Outage Probability of a Multi-Relay Cognitive Network with an Uncertain Number of Forwarding Relays

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1 Outage Probability of a Multi-Relay Cogitive Network with a Ucertai Nuber of Forwardig Relays Yuli Hu, aes Gross ad Ake Scheik RWTH Aache Uiversity, Geray, Eail: {hu scheik}@uic.rwth-aache.de KTH Royal Istitute of Techology, Swede, Eail: jaes.gross@ee.kth.se Abstract I this paper, we focus o a Cogitive Relay Network (CRN where oly the iforatio of the average chael gais is available at the secodary source. A distributed ultiple relay selectio schee is proposed for the secodary trasissio. Sice the secodary source does ot fully cotrol the selectio but lets each secodary relay decide by itself, it does t kow which ad how ay relays will participate i the forwardig. This brigs a serious ucertaity i the uber of forwardig relays. We aalyze this ucertaity ad derive the closed-for expressio of the outage probability of the secodary trasissio uder the ucertaity. Fially, we evaluate the proposed relay selectio schee by eas of siulatio. The siulatio shows the appropriateess of our aalytical odel. I additio, although havig lower iterferece costrait at each relay, the perforaces of the proposed ulti-relay CRN schee are show to be strictly superior to the sigle relay CRN. Moreover, the siulatio results suggest that it is ecessary to sped the geeral overhead for secodary etworks o kowig the exact QoS requireet of the priary user (i.e. the iterferece violatio probability to apply our proposed relay selectio schee with the statistical total iterferece costrait. Idex Ters cogitive relay etworks, outage probability, relay selectio, decode-ad-forward, average chael gai I. INTRODUCTION Cogitive radio is eergig as a proisig techology that eables secodary users (SU to receive sigals fro the secodary source (SS over licesed bads through detectig spectru holes. I cogitive radio etworks, SUs ca be serviced over the spectru of priary etworks as log as the secodary trasissio safeguards the quality of service of the priary users (PU [1]. O the other had, relayig is well kow as a effective way to itigate the wireless chael-fadig by exploitig the spatial diversity gai [2], [3]. Specifically, whe ultiple relays are available to assist the service, the perforace of the trasissio ca be sigificatly iproved [4], [5]. As a result, cogitive relay etworks (CRN have bee of huge iterest [6] [11]. The outage probabilities of the CRNs have bee derived with cosiderig the ipact of the spectru sesig accuracy i overlay coexistece [6], [12]. I additio, the perforaces of both PUs ad SUs are studied i a CRN i copariso to the covetioal relay etworks [7]. For CRNs with ultiple /14/$31.00 c 2014 IEEE secodary relays (SR, oly selectig best SRs for assistig the secodary trasissio ca sigificatly reduce the iterferece to the PU. The outage probability of the secodary trasissio based o the N th best-sr selectio is studied i [8]. A eergy efficiet relay selectio is proposed for CRNs based o the perfect chael state iforatio (CSI at SS ad SU [9]. To select a sigle SR which is expected to have a strog lik to SS but weak lik to PU, [10] applies a ax-i relay selectio schee for a CRN where all the istataeous CSI are assued to be available at the SS. Also with the istataeous CSI at the SS, the best sigle SR with the highest sigal-to-iterferece-ratio is selected for the CRN [11]. I order to select best relays for SU while safeguardig the quality of PU s service, the above existig solutios assue istataeous CSI of all the liks are available at the SS. Ufortuately, this is over-optiistic i practice, especially for a CRN with larger uber of SRs. Hece the study of low-overhead-based distributed ultiple relay selectio will be very beeficial for the desig of realistic CRN. I particular, it is still a ope proble how the secodary trasissios of CRNs safeguard the quality of the PU s service while the SS does ot have the istataeous CSI. I this paper, we cosider a CRN syste where the SS oly has the kowledge of the average chael gais of all the liks. We assue each SR has the istataeous CSI of the lik fro itself to the PU but does o share the CSI with other relays or feedback it to the SS. I order to cotrol the su of (SR iterferece to PU, we itroduce a distributed ultiple relay selectio schee for the CRN to guaratee the PU service. We cosider two differet iterferece costrait policies "absolute total iterferece costrait" (AIC ad "statistical total iterferece costrait" (SIC i the proposed relay selectio schee. I the process of the proposed distributed relay selectio, the SS does t kow which ad how ay SRs will be selected ad participate i the forwardig frae. As oe cotributio of this paper, we derive the closedfor expressio of the outage probability of the secodary trasissio uder this ucertaity. Furtherore, the iial outage probability of the ulti-relay CRN syste is studied. We fid that the iial outage probability of a relay-assisted secodary trasissio is oly subject to the iterferece liks.

