Economic-Robust Transmission Opportunity Auction in Multi-hop Wireless Networks

Transcription

1 Economc-Robust Transmsson Opportunty Aucton n Mult-hop Wreless Networks Mng L, Pan L, Mao Pan, and Jnyuan Sun Department of Electrcal and Computer Engneerng, Msssspp State Unversty, Msssspp State, MS Department of Computer Scence, Texas Southern Unversty, Houston, Texas Department of Electrcal Engneerng and Computer Scence, Unversty of Tennessee, Knoxvlle, TN Emal: ml845@msstate.edu, l@ece.msstate.edu, panm@tsu.edu, jysun@eecs.utk.edu Abstract The rapd growth of wreless devces and servces exacerbates the problem of spectrum scarcty n wreless networks. Recently, spectrum aucton has emerged as one of the most promsng technques to enhance spectrum utlzaton and mtgate ths problem. Although there exst some works studyng spectrum aucton, most of them are desgned for sngle-hop communcatons, and t s usually not clear whom a wnnng user communcates wth. Moreover, most prevous aucton schemes only focus on satsfyng the ncentve compatblty property, also called truthfulness, but gnore another two crtcal propertes: ndvdual ratonalty, and budget balance. Thus, they may not be economc-robust. In ths paper, we propose a transmsson opportunty aucton scheme, called TOA, whch can support mult-hop data traffc, ensure economc-robustness, and generate hgh revenue for the auctoneer. Specfcally, n TOA, nstead of spectrum bands as n tradtonal spectrum aucton schemes, users bd for transmsson opportuntes TOs). A TO s defned as the permt of data transmsson on a specfc lnk usng a certan band,.e., a lnk-band par. The TOA scheme s composed of three procedures: TO allocaton, TO schedulng, and prcng, whch are performed sequentally and teratvely untl the aforementoned goals are reached. We prove that TOA s economc-robust, and conduct extensve smulatons to show ts effectveness and effcency. I. INTRODUCTION The past few years have wtnessed substantal growth of wreless devces and servces, whch, on the other hand, makes spectrum an even more scare resource n wreless networks. Tradtonal spectrum allocaton was conducted n a statc manner, resultng n neffcent spectrum utlzaton. Recently, spectrum sharng through a dynamc real-tme secondary spectrum aucton market has been proposed to enhance spectrum utlzaton and mtgate the problem of spectrum scarcty. In such a market, a spectrum owner or prmary user PU) leases ts dle lcensed spectrum bands to secondary users SUs) to gan profts. SUs, who do not have ther own spectrums but need to delver data traffc, compete for spectrum bands and pay for them f they succeed n the spectrum aucton. In the lterature, there have been some works studyng spectrum aucton n wreless networks. Unfortunately, most of them [1] [11] are only sutable for sngle-hop data transmsson. In partcular, n these schemes, each user bds and s allowed to use the purchased spectrum for communcatons f t wns. However, there are two problems: frst, ths s only Ths work was partally supported by the U.S. Natonal Scence Foundaton under grants CNS CAREER), ECCS , CNS , and HRD for sngle-hop communcatons, and second, t s not clear whom a wnnng user communcates wth the recever s not clearly specfed). Thus, the network performance can be poor. Zhu et al. [12] dscuss spectrum aucton for mult-hop data delvery. But they assume that each secondary network only has one flow, and do not consder tme doman schedulng when utlzng the spectrums. Moreover, n addton to fulfllng SUs traffc demands, aucton schemes need to satsfy certan economc propertes. Specfcally, ncentve compatblty IC) also called truthfulness or strategy-proof), ndvdual ratonalty IR), and budget balance BB) are three of the most crtcal propertes n aucton desgn. An aucton s called economc-robust [8], [13] f all these three propertes are preserved. It has been shown both theoretcally and practcally that an aucton could be vulnerable to market manpulaton and produce very poor outcomes f those propertes are not guaranteed [14]. Most prevous aucton schemes focus on IC only, but do not necessarly satsfy the other two propertes. In ths paper, we am to desgn an economc-robust aucton scheme for mult-hop wreless networks. In partcular, we consder an aucton market where a PU acts as an auctoneer and leases ts dle lcensed spectrum bands to some SUs, whch are deployed by a secondary servce provder SSP) to fulfll certan purposes, such as data delvery, data collecton, and object trackng. SUs may need to transmt data to ther destnatons that are multple hops away. To delver the data traffc, the SSP asks all the SUs to submt bds to the auctoneer. If some SUs wn, they pay a prce to the auctoneer and relay data traffc for each other usng the spectrum purchased. The SSP fnally pays back all the wnnng SUs, and lets them gan some profts so that they are motvated to partcpate n the aucton. To support mult-hop data traffc, ensure economcrobustness, and generate hgh revenue for the auctoneer, we propose a transmsson opportunty aucton scheme, called TOA. In TOA, nstead of spectrum bands as n tradtonal spectrum aucton schemes, SUs bd for transmsson opportuntes TOs). A TO s defned as the permt of data transmsson on a specfc lnk usng a certan band,.e., a lnk-band par. The TOA scheme s manly composed of three procedures: TO allocaton, TO schedulng, and prcng. These three procedures are performed sequentally and teratvely untl the aforementoned goals are reached. Specfcally, n TO allocaton, n

