Energy-efficient Nonstationary Spectrum Sharing

Transcription

1 1 Energy-effcent Nonstatonary Spectrum Sharng Yuanzhang Xao and Mhaela van der Schaar, Fellow, IEEE Abstract We develop a novel desgn framework for energyeffcent spectrum sharng among autonomous users who am to mnmze ther energy consumptons subject to mnmum throughput requrements. Most exstng works proposed statonary spectrum sharng polces, n whch users transmt at fxed power levels. Snce users transmt smultaneously under statonary polces, to fulfll mnmum throughput requrements, they need to transmt at hgh power levels to overcome nterference. To mprove energy effcency, we construct nonstatonary spectrum sharng polces, n whch the users transmt at tme-varyng power levels. Specfcally, we focus on TDMA (tme-dvson multple access polces n whch one user transmts at each tme (but not n a round-robn fashon. The proposed polcy can be mplemented by each user runnng a low-complexty algorthm n a decentralzed manner. It acheves hgh energy effcency even when the users have erroneous and bnary feedback about ther nterference levels. Moreover, t can adapt to the dynamc entry and ext of users. The proposed polcy s also devaton-proof, namely autonomous users wll fnd t n ther self-nterests to follow t. Compared to exstng polces, the proposed polcy can acheve an energy savng of up to 90% when the number of users s hgh. I. INTRODUCTION A key challenge n wreless networks s determnng effcent solutons for the autonomous users to share the spectrum. In cogntve rado networks where the users are dfferentated as prmary users (PUs and secondary users (SUs, we also requre SUs to access the spectrum wthout degradng PUs qualty of servce (QoS [1][34] [36]. To be more general, we consder cogntve rado networks n ths work, and desgn spectrum sharng polces that acheve effcent spectrum usage and protect PUs QoS. Our work can be easly appled to a wreless network n whch users are not dfferentated as PUs and SUs (whch can be consdered as a specal cogntve rado network wth no PUs. Spectrum sharng polces, whch specfy the PUs and SUs transmsson schedules and transmt power levels, are essental to acheve spectrum and energy effcency []. Research on desgnng spectrum sharng polces can be roughly dvded n two man categores. The research n the frst category formulates the spectrum sharng problem as a utlty maxmzaton problem subject to the users maxmum transmt power constrants [3] [1][] [5][3]. Many works n ths category [3] [9][] [5][3] defne the utlty functon as an ncreasng functon of the sgnal-to-nterference-and-nose-rato (SINR, whle neglectng to consder the energy consumpton of the Manuscrpt receved March 19, 013; revsed November 30, 013; accepted January 6, 014. Ths work was supported by NSF Grant No Y. Xao and M. van der Schaar are wth the Electrcal Engneerng Department, Unversty of Calforna, Los Angeles, CA USA (e-mal: yxao@seas.ucla.edu; mhaela@ee.ucla.edu. resultng spectrum sharng polces. Some other works n ths category [10] [1] defne the utlty functon as the rato of throughput to transmt power, n order to maxmze the spectrum effcency per energy consumpton. Research n the second category [13] [1] formulates the spectrum sharng problem as an energy consumpton mnmzaton problem subject to the users mnmum throughput requrements. In ths formulaton, the users throughput requrements can be explctly specfed. Hence, the spectrum effcency s guaranteed wth the mnmal energy consumpton. The work n ths paper pertans to ths second category of research works. One major lmtaton of exstng works n the second category [13] [1] s that they restrct attenton to a smple class of spectrum sharng polces that requre the users to transmt at fxed power levels as long as the envronment (e.g. the number of users, the channel gans does not change 1. We call ths class of spectrum sharng polces statonary. The statonary polces are not energy effcent, because due to mult-user nterference, the users need to transmt at hgh power levels to fulfll the mnmum throughput constrants. To mprove energy effcency, we study nonstatonary spectrum sharng polces. Specfcally, we focus on TDMA (tmedvson multple access spectrum sharng polces, a class of nonstatonary polces n whch the users transmt n a TDMA fashon. TDMA polces can acheve hgh spectrum effcency that s not achevable under statonary polces, and greatly mprove the energy effcency of the statonary polces, because of the followng two reasons. Frst, there s no multuser nterference n TDMA polces. Second, TDMA polces allow users to adaptvely swtch between transmsson and dormancy, dependng on the average throughput they have acheved, for the purpose of energy savng. Note that n the optmal TDMA polces we propose, users usually do not transmt n the smple round-robn fashon, because of the heterogenety n ther mnmum throughput requrements and channel condtons (see Secton IV for a motvatng example that shows the sub-optmalty of round-robn TDMA polces. Another lmtaton of exstng works n the second category [13] [1] s the assumpton that each user s recever can perfectly estmate the local nterference temperature (.e. the nterference and nose power level, and can accurately feed t back to ts transmtter. However, n practce, users cannot perfectly estmate the nterference temperature, and can only 1 Although some spectrum sharng polces [13] [1] go through a transent perod of adjustng the power levels before convergng to the optmal power levels, the users mantan the fxed power levels after the convergence. We use nonstatonary, nstead of dynamc, to descrbe the proposed polcy, because dynamc spectrum sharng has been extensvely used to descrbe general spectrum sharng polces n cogntve rado, where SUs access the channel opportunstcally. In ths sense, our polcy s dynamc. However, our nonstatonary polcy s dfferent from other dynamc spectrum sharng polces, n that the power levels are tme-varyng.

