backlog is increasing, so that the sparse trans-mission constraint can be guaranteed.Otherwise,highattempt probabilities could decrease the successful trans-mission probabilities and consequently cause congestions.And the congestions may significantly increase the queue,causingevenhigher attempt probabilities due totheretransmissions.Toaddressthisproblem, welimit thequeuelengthto a finite number. When the queue length reaches itsupper bound, all newly arriving packets are dropped. LetQmax be the maximum queue length. Then, the maximumbacklog Dmax is equal to Qmax. The maximum attemptprobability is thus bounded byPmaxa= 1 O_1Qmax_. (14)Given a Qmax,we can have different values for Pmaxadepending on O_1Qmax_. To find a better Pmaxa, we con-sider the scenario where all queues in all stations reachQmax. In such a situation, an optimal Pmaxashould maxi-mize the total throughput among the network. In [34,35],Le et al. EURASIP Journal on Wireless Communications and Networking(2015) 2015:49Page 5 of 14it has beenshownthat ina saturatednetwork, theaggregated throughput can be maximized by finding theoptimal attempt probabilities. By adopting the analysis in[35], we can calculate the optimal attempt probability Poptagiven the maximum collision duration Tc, the slot dura-tion Tslot, and the number of stations N. The relationshipbetween N and Poptacan be expressed asPopta=0N , (15)where 0 is the root of the equation1 =_1 TslotTc_e.For a given CRN, Tc and Tslot are predefined constants;thus, Poptacan be calculated by each station. By settingPmaxa= Popta, we havePmaxa= 1 O_1Dmax_=0N . (16)Let O_1Di_be a function of the formc1Di+c2. Notice thatwhen the maximum backlog Diis equal to 0, the stationshould not attempt to transmit. This can be expressed as:___1 c10 +c2= 0 (17)1 c1Dmax +c2=0N . (18)By solving the above equations, we havec1 = c2 = Dmax (N 0)0, (19)and thus the corresponding contention windowsizeCW_Di_can be expressed asCW_Di_= 1 +2 Dmax (N 0)0 Di. (20)4.2 Approximation analysisInthissubsection, weshowtheapproximationanaly-sis of our algorithm, comparing to the optimal solution.AsshowninEquation11, thedifferencebetweentheattempt probability Pia and the transmission probability Pi,denoted by the collision probability Pifcan be expressedas:Pif = Pia Pi = Pia __1

j=i_1 Pja___. (21)Recall that the attempt probability is upper-bounded bythe maximum attempt probability Pmaxa, thus we havePif = Pia __1

j=i_1 Pja___ Pmaxa__1

j=i_1 Pmaxa___= Pmaxa_1 _1 Pmaxa_N1_=0N_1 _1 0N_N1_.(22)Considering the impact of a collision, it is trivial thata collision can lead to a retransmission. If we considersuch additional transmissions as newly arrived packets,the equivalent arrival rate for these retransmissions can beexpressed as

Ni=1 Pif. Thus, comparing to the ideal solu-tion with zero collision probability, our proposed algo-rithm decreases the feasible arrival rate for an amountof

Ni=1 Pif. Hence, the difference between the maximumcapacity region and the feasible capacity region of ourproposed algorithm is no larger thanN