2 I additio, the sigle SR ad ultiple SRs assisted secodary trasissios have the sae iial outage probabilities uder the AIC strategy. At last, we evaluate the relay selectio schee ad show the appropriateess of our aalytical odel by eas of siulatio. We show that the SIC is superior to the AIC oly i ulti-relay CRNs. The results suggest these ulti-relay CRNs (where oly average CSI is available at the SS to sped the geeral overhead for secodary etworks o kowig the exact iterferece violatio probability requireet of the PU. Moreover, it is also show that the ulti-relay CRN perforaces are absolute superior to the sigle relay CRN while the advatage of ulti-relay CRN is lower whe the PU is gettig close to the SRs. The rest of the paper is orgaized as follows. Sectio II itroduces the ulti-relay CRN odel. I Sectio III, a distributed ulti-relay selectio schee is proposed. I additio, the outage probability of the ulti-relay selectio schee is studied with two iterferece costrait policies. By eas of siulatios, Sectio IV evaluates the relay selectio schee ad shows the appropriateess of our aalytical odel. Fially, we coclude the paper i Sectio V. II. SYSTEM MODEL We cosider a CRN i which a PU coexists with several secodary odes as scheatically show i Fig. 1. Aog the Secodary Source Packet Broadcastig Chael Forwadig Chael Iterferece Chael Secodary Relay Group Fig Packet Secodary User A cogitive relay etwork Priary User Selected relay Uselected relay secodary odes, a SS is required to trasit a data packet of size ρ to its destiatio SU. We assue that the SS is far alway fro both SU ad PU. Therefore, the sigals fro the SS are very weak for both the PU ad the SU. A group of secodary decode-ad-forward relays are deployed to help the secodary trasissio, which brig cosiderable iterfereces to the PU. The group {R 1,...R } has relays which are radoly deployed i a certai area while the radius of the area is sigificatly saller tha the distace either fro the SS to the relay group or fro the relay group to the SU or the PU. I the secodary trasissio, tie is divided ito fraes of legth T f. A sigle secodary trasissio requires two fraes o trasittig the packet fro the SS to the SU, which is referred to as broadcastig frae ad forwardig/relayig frae. Durig a broadcastig frae, the SS trasits the data packet to all SRs. Ad afterwards, soe selected SRs forward the packet to the SU i a relayig frae. We assue that the chael states vary radoly due to a Rayleigh-distributed block-fadig process. All the secodary trasitters apply fixed trasit powers deoted as P S at the SS ad P R at each SR. The oise power is deoted as σ 2. We assue that each relay has the istataeous CSI of the chael fro itself to the priary user while the source oly has the kowledge of the average chael gais. Deote the chael gai durig a broadcastig frae fro the SS to R j by h 2 S,j. The, the sigal-to-oise-ratio (SNR of the broadcastig frae is give as γ S,j = P S hs,j/ 2 σ 2. Correspodigly, the chael gai ad SNR fro R j to SU are give by h 2 j,d ad γ j,d = P S h 2 j,d /σ2. If the gai of the chael fro a selected relay R j to the PU is give by h 2 j,p, the correspodig iterferece power ca be obtaied by P R h 2 j,p. Regardig the iterferece costrait, which its the total iterferece fro all relays to PU, we cosider it i the relay selectio process ad do ot assue ay power cotrol at each relay. As the distace aog the relays is fairly sall copared to the distaces fro SRs to the SS or to the SU ad PU, we ake a topology siplificatio that all the relays have the sae "average chael gais" for broadcastig chaels deoted by h 2 B, for relayig chaels deoted by h 2 R ad for iterferece chaels deoted by h 2 I. All the three average chael gais are available at the SS. Give a istataeous SNR γ durig oe frae (with N sybols, at ost N log 2 (1 + γ bits ca be coveyed correctly. Hece, a curretly trasitted packet of size ρ is successfully received if the SNR of the lik is above the threshold γ = 2 ρ/n 1. III. DISTRIBUTED MULTIPLE RELAY SELECTION SCHEME I this sectio, a distributed ulti-relay selectio schee is proposed. Based o it, we aalyze the outage probability of the secodary trasissio. A. The Process of Selectio The distributed ultiple relay selectio process has two steps. A paraeter plays the role of a bato i the selectio process, which is called sigle relay iterferece costrait (SRIC ad deoted by I R. I the first step, the SS is required to decide the value of SRIC