2 send lmted (quantzed feedback. In ths paper, we provde a novel desgn framework to construct nonstatonary spectrum sharng polces that acheve PUs and SUs mnmum throughput requrements wth mnmal energy consumptons, even when the users have erroneous and very lmted (only bnary feedback about ther local nterference temperatures. We frst prove a key property of the optmal TDMA spectrum sharng polcy: each user should choose the same power level whenever t transmts. Ths property enables us to solve the polcy desgn problem n two tractable steps: frst determne the optmal power levels before run-tme, and then determne the transmsson schedule at run-tme. We then propose a low-complexty dstrbuted nstantaneous throughput selecton (ITS algorthm for the users to determne ther optmal power levels before run-tme, and a low-complexty dstrbuted longest-dstance-frst (LDF schedulng algorthm to determne the transmsson schedule at run-tme. We prove that both algorthms converge lnearly ndependent of the number of users (.e. the dstance from the optmal soluton decreases exponentally, resultng n a logarthmc convergence tme. The proposed polcy can also adapt to the dynamc entry and ext of users wthout affectng the convergence of exstng users. Moreover, t s devatonproof, meanng that a user cannot mprove ts energy effcency over the proposed polcy whle stll fulfllng the throughput requrement. In ths way, autonomous users wll fnd t n ther self-nterest to adopt the polcy. The rest of the paper s organzed as follows. We gve detaled comparsons aganst exstng works n Secton II. Secton III descrbes the system model for spectrum sharng. Secton IV gves a motvatng example to show the performance gan acheved by nonstatonary polces and the necessty of devaton-proof polces. We formulate and solve the polcy desgn problem n Secton V and Secton VI, respectvely. Smulaton results are presented n Secton VII. Fnally, Secton VIII concludes the paper. II. RELATED WORKS In ths secton, we provde a comprehensve comparson between the proposed scheme and exstng works. The reader could skp ths secton and go drectly to the system model, f not nterested n the detaled comparsons. Although only some works [13] [1] use the same problem formulaton as ours, we compare aganst a wde range of related works [3] [3] to hghlght the techncal novelty of our work, and to llustrate that the works [3] [1][] [3] proposed under dfferent problem formulatons cannot be adapted to our settng. A. Statonary Spectrum Sharng Polces Table I categorzes exstng statonary spectrum sharng polces based on four crtera: whether the polcy consders energy effcency, whether the polcy s devaton-proof, what are the feedback requrements and the correspondng overhead, and whether they can accommodate a varyng number of users. Throughout ths secton, the feedback s the nformaton on nterference and nose power levels sent from a user s recever to ts transmtter. TABLE I. COMPARISONS AGAINST STATIONARY POLICIES. Energy Feedback User Devaton -effcent (Overhead number -proof [3] [7] No Error-free, unquantzed Fxed No [8][9] No Error-free, unquantzed (Large Fxed Yes [10] [19] Yes Error-free, unquantzed (Large Fxed Yes [0][1] Yes Error-free, unquantzed (Large Varyng Yes [] [4] No Error-free, unquantzed (Large Fxed Yes Proposed Yes Erroneous, bnary (One-bt Varyng Yes TABLE II. COMPARISONS AGAINST NONSTATIONARY POLICIES. [5] [6] [7] [9] Proposed Energy -effcent No No No Yes Power control Yes No No Yes Users Heterogenous Homogenous Homogenous Heterogenous Feedback Error-free Erroneous Error-free Erroneous (Overhead unquantzed bnary bnary bnary (Large (One-bt (One-bt (One-bt User number Fxed Fxed Fxed Varyng Devatonproof Yes No No Yes B. Nonstatonary Spectrum Sharng Polces There have been some works that develop nonstatonary polces usng repeated games [5], Markov decson processes (MDPs [6], and mult-art bandt [7] [9]. We summarze the major dfferences between the exstng nonstatonary polces and our proposed polcy n Table II. C. Comparson Wth Our Prevous Work Most related to ths work s our prevous work [3]. However, the desgn frameworks proposed n [3] and n ths work are sgnfcantly dfferent because the desgn objectves are dfferent. In [3], we amed to desgn TDMA spectrum sharng polces that maxmze the users total throughput wthout consderng energy effcency. Under ths desgn objectve, each user wll transmt at the maxmum power level n ts slot, as long as the nterference temperature constrant s not volated. Hence, what we optmzed was only the transmsson schedule of the users. In ths work, snce we am to mnmze the energy consumpton subject to the mnmum throughput requrements, we need to optmze both the transmsson schedule and the users transmt power levels, whch makes the desgn problem more challengng. Moreover, ths work consders the scenaro n whch users enter and leave the network, whch s not consdered n [3]. D. Comparson Wth Theoretcal Frameworks Our results on nonstatonary polces buld on the concept of self-generatng sets proposed n the game theory lterature [30]. Self-generatng sets are used to analyze repeated games TABLE III. RELATED THEORETICAL FRAMEWORKS. Constructve Feedback User number [30] No N/A Fxed [31] No Erroneous, hgh-granularty Fxed Proposed Yes Erroneous, bnary Varyng

3 3 wth mperfect montorng. For example, the Folk Theorem n repeated games wth mperfect montorng n [31] bulds on the concept of self-generatng sets. However, we cannot apply ths concept straghtforwardly or n a way smlar as n [31] for the followng reasons. The self-generatng set s defned as a fxed pont of a set-valued mappng. The work [30] defned the set-valued mappng, and proved an mportant property of the fxed pont of ths set-valued mappng (.e. the self-generatng set: every payoff vector n the self-generatng set can be acheved at an equlbrum. However, although [30] dscovered ths mportant property, t dd not show how to construct a self-generatng set. Wthout constructng the selfgeneratng set, we do not know what payoff vectors can be acheved at the equlbra or how to acheve them. The concept of self-generatng sets s appled n [31] to prove the Folk theorem n repeated games wth mperfect montorng. However, our work s fundamentally dfferent from [31] n two aspects. Frst, the results n [31] are not constructve: they focus on what payoff vectors can be acheved, but not how to acheve them. In contrast, gven a target payoff vector, we explctly construct the polcy to acheve t. Second, the results n [31] requre a hgh-granularty feedback sgnal, namely the cardnalty of feedback sgnals should be proportonal to the number of power levels a user can choose. In contrast, by explotng the structure of the spectrum sharng problem, we prove that bnary feedback s suffcent to acheve optmalty n the consdered scenaros. In Table III, we summarze the key dfferences between our work and [30][31]. III. SYSTEM MODEL A. Model For Spectrum Sharng n Cogntve Rado Networks We consder a cogntve rado network that conssts of M prmary users and N secondary users transmttng n a sngle frequency channel. The set of PUs and that of SUs are denoted by M {1,,..., M} and N {M + 1, M +,..., M + N}, respectvely. A wreless network n whch users are not dfferentated as PUs and SUs s a specal case of our model wth M = 0. Each user 3 has a transmtter and a recever. The channel gan from user s transmtter to user j s recever s g j. Each user chooses ts power level p from a compact set P R +. We assume that 0 P, namely user can choose not to transmt. The set of jont power profles s denoted by P = M+N =1 P, and the jont power profle of all the users s denoted by p = (p 1,..., p M+N P. Let p be the power profle of all the users other than user. Each user s throughput s a functon of the jont power profle, namely r : P R +. Snce the users cannot jontly decode ther sgnals, each user treats the nterference from the other users as nose, and obtans the followng throughput at the power profle p [] [4]: p g r (p = log (1 + j M N,j p jg j + σ, 