2 each teraton the auctoneer solves a TO allocaton TO-AL) optmzaton problem to fnd out the lnk-band pars.e., TOs) that can be actve at the same tme and have the hghest total bd. It consders the set of the transmtters n these TOs as a wnnng vrtual bdder group VBG). In TO schedulng, the auctoneer formulates a mnmum length schedulng problem, called TO schedulng TO-SC), to see f the wnnng VBGs found so far can support the traffc demand n the network by explorng schedulng n both tme and frequency domans) and routng. If the mnmum schedulng length s larger than 1, t means that the current wnnng VBGs cannot support the traffc demand, and the auctoneer needs to fnd another VBG through TO-AL agan. Otherwse, the auctoneer can then determne the clearng prce for each wnnng VBG and SU, and computes ts own revenue. The auctoneer fnally chooses the teraton,.e., the wnnng VBGs, that can generate the hghest revenue among the results t obtans. Moreover, notce that our aucton scheme TOA s developed based on second-prce sealed-bd aucton [15]. We prove that TOA s economc-robust for VBGs and ndvdual SUs. The rest of ths paper s organzed as follows. We dscuss related work n Secton II. The problem formulaton s presented n Secton III. We detal the proposed transmsson opportunty aucton TOA) scheme n Secton IV, and prove the economcrobustness of TOA n Secton V, respectvely. We conduct smulatons n Secton VI to evaluate the performance of TOA. We fnally conclude ths paper n Secton VII. II. RELATED WORK Aucton has been employed by the Federal Communcatons Commsson FCC) to effcently allocate spectrum resources [16]. Based on ths dea, some works propose to apply aucton to spectrum sharng n wreless networks. Kloeck et al. [1] consder a mult-unt sealed-bd aucton for effcent spectrum allocaton. Huang et al. [11] propose an aucton mechansm whch allows users to bd for ther transmsson power to effcently share the spectrum. Gandh et al. [2] desgn an aucton scheme consderng spectrum reuse n wreless networks. However, all the above works gnore the possble strategc behavor of bdders. Zhou et al. [10] then propose a truthful spectrum aucton scheme VERITAS wth greedy channel assgnment and crtcal value based prcng. Ja et al. [3] dscuss how to generate maxmum expected revenue, whch s an alternate goal of maxmum socal welfare, n spectrum aucton whle satsfyng the truthfulness property. In order to further mprove the expected revenue, Al-Ayyoub et al. [6] desgn a color-based channel allocaton scheme. Takng farness n channel allocaton nto account, Gopnathan et al. [7] develop a truthful aucton protocol by applyng lnear programmng technques to balance the socal welfare and max-mn farness n secondary spectrum markets. In addton to sngle-sded aucton, some works employ double aucton n spectrum market, where multple spectrum owners compete wth each other to sell dle spectrums for proft. Zhou and Zheng [8] propose a framework TRUST for truthful double spectrum aucton enablng spectrum reuse. Wang et al. [9] desgn a truthful double aucton scheme consderng that spectrums are tradable only wthn ther lcensed areas. However, these two works assume each seller can only sell one channel and each buyer can buy one channel at most. Ths lmts the utlty of both buyers and sellers, as well as the revenue of the auctoneer. Most mportantly, all the above works are only sutable for sngle-hop data transmsson. Although Zhu et al. [12] dscuss spectrum aucton for mult-hop data delvery, they assume that each secondary network only has one flow, and do not consder tme doman schedulng when utlzng the spectrums. A. System Model III. PROBLEM FORMULATION We consder an aucton market where a spectrum owner or prmary user PU) acts as an auctoneer and leases ts dle lcensed spectrum bandsm = {1,2,...,m,...,M} to secondary users SUs) N = {1,2,...n,...,N}. The SUs are deployed by a secondary servce provder SSP) to fulfll some purposes such as data delvery, data collecton, and object trackng. In ths study, we assume that each SU s equpped wth one rado, whch means t cannot transmt and receve smultaneously. Suppose there are a set of L = {1,2,...,l,...L} sessons n the secondary network. We let sl), dl), and rl) denote the source node, destnaton node, and traffc demand of sesson l L, respectvely. To delver the traffcs, the SSP asks all the SUs to submt bds to the auctoneer for transmsson opportuntes TOs), each of whch s defned as the permt of data transmsson on a specfc lnk usng a certan band,.e., a lnk-band par. If some SUs wn, they pay a prce to the auctoneer and relay data traffc for each other wth obtaned TOs. The SSP fnally pays back all the wnnng SUs and lets them gan some profts. Gven the network topology, the PU can