based o the iforatio of the average chael gais, the the SS broadcasts the value I R to all SRs oce at the begiig of each tie relay selectio. After that, the source broadcasts the data packet to all the SRs. Due to the fadig, a varyig aout of SRs could decode the packet successfully durig the broadcastig phase. We call this set the survivig relay set ad deote it by Θ s. Deote by Pr 1 the outage probability of the lik fro the SS to relay R j. Therefore, Pr 1 ca be give as: Pr 1 = 1 exp ( γ σ 2 0/ 2 h2 B P S. (1 Hece the probability that Θ s has relays ca be give by: Pr B (;, Pr 1 = (1 Pr 1 (Pr 1. (2 I the secod step of the selectio, each SR i Θ s decides whether to participate i the forwardig based o the SRIC fro the SS together with chael quality of the iterferece lik fro itself to the PU. More precisely, a relay R j Θ s decides whether to joi the subsequet forwardig frae based

3 Pr IV (IR s = (1 Pr 1 (Pr 1 (1 Pr e(ir s (Pr e(ir s 1 PI su (IR s. (3 =1 =1 sec,ulti = (1 Pr 1 (Pr 1 (1 Pr e(i R (Pr e(i R Pr 2 (. (4 =1 =1 o the result of coparig h 2 j,p to the SRIC. For istace, if P R h 2 j,p > I R, the this relay R j will keep silet i the subsequet relayig frae. The deteriatio of I R will be discussed i the ext subsectio. Based o the topology siplificatio itroduced i Sectio II, the probability that the iterferece power of a secodary relay is exceedig the SRIC ca be obtaied by: Pr e (I R = exp ( I R / 2 h2 I P R. (5 We deote the fial forwardig relay set as Θ f, so the total iterferece power to PU ca be obtaied by: P I su = R j Θ f P R h 2 j,pu. (6 If the uber of survivig relays is, the distributio of uber of relays i the Θ f, deoted by, ca give as: Pr B (;, Pr e = (1 Pr e (Pr e. (7 B. The Value of Sigle Relay Iterferece Costrait It reais to aswer how the SS deteries I R. Obviously, the SRIC is strogly depedet o the total iterferece costrait (deoted by I su to the PU. I the followig we study I R uder two iterferece costrait policies: 1 Absolute total iterferece costrait (AIC: This is a coo iterferece costrait i cogitive etworks. Uder AIC, the su of iterferece should be saller tha the threshold I su with the probability of 1. I our distributed relay selectio process, the SS does t kow the exact uber of forwardig relays. Therefore, i order to esure the total iterferece costrait, the SS should decide the I R based o the axiu forwardig relay uber which is actually the uber of relays deployed i the syste. Hece the correspodig SRIC uder AIC ca be give by: I a R = I su /. (8 A SR R j keeps silet if P R h 2 j,p > Ia R. Therefore, the actual su iterferece to the PU has the upper boud I su : P R h 2 j,p IR a IR a = I su, (9 R j Θ f where Θ f is the forwardig SR set with size,. 2 Statistical Total Iterferece Costrait (SIC: Differet fro the absolute costrait, the SIC has ore accurate requireets fro a secodary trasissio of iterferece costrait by itroducig a ew costrait paraeter called iterferece-violatio-probability costrait (IVPC. I other words, with a certai probability costrait SIC allows the iterferece fro the secodary trasissio accidetally exceedig the threshold. I practice, the trasissio of PU ay have certai Quality-of-Service (QoS requireets, e.g., outage probability or delay. If the IVPC is oe agitude lower tha the outage probability requireet of PU, the egative ipact fro the secodary trasissio o the QoS of PU is still egligible. Deote the SIC as IR ac, hece the Pr iv(ir s, which deotes the iterferece violatio probability of the syste o a give SIC, ca be expressed by (3. I (3, the 1 PI su (IR s is a idicator fuctio o whether the iterferece power received by PU P I su exceeds the threshold I su ad give by: { 1 PI su (IR s 1, if PI = - su (IR s > I su; 0, if P I - su (IR s I (10 su. Hece, for a give IVPC, deoted by Pr c, the SS is able to decide the value of IR s by solvig: Pr iv (I s R = Pr c. (11 As we kow, the high IR ac activates ore relays i forwardig frae, ad thus, geerates ore iterfereces to the PU. As a result, this leads to a high iterferece violatio probability. Therefore, the Pr IV (IR s is icreasig i Iac R. Hece, (11 has a uique solutio o IR s which ca be foud uerically by iterative searchig. C. Outage Probability Aalysis I a relayig frae, all relays i set Θ f are able to forward the packet to the destiatio. The, the SNR at the destiatio is siply the su of the SNR of the idividual liks. Therefore, we have γ D = j Θ f γ j,d. Based o our previous work [13], γ D is a gaa / distributed variable with the scale paraeter β = 2P R h2 R σ 2. The outage probability of the relayig frae, deoted by Pr 2 (, is a cuulative distributio fuctio of the gaa distributio. For a Θ f with relays, the outage probability of relayig frae is: 1 1 γ j 1 e γ β ; > 0 Pr 2 ( = j! β, (12 j=0 1; = 0 where γ is SNR threshold itroduced i Sectio II. Cobiig (2 ad (7 with (12, the expected outage probability of the two frae secodary trasissio is obtaied by (4 i which I R is either I s R or Ia R. I particular, if the secodary etwork oly has oe relay, the outage probability ca be give by: sec,sigle = (1 Pr 1 (1 Pr e (I R. (13