3 We refer to a prmary user or a secondary user as a user n general, and wll specfy the type of users only when necessary. where σ s the nose power at user s recever. We defne user s local nterference temperature I (p as the nterference and nose power level at ts recever, namely I (p j p jg j + σ. Each user s recever measures the nterference temperature wth errors and feedback the quantzed measurement to ts transmtter. We assume that each user uses a unbased estmator wth an addtve estmaton error to obtan the estmate Î I + ε, where ε s the estmaton error wth zero mean, whose probablty dstrbuton functon f ε s known to user. We also assume that each user uses the followng smple two-level quantzer Q : { Ī, f Q (Î(p = Î(p > θ, (1 I, otherwse where θ s user s quantzaton threshold, and Ī and I are two reconstructon values. We assume that the quantzer preserves the mean value of Î(p when there s no multuser nterference. In other words, when p = 0 (.e. when I (p = σ, the quantzer should satsfy E ε {Q (Î(p p =0} = E ε {Î(p p =0} = σ. Ths property can be easly satsfed by settng Ī = x σ supp(fε, x f x θ ε (x σ dx I = x σ supp(fε, x f x<θ ε (x σ, ( dx where supp(f ε s the support of the dstrbuton functon f ε. In practce, t s easy to mplement an unbased estmator and a smple two-level quantzer as n (1 and (. As we wll show later, such an estmator and a quantzer are suffcent to acheve the optmal performance. Remark 1: Here s an ntuton why an unbased estmator and the two-level quantzer n (1 and ( are good enough for us. For user to acheve a mnmum throughput r, gven the feedback Q (Î, ts transmt power level ˆp should be ˆp = ( r 1 Q (Î/g. In a TDMA polcy, there s no mult-user nterference (.e. p = 0 when user transmts. Hence, usng an unbased estmator and the quantzer n (1 and (, user s expected transmt power level s { } E ε {ˆp } = E ε ( r 1 Q(Î/g = ( r 1E ε {Q(Î}/g = ( r 1σ /g, whch s exactly the transmt power level when user perfectly knows the nterference temperature σ. In contrast, under a non-tdma polcy, there s mult-user nterference. In ths case, one user s erroneous and quantzed feedback affects ts own transmt power level, whch n turn affects the others transmt power levels through the nterference. Hence, an unbased estmator and a smple two-level quantzer n (1 and ( may result n performance loss under non-tdma polces. Snce each user adopts a two-level quantzer, ts feedback from the recever to the transmtter s bnary. Then we can further reduce the feedback overhead as follows. Each user s recever nforms ts transmtter of the two reconstructon values Ī and I only once, at the begnnng, after whch the recever sends a sgnal, probably n the form of a smple probe,

4 4 only when the estmated nterference temperature Î exceeds the quantzaton threshold θ. The event of recevng or not recevng the probng sgnal, whch s sent only when Î > θ, s enough to ndcate user s transmtter whch one of the two reconstructon values t should choose. Snce the probng sgnal ndcates hgh nterference temperature, we call t the dstress sgnal as n [14],[1]. Wth some abuse of defnton, we denote user s dstress sgnal as y Y = {0, 1} wth y = 1 representng the event that user s dstress sgnal s sent (.e. Î > θ. We wrte ρ (y p as the condtonal probablty dstrbuton of user s dstress sgnal y gven power profle p, whch s calculated as ρ (y = 1 p = f ε (xdx. B. Spectrum Sharng Polces x>θ I (p The system s tme slotted at t = 0, 1,,.... At the begnnng of tme slot t, each user chooses ts transmt power p t, and acheves the throughput r (p t. At the end of tme slot t, each user j who transmts (p t j > 0 sends ts dstress sgnal yj t = 1 f the estmate Îj exceeds the threshold θ j. We defne y Y as the system dstress sgnal, ndcatng whether there exsts a user who has sent ts dstress sgnal, namely y = 1 f there exsts j such that p j > 0 and y j = 1, and y = 0 otherwse. The condtonal dstrbuton s denoted ρ(y p, whch s calculated as ρ(y = 0 p = Π j:pj>0ρ j (y j = 0 p. Note that the system dstress sgnal s not a physcal sgnal sent n the system, but rather a logcal sgnal summarzng the status of the system. From now on, we refer to the system dstress sgnal smply as the dstress sgnal. Each user determnes the transmt power level p t based on the hstory of dstress sgnals. The hstory of dstress sgnals s h t = {y 0 ;... ; y t 1 } Y t for t 1, and h 0 = for t = 0. Then each user s strategy π s a mappng from the set of all the possble hstores to ts acton set, namely π : t=0y t P. The spectrum sharng polcy, denoted by π = (π 1,..., π M+N, s the jont strategy profle of all the users. Hence, user s transmt power level at tme slot t s determned by p t = π (h t, and the users jont power profle s determned by p t = π(h t. We classfy all the spectrum sharng polces nto two categores, statonary and nonstatonary polces. As n [39, pp. ] and [40, Sec. 5.5.], statonary polces always choose the same acton under the same state, whle nonstatonary polces may choose dfferent actons under the same state. In our model, the state can be consdered as the system parameters (e.g. the number of users, the channel condtons, etc.. Hence, a spectrum sharng polcy π s statonary f and only f for all N, for all t 0, and for all h t Y t, we have π (h t = p stat, where p stat P s a constant. A spectrum sharng polcy s nonstatonary f t s not statonary. In ths paper, we restrct our attenton to a specal class of nonstatonary polces, namely TDMA polces (wth fxed transmt power levels. A spectrum sharng polcy π s a TDMA polcy f at most one user transmts n each tme slot. TDMA polces are optmal when the nterference among the users s strong [37], whch s often the case when the number of users s large. We wll llustrate how TDMA polces outperform statonary polces through a smple example n Secton IV and through extensve smulatons n Secton VII. Remark : In the formal defnton of a nonstatonary polcy, t seems that each user needs to keep track of the hstory of all the past dstress sgnals at each tme slot. However, as we wll see from the longest-dstance-frst schedulng algorthm that mplements the proposed polcy, each user only needs a fnte memory. C. Defnton of Spectrum and Energy Effcency We characterze the spectrum and energy effcency of a spectrum sharng polcy by the users dscounted average throughput and dscounted average energy consumpton, respectvely. Each user dscounts ts future throughput and energy consumpton because of ts delay-senstve applcaton (e.g. vdeo streamng [] [5][3]. A user runnng a more delay-senstve applcaton dscounts more (wth a lower dscount factor. Assumng as n [] [5][3] that all the users have the same dscount factor δ [0, 1, user s average throughput s { } U (π = E h 0,h 1,... (1 δ δ t u (π(h t. t=0 Smlarly, user s average energy consumpton s the expected dscounted average transmt power per tme slot, wrtten as { } P (π = E h0,h 1,... (1 δ δ t π (h t. t=0 Each user ams to mnmze ts average energy consumpton P (π whle fulfllng a mnmum throughput requrement R mn. From one user s perspectve, t has the ncentve to devate from a gven spectrum sharng polcy, f by dong so t can fulfll the mnmum throughput requrement wth a lower average energy consumpton. Hence, we can defne devatonproof polces as follows. Defnton 1: A spectrum sharng polcy π s devatonproof f for all M N, we have π = arg mn P (π, π, subject to R (π, π R mn π, where π s the strategy profle of all the users except. IV. MOTIVATION FOR OPTIMAL NONSTATIONARY TDMA POLICIES Before formally descrbng the desgn framework, we provde a motvatng example to show the advantage of the proposed optmal nonstatonary TDMA polcy, compared to the statonary polcy and round-robn polces, n terms of both the energy effcency and the computatonal complexty. Consder a smple network wth three symmetrc SUs. They have the same drect channel gan of g = 1, the same cross channel gan of g j = 0.5, the same nose power σ = 5 mw, the same mnmum throughput requrement of R mn = 1.5 bts/s/hz, and the same dscount factor of δ = 0.6.

