Energy Efficient Resource Allocation for Quantity of Information Delivery in Parallel Channels

Transcription

1 TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIES Trans. Emergng Tel. Tech. 0000; 00: 6 RESEARCH ARTICLE Energy Effcent Resource Allocaton for Quantty of Informaton Delvery n Parallel Channels Jean-Yves Baudas, Andrea M. Tonello 2 and Abdallah Hamn 3 Natonal Center for Scentfc Research CNRS, Insttute of Electroncs and Telecommuncatons of Rennes IETR, UMR 664, F Rennes, France emal: jean-yves.baudas@nsa-rennes.fr, 2 Dpartmento d Ingegnera Elettrca, Gestonale e Meccanca, Unverstà d Udne, Italy e-mal: tonello@unud.t, 3 Unversté européenne de Bretagne, France, INSA, IETR, UMR 664, F Rennes. ABSTRACT Ths paper deals wth the problem of mnmzng the energy requred to transmt a certan amount of nformaton bts. If there s no constrant, n a Gaussan channel, the soluton leads us to the use an nfnte transmsson tme or an nfnte bandwdth. In partcular, the mnmum requred sgnal-to-nose rato to transmt one bt s equal to.59 db. To allow the development of new green communcaton servces, a fnte bandwdth and a fnte tme have to be used n practce. Thus, we focus on the transmsson of a gven number of bts over a set of parallel Gaussan channels when there s an energy constrant and we have the goal of mnmzng ether the hghest transmsson tme among the channels or the average channel occupancy tme. Ths s a resource allocaton problem that s formulated by targetng a certan energy consumpton factor, heren defned as the rato between the energy requred n the asymptotc regme and the energy requred to transmt a certan quantty of nformaton wth lmted tme, bandwdth resources. The Pareto front for the jont optmzaton of the maxmum transmsson tme and the average channel occupancy tme s also derved. Several explcatve numercal results of the derved resource allocaton algorthms for energy effcent communcatons are also reported. Copyrght 0000 John Wley & Sons, Ltd.. INTRODUCTION Energy consumpton of data communcaton systems s becomng an mportant ssue due to the large use of devces. The evaluaton of energy consumpton can be done from varous perspectves whch nclude mplementaton and hardware ssues as crcut consumpton and non deal performng sgnal processng algorthms for data recovery [, 2]. In fact, t s well-known that the mnmum sgnal energy per nformaton bt that s requred for relable Part of ths work was presented at the IEEE Wreless Communcatons, Networks Conference, Cancun, Mexco, Mar. 20 and at the IEEE Internatonal Symposum on Power Lne Communcatons and Its Applcatons, Udne, Italy, Apr. 20. communcatons n a Gaussan channel can be obtaned from the mnmum sgnal-to-nose rato SNR that s equal to.59 db. Ths result was frstly derved n [3] n the asymptotc regme assumng that the transmsson of the nformaton requres an nfnte amount of tme. More recently t has been extended to a general class of channels n [4] and t has been shown that t can be acheved as the bandwdth goes to nfnty. In the green communcatons context, ths lower lmt gves the mnmal transmt energy per bt requred for relable communcaton. Consequently, the effcency of green communcaton systems can be measured usng the bt-per-joule bt/j metrc [, 5]. Ths performance metrc has been studed takng nto account varous aspects both wth pragmatc and nformaton Copyrght 0000 John Wley & Sons, Ltd. [Verson: 202/06/9 v2.]