construct a conflct graph denoted by GV,E), where V s the vertex set and E s the edge set. In partcular, each vertex corresponds to a lnk-band par denoted by,j),m), where N, j T m, and m M. Here, T m s the set of SUs wthn SU s transmsson range on band m. Besdes, two vertces n V are connected wth an undrected edge f the correspondng lnk-band pars nterfere wth each other,.e., f any of the followng condtons s true: The recevng SU n one lnk-band par s wthn the nterference range of the transmttng SU n another lnkband par, gven that the both of them are usng the same band; The two lnk-band pars have at least one node n common. In ths conflct graph, an ndependent set IS) s a set n whch each element s a lnk-band par standng for a transmsson, and all the elements or transmssons) can be carred out successfully at the same tme. If addng any more lnk-band pars nto an IS results n a non-ndependent one, ths IS s defned as a maxmum ndependent set MIS). We denote the set of all the MISs by I = {I 1,I 2,...I q,...,i Q }, where Q = I, and I q V for 1 q Q. We wll show later that we do not really need to fnd all the MISs. We denote the

3 MIS I q s tme share out of unt tme 1) to be actve by λ q λ q 0). Therefore, f all the data traffcs n the network can be supported, we have Q λ q 1. Besdes, we let c m j I q) be the nstantaneous transmsson rate of the lnk-band par,j),m) when MIS I q s actve. Thus, c m j I q) s equal to 0 when,j),m) I q, and the capacty of,j),m),.e., c m j, otherwse, whch wll be ntroduced soon. Moreover, we denote an SU s real valuaton of and bd prce for unt nstantaneous transmsson rate by v and p, respectvely. Note that v can be the rewards SU receves from the SSP f t wns. In an aucton, SUs submt ther bds p s n a sealed manner, so that no one has access to any nformaton about the others bds. After the auctoneer receves all the bds, t dvdes the bdders nto dfferent vrtual bdder groups VBGs), each of whch s the set of transmtters of all lnk-band pars n one MIS. Smlarly, we denote the set of all the VBGs by G = {G 1,G 2,...,G q,...,g Q }. Obvously, we have I = G = Q. Then, wth SUs unt valuatons and unt bd prces, the auctoneer can then calculate SUs equvalent valuatons of and equvalent bds for dfferent TOs. Notce that n one MIS, any SU can have at most one TO. Let v q denote SU s equvalent valuaton of the TO t can obtan from I q. Then, we can have v q = v j T c m m j I q). Accordngly, SU s equvalent bd for the same TO, denoted by b q, can be calculated as bq = p j T c m m j I q). The auctoneer consders VBGs as vrtual bdders n the aucton.the vrtual bd from a VBG s the sum of all SUs equvalent bds n the group. In partcular, let B q denote the vrtual bd from VBG G q 1 q Q). Then, we have B q = G q b q. Denote by B q the vector of the vrtual bds from the other VBGs G/G q. Thus, the entre bd prce set, denoted by B, s B = B q,b q ). Besdes, denote by G W the set of the wnnng VBGs, and I W the set of the correspondng wnnng MISs. Notce that an SU can be nvolved n multple wnnng VBGs. Thus, SU s total equvalent bd for the TOs t obtans, denoted by b, s equal to b = G q G W b q. Note that b q = 0 f G q. B. Objectve of Aucton Desgn The desgn of aucton schemes heavly depends on the desred propertes. In ths paper, we assume that all SUs are strategc n the sense that they may manpulate ther bds to obtan favorable outcomes. Denote by c N ) the clearng prce the auctoneer charges SU for unt nstantaneous transmsson rate. We am to desgn an aucton scheme that can satsfy three of the most mportant economc requrements: Incentve Compatblty IC), Indvdual Ratonalty IR) and Budget Balance BB), whch are defned as follows: Incentve Compatblty IC): The utlty functon of SU N ), s a functon of all the bds: u p,p ) = [G q G W j T c m m j I q) ] v c ), f wns wth unt bd p, 0, otherwse. 1) where p denotes the vector of bds from the other SUs. Thus, an aucton s IC f for any SU N ) wth any p v whle the others bds are fxed, we have u p,p ) u v,p ). 2) Indvdual Ratonalty IR): An aucton s IR, f no bdder s charged hgher than ts bd n the aucton,.e., c p for all N. Budget Balanced BB): To make the aucton selfsustaned wthout any external subsdes, the generated revenue of the auctoneer,.e., the PU, s requred to be non-negatve. We say an aucton s economc-robust [8], [13] f t s ncentve compatble, ndvdually ratonal and budget balanced. Snce n ths paper, we consder that the PU leases ts own dle spectrum bands wthout causng qualty degradaton to ts own servces, the PU s revenue s the total payment receved from the wnnng SUs, whch s always non-negatve. Thus, our aucton scheme s always BB. We wll focus on achevng IC and IR n our aucton scheme desgn. Moreover, an aucton scheme s sad to be system-effcent f the revenue of auctoneer s maxmzed. Unfortunately, accordng to the mpossblty theorem demonstrated n [17], an aucton cannot be economc-robust and system-effcent at the same tme. Therefore, n ths study we am to desgn an economc-robust aucton, whle try to generate hgh revenue for the auctoneer. C. Transmsson Opportunty s Capacty Suppose the power spectral densty of