5 5 TABLE IV. Polces Optmal statonary [13] [1] Optmal round-robn (cycle length L = 3 Optmal round-robn (cycle length L = 4 Optmal nonstatonary (proposed A. Energy Effcency ILLUSTRATION AND PERFORMANCE OF POLICIES. Power levels Transmsson schedule Average (mw (frst 1 tme slots energy (186,186,186 smultaneous transmsson 186 mw (33,144, mw (43,1, mw (108,108, mw We llustrate the polces and ther performances n Table IV. The power levels are the transmt power levels of the 3 users whenever they transmt. In the optmal constant polcy, users 1,,3 all transmt all the tme, at the same power level of 186 mw. In TDMA polces, whether round-robn or the proposed optmal polcy, users do not all transmt all the tme. For round-robn polces, we compute the optmal polcy gven the cycle length by determnng the optmal (n terms of average long-term energy consumpton across users order of transmsson n a cycle and the correspondng power levels. In the optmal round-robn polcy wth cycle of length 3, user 1 transmts frst at a low power level (33 mw, user transmts after user 1 at a hgher power level (133 mw to compensate for havng to wat for transmsson and user 3 transmts last at a stll hgher power level (143 mw to compensate for havng to wat stll longer. In the cycle of length 4, agan user 1 transmts at the lowest power level, user transmts at a mddle power level, and user 3 transmts at the hghest power level, but the last two power levels are closer together (than n the cycle of length 3 because user 3 transmts more often. In the optmal nonstatonary polcy, the users all transmt at the same constant power level (108 mw whenever they transmt; ths works because the order n whch they transmt s constantly changng. In the last column of Table 1, the dscounted average energy per user per tme slot s calculated. Notce that the cycle of length 3 s slghtly more effcent than the constant polcy, the cycle of length 4 s much more effcent, but the optmal nonstatonary polcy s more effcent stll. Indeed, the optmal polcy acheves 80%, 67% and 5% energy savngs compared to the optmal constant polcy, the optmal round-robn polcy wth cycle of 3 and wth cycle of 4, respectvely. Importantly, the energy savngs are even more sgnfcant when the number of users s large (see Sec. VII. B. Computatonal Complexty Remarkably, not only s the proposed optmal nonstatonary polcy much more effcent than round-robn polces, t s much easer to compute. To get a hnt of why ths s so, note that n a round-robn polcy, the user s performance s determned not only by the number of slots n a cycle but also by the postons of the slots snce users are dscountng ther future throughput (due to delay senstvty. For a gven number of users M + N and a gven cycle length L, the number of nontrval round-robn schedules (the ones n whch each user gets at least one slot s greater than (M + N L (M+N. So searchng among these schedules wll be totally mpractcal even f L s moderately larger than M + N - but n order to acheve effcency close to the optmal non-statonary polcy, the cycle length L must be much larger than M + N. For nstance, for the 3-user case above, achevng energy effcency wthn 10% of the optmal nonstatonary polcy requres that the cycle length L be at least 8, and so requres searchng among the thousands (5796 of dfferent nontrval schedules of cycle length 8 and fndng power levels for each user. Even ths small problem s computatonally ntensve. For a moderate number of users - say 10 - and a cycle length of 0 - ths means searchng more than ten bllon (.e schedules and fndng power levels for each user - a completely ntractable problem. However, we wll propose a smple algorthm to compute the optmal nonstatonary polcy - both the schedule and the power levels - whose complexty grows only lnearly wth the number of users. V. THE DESIGN PROBLEM FORMULATION Our goal s to construct a devaton-proof TDMA polcy that fulflls all the users mnmum throughput requrements and optmzes a certan energy effcency crteron. The energy effcency crteron can be represented by a functon defned on all the users average energy consumptons, E(P 1 (π,..., P M+N (π. Note, mportantly, that the energy effcency crteron can also reflect the prorty of the PUs over the SUs. For example, the energy effcency crteron can be the weghted sum of all the users energy consumptons,.e. E(P 1 (π,..., P M+N (π = M N w P (π wth w 0 and M N w = 1. Each user s weght w ndcates the mportance of ths user. We can set hgher weghts for PUs and lower weghts for SUs. Gven each user s mnmum throughput requrement R mn, we can formally defne the polcy desgn problem as mn π s.t. E(P 1 (π,..., P M+N (π (3 π s a devaton proof TDMA polcy, R (π R mn, M N. In the above problem formulaton, the usual constrants on the nterferences caused by SUs to PUs are satsfed by restrctng to TDMA polces, n whch there s no mult-user nterference. VI. A DESIGN FRAMEWORK FOR SPECTRUM AND ENERGY EFFICIENT POLICIES We frst outlne the procedure to solve the polcy desgn problem (3. Then we show n detal how to solve the desgn problem, and dscuss mplementaton ssues. Fnally, we adapt the proposed polcy to the dynamc entry and ext of users. A. Outlne of The Desgn Framework The protocol desgn problem (3 s dffcult to solve drectly, because the decson varable π s the spectrum sharng polcy, whch s a mappng from the set of all hstores to the set of actons. We frst unravel an mportant property of the optmal TDMA polcy, namely each user should adopt the