4 Obviously, the outage probability of the secodary trasissio is icreasig i the packet size. The related iial outage probabilities are achieved as log as the packet size ρ goes to 0. Therefore, the ulti-relay assisted secodary trasissio satisfies: Pr 1 = ρ 0 γ 0 Pr 2( = ρ 0 { 1 exp ( γ σ 2 0/2 h 2 BP S } = 0, ( γ 0 j=0 1; = 0 1 j! ( γ β j e γ β = 0; > 0 (15 Equatio (14 shows that all the relays are able to decode the packet successfully. I additio, it is idicated by (15 that the SU is able to decode the packet correctly as log as the fial forwardig relay uber is ot zero. I other words, the iial outage probability of the secodary trasissio is equal to the probability of all the SR-PU iterferece liks violatig the sigle relay iterferece costrait. As a result, the iial outage probability of the studied ulti-relay assisted secodary trasissio satisfies: ρ 0 Prout sec,ulti = Pr e (I R. (16 Hece, we have the followig propositio: Propositio 1. For a CRN where oly the average CSI is available at the SS, the iial outage probability of the secodary trasissio is aily subject to ot the secodary trasissio liks but the iterferece liks. I particular, the iial outage probability of ulti-relay schee uder the AIC policy is obtaied by: ρ 0 Prout sec,ulti =Pr e (I su / =exp ( I su /2 h 2 I P R. (17 O the other had, the iial outage probability of a sigle SR assisted trasissio uder the AIC policy: ρ 0 Prout sec,sigle = Pr e (I su = exp ( I su /2 h 2 I P R. (18 Obviously, the followig relatioship holds: Propositio 2. Uder the AIC policy, the sigle relay ad ulti-relay assisted secodary trasissios have the sae iial outage probability if the SS oly has the average CSI: ρ 0 Prout sec,ulti = ρ 0 sec,sigle. (19 IV. NUMERICAL RESULTS AND DISCUSSION I this sectio, we evaluate the proposed relay selectio schee ad show the appropriateess of our aalytical odel by siulatios. We radoly deploy 10 relays i a circular area with radius 10 while the distaces of the SS-SRs, SRs- SU ad SRs-PU liks are set as 100, 100 ad 300. We assue that the ceter frequecy is 2GHz ad the frae legth is set to T f = 10 s. Besides, we set P S = 30 db, P R = 20 db ad σ 2 = 90 db. For calculatig the path-loss, we utilize the well-kow COST231 odel. I the siulatios,. we obtai the outage probabilities of the ulti-relay CRN by varyig the packet sizes ad syste topologies. First, the theoretical ad siulatio values of the outage probabilities of the secodary trasissio versus packet sizes are show i Fig. 2. The theoretical values are calculated by the above equatios while the siulatio values are obtaied statistically based o radoly geerated chaels. As a result, Fig. 2 shows the appropriateess of our aalytical odel. The theoretical values icely atch the siulatio values while slight isatch is caused by the topology siplificatio Siulatio AIC Theoretical AIC Siulatio SIC 10 2 Theoretical SIC Packet size (bit Fig. 2. The theoretical ad siulatio values of the outage probability of the secodary trasissio. Furtherore, we show the outage probability perforaces uder differet iterferece costrait policies ad copare the ulti-relay trasissio to sigle relay schees i Fig. 3. As refereces, the sigle relay schees have the sae total iterferece costraits as the ulti-relay schees. Therefore, the sigle relay schees have sigificatly loose (high threshold iterferece costraits for (each relay i copariso to ulti-relay trasissio. We lear the followig itriguig re Sigle relay AIC Sigle relay SIC 10 3 Multiple relay AIC Multiple relay SIC 10 2 Multiple relay SIC Packet size (bit Fig. 3. The outage probability of the secodary trasissio uder differet iterferece costraits. latioships aog the curves. Firstly, uder AIC policy, ultirelay ad sigle relay assisted secodary trasissios alost have the sae outage perforace with extree short packets.