SU on band m s a constant and denoted by P m. A wdely used model [18] for power propagaton gan between SU and SU j, denoted by g j, s g,j = C [d,j)] γ, where and j also denote the postons of SU and SU j, respectvely, d,j) refers to the Eucldean dstance between and j, γ s the path loss factor, and C s a constant related to the antenna profles of the transmtter and the recever, wavelength, and so on. We assume that the data transmsson s successful only f the receved power spectral densty at the recever exceeds a threshold PT m. Meanwhle, we assume nterference becomes non-neglgble only f t produces a power spectral densty over a threshold of PI m at the recever. Thus, the transmsson range of SU on band m s R,m T = CP m /PT m)1/γ, whch comes from CR,m T ) γ P m = PT m. Smlarly, based on the nterference threshold PI mpm I PT m ), the nterference range of SU s R,m I = CP m /PI m)1/γ, whch s no smaller than R,m T. Thus, dfferent SUs may have dfferent transmsson ranges/nterference ranges on dfferent channels wth dfferent transmsson power. In addton, accordng to the Shannon-Hartley theorem, f SU sends data to SU j on lnk,j) usng band m, the capacty of the TO,.e., lnk-band par,j),m), s c m j = W m log 2 1+ g jp m η ), 3) where η s the thermal nose at the recever. Note that the denomnator nsde the log functon only contans η. Ths s because of one of our nterference constrants,.e., when node s transmttng to node j on band m, all the other neghbors of node j wthn ts nterference range are prohbted from

4 usng ths band. We wll address the nterference constrants n detals n the followng secton. IV. TRANSMISSION OPPORTUNITY AUCTION In ths secton, we ntroduce our proposed transmsson opportunty aucton scheme, called TOA. Recall that n the network there are SUs who need to delver data traffc to ther destnatons that are multple hops away. Thus, the objectve of TOA s to choose MISs, and hence VBGs, whch can support such traffcs and brng hgh revenue to the auctoneer. Meanwhle, TOA should be economc-robust. In general, the TOA scheme s composed of three procedures: TO allocaton, TO schedulng, and prcng. These three procedures are performed sequentally and teratvely untl our goals are reached. In what follows, we detal the desgn of the three procedures, respectvely. A. Transmsson Opportunty Allocaton At the begnnng of TO aucton, each SU N ) submts ts unt bd prce p to the auctoneer. Then, as mentoned before, the auctoneer can calculate SU s equvalent bds b q for the TO t obtans from a VBG G q, and the vrtual bd from G q, whch s B q = b q = c m j I q) p. 4) G q G q Note that as explaned above, an aucton cannot be economcally robust and system-effcent at the same tme, and n ths study we am to desgn an economc-robust aucton. Thus, the objectve of TO allocaton s to fnd out one wnnng MIS, whch corresponds to a wnnng VBG, that maxmzes the vrtual bd B q n each teraton n a monotonc manner. In partcular, we wll fnd the VBG wth the hghest vrtual bd n the frst teraton, the one wth the second hghest vrtual bd n the second teraton, and so on and so forth untl the teraton ends. Such VBGs MISs) are consdered as wnnng VBGs MISs) denoted byg W I W ). We wll show n SectonV-A that a monotonc TO allocaton procedure s crtcal n achevng the IC and IR propertes. Before formulatng the optmzaton problem, we frst lst several constrants as follows. Notce that n the procedure of TO allocaton, we do not assume that we know all the MISs, fndng whch s n fact an NP-complete problem. We denote { s m 1, f can transmt to j on band m, j = 0, otherwse. Snce an SU s not able to transmt to or receve from multple SUs on the same frequency band, we have 1, and 1. 5) s m j { } Besdes, an SU cannot use the same frequency band for transmsson and recepton, due to self-nterference at physcal layer,.e., s m j + s m jq 1. 6) { } q T m j s m j Moreover, recall that n ths study, we consder each SU s only equpped wth a sngle rado, whch means each SU can only transmt or receve on one frequency band at a tme. Thus, we can have s m j + s m jq 1. 7) { } q T m j Notce that 5)-6) wll hold whenever 7) holds. In addton to the above constrants at a certan SU, there are also constrants due to potental nterference among the SUs. In partcular, for a frequency band m, f SU uses ths band for transmttng data to a neghborng SU j T m, then any other SUs that can nterfere wth SU j s recepton should not use ths band. To model ths constrant, we denote by Pj m the set of SUs that can nterfere wth SU j s recepton on band m,.e., Pj m = {p dp,j) R p,m I,p j,tp m }. The physcal meanng of Tp m n the above defnton s that SU p has at least one neghbor to whch t may transmt data and hence cause nterference to SU j s recepton. Therefore, we have s m j + s m pq 1 p Pm j ). 