6 6 Pareto optmal throughput achevable by statonary polces User 's Step 1: Characterze the set throughput of feasble nstantaneous throughput vectors Step : Select the optmal nstantaneous throughput vector (ITS algorthm Step 3: Construct the optmal TDMA polcy (LDF schedulng User 1's throughput Before run-tme At run-tme Feasble nstantaneous throughput vector Optmal nstantaneous throughput vector Mnmum throughput requrements between the average throughput and the average energy consumpton: P (π R (π = = (1 δ t=0 δt 1 {π(t>0}p tdma (1 δ ( t=0 δt 1 {π(t>0} log 1 + gptdma p tdma ( = σ log 1 + gptdma σ r tdma 1 g r tdma σ, (4 Fg. 1. The desgn framework to solve the polcy desgn problem. same power level whenever t transmts (see Lemma 1. Ths greatly reduces the dmenson of the decson varable; now we only need to fnd the sngle transmt power level (or equvalently, the nstantaneous throughput of each user and the transmsson schedule. We propose a three-step desgn framework, llustrated n Fg. 1, to solve the desgn problem. Frst, we characterzaton of the set of feasble nstantaneous throughput vectors under whch the users can fulfll ther throughput requrements (see Theorem 1. Based on ths, we then reformulate the orgnal problem (3 nto a problem of fndng the optmal nstantaneous throughput vector, and propose a dstrbuted nstantaneous throughput selecton (ITS algorthm to solve the reformulated problem (see Theorem. Fnally, gven the optmal nstantaneous throughput vector, we propose a longest-dstance-frst (LDF schedulng algorthm to determne the transmsson schedule, whch results n the optmal TDMA polcy that solves the desgn problem (3 (see Theorem 3. We llustrate the desgn framework n Fg. 1. B. Solvng The Polcy Desgn Problem We frst prove a key property of the optmal energy-effcent TDMA protocol: each user should choose the same power level whenever t transmts. Lemma 1: The optmal soluton π to the desgn problem (3 must satsfy that each user chooses the same power level whenever t transmts, namely π (t 1 = π (t for all t 1 and t such that π (t 1 > 0 and π (t > 0. Proof: See Appendx A. Lemma 1 greatly smplfes the desgn problem: now we only need to fnd a sngle optmal power level p for each user to choose whenever t transmts, nstead of solvng for ts optmal power levels n all ts transmssons. In the followng, we frst fnd the optmal power levels {p } M N durng the users transmssons (whch s equvalent to fndng ( each user s optmal nstantaneous throughput, r log 1 + gp. σ Then gven {p } M N (or {r } M N, we fnd the transmsson schedule that acheves the mnmum throughput requrements. 1 Step 1 Characterzng feasble nstantaneous throughput vectors: Now we formulate the problem of fndng the users optmal nstantaneous throughput {r } M N. Frst, the structure of the optmal TDMA protocol dscovered n Lemma 1 enables us to establsh the followng relatonshp where 1 { } s the ndcator functon, p tdma s user s power level when t transmts n the TDMA protocol, and r tdma s the correspondng nstantaneous throughput. We can see from (4 that gven r tdma, the average energy consumpton P (π s proportonal to the average throughput R (π. Hence, to mnmze the energy consumpton, we should let R (π = R mn for all. Then based on (4, we can rewrte the objectve functon E(P 1 (π 1,..., P M+N (π M+N of the desgn problem (3 as a functon of the nstantaneous throughput {r tdma } M N : ({ } σ r tdma 1 E R mn. g r tdma M N An nstantaneous throughput vector {r tdma } M N s feasble, f there exsts a TDMA protocol π that has the nstantaneous throughput {r tdma } M N and can acheve the mnmum average throughput {R mn } M N. Before characterzng the feasble nstantaneous throughput vectors, we wrte p = (p tdma (r tdma, p = 0 as the jont power profle when user transmts n a TDMA polcy. Now we state Theorem 1. Theorem 1: An nstantaneous throughput vector {r tdma } M N s feasble for the mnmum throughput requrements {R mn } M N, f the followng condtons are satsfed: Condton ( 1: the dscount factor δ satsfes δ 1 M N δ 1/ 1 + µ, M+N 1+ M N where b j = sup pj P j,p j p j 1 ρ(y=1 p and µ max j b j. /r tdma ( ρ(y=1 j p /b j ρ(y=1 p ρ(y=1 p j, p j r j(p j,, p j / rj Condton : M N Rmn R mn /µ. = 1, and r tdma Proof: See Appendx B. The problem of fndng the optmal nstantaneous throughput {r } M N can then be formulated as {r tdma mn } M N E s.t. ({ σ M N 0 < r tdma tdma r 1 g r tdma R mn } M N R mn r tdma = 1, (5 r R mn /µ, M N. Step Select the optmal nstantaneous throughput vector: We solve the above optmzaton problem (5 for the optmal nstantaneous throughput vector {r } M N usng

7 7 the dstrbuted ITS algorthm, whch s proved to converge n logarthmc tme n Theorem. The ITS algorthm essentally solves the followng equaton (derved from the KKT condton n a dstrbuted fashon: E P P σ = Rmn g r 1 r ( r 1 r r ln σ g = λ, (6 where λ s the Lagrangan multpler for the constrant R mn = 1 n (5, and should be chosen such that r tdma R mn r n (6 s the dervatve of the energy effcency crteron E( wth respect to user s average energy consumpton. If the energy effcency crteron s the weghted sum of all the users energy con- = 1. The term E P sumptons, we have E P P σ = Rmn r 1 = w, r. If the g r energy effcency crteron s the weghted proportonal farness M N w log(p, we have E P P σ = Rmn r 1 = g r g w r E. Each user selects the term P n the σ Rmn r 1 ITS algorthm based on the energy effcency crteron chosen by the protocol desgner. Algorthm 1 Instantaneous Throughput Selecton (ITS algorthm run by user. Requre: Mnmum throughput requrement R mn, precson e 1: Set λ = 0, λ = 1, λ = λ. : Solve (6 for r, set r mn{r, r } 3: Broadcast R mn /r, and receve Rmn j /rj for all j 4: whle j M N Rmn j /rj > 1 do 5: λ λ, λ λ 6: Solve (6 for r, set r mn{r, r } 7: Broadcast R mn /r, and receve Rmn j /rj for all j 8: end whle 9: whle j M N Rmn /rj 1 > e do 10: λ λ+ λ 11: Solve (6 for r, set r mn{r, r } 1: Broadcast R mn j /r 13: f j M N Rmn j 14: λ λ 15: else 16: λ λ 17: end f 18: end whle, and receve Rmn j /rj < 1 then 19: Normalze r r / ( j M N Rmn j /r j /r j for all j Theorem : The problem (5 of fndng the optmal nstantaneous throughput vector can be converted nto a convex optmzaton problem, whose soluton {r } M N can be found by each user runnng the dstrbuted ITS algorthm. The algorthm converges lnearly 4 at rate 1. Proof: See Appendx C. Algorthm The Longest-Dstance-Frst (LDF schedulng run by user. Requre: {Rj mn /rj } j M N, r Intalzaton: Set t = 0, r j (0 = Rmn j /rj for all j M N repeat Calculates the dstance from the optmal operatng pont d j (t = r j (t µ j 1 r j (t ρ(y = 1 pj, j Fnd the user wth the largest dstance arg max j M N d j (t f = then Transmt at power level p tdma (r end f Updates r j (t + 1 for all j M N as follows: f No Dstress Sgnal Receved At Tme Slot t then r (t+1 = 1 δ r (t ( 1 δ 1 (1+ ρ(y=1 p j b j r j (t + 1 = 1 δ r j (t + ( 1 δ 1 ρ(y=1 p b j, j else r (t+1 = 1 δ r (t ( 1 δ 1 (1 ρ(y=0 p j b j r j (t + 1 = 1 δ r j (t ( 1 δ 1 ρ(y=0 p b j, j end f t t + 1 untl 3 Step 3 Construct the optmal devaton-proof polcy: Gven the optmal nstantaneous throughput vector, each user runs the longest-dstance-frst schedulng algorthm n a decentralzed manner. On one hand, the transmsson schedule can be vewed as a smple largest-dstance-frst schedulng, namely the user farthest away from ts throughput requrement transmts. On the other hand, t s nontrval to defne the dstance from ts throughput requrement. As we wll prove later, user j s dstance from ts throughput requrement can be r j defned as d j (t = (t µ j 1 r j (t+, where ( ρ(y=1 k j pj /b jk r j (t s the future throughput to acheve startng from tme slot t normalzed by rj. The normalzed future throughput r j (t can be also nterpreted the future transmsson opportunty. If user j transmtted all the tme n the future, t would have an average throughput rj. If t transmts n a fracton r j (t of tme after tme t, t has an average future throughput of r j (t r j. Theorem 3 proves the desrable propertes of the LDF schedulng algorthm. Theorem 3: If each user M N runs the LDF schedulng algorthm, then we have each user can acheve ts mnmum throughput requrement R mn wth an energy consumpton P that mnmzes the energy effcency crteron E(P 1,..., P M+N ; 4 Followng [38, Sec ], we defne lnear convergence as follows. Suppose that the sequence {x k } converges to x. We say that ths sequence converges lnearly at rate c, f we have lm k x k+1 x x k x = c.

8 8 f a user does not follow the algorthm, t wll ether fal to acheve the mnmum throughput requrement, or acheve t wth a hgher energy consumpton; the dstance between each user s average throughput at tme t and ts throughput requrement decreases exponentally wth tme, namely (1 δ t τ=0 δ τ r τ R mn r δ t+1. (7 Proof: See Appendx D. Theorems and 3 establsh the convergence results of our proposed scheme. Theorem proves that the process of fndng the optmal nstantaneous throughput vector converges n logarthmc tme, and Theorem 3 proves that the LDF schedulng acheves the mnmum throughput requrements n logarthmc tme. Hence, the overall convergence speed s fast. Note that our convergence results are very dfferent from the convergence results n some recent works on power control n cogntve rado [6] and wreless networks [7]. These works [6][7] belong to the statonary spectrum sharng polces, namely they am to fnd the optmal fxed power levels of the users that maxmze the network utlty. The convergence results n [6][7] dffer from our results n two mportant ways. Frst, snce our work studes nonstatonary spectrum sharng wth tme-varyng power levels, we need to determne not only the optmal power levels of the users, but also the transmsson schedule of the users. We prove that the average throughput obtaned by adoptng the proposed LDF schedulng converges lnearly. Such a result does not appear n [6][7]. Second, the technques used n provng the convergence to the optmal power levels are dfferent. In [6][7], the algorthms are akn to the celebrated dstrbuted power control algorthm [13], and hence the proofs use and extend the standard nterference functon argument. Such an argument s not used n our work snce there s no nterference among the users under the proposed TDMA spectrum sharng polcy. C. Implementaton We dscuss the total overhead of nformaton exchange and feedback and the computatonal complexty of the proposed scheme. 1 Overhead of ntal nformaton exchange and feedback: In Table V, we compare the overhead of nformaton exchange and feedback of the proposed framework wth the energy effcent spectrum sharng polces proposed n [13] [0] and [1] for wreless networks and cogntve rado networks, respectvely. Before run-tme, the nformaton exchange n the proposed framework comes from the ITS algorthm ((M + N O(log (1/e wth e beng the performance loss tolerance and the exchange of b j for the LDF schedulng. The exchange of b j s for devaton-proofness. However, n the run tme, the feedback overhead of the proposed polcy s sgnfcantly lower than that of [13] [1]. Specfcally, n [13] [1], each user s recever needs to feedback the nterference temperature I n each tme slot. Hence, the total amount of feedback n [13] [1] grows lnearly wth tme. In concluson, ITS 0 1 LDF ITS t 1 ENTER sgnal receved, because users enter at t1 LDF ITS t EXIT sgnal receved, because users leave at t epoch 0 epoch 1 epoch Fg.. The proposed protocol mplemented by user when t receves ENTER sgnal at t 1 and EXIT sgnal at t. our proposed framework has a much lower total overhead than [13] [1]. Computatonal complexty: The mplementaton of the proposed polcy ncludes the ITS algorthm before run-tme and the LDF schedulng at run-tme. Frst, both the ITS algorthm and the LDF schedulng converge fast n logarthmc tme as proved n Theorems and 3. Second, each teraton n the ITS algorthm nvolves solvng the equaton (6, whch can be done effcently usng the Newton method. Each teraton n the LDF schedulng nvolves computng M + N ndces {d j (t} j M N and M +N normalzed values {r j (t} j M N, all of whch are determned by analytcal