i=1Pif N 0N_1 _1 0N_N1_. (23)4.3 Protocol descriptionOur protocol consists of four major steps: i) flowselection,ii) random backoff period, iii) transmission and acknowl-edgement exchange, and iv) forwarding decision. Similarto the originalopportunistic routing protocol[36],weassume that a distance matrix is maintained by each sta-tion, whichisusedtocalculatethedistancetootherstations.Whenastationisreadytotransmit, it first selectsthe flow with the maximum backlog.After the flow isselected, the CW value is calculated based on the maxi-mum backlog, as described in Section 4.1, and a randombackoff period is generated. The packet is then broadcastwhen the backoff period is passed. The distance betweenthestationandthedestinationof thepacket, aswellas the current queue length of the flow, is appended tothe packet header. Each station that receives the packetshould compare the distance and queue length from theheader with its own values.If the receiver is closer tothe destination and the queue length of the correspond-ing flow is smaller than that of the sender, the receiversends an acknowledgement back and considers itself as acandidate relay. Candidate relays decide whether or notLe et al. EURASIP Journal on Wireless Communications and Networking(2015) 2015:49Page 6 of 14to forward the received packet after all the acknowledge-ments are received. This procedure is repeated until thepacket reaches the destination station.4.3.1 FlowselectionIn our protocol,each station selects the flow with themaximum backlog to serve. To calculate the backlog, eachstation maintains a matrix that includes the approximatequeue length of its neighbors. When the station receives apacket or an acknowledgement message, the correspond-ing queue length value in the matrix is synchronized to thevalue included in the header.4.3.2 Randombackoff periodAfter a flow is selected, the maximum backlog Diis usedto calculate the CWsize as defined in Equation 20. Duringthe backoff period, the station continues monitoring thespectrum. If a packet transmission is sensed or a PU occu-pies the spectrum, the station should pause the backofftimer and wait until the channel becomes available again.4.3.3 Transmission and acknowledgementWhen the backoff period is passed,the station pops apacket from the flow queue and immediately starts trans-mitting. The distance between the current station and thedestination station and the current queue length of theflow are both stored in the header. Then, the station waitsfor acknowledgement messages from its neighbors. Dif-ferent from the default 802.11 MAC algorithm, we reservemultiple time slots for ACK exchanges in our protocol.In the ith slot, the recipient with the ith smallest queuelength sends the ACK. This order can be calculated fromthe queue length matrix stored at each station. In the casewhere multiple stations have the same queue length, thestations send their ACKs in the order of their station IDs.To indicate that a packet has been received successfully,the ACK also includes the station ID and the queue lengthof the best candidate relay known to the ACK sender. Forinstance, assume the queue length values of the flow c instations A, B, and C are 2, 4, and 5, respectively. In thiscase, station A sends the ACK message first with its ownID and queue length value. In the second time slot, sta-tion B sends its ACK, reporting that station A, with queuelength value 2, is the best candidate relay. In the last timeslot, station C sends its ACK carrying the same best can-didate relay information. This acknowledgement mech-anism can prevent duplicate forwarding when an ACKmessage from a higher priority recipient is not received bya lower priority one. As shown in Figure 1, assume thatthe ACK message from station A is not received by sta-tion B due to the hidden terminal problem. At the secondtime slot, station B considers itself as the best candidateand sends an ACK message reporting B as the best can-didate with queue length 4 as the minimum queue length.Figure 1 Illustration of sequential ACKs.Station C receives the two ACKs and compares the twoqueue lengths. The smaller one, which is from station A inthis example, is chosen as the best candidate and is broad-casted with the ACK message. Finally, station B receivesthe ACK from station C and updates its best candidateinformation.4.3.4 Forwarding decisionAfter exchanging the ACK messages, the stations makethedecisionregardingwhetherornottoforwardthereceived packet. Specifically, after all ACK messages arereceived, if a station does not receive any ACK messagewith a smaller queue length value than itself, it pushes thereceived packet into the corresponding queue and laterforwards it toward its destination. Otherwise, the stationdiscards the received packet.5 Performance evaluationIn this section, we evaluate the performance of our pro-posed transmission protocol using the OMNET++ plat-form [37].The average queue length,packet loss rate,throughput, and end-to-end delay are all evaluated in oursimulation tests.5.1 Simulation settingsWe adopt the INETMANETframework in OMNET++[37] as our cornerstone. The MAC and IP modules aremodifiedtoimplement our transmissionprotocol. Inoursimulation, theparametersof theMACandPHYlayer adopt the default values same as those inthe802.11b standard. Additionally, the transmit power andsignal-to-interference-plus-noise ratio (SINR) thresholdsaredefined so that thetransmission range isequalto250 m. Table 1 lists the basic parameters employed in oursimulation.In our network, multiple stations are placed in the sim-ulation area and set to be static. We assume the stationsalways transmit packets using the maximum transmis-sion rate, which implies that rate adaptation is disabled.Le et al. EURASIP Journal on Wireless Communications and Networking(2015) 2015:49Page 7 of 14Table 1 The 802.11b parameters in Omnet++Slot time (s) SIFS (s) DIFS (s)20 10 500MTU (bytes) ACK length (bits)0.1622 1,500 112IP header (bits) MAC header (bits) PHY header (bits)160 272 192Bitrate Bitrate of ACK Bitrate of PHY11 Mbps 2 Mbps 1 MbpsTransmit power (mW) SINR threshold (dB) Sensitivity (dBm)2 4 85The flows are defined at the beginning of the simula-tion,and their arrivalrates remain the same through-out the entire simulation period. Additionally, we disableRTS/CTS exchanges. The maximum queue length of eachqueue is set to 15 in our simulation.5.2 MethodologyOur evaluation is conducted under two different networkscenarios. We first consider a simple multi-hop networkwith four stations located at the four vertices of a square.The edge length of the square is set to 200 m; thus, thetwo diagonal vertices cannot communicate directly. Wedefine two flows in this network: the first flow transmitspackets from the upper-left station to the lower-right sta-tion; and the other flow goes from the lower-right stationto the upper-left one. Each flow can have two paths fromtheir source nodes to their destinations. To evaluate theimpact of link availability on network performance, weconsider a simple PU deploy model, in which each stationcan be interrupted by one and only one PU. In addition,each PU individually occupies the channel with a certainoccupancy rate. Please notice that the random PU deploymodel is not adopted in this scenario because all the fourstationsareclosetoeachother, andthusarandomlydeployed PU could simultaneously interfere with all thestations at a high probability. The impact of the packetarrival rate and that of the primary user occupancy rateaon the network performance are evaluated for this sce-nario. The network topology and the flows are illustratedin Figure 2.In our second scenario, we randomly deploy N stationsand five primary users in a 1,500 by 1,500 m simulationarea. The interference range of the primary users is set to300 m; thus when a primary user occupies the channel, thestations within its interference range cannot transmit orreceive any packets. In addition, L flows are defined in thenetwork by randomly selecting L pairs of sources and des-tinations from the N stations. In this scenario, we studyFigure 2 Network topology and the two flows for scenario I.the impact of the packet arrival rate A, the total num-ber of stations N, and the total number of flows L, on theperformance of our protocol.5.3 Evaluation resultsThe evaluation results are detailed in terms of the averagequeue length, the average end-to-end delay, the averagepacket loss rate, and the average throughput. All of oursimulation results are the averaged value of ten runs.5.3.1 Average queue lengthFigure 3 shows the impact of the packet arrival rate on theaverage queue length in the simple network. The queuelength calculated from Equations 10 and 19 is shown inthe figure as well (labeled by ideal queue length). Fromthis figure, we observe that the average queue length curveis comprised of three distinct phases. In the first phase,the time gap between the arrivals of two packets is longerthan the length of the maximum backoff period; and thus,Figure 3 Average queue length vs. arrival rate.Le et al. EURASIP Journal on Wireless Communications and Networking(2015) 2015:49Page 8 of 14the first packet is transmitted before the second packetarrives, althoughthebackoff periodof eachpacketislong. As a result, the average queue length remains 0 or1 for the majority of the time. In the second phase, theaverage queue length grows linearly while the arrival rateincreases. In this phase, packets reach their destinationsat approximately the same rate as the arrival rate; thus, thetransmission probability is proportional to the arrival rate.In the last phase, the low transmission attempt probabil-ity condition does not hold after the arrival rate exceedssome threshold. Accordingly, packets collide with othersfrequently, and the average queue length grows quickly tothe upper bound. We also evaluate the variation of theaverage queue length for the second phase in Figure 4.From the figure, we observe that at the beginning, thequeue length of the source station rapidly increases. Theincreasing queue length then reduces the CW size andthus increases the transmission probability. As a result,the queue length becomes stable after the transmissionprobability reaches the arrival rate.In Figure 5, we evaluate the impact of the PU occupancyrate on the average queue length. In this evaluation, thearrival rate is set to 150 packets per second. We observethat the curve is similar to that of Figure 3. We can inferthis result by considering the impact of PU occupancy asa reduction of the arrival interval. In other words, increas-ing the PU occupancy rate is approximately equivalent toincreasing the packet arrival rate, although the increas-ingratiomaynotnecessarilybeequal. Similarresultsare observed in the simulation of the other three met-rics as well; thus, the impact of the PU occupancy ratewill not be reported in the following subsections. Further-more, the PU occupancy rate is set to 0.4 without furtherclarification.The performance of our protocol in the large networkis shown in Figures 6 and 7. From the figures, we canFigure 4 Average queue length vs. time.Figure 5 Average queue length vs. PU occupancy rate.observe that the three phases exist in this scenario as well.In addition, when the arrival rate is low, the increasednumber of stations does not obviously impact on the per-formance, as the attempt probabilities of all the stationsare very low, and thus the increasing number of stationsdoes not obviously enlarge the collision probability. Whenthe arrival rate grows, the impact of the amount of stationson the average queue length becomes more apparently.This is because the increasing number of stations lead toa higher collision probability and thus increase the queue-ing time for each packet. On the other hand, by increasingthe number of flows, the threshold to transfer to phasethree can be reached more quickly, and the average queuelength grows faster. This result can