2 theoretc approaches [6]. In ths paper we only focus on the energy effcency n relaton to capacty. Wth delay tolerant applcatons and networks, the constrant on the tme of transmsson can be relaxed [7,8] and the soluton to the problem of mnmzng the transmt energy leads to the use of an nfnte tme of transmsson. However, from a nformaton theoretc pont of vew, t becomes nterestng to study and develop resource allocaton schemes that ensure relable communcatons wth a gven energy constrant near to the asymptotc energy lmt but assumng a fnte bandwdth and tme of transmsson. The objectve s then to fnd the feasble and bounded set of parameters that provde a gven energy effcency. From an operatonal pont of vew, focusng on energy effcency, nstead of spectrum effcency, can lead to the development of new and green communcaton servces that explot the underloaded usage perods of the networks and the mechansms of load-sheddng [9]. More than two orders of energy gan can then be expected [, ]. In ths paper, to focus on the system energy effcency and formulate the objectve, we start by defnng the energy effcency factor as the rato between the asymptotc energy lmt and the energy requred to transmt an amount of bts wth lmted tme, bandwdth resources. Then, the optmzaton problem becomes a resource allocaton problem under the constrant gven by the defned energy effcency factor and under the constrant gven by the quantty of nformaton to be transmtted. The problem s solved for parallel ndependent Gaussan channels. The focus s gven on partcular solutons that mnmze the tme of transmsson defned as the maxmum among the transmsson perods n the set of channels referred to as transmsson tme, or on the solutons that mnmze the average of the transmsson perods n the set of channels referred to as average channel occupancy tme. The remander of the paper s organzed as follows. Secton 2 ntroduces the model of the communcaton system. Secton 3 recalls some results on energy mnmzaton. In ths secton, we also derve the energy lower bound for parallel channels and we analyze the new energy effcency metrc. The general problem s formulated n Secton 4 and two smple cases are analyzed. Partcular solutons are analysed n Secton 5. These solutons address the problem of mnmzng the transmsson tme and the problem of mnmzng the average channel occupancy tme, two problems that are stated and studed n ths Secton 5. The jon optmzaton of these two tme parameters s studed n Secton 6. Secton 7 concludes the paper. 2. SYSTEM MODEL We consder n parallel ndependent channels. On the th channel, the nput-output relatonshp s Y = h X + W, =,, n where X denotes the transmtted data sgnal wth power P, Y s the receved sgnal, and h s the complex scalar channel gan. The addtve nose sample W s assumed to be a complex Gaussan random varable wth zero-mean. The nose s assumed ndependent across the channels and whte n the transmsson band B, wth power spectral densty N. Wth a Gaussan nput sgnal, the maxmum nputoutput mutual nformaton n channel, s the well-known channel capacty measured n bt-per-second-per-hertz C = log 2 + h 2 P. 2 B N It should be noted that 2 can be used to derve a tght approxmaton of the maxmal number of bts that can be transmtted when the number of bts s large. Other bounds can be obtaned wth fnte and low number of bts to be transmtted [2]. The maxmum quantty of nformaton expressed n bt that can be transmtted n T seconds n the channel wth bandwdth B s Q = C B T. The correspondng consumed energy, whch s the transmt energy durng the transmsson perod T, s J = P T = 2 Q T B BT, 3 where = h 2 N s the normalzed SNR n channel obtaned wth unt transmt power and bandwdth. Ths energy s requred to transmt the nformaton bts Q durng the transmsson perod T n channel when the channel state nformaton s known at both the transmtter and recever sdes. 2 Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd.

3 Now, wth n parallel channels, the total energy requred to transmt Q = Q bts becomes J = J = 2 Q T B BT. 4 In ths paper, we provde results n the asymptotc regme assumng that Q and T are suffcently large so that 2 and 3 gve vald and suffcently tght results. In the non-asymptotc regme, bounds tghter than the one gven by the capacty formula 2 can be used [3]. 3. ENERGY CONSIDERATIONS AND PRELIMINARIES Ths secton recalls some known results on energy consumpton adapted to our system model. An energy effcency metrc, called energy effcency factor, s also ntroduced and compared to the conventonal energy effcency metrc. 3.. Energy consumpton mnmzaton We frst formulate the allocaton problem where the objectve s the quantty of nformaton to be transmtted and the resource s the energy to be mnmzed. Ths s obtaned startng from the power allocaton problem to maxmze the transmsson rate n Gaussan parallel channels. Proposton. Under a certan transmsson perod allocaton {T } n, the nformaton bt allocaton {Q } n whch mnmzes the energy needed to transmt Q bts of nformaton s wth λ such that See Appendx A. Q = 0, λ, Q = B T log 2 λ, λ >, Q = Q. Ths result s a water-fllng soluton appled to the energy mnmzaton. If all B = B for all [, n], ths soluton s also the soluton of the power mnmzaton problem under a bt-rate constrant C C = Q BT, and where Q and T go to nfnty Lower bound of energy = C wth It s well known that the mnmal needed energy to transmt a certan amount of bts can be obtaned n the ultra wde band regme [4] wth partcular condtons. The same result can also be obtaned n the ultra wde tme regme. Ths concept of ultra wde tme s smlar to the ultra wde band case when the bandwdth s replaced by the tme of transmsson. In both ultra wde cases, the product B T proposton [4]. goes to nfnty. We recall here the followng Proposton 2. The asymptotc lmt of energy needed to transmt Q bts over one channel of normalzed SNR s J,0 = lm B T J = Q log e 2. Ths proposton s another formulaton of the mnmum receved sgnal energy per nformaton bt requred for relable communcatons. It should be noted that the channel gan or attenuaton s taken nto account through the coeffcent. The use of ultra wde band or ultra wde tme s not a suffcent condton to ensure the asymptotc regme. The correct condton to verfy Proposton 2 s such that the product B T must go to nfnty,.e., f only one parameter goes to nfnty, the other shall not go to 0. In fact, f all the nformaton bts Q are transmtted n only one symbol, the tme T and the bandwdth B can be such that the product B T s a certan constant. In ths case B T = α and the needed energy J = 2 Q α α 5 s hgher than J,0 for all fnte α even f ether B or T goes to nfnty. The result of Proposton 2 s now extended to the case of parallel ndependent channels. Proposton 3. The asymptotc lmt of energy needed to transmt Q bts through n ndependent parallel channels wth normalzed SNR {} n s obtaned when all the nformaton bts are transmtted over the best channel, so Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd. 3