8) { } q T m p Moreover, recall that we need fnd the t-th hghest vrtual bd n the t-th teraton. Thus, n the t-th t 2) teraton, we need fnd the VBG gvng the hghest vrtual bd wth the prevously found t 1 VBGs beng excluded. Lettng I W,t and G W,t denote the MIS and the correspondng VBG that we fnd n the t-th teraton, respectvely, we have,j),m) I W,τ s m j < I W,τ, 1 τ t 1, 9),j),m) I W,τ s m j 1, 1 τ t 1, 10) where I W,τ s the number of elements contaned n I W,τ. 9) means that all the lnk-band pars n any of the prevously found t 1 MISs cannot be selected at the same tme n the t-th teraton, whch excludes the prevous t 1 MISs. 10) means that the newly found MIS should contan at least one dfferent lnk-band par from any of the prevously found t 1 MISs. Consequently, accordng to the above constrants, the TO allocaton TO-AL) optmzaton problem fndng the VBG wth the t-th hghest vrtual bd n the t-th teraton can be formulated as follows: TO-AL: Maxmze c m j sm j p N j T M j s.t. Equatons 7) 10) s m j = 0 or 1 where s m j s are the optmzaton varables, cm j s are calculated accordng to 3), p are known constants receved from the

5 SUs. Note that 9) and 10) make sure the newly found IS n t- th teraton s an MIS and t s dfferent from any MIS found n prevous t 1 teratons. Besdes, 9) s n fact always satsfed as long as 10) holds. Snce s m j can only take value of 0 or 1, TO-AL s a Bnary Integer Programmng BIP) problem, whch can be solved by applyng the tradtonal branch-andbound or branch-and-cut [19] approach. B. Transmsson Opportunty Schedulng In ths paper, we assume strct allocaton [3] n TO aucton,.e., a source node pays the auctoneer only f ts traffc demand s fully satsfed. Thus, the auctoneer needs to fnd an optmal way to utlze those wnnng MISs, tryng to delver all source nodes traffc by explorng jont schedulng and routng. Denote the set of the wnnng MISs found up to the t-th teraton by IW t = t τ=1 I W,τ. Note that IW t = t. Lettng f j l) denote the flow rate of traffcl over lnk,j), where N, l L, and j T gven T = T m, the schedulng of the MISs should satsfy the followng: t f j l) λ q c m j I q). 11) l L We then gve routng constrants n the followng. Recall that a source SU may need a number of relay nodes to relay ts data packets toward the ntended destnaton node. Snce routng packets along a sngle path may not be able to fully take advantage of the local avalable channels, n ths study, we employ mult-path routng to delver packets more effectvely and effcently. In partcular, f SU s the source of sesson l,.e., = sl), then we have the followng constrants: j sl),sl) T j f jsl) l) = 0, 12) j sl),j T sl) f sl)j l) = rl). 13) The frst constrant means that the ncomng data rate of sesson l at ts source node s 0. The second constrant means that the traffc for sesson l may be delvered through multple nodes on multple paths, and the total data rates on all outgong lnks are equal to the correspondng traffc demand rl). If SU s an ntermedate relay node for sesson l,.e., sl) and dl), then f j l) = f p l), 14) j sl),j T p dl), T p whch ndcates that the total ncomng data rates at a relay node are equal to ts total outgong data rates for the same sesson. Moreover, f SU s the destnaton node of sesson l,.e., = dl), then we have j dl),j T dl) f dl)j l) = 0, 15) p dl),dl) T p f pdl) l) = rl). 16) The frst constrant means the total outgong data rate for sessonlat ts destnatondl) s 0, whle the second constrant ndcates that the total ncomng data rate for sesson l at the destnaton dl) s equal to the correspondng traffc demand rl). Thus, based on the constrants mentoned above, the TO schedulng TO-SC) optmzaton problem n the t-th teraton can be formulated as follows: t TO-SC: Mnmze s.t. λ q Equatons 11) 16) λ q 0 1 q t) f j l) 0 N,j T,l L) The formulated optmzaton problem s a lnear programmng LP) problem, whch can be easly solved by usng the smplex method [20]. The optmal result of TO-SC ndcates whether the current wnnng MISs are enough to support the traffc demand. Specfcally, If the optmal objectve functon s no larger than 1, then the traffc can be supported. The soluton also shows how to schedule the MISs and route the traffcs. Then, the auctoneer contnues to perform prcng as ntroduced next. Otherwse, t means that the current wnnng MISs cannot satsfy the traffc demand. Thus, the auctoneer does not need to perform prcng and another wnnng MIS s needed from TO-AL. C. Prcng In an teraton, f the mnmum schedulng length t λ q s no larger than 1, gven the wnnng MISs I W and ther schedules, the auctoneer can then determne the clearng prce for each SU. The prcng procedure conssts of two steps: determnng the clearng prce for each wnnng VBG, and determnng the clearng prce for each wnnng SU. Denote the number of teratons the auctoneer takes to get t λ q 1 for the frst tme by T 0. To mantan the economc propertes and take spectrum utlzaton nto consderaton, we determne the clearng prce for each wnnng VBG n the t-th teraton, denoted by C t, as follows: { t C t = max B t λ q, B t+1 }, for t T 0, where B t s the vrtual bd from the VBG found n the t-th teraton,.e., the lowest bd among all the wnnng VBGs bds, and B t+1 s the vrtual bd from the VBG found n the t+1)-th teraton,.e., the hghest bd among all the losng VBGs bds. Notce that t λ q ndcates the spectrum utlzaton. When t s less than 1, t means that the auctoneer can launch another aucton to rent the unutlzed spectrum, and hence t s reasonable to consder t n the clearng prce. Wth each wnnng VBG s clearng prce defned as above, the prce a wnnng SU needs to pay, denoted by C t,, s gven as follows: t b q ) C t, = C t, for t T 0. B q

