Unicast Barrage Relay Networks: Outage Analysis and Optimization

Transcription

1 Uncast Barrage Relay Networks: Outage Analyss and Optmzaton Salvatore Talarco, Matthew C. Valent, and Thomas R. Halford West Vrgna Unversty, Morgantown, WV, USA. TrellsWare Technologes, Inc., San Dego, CA, USA. arxv: v3 [cs.ni] 23 Oct 2015 Abstract Barrage relays networks (BRNs) are ad hoc networks bult on a rapd cooperatve floodng prmtve as opposed to the tradtonal pont-to-pont lnk abstracton. Controlled barrage regons (CBRs) can be used to contan ths floodng prmtve for uncast and multcast, thereby enablng spatal reuse. In ths paper, the behavor of ndvdual CBRs s descrbed as a Markov process that models the potental cooperatve relay transmssons. The outage probablty for a CBR s found n closed form for a gven topology, and the probablty takes nto account fadng and co-channel nterference (CCI) between adacent CBRs. Havng adopted ths accurate analytcal framework, ths paper proceeds to optmze a BRN by fndng the optmal sze of each CBR, the number of relays contaned wthn each CBR, the optmal relay locatons when they are constraned to le on a straght lne, and the code rate that maxmzes the transport capacty. I. INTRODUCTION A sgnfcant research nvestment has been made over the past decade on the topc of cooperatve communcatons for wreless networks n general, and moble ad hoc networks (MANETs) n partcular. The mprovements provded by t have been wdely noted n the lterature [1], [2]. See, for nstance, [3] [7] for example protocols nvolvng sngle-antenna mobles that cooperatvely transmt and/or receve. Barrage relay networks (BRNs) are a knd of cooperatve MANET desgned for use at the tactcal edge [8]. BRNs utlze tme dvson multple access (TDMA) and cooperatve communcatons as the bass of an effcent floodng protocol wheren packets rpple out from sources n ppelned spatal waves. In a BRN, smultaneous transmssons of the same packet are not suppressed but rather exploted for the resultng dversty gans. The spatal extent of uncast and multcast transmssons can be contaned n BRNs va controlled barrage regons (CBRs). Brefly, a CBR s establshed by dentfyng a rng of buffer nodes around a porton of the network contanng a source and ts destnaton(s). Wthn the CBR, the barrage floodng prmtve s used to transport data. The buffers suppress ther relay functon, thereby enablng multple CBRs to be actve at the same tme. The ncrease n network capacty afforded by ths spatal reuse was analyzed n [9] under standard nformaton theoretc assumptons. BRNs resemble the opportunstc large arrays (OLAs) ntroduced by Scaglone and Hong n [6]. Indeed, analogs to CBRs n OLAs have been proposed [10]. Both OLAs and BRNs explot the dversty that can be obtaned when multple nodes transmt dentcal packets. BRNs can be dstngushed from early OLA descrptons by the tme synchronzaton and recever sgnal processng employed n BRNs whch enables packets to be longer and data rates to be larger than the nverse of the maxmum relatve delay spread between cooperatng transmtters. The most mportant dfference between more recent OLA descrptons (e.g., [11]) and BRNs s the latter s use of autonomous cooperatve communcatons [7] as opposed to dstrbuted space-tme codng. Ths feature makes BRNs more suted for use n hghly dynamc, tactcal envronments. In ths paper, uncast data transport n a CBR s modeled as a Markov process. Snce relayng n BRNs s entrely opportunstc, a Markov chan s used to track every possble cooperatve lnk transmsson. The transton probabltes are evaluated for a gven network topology va a closed-form expresson for the outage probablty of each cooperatve transmsson [12], whch takes nto account path loss, Raylegh fadng, nose, and co-channel nterference (CCI) from adacent CBRs. A key analytcal challenge s that on the one hand, CCI nfluences whch nodes successfully receve and therefore affects the transton probabltes of the Markov process, yet on the other hand, the transton probabltes of the Markov process determne whch nodes transmt and therefore affect the nterference. Ths couplng between the transton and outage probabltes s solved usng an teratve approach, whereby the transton probabltes of the