4 that [] See Appendx B. J 0 = Q log e 2. max Ths soluton s not the water-fllng one or, n other words, t can be consdered as a lmtng case of waterfllng: all the nformaton bts are transmtted over the channel wth the hghest normalzed SNR, called hereafter the best channel. Ths result s not surprsng. The best way to mnmze the needed energy s to use the channel whch needs the least energy. The mnmal energy lmt can not be reached under fnte tme or bandwdth constrants. If we try to mnmze the needed energy, we obtan only nfnte solutons for the transmsson tme and bandwdth. Therefore, t s nterestng to relax the problem and to fnd a resource allocaton soluton when the goal s to transmt wth energy that s close, but not dentcal, to the one obtaned n the asymptotc regme Energy effcency factor Conventonally, the energy effcency η EE s expressed n bt/j [] η EE = Q C = γ J 2 C, 6 where η EE decreases monotoncally wth C and the communcaton s spectrally effcent for hgh C energetcally neffcent. but The energy effcency η EE depends on the channel gan and, n some cases, t can be domnated by the lnk budget. The trade-off between the capacty and the energy has been wdely nvestgated. In ths paper, we focus on the energy effcency to derve new resource allocaton algorthms. To do ths, we start by defnng the energy effcency factor as the rato between the requred energy for a certan quantty of nformaton transmsson and the asymptotc energy lmt. We use the defnton proposed n [, ]. Defnton. The energy effcency factor s the rato between the asymptotc lmt and the needed energy wth fnte tme and frequency resource. Usng 3 and Proposton 2, the energy effcency factor of channel s therefore = J,0 J = Q log e Q B T B T Ths energy effcency factor s ndependent from the normalzed SNR,.e., ndependent from the channel gan. Ths means that ths energy effcency factor does not depend on the lnk budget and t characterzes the ablty of a transmsson to explot the energy capacty of a channel. It s then an effectve measure to compare dfferent communcaton systems ndependently from the channel effect. For a gven tme-bandwdth product, the energy effcency factor depends on the amount of nformaton bts and decreases wth Q. However, ths energy effcency factor becomes ndependent of Q f the tme used to send nformaton grows lnearly wth the amount of nformaton bts. The energy effcency factor verfes the followng propertes. Property. 0 <. Property 2. Property 3. lm =. B T + lm = 0. B T 0 The transmsson of a certan amount of nformaton bts s then effcent f the energy effcency factor s near to and t s neffcent f the energy effcency factor s near to 0. Ths energy effcency factor s lnked to the spectral effcency wth the followng property. Property 4. = C log e 2 2 C. In ths property, C s the spectral effcency expressed n bt/s/hz assumng to transmt over the perod T. The spectral-energy effcency factor trade-off s drawn n Fg.. For example, the energy effcency factor = 3 db s obtaned wth the spectral effcency C =.8 b/s/hz. The energy effcency factor of a communcaton system decreases wth the spectral effcency and the hgher the spectral effcency, the lower the energy effcency s. Ths concluson s the same as the one obtaned wth the conventonal defnton of energy effcency η EE. The total energy effcency factor n a system usng n parallel channels s not the sum of the energy effcences 4 Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd.