be easily inferred asthe aggregated arrival rate is proportional to the numberof flows.Figure 6 Average queue length vs. arrival rate with differentnumber of stations.Le et al. EURASIP Journal on Wireless Communications and Networking(2015) 2015:49Page 9 of 14Figure 7 Average queue length vs. arrival rate with differentnumber of flows.5.3.2 End-to-end delayWe first illustrate the end-to-end delay in the simple sce-nario in Figure 8. From the figure, one can see that whenthe packet arrival rate is lower than 50 packets per second,the end-to-end delay is approximately equal to 20 ms. Themajor time cost in this case comes fromthe maximal back-off period. As the arrival rate increases, the end-to-enddelaydecreasesrapidly, whichiscausedbytheshort-ened backoff period corresponding to the increased queuebacklog. Followingthis, thedelayremains unchangeduntil the arrival rate exceeds the threshold for phase three.Once this threshold is reached, the end-to-end delay thenconverges to infinity.As the queue length exceeds theupper bound, packets start being dropped.Figures 9 and 10 illustrate the end-to-end delay of ourprotocol in the second scenario. We notice that the end-to-end delay increases when Ngrows since the averageFigure 8 End-to-end delay vs. arrival rate.Figure 9 End-to-end delay vs. arrival rate with different numberof stations.number of hops for eachflowis relatedtothenet-work size, and the backoff period for each transmissionis related to the number of stations within its transmis-sion range. Also, the end-to-enddelayincreases whenL becomes larger, as the queueing time in each hop isincreased. Similar to the average queue length, the thresh-old to reach phase three becomes smaller while L grows.5.3.3 Packet loss ratioThepacketlossratioismeasuredbytheratioof theamount of lost packets to the total number of transmit-ted packets. Similar to the average queue length, we canobserve the existence of a threshold and divide the curveintothreephases. AsshowninFigure11, thepacketlossratioremainslowuntil thethresholdisreached.Then, the ratio increases rapidly to 1 as the entire net-work becomes congested; and consequently, most packetsFigure 10 End-to-end delay vs. arrival rate with differentnumber of flows.Le et al. EURASIP Journal on Wireless Communications and Networking(2015) 2015:49Page 10 of 14Figure 11 Packet loss ratio vs. arrival rate.either are dropped by the stations or have collided withother packets.The impacts of Nand L on the packet loss ratio arereportedinFigures12and13, respectively. It canbeobserved that the impact of N and L on the packet lossratio is in accordance to that on the average queue length.5.3.4 ThroughputThe throughput of the networkcanbe measuredasthetotal number of packets arrivedat their destina-tionswithinaunitof time. TheresultsareshowninFigures 14, 15, and 16. For the simple scenario, we noticethat the throughput is equal to the aggregated arrival ratebefore the threshold is exceeded. After that, the through-put drops quickly to 0, as no packet can be delivered to thedestination.Figure 12 Packet loss ratio vs. arrival rate with different numberof stations.Figure 13 Packet loss ratio vs. arrival rate with different numberof flows.For the second scenario,from Figure 15,we can seethat the thresholds are not affected by the total number ofstations. When the arrival rate approaches to the thresh-old, the collision probability becomes higher, so some ofthe packets are dropped after multiple retransmission fail-ure. A higher N can lead to a higher collision probabilityand thus can cause more packets being dropped. On theother hand, the increasing stations provide more queueingspace for storing packets when the network is congested,as well as more feasible paths to the destination. So whenthe network is not fully congested, the performance ofour algorithm does not have much difference when Nincreases. However, when the arrival rate becomes largeenough, a network with a greater N is easier to be fullycongested, as the collision probability grows faster than anetwork with a smaller N. In contrast, Figure 16 shows theFigure 14 Throughput vs. arrival rate.Le et al. EURASIP Journal on Wireless Communications and Networking(2015) 2015:49Page 11 of 14Figure 15 Throughput vs. arrival rate with different number ofstations.impact of L on the performance of our algorithm. It is clearthat by increasing the number of flows, a higher aggre-gated throughput can be achieved when the arrival rateis low, because the aggregated arrival rate is proportionalto the number of flows. At the same time, the networkbecomes much more congested, and thus the threshold isreached earlier.5.3.5 SummaryThe above evaluation results demonstrate that our pro-posed algorithm performs well when the arrival rate issmaller than a certain threshold. This threshold dependson the number of flows in the network but is insensitive tothe network size.Figure 16 Throughput vs. arrival rate with different number offlows.6 ConclusionsInthispaper, wehaveproposedaframeworkof dis-tributed back-pressure scheduling with opportunisticrouting for CRNs and studied the sufficient condition forthe proposed framework to achieve throughput optimal-ity. To the best of our knowledge, we are the first to com-bine opportunistic routing with back-pressure schedulingin a distributed fashion. We also have proposed a noveldistributed routing and medium access control protocolfor end-to-end communications in a CRN. We have eval-uated the performance of our protocol in a simulation-based study, and our evaluation results demonstrate thatthe proposed protocol is able to achieve both through-put optimality and reliability with the existence of PUs. Aspart of our future work, we plan to improve the end-to-end delay of our proposed protocol.EndnoteaThe PU occupancy rate refers to the probability of aPU occupying its channel at each time slot.Appendix[Proof of Lemma 1]Proof Similar tothe proof in[12], we consider aLyapunov functionV(X(t)) =F