5 This atches our aalysis that the above two trasissios have the sae value of the iial outage probabilities uder the AIC policy. Secodly, copared to the sigle relay CRN, although havig a lower iterferece costrait at each relay, ulti-relay CRN perforaces lie i the optial positios (for ozero packet size. Therefore, for a ulti-relay CRN where the SS oly has the average chael gais iforatio, a appropriate policy of guarateeig the priary trasissio does ot let oly oe SR assist the secodary trasissio but keeps utilizig the ulti-relay ad gives each SR a relatively lower/striget iterferece costrait. Thirdly, the SIC schees are ore reliable tha the AIC schee for ultirelay trasissios. Eve if the IVPC is as low as 10 3, the SIC is still sigificatly superior to AIC. At the sae tie, SIC ad AIC have the sae perforaces whe the secodary etwork oly has oe relay. I other words, SIC is oly proisig i ulti-relay CRN. Aother copariso betwee the sigle-relay ad ultirelay schees is show i Fig. 4 by varyig the SRs-PU distace fro 300 to 100 ad SRs-SU distace fro 100 to 300. Whe the PU is close to the SR group, Sigle relay schee Multiple relay AIC Multiple relay SIC 10 2 Multiple relay SIC Distrace (SRs PU distace ius SRs SS distace Fig. 4. The outage probability of the secodary trasissio while the syste topology is varied. The Packet size is set as 100 bits. the SRs-PU iterferece chaels becoe strog. I order to protect the priary trasissio, the secodary trasissio is severely ited, i.e., relays caot forward the packet to the SU with a high probability. O oe had, this itatio akes the outage probability icrease as the distace gap (SRs-PU distace ius SRs-SS distace icreases. O the other had, the flexibility of the ulti-relay schees are weakeed by icreasig the gap. As a result, the curves of the sigle relay schee ad the ulti-relay AIC schee coverge whe the PU is uch closer to SRs tha to the SU. At the sae tie, ultirelay SIC schees are still sigificatly superior to the ultirelay AIC schee. Hece, it is ecessary to sped the geeral overhead for a secodary ulti-relay etwork o kowig the exact QoS requireet of the PU (i.e. the iterferece violatio probability for applyig the SIC policy. V. CONCLUSION I this paper, we focus o a ulti-relay CRN syste where the SS oly has the kowledge of the average CSI. A siple distributed ulti-relay selectio schee is proposed while two iterferece costrait policies (AIC ad SIC are cosidered for the selectio. The proposed selectio schee ideed protects the PU while providig ice perforaces for the secodary trasissios. Moreover, as the costrait fro the PU gets striget, i copariso to AIC policy, SIC policy iproves the syste perforace drastically by exploitig the spatial diversity of the ultiple relays. This is iterestig ad differet fro the traditioal relayig. Therefore, our work suggests to sped the geeral overhead for secodary etworks o kowig the exact QoS requireet of the PU. I additio, we aalyze the outage probability as well as the iial outage probability of the ulti-relay CRN. We fid that the iial outage probability of a CRN is oly subject to the iterferece liks ad that the sigle ad ultiple relays assisted secodary trasissios have the sae iial outage probabilities uder the AIC policy. At last, we show that although havig ucertai uber of forwardig relays ad ucertai iterferece to the PU, the proposed relay selectio schee let the perforace of ulti-relay secodary trasissios be absolutely superior to the sigle-relay. REFERENCES [1] A. Goldsith, S. afar, I. Maric, ad S. Sriivasa, Breakig spectru gridlock with cogitive radios: A iforatio theoretic perspective, Proc. IEEE, vol. 97, o. 5, May 2009, pp [2] Y. Ki ad H. Liu, Ifrastructure relay trasissio with cooperative io, IEEE Tras. o Vehicular Techology,, vol. 57, o. 4, ul. 2008, pp [3] A. 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