local CBR are used to seed the transton probabltes of neghborng CBRs, assumed to be smlarly confgured. On the best of our knowledge, Markov processes were also used to analyze a cooperatve system n other papers, but the nterference was neglected, e.g [13], [14], or, when t s consdered, smplfed assumptons, whch are accurate only for hgh densty networks, were made [15] n order to facltate the analyss. Havng establshed an analyss that can obtan exact expressons for the throughput of a gven network confguraton, ths paper proceeds to optmze the network wth respect to the transport capacty (TC), whch s a measure of forward progress. In partcular, the optmal CBR sze s dentfed, along wth the optmal confguraton (number and locaton) of relays wthn each CBR, and the optmal code rate for each transmsson. II. BARRAGE RELAY NETWORK A BRN s a smplfed abstracton of a tactcal MANET archtecture that s currently beng used n the feld [8], [9]. BRNs employ TDMA and cooperatve communcatons. All nodes utlze a common frame format that requres coarse

2 Fg. 1. Example of BRNs, composed of multple CBRs. Each CBR s composed a four-node network. slot-level synchronzaton. Ths can be accomplshed va a dstrbuted network tmng protocol when a satellte reference s unavalable. BRNs use autonomous cooperatve communcatons rather than dstrbuted beamformng or space-tme codng to () mnmze the communcatons overhead requred for cooperaton and () enable multple transmtters to partcpate n cooperatve lnks to multple destnatons wthout knowng whch other nodes are transmttng. As detaled n [7], each node apples an ndependent, random phase-dtherng pattern to each relayed message. Ths nduces a tme-varyng fadng characterstc at each recever. By changng the phase dther wthn a packet, destructve nterference across an entre packet can be avoded. When modern codes wth long block lengths are used, the phase-dtherng technque effectvely translates spatal dversty from multple transmtters nto tme dversty that can be captured va maxmal-lkelhood sequence estmaton (MLSE) equalzaton and teratve detecton. The BRN floodng prmtve works as follows. Durng the frst slot of a frame, the source broadcasts ts message. Durng the second slot, any node that receved and successfully decoded the ntal transmsson wll rebroadcast t. If more than one node concurrently transmts the message, then the superposton of ther sgnals s receved. The dverse sgnal components are effectvely maxmal-rato combned at each recever. Durng each subsequent slot, all messages that were successfully decoded durng the prevous slot are rebroadcast. The process repeats untl ether the destnaton s reached, no recever successfully decodes a transmsson, or a maxmum number of transmssons s reached. Packets thus propagate outward from the source va a decode-and-forward approach. To prevent relay transmssons from propagatng back towards the source, each node relays a gven packet only once. III. CONTROLLED BARRAGE REGIONS In a tradtonal network, a lnk s defned by a transmt/receve rado par that share a sutably relable pontto-pont communcatons channel. A uncast route s smply a seres of these lnks connectng a source-destnaton node par. In a BRN, however, lnks are cooperatve and comprse multple transmtters and recevers. A cooperatve path s made up of seres of cooperatve lnks between the source and destnaton nodes. A CBR s smply the unon of one or more such cooperatve paths wthn some subregon or zone of the overall network. CBRs afford a mechansm for uncast transport that s more robust than both tradtonal uncast routng and non-cooperatve multpath varants [16]. CBRs can be establshed by specfyng a set of buffer nodes around a set of cooperatng nteror nodes. Each buffer node acts as the source for transmssons nto a CBR and the destnaton for transmssons emanatng from wthn the CBR. External packets may enter the CBR only through a buffer node, and packets nternal to a CBR may only propagate to the rest of the network through a buffer node. In ths way, multple uncast transmssons may be establshed n dfferent portons of the network. Ths s referred to as spatal reuse. The buffer and nteror nodes correspondng to a gven sourcedestnaton par can