6 Note that bq B q C t s the prce SU needs to pay n VBG G W,q, and hence C t, s the total clearng prce for SU. Thus, a wnnng SU s clearng prce, denoted by c t,, s c t, = t C t, D. Iteraton Termnaton Condton j T c m m j I W,q), for t T 0. As mentoned before, the auctoneer performs the above three procedures sequentally and teratvely. Here, we dscuss when the auctoneer stops and fnshes the aucton process. Notce that the auctoneer s revenue obtaned n the t-th teraton, denoted by Rt), s Rt) = C t t. Accordng to Moon and Moser s result [21], any graph wth n vertces has at most 3 n 3 MISs. Thus, the number of teratons does not need to exceed 3 V 3, whch we denote by T a.e., t T a. Recall that we denote the number of teratons the auctoneer takes to get t λ q 1 for the frst tme by T 0. We also defne a control parameter T b to be the maxmum number of teratons the auctoneer runs beyond T 0 to calculate for ts maxmum revenue under the proposed aucton scheme. Therefore, we have t T 0 +T b, and hence t mn{t 0 +T b,t a }. We wll show n our smulatons that we usually only need a small number of teratons n practce. Moreover, n the case that the auctoneer fnds that the SUs traffc demands cannot be supported, the auctoneer wll drop one of them each tme untl the remanng traffc demands can be satsfed. After the teraton ends, the auctoneer fnds the optmal teraton t that gves the maxmum revenue Rt) among all the t T 0 +1 from T 0 to t) outcomes. Note that Rt) s not equal to the maxmum revenue the auctoneer can possbly get under system-effcent aucton scheme as we explaned before. Then, SU s clearng prce wll be c t,. V. PROOF OF ECONOMIC PROPERTIES In ths secton, we frst prove that our proposed aucton scheme TOA s IC and IR for VBGs, and then show that those two economc propertes also hold for ndvdual SUs. A. Proof of Economc-robustness for VBGs Accordng to Myerson s characterzaton of IC and IR sustaned aucton [22], f the tem n the aucton s monotoncally allocated and the wnners are charged wth crtcal value, then the aucton satsfes the IC and IR propertes. Defnton 1: Monotonc Allocaton: When others bds,.e., B q are fxed, f one bdder wns by bddng B q, then t also wns by bddng B q > B q. Defnton 2: Crtcal Value: Crtcal value s such a value that f bdders bd hgher than t, then they wn, and f bdders bd lower than t, then they lose. Lemma 1: The aucton tems,.e., TOs, are monotoncally allocated n our aucton scheme. Proof: Snce the TO allocaton procedure determnes a wnnng VBG each tme by fndng the one wth the hghest bd, the lemma drectly follows. Lemma 2: The clearng prce C t for each wnnng VBG s a crtcal value. Proof: Recall that B t s the vrtual bd from the VBG found n the t-th teraton,.e., the t-th wnnng VBG, and the clearng prce s C t = max{b t t λ q,b t+1 }. Frst, f B t t λ q B t+1, then C t s equal to B t+1, whch s obvously a crtcal value snce bdders wth hgher bds than B t+1 wn and those wth lower bds lose. Second, f B t t λ q > B t+1, then C t s equal to B t t λ q, whch s also a crtcal value. Thus, the clearng prce s always a crtcal value. Thus, from Lemma 1 and Lemma 2, we have the followng theorem. Theorem 1: The proposed aucton scheme TOA s IC and IR, and hence economcally robust for VBGs. B. Proof of Economc-robustness for Indvdual SUs Although n the prevous secton, we have shown that our proposed aucton scheme TOA preserves IC and IR propertes and hence s economcally robust for VBGs, we need further prove that t also has these propertes for ndvdual SUs. The followng lemma demonstrates the monotonc allocaton for SUs. Lemma 3: When the other SUs bds,.e., p are fxed, f SU wns by bddng p, then t also wns by bddng p > p. Proof: Consder an arbtrary teraton t. If SU s a wnner up to ths teraton wth bd p, t means that s n at least one of the t wnnng VBGs. Denote the wnnng VBG that contans SU and wns n the q-th teraton by G q 1 q t). Then, ts vrtual bd s B q = b q j = b q j +bq j G q j G q\) where b q = p j T c m m j I W,q). When SU bds p > p, the VBG G q s new vrtual bd, denoted by B q, s B q = b q j +p c m ji W,q ) > B q. j G q\) Denote the set of the wnnng VBGs found up to the q-th teraton by G q W,.e., Gq W = q τ=1 G W,τ. For any VBG G s that does not contan SU and loses n all q teratons when bds wth p,.e., G s and G s G \ G q W, we denote ts vrtual bd when SU bds wth p and wth p by B s and B s, respectvely. Snce the other SUs bds reman the same, we have B s = B s B q < B q. Therefore, the VBGs whch do not contan SU and lose n all q teratons when bds wth p wll stll lose when bds wth p. Consequently, when bds wth p, snce the vrtual bds of the VBGs contanng SU become larger, the number of VBGs contanng SU n the q wnng VBGs n all q teratons gets no smaller than that when bds wth p. Thus, SU stll wns. Usng the above lemma, we are able to prove the IC property for ndvdual SUs as follows. Theorem 2: The proposed aucton scheme TOA s IC for SUs.