c=1N

i=1(Xci (t))2.To prove the throughput optimality, we need to showthat the following relationships hold:E [V(X(t +1)) V(X(t))|X(t)] < , X(t), (24)and for any given > 0 and arrival rate a C, there existsa positive number b, such thatE [V(X(t +1)) V(X(t))|X(t)] < , for V(X(t)) > b(25)Inthefollowing, wefirst provethat theinequality(Equation 24) holds.E [V(X(t +1)) V(X(t))|X(t)]= E_F

c=1N

i=1__Xci (t +1)_2_Xci (t)_2_|X(t)_= E_F

c=1N

i=1_Xci (t +1) Xci (t)_2|X(t)_()+ 2 E_F

c=1N

i=1_Xci (t +1) Xci (t)_ Xci (t)|X(t)_()Le et al. EURASIP Journal on Wireless Communications and Networking(2015) 2015:49Page 12 of 14For part (), it is trivial to show that the amount of datadirected in or out of a queue is bounded by the total capac-ity of the links connected to the queue. Thus, part isbounded by a constant, say, b1.Considering part (), we haveE_F

c=1N

i=1_Xci (t +1) Xci (t)_ Xci (t)|X(t)_= E_F

c=1N

i=1_L

l=1rcil bl pcl +aci_ Xci (t)|X(t)_= E_F

c=1N

i=1L

l=1Xci (t) rcil bl pcl|X(t)_+E_F

c=1N

i=1Xci (t) aci|X(t)_=F

c=1L

l=1Dcl(t +1)pcl +F

c=1(Xc)TAc(26)Clearly, the term

Fc=1

Ll=1 Dcl(t + 1)Pcl is boundedby 0.The proof in [12] indicates that Fc=1(Xc)TAcisboundedbyD(t + 1)T