be specfed va the broadcast of requestto-send (RTS) / clear-to-send (CTS) packets [16]. An example of lnear BRN s shown n Fg. 1. Each CBR n the example s made up of four nodes. The two nteror nodes of each CBR, R 1 and R 2, act as relays, whle the other two nodes are buffers, actng as the source S and destnaton D of the transmssons wthn the CBR, respectvely. In the case of a lnear BRN, each CBR s defned to be a segment of length d, here assumed to be the same for all CBRs. Let N be the number of nteror nodes n a CBR. It s assumed that the transmsson from a CBR s source to ts destnaton must take place wthn one rado frame or else an outage occurs 1. Snce each node transmts each packet at most once, the maxmum number of transmssons per frame s F = N + 1. Snce each transmsson requres one slot, the maxmum number of slots per rado frame s also N +1. Whle varable-length frames could be supported, ths would complcate the synchronzaton of neghborng CBRs. It s thus assumed that all CBRs have N nteror nodes and F = N+1. A packet may be broadcast by the source of a CBR durng only the frst tme slot of a frame. In deal channel condtons, t s possble that the destnaton receves the ntal source transmsson n the frst slot, though more generally, addtonal transmssons from the nteror nodes may be requred for successful recepton. The buffer node actng as the source for one CBR also acts as the destnaton for the prevous CBR. Because the nodes operate n a half-duplex mode, t s not possble for the buffer to smultaneously transmt nto one CBR whle recevng from another CBR. Thus, t s advsable to only allow one of the two CBRs servced by a gven buffer node to be actve;.e., ether the CBR that the buffer node s transmttng nto s actve or the CBR that the buffer node s recevng from s actve, but not both. More generally, ths suggests that CBRs can be dvded nto two types durng a partcular rado frame: 1) actve zones and 2) slent zones. 1 Note that n practce, BRNs often employ a fxed frame length of F = 4 and a fxed maxmum number of relays that s greater than F [8]. In such networks, spatal ppelnng enables multple packets to be actve wthn a gven CBR at the same tme. In the nterest of analytcal tractablty, ths paper consders only nterference from transmssons n dfferent CBRs. Ths s motvated by the observatons that () n practce, most tactcal uncast traffc s local so that CBRs wll typcally be less that 4 hops long and () nter-cbr nterference s more problematc than ntra-cbr nterference when F = 4.

3 Durng a gven rado frame, only the nodes n actve zones may transmt. As llustrated n Fg. 1, zones wll typcally alternate between actve zones and slent zones, though t s feasble to have even further spatal separaton between actve zones. A key advantage to alternatng between actve and slent zones s that t provdes spatal separaton between CCI, as the CCI n a gven actve zone s due to transmssons n other actve zones, whch are at least two zones away. IV. NETWORK MODEL Consder a BRN composed of K CBRs. The BRN s a set of M moble rados or nodes X = {X 1,...,X M }. The varable X represents both the th node and ts locaton. In the followng, X s typcally used to denote a transmttng node and X used to descrbe a recevng node. Durng the t th tme slot of a frame,t {1,...,F}, nodex transmts wth probabltyp (t). Any node n a slent zone wll be characterzed by p (t) = 0 for all slots of the frame. Snce the source of an actve CBR may only transmt n the frst slot of the frame, t s characterzed by p (1) = 1 and p (t) = 0 for t {2,...,F}. Let X (t) X be the set of cooperatng or barragng nodes that transmt dentcal packets to X durng the t th tme slot, X (t) the number of barragng transmtters, and G (t) the set of the ndexes of the nodes n X (t). Nodes wthn the same CBR as X but not n X (t) do not transmt;.e., there s no ntra-cbr nterference. However, nodes n other actve CBRs are potental sources of nterference, and each such node wll generate CCI wth probablty p (t). The value of p (t) for each of these nodes wll depend on the dynamcs of the protocol, as wll be descrbed shortly. When X transmts, t broadcasts a sgnal whose average receved power n the absence of fadng s P at a reference