7 Proof: Recall that to prove the IC property, we need to show that for any SU wth any p v whle the others bds are fxed, the condton n 2) holds. Let u p,p ) and u v,p ) denote SU s utlty when SU bdsp andv, respectvely. We frst consder the scenaro where p > v. Case 1: SU loses wth both v and p. In ths case, u p,p ) = u v,p ) = 0 accordng to our defnton n 1). Thus, 2) holds. Case 2: SU loses wth v but wns wth p. In ths case, obvously we have u v,p ) = 0. Snce SU wns wth p, n an arbtrary t-th teraton t T 0 ), we can obtan u p,p ) t = v c ) = v t = t j T m v q b q c m j I W,q) c m j I W,q) ), +bq t b q ) C t B q where b q = j G q,j bq j. Besdes, snce SU s a wnnng SU, there must be at least one out of the t wnnng VBGs that contans. Denote the set of the ndexes of such VBGs by H. Then, we get v q = bq = 0 for q H, and hence u p,p ) = v q b q ). q H +bq In addton, snce v < p, we can have that for any q H, v q < bq and thus b q v q < v q +bq. 17) +vq Furthermore, for any wnnng VBG that contans SU, say G q q H), ts vrtual bd satsfes B q = b q +bq C t. Due to the fact SUloses by bddngv, thetwnnng VBGs, denoted by G k 1 k t) wth vrtual bd B k, do not contan SU when SU bds v. Thus, we have b q +vq B t. Snce at least one VBG contanng SU becomes a wnner when SU bds p, G t must lose, and hence B t C t. Thus, we have C t b q +vq for q H. As a result, we fnally get u p,p ) < q Hv q v q ) < 0, +vq whch leads to u p,p ) u v,p ) as well. Case 3: SU wns wth v and loses wth p. Snce p > v, accordng to the monotoncty property we have proved n Lemma 3, ths wll not happen. Case 4: SU wns wth both v and p. In an arbtrary t-th teraton t T 0 ), we denote the set of the ndexes of the wnnng VBGs contanng SU when SU bds wth p and that when U bds wth v by H and H, v q respectvely. We also denote the clearng prces when bds wth p and v by C t and C t, respectvely. Notce that 1) f the set of wnnng VBGs when SU bds wth p and that when SU bds wth v, denoted by G q W p ) and G q W v ), respectvely, are the same, we have C t C t accordng to 17) snce the VBGs vrtual bds are larger when SU bds wth p and λ q s reman the same; 2) f G q W p ) and G q W v ) are dfferent, t means at least one of the wnnng VBGs when SU bds wth v loses when SU bds wth p. Snce ths VBG s vrtual bd when SU bds wth p, denoted by B x, s no smaller than that when when SU bds wth v, denoted by B x, we have C t B x B x C t. Consequently, we always have C t C t. Besdes, smlar to that n Case 2, we get u p,p ) = v q b q ) q H +bq u v,p ) = v q v q ) b q C +vq t q H When SU bds wth p, denote ts utlty attrbuted to the common VBGs between G q W p ) and G q W v ) by u 1 p,p ) and the utlty attrbuted to the other VBGs by u 2 p,p ). Smlarly, when SU bds wth v, denote ts utlty attrbuted to the common VBGs between G q W p ) andg q W v ) byu 1 v,p ) and the utlty attrbuted to the other VBGs by u 2 v,p ). Then, we have the followng results. Frst, for those common VBGs between G q W p ) and G q W v ), we have u 1 p,p ) u 1 v,p ) v q b q ) q H H ) +bq v q v q ) +vq q H H ) whch s less than 0 accordng to 17). Second, for any VBG n G q W p ) but not n G q W v ), we have b q v q < v q +bq. +vq Snce ths VBG loses when SU bds wth v, we have C t C t bq +vq. Thus, we get u 2 p,p ) = v q b q ) 0. +bq q H\H H ) Thrd, for any VBG n G q W v ) but not n G q W p ), we have v q vq C b q +vq t 0 snce ths VBG wns when SU bds wth v and C t bq +vq. Thus, we obtan u 2 v,p ) = v q b q ) 0. +bq q H \H H ) v q

8 As a result, we can get u p,p ) u v,p ) =u 1 p,p ) u 1 v,p )+u 2 p,p ) u 2 v,p ) 0. The proof s smlar when p < v, whch s omtted due to space lmt. In general, u p,p ) u v,p ) always holds, and hence the theorem drectly follows. Theorem 3: The proposed aucton scheme TOA s IR for SUs. Proof: In an arbtrary t-th teraton t T 0 ), snce TOA s IR for VBGs, we have C t B q for 1 q t, and hence t b q ) B q C t c = t c m j I W,q) = t t p t t bq j T m c m j I W,q) c m j I W,q) c m j I = p. W,q) Therefore, TOA s IR for SUs. From Theorem 2 and Theorem 3, we can have the followng theorem. Theorem 4: The proposed aucton scheme TOA s economc-robust for SUs. VI. SIMULATION RESULTS In ths secton, we conduct smulatons to evaluate the performance of our proposed aucton scheme TOA. Smulatons are carred out n CPLEX 12.4 on a computer wth a 2.27 GHz CPU and 24 GB RAM. We randomly deploy SUs n a square network of area 1000m 1000m. There are totally 5 mult-hop sessons n the network, each of whch has traffc demand of 1Mbps. We assume that each bdder s true valuaton of and hence ts bd for) unt nstantaneous transmsson rate s unformly dstrbuted over[10 6,10 5 ]. In addton, assume the PU has 3 dle spectrum bands to lease to the SUs, wth ther bandwdths beng 1.0MHz, 1.5MHz and 2.0MHz, respectvely. Some other mportant smulaton parameters are lsted as follows. The path loss exponent s 4 and C = The nose power spectral densty s η = W/Hz at all nodes. The transmsson power spectral densty of nodes s η, and the recepton threshold and nterference threshold are both 8.1η on each spectrum band. Thus, the transmsson range and the nterference range on each frequency band are both equal to 500m. Snce we have proved that our aucton scheme s economc-robust n the prevous secton, we demonstrate the aucton effcency and the auctoneer