|S|i=1 isi. Here, S ={si, i =1, 2, . . . , |S|} is the set of all feasible activation vectors, and{i} is a set of positive factors that satisfy |S|i=1 i b. In the previous proof, we have shown thatthe following inequalities hold:F

c=1_Xc_TAc D(t +1)T|S|

i=1isi, (27)E_F

c=1N

i=1__Xci (t +1)_2_Xci (t)_2_|X(t)_< b1(28)Assume that there exists an activation vector s that canmaximize D(t +1)Ts. Let scl be the indicator for flow c onlink l. Then, equation scl = 1 holds if c is the correspondingflow to Dl and sl is equal to 1. Considering the differencebetween

Fc=1

Ll=1 Dcl(t +1)pcl and D(t +1)T s, we have:D(t +1)T s F

c=1L

l=1Dclpcl=

c

lsclDcl

l

cDclpcl

c

lDl_scl pcl_=

lDl

c_scl pcl_(29)If we represent a link l by its source and destination nodepair, we can rewrite

l Dl

c_scl pcl_as:

lDl

c_scl pcl_=N

i=1

jN(i)Dij

c_scij pcij_,(30)where N(i) is the set of neighboring nodes of i. Then, wehave:D(t +1)T s F

c=1L

l=1Dcl(t +1)pclN

i=1

jN(i)Dij

c_scij pcij_

i

jDi

c_scij pcij_=

iDi

j

c_scij pcij_.(31)Notice that for any node i, the total number of links thatcan transmit simultaneously is no more than 1. Thus, wehave

j

cscij 1. (32)From Equations 31 and 32, we obtain:D(t +1)T s F

c=1L

l=1Dcl(t +1)pcl

iDi__1

j

cpcij__=

iDi(1 Pi) .By setting Pi to_1 O_1Di__, we haveF

c=1L

l=1Dcl(t +1)pcl D(t +1)T s +C0 N, (33)Le et al. EURASIP Journal on Wireless Communications and Networking(2015) 2015:49Page 13 of 14whereC0isaconstantthatisrelatedtothefunctionO_1Di_. Then, with Equations 26, 27, 28, and 33, we haveE [V(X(t +1)) V(X(t))|X(t)] b1 +2__F

c=1L

l=1Dcl(t +1)pcl +D(t +1)T|S|

i=1isi__ b1 +2_D(t +1)T s +C0 N_+2D(t +1)T|S|

i=1isi b1 +2C0 N 2D(t +1)T s +2D(t +1)T s|S|

i=1i b1 +2C0 N 2D(t +1)T s__1 |S|

i=1i__.As shown in [12], we have:(D(t +1)T s) 1N_bFN minl{bl},when V(X(t)) > b(34)Thus, when V(X(t)) > b, we haveE [V(X(t +1)) V(X(t))|X(t)] b1 +2C0 N 2____1 |S|

i=1i__ 1N_bFN minl{bl}__.(35)By setting b = NF_N(+b1+2C0N)2_1

|S|i=1 i_minl{bl}_2, we prove thatthe inequality (Equation 25) holds. Thus, the throughputoptimal is achieved when Equation 10 holds for all thenodes.Competing interestsThe authors declare that they have no competing interests.AcknowledgementsThis work is supported in part by the National Science Foundation of the USunder grants CNS-0831939, CNS-1162057, CNS-1162159, and CNS-1265311.The authors would also extend their appreciation to the Deanship of ScientificResearch at King Saud University for funding this work through research groupno. RGP-VPP-264.Author details1Department of Computer Science, The George Washington University,Washington, DC, USA. 2Uniformed Services University of the Health Sciences,Bethesda, MD, USA. 3Department of Computer Science, University ofPittsburg, Pittsburgh, PA, USA. 4College of Computer and InformationSciences, King Saud University, Riyadh, Saudi Arabia.Received: 29 August 2014Accepted: 9 February 2015References1. 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