dstanced 0. For ease of exposton, t s assumed that all thex that transmt do so wth a common power P = P. However, ths assumpton could be relaxed at the expense of slghtly complcatng the notaton and analyss. X s power at recever X durng tme slot t s ρ (t), = Pg (t), f(d,) (1) where g (t), s the power gan due to fadng, d, = X X s the dstance from X to X, and f( ) s a path-loss functon. The {g (t), } are ndependent and exponentally dstrbuted wth unt mean, correspondng to Raylegh fadng. It s assumed that the {g (t), } reman fxed for the duraton of a tme slot, but vary ndependently from slot to slot. For d d 0, the path-loss functon s expressed as the attenuaton ( ) power law α d f (d) = (2) where α > 2 s the attenuaton power-law exponent, and d 0 s suffcently large that the sgnals are n the far feld. A commonly accepted approach to analyzng dversty combnng s to assume that the nterference realzatons n the dfferent branches are the same;.e., the nterference s fully-correlated among the branches (cf., [17]). Whle strctly speakng, ths condton s not always met, the assumpton d 0 makes the analyss manageable. Furthermore, as a worstcase scenaro, the analyss makes t possble to obtan an upper bound on outage probablty, whch happens to be tght [18]. Under ths assumpton and by usng (1) and (2), the nstantaneous sgnal-to-nterference-plus-nose rato (SINR) at moble X durng slot t s g (t) k, Ω k, γ (t) = k G (t) Γ 1 + G (t) I (t) g (t), Ω, where Γ = d α 0P/N s the sgnal-to-nose rato (SNR) of a unt-dstance transmsson when fadng s absent, Ω, = d α, s the relatve path gan, I (t) s a Bernoull varable ndcatng the X s a source of nterference durng slot t, and P[I (t) = 1] = p (t). Letβ denote the mnmum SINR requred byx for relable recepton and Ω = {Ω 1,,...,Ω M, } represent the set of relatve path gans from all {X } to X. An outage occurs when the SINR falls below β. Condtonng on the path gans Ω and the set of barragng nodes X (t), the outage probablty of moble X durng slot t[ s ] ǫ (t) = P γ (t) β Ω,X (t). (4) Because t s condtoned on Ω, the outage probablty depends on the partcular network realzaton, whch has dynamcs over tmescales that are much slower than the fadng. From [12], the outage probablty) n Raylegh fadng s ǫ (t) = 1 k G (t) / G (t) exp ( β Ω k, Γ ( Ω k, +β 1 p (t) s G (t),s k ) Ω, Ω k, Ω k, Ω s, (3) Ω k, +βω,. (5) The outage probablty s condtoned on the node locatons (represented by the {Ω, }) and by the partcular set of barragng transmtters (represented by X (t) ). Notce that f X (t) = 1, (5) concdes wth (30) n [19] and (13) n [20]. V. CBR AS A MARKOV PROCESS Consder a sngle packet beng transmtted through a sngle CBR composed of a source (denoted S), a destnaton (D) and N relays ({R 1,...,R N }). At the boundary between any two tme slots, each node can be n one of the three followng states, whch we refer to as the node state: Node state 0: The node has not yet successfully decoded the packet. Node state 1: It has ust decoded the packet receved durng prevous slot, and t wll transmt on next slot. Node state 2: It has decoded the packet n an earler slot and t wll no longer transmt or receve that packet. The state of the CBR s the concatenaton of the states of the ndvdual nodes wthn the CBR. Hereafter, we refer to ths as the CBR state. The CBR state can be compactly represented by a vector of the form[s,r 1,...,R N,D] contanng the states of the source, relays, and destnaton.

4 The behavor of the CBR can be descrbed as an absorbng Markov process, whch descrbes the dynamcs of how the CBR states evolve over a rado frame. Defne s = {s 1,s 2,...,s }, whch s the state space of the process composed of Markov statess. An absorbng Markov process s characterzed by τ transent states and r absorbng states. A state s of a Markov chan s called absorbng f once n that state, t s mpossble to leave t, whle a state that s not absorbng s called a transent state. We note that t s often possble to group several CBR states nto a sngle Markov state. For nstance, we group all of the CBR states correspondng to successful decodng by the destnaton nto a sngle absorbng Markov state. The probablty that the process moves from (Markov) state s to