s revenue n what follows. Note that aucton effcency s defned as the rato of the number of fnally successfully delvered traffc flows to the total number of traffc flows demanded by the SUs. We frst compare the aucton effcency of the proposed TOA scheme wth those of two other aucton schemes: one for sngle-hop data transmsson [10], and the other for mult-hop Aucton Effcency Aucton Effcency hop Aucton, M=1 1 hop Aucton, M=3 TO Aucton, M=1 TO Aucton, M= Number of SUs Bdders) a) Smple Mult hop Aucton, M=1 Smple Mult hop Aucton, M=3 TO Aucton, M=1 TO Aucton, M= Number of SUs Bdders) b) Fg. 1. Aucton effcency comparson wth 1-hop aucton scheme and greedy mult-hop aucton scheme. a) Sngle-hop data transmsson scenaro. b) Mult-hop data transmsson scenaro. data transmsson [12] whch greedly assgns spectrum bands to lnks. We call these two schems 1-hop aucton and greedy mult-hop aucton, respectvely, n our smulatons. To make far comparsons, we compare TOA wth these two schemes n sngle-hop and mult-hop scenaros, respectvely. In the sngle-hop scenaro, each source SU can reach ts ntended destnaton SU n one hop, and hence the data traffc can be delvered n one-hop as well. Fg. 1a) gves the results when the number of SUs ranges from 10 to 30 and the number of avalable spectrums M s equal to 1 and 3. We can fnd that TOA can acheve much hgher aucton effcency than 1-hop aucton. Partcularly, n the case that there s only one avalable spectrum band, TOA can support two and three traffc flows when the number of SUs s 10 and 15, respectvely, whle 1- hop aucton cannot support any of the traffc flows. When there are more SUs n the network, TOA can support four traffc flows whle 1-hop aucton can only support one of them. In the case that there are three avalable spectrum bands, TOA can support four flows when there are 10 SUs and all the fve flows when there are more SUs, whle 1-hop aucton can only support one flow, two flows, and three flows, when there are 10, 15 and 20, and more SUs, respectvely. As we mentoned before, ths s because n 1-hop aucton, t s not clear whom a wnnng SU communcates wth and there can be a lot of collsons n the network. In the mult-hop scenaro, each source node needs to delver data to ts destnaton va multple hops. The aucton effcency s shown n Fg. 1b) when the number of SUs ranges from 10 to 30 and the number of avalable spectrums M s equal to 1 and 3. In partcular, n the case that there s only one avalable spectrum band, TOA can support three traffc flows when the

9 number of SUs s 10, and all the fve traffc flows when there are more SUs n the network. On the other hand, greedy multhop aucton cannot support any traffc flows when there are 10 SUs, and only two flows when there are more SUs. Besdes, n the case that there are three avalable spectrum bands, TOA can support four flows when there are 10 SUs and fve flows when there are more SUs, whle greedy mult-hop aucton can only support one flow, two flows, and three flows, when there are 10, 15, and more SUs, respectvely. Ths s because that we consder transmsson opportuntes n auctons as well as spectrum schedulng n both frequency and tme domans. Fg. 2. Revenue M=2 M= Iteraton Number t Computaton performance under dfferent number of spectrum band. We then llustrate the revenues TOA generates for the auctoneer. Note that n TOA, SUs bd for TOs based on unt nstantaneous transmsson rate, whle the other aucton schemes bd for spectrum bands based on bandwdth. Thus, SUs valuatons and bds have very dfferent meanngs from those n prevous schemes, and we cannot compare wth ther revenues here. In ths case, we consder the scenaros where there are 20 SUs and the number of avalable spectrum bands M s equal to 2 or 3. The results are shown n Fg. 2. We can see that the auctoneer s revenue frst grows and then declnes as the teraton number ncreases. We can also fnd that the number of teratons does not need to be very large. For example, when M = 2, T 0 s around 55, and the maxmum revenue s acheved when the teraton number s approxmately 310. When M = 4, the maxmum revenue s acheved when the teraton number s about 390. Moreover, the auctoneer s revenue s hgher when there are more spectrum bands, whch fts our ntuton snce t sells more resources. On the other hand, the revenue when M = 3 does not exceed much than that when M = 2 snce the auctoneer does not need to fully utlze all the spectrum resources to support the traffc and can save some for other applcatons. VII. CONCLUSIONS In ths paper, we have proposed a novel spectrum aucton scheme, called transmsson opportunty aucton TOA), based on TOs. The TOA scheme s manly composed of three procedures: TO allocaton, TO schedulng, and prcng. In TO allocaton, n each teraton the auctoneer fnds out the VBG that has the hghest vrtual bd. In TO schedulng, the auctoneer checks f the wnnng VBGs found so far can support the traffc demand n the network by solvng a mnmum length schedulng problem. In prcng, the auctoneer determnes the clearng prce for each wnnng VBG and SU, and computes ts own revenue. The auctoneer fnally chooses the wnnng VBGs whch can generate the hghest revenue among the results t obtans. We have proved that TOA s IC, IR, and BB, and hence economc-robust. 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