expressons. Fnally, although the orgnal defnton of the polcy requres each user to memorze the entre hstory of dstress sgnals, n the LDF schedulng, each user only needs to know the current dstress sgnal y t and memorze M + N normalzed values {r j (t} j M N. In concluson, the overall computatonal complexty of each user n mplementng the proposed polcy s small. D. Users Enterng and Leavng the Network We adapt the protocol to the scenaro where users enter and leave the network. We dvde tme nto epochs, where a new epoch begns when users enter or leave. The system starts at epoch 0, and we denote the optmal nstantaneous throughput n epoch 0 by r (0. When new users enter or exstng users leave at t 1, each of them broadcasts a ENTER or EXIT sgnal, respectvely. Upon recevng such a sgnal, the users run the ITS algorthm agan to determne the optmal nstantaneous throughput n epoch 1, r (1. Note that for each exstng user, the nput to the ITS algorthm s the contnuaton throughput at t 1, namely γ (t 1 ; whle for each new user j, the nput should be ts mnmum throughput Rj mn. Then they run the LDF schedulng wth the new nstantaneous throughput, untl a new epoch begns when the ENTER or EXIT sgnals are broadcast by some users at t. We llustrate how to adapt the protocol n Fg.. One nce property of the proposed protocol s that, the convergence of the LDF schedulng s not affected by users comng or leavng. Theorem 4: In the proposed spectrum sharng protocol, each user s average throughput converges to the mnmum throughput requrement n logarthmc tme, even wth users enterng and leavng the network. Proof: See Appendx E. LDF t

9 9 TABLE V. COMPARISON OF THE TOTAL OVERHEAD OF INITIAL INFORMATION EXCHANGE AND FEEDBACK. Informaton exchange before run-tme Feedback at run-tme [13] [0] N/A Each user : I each tme slot Amount: M + N reals per tme slot [1] Spectrum coordnator to each user: degradaton of ts mnmum throughput requrement Each user : I each tme slot, dstress sgnal f necessary Amount: M + N reals Amount: M + N reals per tme slot, dstress sgnal f necessary Each user broadcasts to other users: ρ(y = 1 p Proposed and {b j} j once, and R mn /r dstress sgnal f necessary at each teraton of the ITS algorthm; Each user s recever to ts transmtter: Ī, I Amount: (M + N + (M + N O(log (1/e reals Amount: dstress sgnal f necessary Note that we can also deal wth the changes of system parameters (e.g. the channel gans n the same way as we deal wth the dynamc entry and ext of users. Specfcally, whenever a user observes a change n the system parameters, t can broadcast a sgnal that trggers the users to run the ITS algorthm and the LDF schedulng agan. The convergence result n Theorem 4 also apples to ths case. In some works [0] for energy effcent power control n wreless networks, the locally stable asymptotc convergence of the proposed algorthm s proved. The locally stable asymptotc convergence guarantees that slght perturbaton from the equlbrum (nduced by, for example, an ncomng user wll not make the algorthm dverge. However, the convergence result n Theorem 4 are dfferent from that n [0]. Specfcally, we study the convergence of not only the transmt power levels, but also the transmsson schedule, whch s not studed n [0]. More mportantly, the nfluence of dynamc entry and ext of users on the convergence and stablty s qute dfferent n our work as compared to [0]. Snce our proposed polcy s TDMA, there s no nterference among the users. Hence, an ncomng user wll not nterfere wth the exstng users when they transmt. In other words, the nfluence of ncomng users s not through the nterference as n [0], but through acqurng the transmsson opportuntes of the exstng users. We show that under such perturbaton (n terms of transmsson opportuntes, the proposed LDF schedulng stll converges to the target throughput at the same rate. VII. PERFORMANCE EVALUATION In ths secton, we demonstrate the performance gan of our spectrum sharng polcy over exstng polces, and valdate our theoretcal analyss through numercal results. Throughout ths secton, we use the followng system parameters by default unless we change some of them explctly. The nose powers at all the users recevers are 0.05 W. For smplcty, we assume that the drect channel gans have the same dstrbuton g CN (0, 1,, and the cross channel gans have the same dstrbuton g j CN (0, 0.5, j. The users have the same mnmum throughput requrement of 1 bts/s/hz. The dscount factor s The nterference temperature threshold s θ = 1 W. The measurement error ε s Gaussan dstrbuted wth zeros mean and varance 0.1. The energy effcency crteron s the average energy consumpton across users. A. Comparsons Aganst Exstng Polces Frst, assumng that the populaton s fxed, we compare the proposed polcy aganst the optmal statonary polcy n [13] [1], and the optmal round-robn polcy wth cycle length Avg. energy consumpton (mw Statonary Round robn Proposed Statonary: nfeasble beyond 4 users 50% energy savng Number of users (a Small numbers of users. Avg. energy consumpton (mw Round robn Proposed 90% energy savng Number of users (b Large numbers of users. Fg. 3. Energy effcency of the statonary, round-robn, and proposed polces under dfferent numbers of users. L = M + N (.e. each user gets one slot n a cycle. We compare the energy effcency of the polces as the number of users ncrease n Fg. 3. Each data pont plotted s the average of 1000 channel realzatons. Frst, we can see that the statonary polcy becomes nfeasble when the number of users s more than 4. In contrast, the round-robn and proposed polces reman feasble when the number of users ncreases. Second, the proposed polcy acheves sgnfcant energy savng compared to the round-robn polcy, especally when the number of users s large. Specfcally, t acheves 50% and 90% energy savng compared to the round-robn polcy when