state s s denoted by p, and the probabltes {p, } are called transton probabltes. Fg. 2 shows the Markov chan for a CBR wth N = 2 relays (a four-node network). The CBR states are shown as a 4-element vector, along wth the transton probabltes. The message s successfully delvered whenever the destnaton receves the message, whch s ndcated by a 1 n the last poston of the CBR-state vector. Such states are marked n green, and hereafter we refer to ths condton as a CBR success. The transmsson fals whenever all entres n the CBR-state vector are ether 0 or 2, ndcatng that none of the nodes that have not yet transmtted have successfully receved the message. Such states are marked n red, and hereafter we refer to ths condton as a CBR outage. Each transent state n the Markov process corresponds to a sngle CBR state, and such states are numbered 1 through 6 n the dagram. The 4 CBR states that correspond to a CBR outage are collapsed nto a sngle absorbng state of the Markov process (state 7), and smlarly the 9 CBR states correspondng to a CBR success are collapsed nto a sngle absorbng state of the Markov process (state 8). Consder how the CBR state may change from tme t to tme t+1. All nodes at tme t wth node state 1 transmt, so at tme t + 1, those nodes wll now be n state 2. All nodes that have node state 0 at tme t wll receve the transmsson, so at tme t + 1 these nodes wll be ether n state 1 (f the transmsson was successful) or 0 (f the transmsson faled). Snce the channels from transmttng to recevng nodes are ndependent, the state transton probablty s the product of the ndvdual transmsson probabltes. For nstance, the probablty of gong from state [1100] to [2210] s the product of the probablty that R 2 successfully decodes the ont transmsson (from S and R 1 ) and the probablty that D does not successfully decode the transmsson. The ndvdual probablty of successful decodng at each node s found usng the outage probablty of (5). Whle there s a one-to-one correspondence between CBR states and the transent Markov states, there are several CBR states assocated wth each of the two absorbng Markov states. Thus, for each absorbng Markov state, the probablty of transtonng nto t s equal to the sum of the probabltes of transtonng nto the consttuent CBR states. Fg. 2. Markov chan for a CBR composed of four nodes (N = 2). Transent states are n whte, whle the CBR success absorbng state s n green and the CBR outage absorbng state s n red. Each of the two absorbng states s the unon of several CBR states. The Markov transton probabltes are placed nto a state transton matrx P, whose (,) th entry s {p, }. The Markov states are ordered such that the τ transent states and ndexed before the r absorbng states. Wth ths orderng, the state transton matrx assumes the followng canoncal form [21] [ ] Q R P = (6) 0 I whereqs the τ τ transent matrx, R s the τ r absorbng matrx and I s an r r dentty matrx. Let b, be the probablty that the process wll be absorbed n the absorbng state s f t starts n the transent state s. The absorbng probablty b, s the (,) th entry of the matrx B, whch can be computed by (see, for nstance [21]) B = NR (7) where N = (I Q) 1 s the fundamental matrx. The two absorbng states are ndexed so that the frst absorbng state corresponds to a CBR outage, whle the second corresponds to a CBR success. Snce the process always starts n state s 1, t follows that b 1,1 s the CBR outage probablty (whch s ndcated by ǫ CBR ) and b 1,2 s the CBR success probablty (whch s ˆǫ CBR = 1 ǫ CBR ). Example # 1 Consder the four-node CBR whose nodes are equally spaced along a lne, as shown n the nset n Fg. 3. In ths example, the CCI from adacent CBRs s neglected, and the path-loss exponent s α = 3.5. Fg. 3 shows the CBR outage probablty as a functon of the SNR Γ for three values of the threshold β. Sold curves are obtaned analytcally accordng to the methodology of ths secton, whle the markers correspond to a smulaton usng the methodology ntroduced n [22]. In partcular, the smulaton works by frst computng the outage probabltes usng (5), usng these probabltes to determne the state-transton probabltes, and smulatng each state transton by drawng a random number.