the number of users s 11 and 15, respectvely. These are exactly the deployment scenaros where mprovements n spectrum and energy effcency are much needed. B. Adaptng to Users Enterng and Leavng the Network We demonstrate how the proposed polcy can seamlessly adapt to the entry and ext of PUs/SUs. We consder a network wth 10 PUs and SUs ntally. The PUs mnmum throughput requrements range from 0. bts/s/hz to 0.38 bts/s/hz wth 0.0 bts/s/hz ncrements, namely PU n has a mnmum throughput requrement of 0. + (n bts/s/hz. The SUs have the same mnmum throughput requrement of 0.1 bts/s/hz. We show the dynamcs of average energy consumptons and throughput of several PUs and all the SUs n Fg. 4. In the frst 100 tme slots, we can see that all the users quckly acheve the mnmum throughput requrements at around t = 50. PUs have dfferent energy consumptons because of ther dfferent mnmum throughput requrements. The two SUs converge to the same average energy consumpton and average throughput. There are SUs leavng (t = 100 and enterng (t = 150, 50, and a PU enterng (t = 00. We can see that durng the entre process, the PUs/SUs that are ntally n the system mantan the same

10 10 throughput and energy consumpton. The new PU (PU 11 has a hgher energy consumpton, because of ts hgher mnmum throughput requrement (0.4 bts/s/hz, and because of the lmted transmsson opportuntes left for t. SU 3, however, does not need a hgher energy consumpton because t occupes the tme slots orgnally assgned to SU, who left the network at t = 100. But SU 4 does need a hgher energy consumpton, because there are more SUs and less transmsson opportuntes n the network after t = 50. VIII. CONCLUSION In ths paper, we proposed nonstatonary spectrum sharng polces that allow the PUs and SUs to transmt n a TDMA fashon. The proposed polcy can acheve hgh spectrum effcency that s not achevable by exstng polces, and s more energy effcent than exstng polces under the same mnmum throughput requrements. The proposed polcy can acheve hgh spectrum and energy effcency even when the users have erroneous and bnary feedback of the nterference temperature. We extend the polcy to the case wth users enterng and leavng the network, whle stll mantanng the spectrum and energy effcency of the exstng users. The proposed polcy s amenable to decentralzed mplementaton and s devaton-proof. Smulaton results demonstrate the sgnfcant performance gans over state-of-the-art polces. Interestng future research drectons nclude how to desgn the optmal polcy when the feedback s fner than bnary and when the users have dfferent delay senstvtes (.e. dfferent dscount factors. APPENDIX A PROOF OF LEMMA 1 Suppose that n the optmal TDMA protocol π, there exsts a user and two tme slots t 1 t, such that 0 < π (t 1 < π (t (note that we do not assume t 1 < t or t 1 > t. We wll fnd another protocol π that fulflls the same mnmum throughput requrements wth lower energy consumptons, whch contradcts the fact that π s optmal. We construct the protocol π as follows. The transmsson strateges of the users other than user reman the same, namely π = π. For user, the transmsson remans the same for the tme slots other than t 1 and t, namely π (t = π (t, t t 1, t. Then we ncrease user s power level at t 1 by ɛ 1 > 0,.e. π (t 1 = π (t 1 + ɛ 1, and decrease ts power level at t by ɛ > 0,.e. π (t = π (t ɛ. To mantan user s average throughput, ɛ 1 and ɛ should satsfy ( ( δ t1 log 1 + gp (t1 + δ σ t log [ [ = δ t1 log 1+ g(p ]+δ (t1+ɛ1 t log σ 1 + gp (t σ 1+ g(p (t ɛ σ Gven ɛ 1, we can calculate ɛ as ɛ (ɛ 1 = ( σ +gp (t σ δ g [1 +gp (t1 t 1 t ] σ +g(p (t1+ɛ1. Then the decrease n average energy consumpton by swtchng to ]. protocol π can be calculated as (ɛ 1 = δ t1 ɛ 1 + δ t ɛ (ɛ 1. Takng the dervatve of (ɛ 1 wth respect to ɛ 1, we have [ ɛ 1 = δ t1 σ +gp (t +gp (t1 t 1 t 1]. ( σ δ σ +g(p (t1+ɛ1 σ +g(p (t1+ɛ1 Snce p (t > p (t 1, we have ɛ 1 > 0 when ɛ 1 = 0. Snce ɛ 1 s contnuous n ɛ 1 when ɛ 1 0, we can fnd a small enough ζ > 0, such that ɛ 1 > 0 for all ɛ 1 [0, ζ]. Hence, the decrease (ɛ 1 n user s average energy consumpton by swtchng to π s postve for any ɛ 1 [0, ζ]. Ths contradcts wth the fact that π s optmal, whch proves the lemma. APPENDIX B PROOF OF THEOREM 1 Due to space lmtaton, we present the proof of a smplfed verson of Theorem 1 n the specal case when the users are not self-nterested. Ths proof wll llustrate the man dea of the complete proof. Please refer to [33, Appendx B] for the complete proof of Theorem 1. Specfcally, we prove the followng lemma on the feasble nstantaneous throughput when the users are obedent. The lemma s a specal case of Theorem 1 by settng b + j = for all, j. Lemma : When the users are obedent, an nstantaneous throughput vector {r tdma } M N s feasble for the mnmum throughput requrements {R mn } M N, f the dscount factor δ satsfes δ 1 1 M+N, M N Rmn /r tdma = 1. Proof: As n dynamc programmng, we can decompose each user s dscounted average throughput nto the current throughput and the contnuaton throughput as follows: R (π = (1 δ δ t (1 {π(t>0} r tdma t=0 ( = (1 δ 1{π(0>0} r tdma }{{} the current throughput at t=0 [ ] + δ (1 δ δ t 1 (1 {π(t>0} r tdma. t=1 }{{} the contnuaton throughput startng from t=1 We can see that the contnuaton throughput startng from t = 1 s the dscounted average throughput as f the system starts from t = 1. In general, we can defne user s contnuaton throughput startng from t as γ (t (1 δ τ=t δτ t (1 {π(τ>0} r tdma. Then the decomposton at tme t can be wrtten as γ (t = (1 δ (1 {π(t>0} r tdma + δ γ (t + 1. Wrte the contnuaton throughput vector as γ = (γ 1,..., γ N. Defnton (Self-generatng set: A set of throughput vectors R s a self-generatng set, f for any throughput vector γ R, there exsts a N and a contnuaton throughput vector γ R such that for all N, γ = (1 δ (1 {= } r tdma + δ γ. (8