5 10 0 Analytcal results Smulatons 10 0 Γ= 0 db 10 1 CBR outage probablty S R 1 R 2 D β=6 db β=3 db β=0 db Γ (n db) Fg. 3. CBR outage probablty as a functon of Γ when CCI s neglected. The sold lnes are obtaned analytcally whle the markers are obtaned from Monte Carlo smulaton. The network topology s a four node lne network shown n the nset, and α = 3.5. The analytcal results concde wth the smulatons as shown n Fg. 3, and any dscrepancy between the curves can be attrbuted to the fnte number of Monte Carlo trals (10 7 trals were executed per SNR pont). VI. INTER-CBR INTERFERENCE Let P k represent the transton matrx of the k th CBR. The transton probabltes n P k are computed usng (5), whch depend on the {p (t) } assocated wth nodes n other CBRs. However, each p (t) represents the probablty that node X transmts durng tme slot t, whch can be found from the P k of the correspondng CBR. Thus, nterference causes a lnkage between the transton probabltes n the gven CBR and the transmsson probabltes of adacent CBRs. Whle ths lnkage could be handled by consderng the overall BRN as one large Markov chan, ths s a cumbersome soluton as the number of states grows exponentally wth the number of nodes. A more effcent approach s an teratve one, whch alternates between computng the {p (t) } and the {P k }. Let S,t be the set of states assocated wth tme slot t and labeled wth a 1 n the poston correspondng to X. These are the states for whch X transmts durng tme slot t. The can be found by addng the probabltes of beng n each of the states n S,t, gven that the system starts from ntal state s 1. The probablty of beng n state s S,t s found by multplyng the probabltes of all transtons n the Markov chan leadng from state s 1 to state s Let P k [ t ] represent the state transton matrx for the k th probablty p (t) CBR correspondng to teraton t, and smlarly let p (t) [ t ] represent the transmsson probablty for the th user durng the t th tme slot correspondng to teraton t. The teratve method can be descrbed as follows: 1) Intalzaton: Set t = 0 and ntalze p (t) [0] = 0,,t, whch corresponds to neglectng nterference. Then compute each P k [0] usng the ntal {p (t) [0]}. CBR outage probablty Γ= 10 db α=3 α=3.5 α= Number teratons Fg. 4. CBR outage probablty for the k th CBR as functon of the number of teratons used. Set of curves at the top: Γ = 0 db. Set of curves at the bottom: Γ = 10 db. Each set of curves s obtaned usng three values of α. 2) Recurson: Increment t. Evaluate each P k [ t ] usng the {p (t) [ t 1]} from the prevous teraton. 3) Decson: Halt the process f P (t) k P (t 1) k F < ξ, k, where the operator F s the Frobenus norm and ξ s a tolerance. Otherwse, go back to step 2. Example # 2 The approach s further smplfed f t s assumed that the BRN s comprsed of an nfnte cascade of CBRs, all havng dentcal topology. The P k and the set of {p (t) } wll be the same n all actve zones. Because of ths, performance can be descrbed by a typcal CBR. In ths example, assume that each CBR s composed of four nodes whch are confgured as n Example #1. The typcal CBR s analyzed assumng t receves CCI from ust the two closest adacent actve zones (the nterference from dstant zones s neglected). Fg. 4 shows the CBR outage probablty for a typcal CBR as a functon of the number of teratons used n the algorthm descrbed above. Curves are shown for two values of Γ and three values of α. All sx curves are obtaned wth β = 6 db. Fg. 4 shows that the CBR outage probablty ncreases as a functon of the number of teratons, and that the teratve method converges rapdly after very few teratons. VII. NETWORK OPTIMIZATION The transport capacty [23] of an ad hoc network quantfes the rate that data can be relably communcated over a unt of dstance, and s typcally expressed n unts of meter-bts-persecond. In the context of a BRN, the TC of a typcal CBR s Υ = Td (8) where T s the throughput of the CBR and d s the dstance between the CBR s source and destnaton. From [24], the throughput can be wrtten as T = (1 ǫ CBR) R (9) 2F where R s the code rate expressed n unts of bts per channel use (bpcu) and the factor 2 n the denomnator s

6 TABLE I OPTIMIZATION RESULTS. Γ (n db) CCI R N d Optmal nodes locaton for a lne network Υ opt a consequence of the buffer nodes operatng n half-duplex mode, or, equvalently, due to alternatng between actve and slent zones. Substtutng (9) nto (8), and relatng the rate and the SINR threshold by R = log 2 (1+β), whch s the Shannon capacty for complex dscrete-tme AWGN channels, yelds Υ = d ˆǫ CBR 2(N +1) log 2 (1+β). (10) Let X k be a vector contanng the poston of the N relays n the k th CBR. The goal of the network optmzaton s to determne the set Θ = (X k,r,n,d) that maxmzes the TC. Because the TC s nonlnear n Θ and the search space s mult-dmensonal, effcently fndng a soluton s a challenge. However, by ntally usng an exhaustve search, we have confrmed that the optmzaton surface s convex over the set (R,N,d). Thus, the optmzaton can be solved through a convex optmzaton over (R, N, d) combned by a stochastc optmzaton over X k. It follows that an effcent approach to fndng the optmal set Θ s as follows: 1) For each of the parameters N, R and d, select a par of endponts and a mdpont (3 3 = 27 ponts). The endponts should ntally be far enough apart to guarantee that the optmal pont les wthn the endponts. 2) Defne a reference vector X ref (N,d), whch s a vector X k for the par(n,d) and a reference set {X ref (N,d)} contanng the reference vectors of all (N, d) consdered by the algorthm. Create three vectors {X k } for each par of (N,d). When (N,d) s chosen for the frst tme, the frst vector of {X k } s obtaned by placng the N relays on the length-d lne connectng the source and the destnaton and ths vector s added to the reference set, consttutng X ref (N,d) for the par (N,d) chosen. The placement s done such that relays are separated by a mnmum dstancer s = d 3N, but are otherwse unformly dstrbuted along the lne. If the par (N,d) has been already used, the frst vector of {X k } s set to be equal to X ref (N,d). The other two vectors are obtaned by mutatng X ref (N,d) as follows: a) For each relay n X ref (N,d), draw a Bernoull random varable ndcatng f the relay should be moved or f t should reman n ts current locaton. b) If the node X s to be moved, the drecton (rght or left) of movement s selected wth equal probablty. The new poston s obtaned by movng the current poston n the selected drecton by a dstance = d n N, where ntally n = 2. 3) Compute the TC for the endponts and the mdponts of the nterval of one of the parameters n the set (R,N,d) and for the dfferent sets of locatons of the relays. For each par (N,d) chosen, determne the vector X k that provdes the hgher TC and replace the X ref (N,d) n the reference set wth ths vector. 4) For the same parameter of the set (R,N,d) used n step 3, move the mdpont of the nterval towards the endpont that gves hgher TC. For each par (N,d), generate the three sets of postons as n step 2. 5) Repeat step 3 and 4 recursvely for all the parameters n the set (R,N,d) and gradually reduce for one parameter at the tme the range between the endponts and the mdpont untl a certan tolerance s reached. If the optmal TC remans the same as n the prevous teraton n s ncreased by one, untl a certan value s acheved. 6) If the optmal TC remans the same as n the prevous teraton, for all the parameters n the set (R, N, d) t s not possble to further reduce the range between the endponts and the mdpont, snce a gven tolerance s acheved, and n reaches ts maxmum value, the algorthm stops. Usng the methodology descrbed above, optmzaton results were obtaned under the same assumptons that were used n Example #2. In partcular, t s assumed that the BRN extends nfntely and CBRs are composed of lne networks wth the same number of relays whch are all equally postoned. For each CBR the only CCI consdered are the ones from the two closest adacent actve zones (the nterference from farther actve zones s neglected, snce they are at least at 3d dstance away). Although ths assumptons can be relaxed

7 usng the analyss and methodology proposed along ths paper, n ths secton they are used for smplcty of exposure. Table I shows the optmal values of N, R, d and the optmal locaton for the N relays. Optmzaton results are provded for three values of Γ and when the CCI s neglected as well as when they are taken nto account. Vertcal red lnes ndcate the optmal poston of the mobles for Γ = 0 db, and they are used to facltate the graphcal analyss and comparson of the optmal locaton of the mobles for dfferent scenaros. For all scenaros consdered, the path loss exponent s fxed to α = 3.5. Table I emphaszes that when there s no CCI, the optmal confguraton for a cooperatve BRN s to use short CBRs characterzed by a pont-to-pont conventonal communcaton, whle fve relays dstrbuted over a larger CBR allow to acheve the optmal TC when there s CCI from the adacent CBRs. As expected, the optmal TC ncreases by ncreasng Γ, snce the lnks experence a more favorable channel. Furthermore, the optmal TC ncreases gong from a scenaro n whch there s CCI to one n whch CCI s neglected, and ths s more promnent as Γ ncreases snce the network becomes more nterference senstve. In agreement wth [25], an ncrease n Γ produces a shft towards the source of the optmal poston of the relays, whch ncreases the lkelhood to cooperate among themselves. A more favorable condton of the channel produces an ncrement n both the optmal code rate R and n the optmal dmenson of the CBR. VIII. CONCLUSION Ths paper presents a new analyss and optmzaton for uncast n a BRN. A BRN s analyzed by descrbng the behavor of each consttuent CBRs as a Markov process. The transton probabltes are computed by usng a new closed form expresson for the outage probablty, whch takes nto account the path loss, Raylegh fadng, and nterference. 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