5 Property 7. The energy effcency factor that can be obtaned usng the only one channel s upper bounded by γ max j Dscusson Fgure. Energy effcency factor versus spectral effcency C. but, wth 4 and Proposton 3, t s = J0 J =. 8 Q max j Q Dfferently from the sngle channel case, depends on the channel state but not n absolute terms. In fact, the channel nfluence s normalzed by the best channel coeffcent through the defnton of. Usng the weghted harmonc mean and the mean nequaltes, verfes the followng property. Property 5. mn max Q Q max max. Q Q In a Gaussan channel, t follows that mn max, and f {Q } are equally dstrbuted γ H then max mn γ H max max wth γ H the harmonc mean of {}. By defnton, satsfes also Property. Property 3 can be extended to : f only n one channel the product B T goes to zero, then goes also to zero. Property 2 s modfed by the followng property. Property 6. Let j = arg max γ, then = f and only f Q j = Q and the product B jt j goes to nfnty. Ths property says that f a very hgh energy effcency factor s needed, then all the nformaton bts must be transmtted through the best channel. In ths case, only one channel s used. If a certan channel, not the best, s used, then the energy effcency factor s lmted. As t s shown n Secton 3.3, the defned energy effcency factor s a relatve measure. Ths measure characterzes the ablty of a communcaton system to explot what s feasble for a gven channel from an energetc pont of vew. The conventonal defnton of energy effcency gven n 6 s a key measure n the green communcaton context to evaluate the energy costs. However, t should be noted that t depends on the lnk budget: the same channel capacty C and bandwdth B lead to dfferent energy effcences η EE dependng on the channel gan and nose level. On the contrary, the energy effcency factor defnton used n ths paper and gven n 7 s more approprate to evaluate the performance of a certan allocaton of resources and to measure ts ntrnsc energetc capacty, as t wll be dscussed below. 4. PROBLEM FORMULATION Our nterest s not to mnmze the energy needed to transmt Q nformaton bts because ths problem s solved wth the use of nfnte transmsson tme or wth an nfnte bandwdth. The goal s to transmt Q bts n a fnte tme and fnte bandwdth wth the effcency factor, as stated n Problem below. Problem. Fnd {Q, T } n such that J = J0, 9 Q = Q, [, n] Q 0, [, n] T 0. 2 The varables are {Q, T } n n ths problem formulaton. There are two possble other formulatons wth varables {Q, J } n or {J, T } n. Note that despte the Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd. 5

6 T* T* 6 Fgure 2. Feasble set {T, T 2} for the 2-channel case. fact that all three sets of varables {Q, T, J } n must be non-negatve, the constrants 2 are suffcent. Let us analyze the case of two channels, n = 2, wth such that Property 7 s satsfed for both channels. In ths case, all the nformaton bts Q can be transmtted n channel or n channel 2. The feasble set of parameters {Q } s defned by Q 2 = Q Q, wth Q [0, Q]. The feasble set of parameters {J } s defned by J 2 = J 0 J, wth J [0, J 0 ]. On the contrary, there s no expresson for the feasble set {T }. The regon of feasble {T } can be obtaned by smulaton solvng Problem and an example s gven Fg. 2. In ths fgure, the feasble set corresponds to the dashed regon and T s the soluton of 3 wth J = J 0 and Q = Q for {, 2}. For T [T, +, T 2 vares from 0 to T2 and for T 2 [T 2, +, T vares from 0 to T. In the followng, we frstly analyze some specfc cases,.e., we add a further constrant to Problem so that we smplfy and reduce the set of solutons. Such constrants are gven by unform tme and unform bt allocatons over the channels as well as unform spectral effcences. The strength of these cases s that they lead to a smpler transmstter or recever structure. 4.. Unform tme and unform nformaton bt allocatons Let us assume that all the channels transmt the same nformaton bts Q = Q durng the same tme T = T n and n the same bandwdth B = B. In ths case, the transmtter does not need to know the channel condtons to allocate the nformaton over the n channels. The problem s then: fnd T such that n T B2 Q nt B = J0. 3 From Property 6, t s clear that 3 cannot be solved for very hgh values of. The transmsson s feasble under certan condtons on as dscussed n the followng. Proposton 4. Under unform tme and nformaton bt allocatons, the energy effcency factor s reachable f and only f See Appendx C. max [,n] γ n. Wth a more flexble allocaton, empty channels can be allowed. In ths case, the energy effcency factor s always feasble wth a varable number of loaded channels. Ths case s analyzed n the next paragraph Unform spectral effcences Let α be the spectral effcency per channel,.e. α = C for all, wth the same bandwdth B = B. Wth ths constrant n pratcal systems, the same modulaton scheme s used for all channels. Wth unform spectral effcences, Q depends on T as Q = αbt for all. Then, the number of varables can be reduced from 2n down to n. Wth n = 2 the solutons are defned by X 2 = X J X where X s Q, J or T, and X s Q, 0 or Q, α respectvely. Wth n hgher than 2, the solutons are hyper planes. 5. TIME MINIMIZATIONS The problem we are now nterested n s not to fnd all the feasble ponts n the regon of solutons of Problem but only partcular ponts that mnmze some defned temporal quanttes. Specfcally, Problem s reformulated so that we obtan the soluton that mnmzes the maxmum transmsson tme or the average occupancy tme, as defned n the followng. 6 Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd.

7 5.. Defntons and new problems Even f the key parameter s the energy effcency factor, t s mportant to mnmze the tme needed to transmt the nformaton bts for a gven energy effcency factor. We frst defne the transmsson tme. Defnton 2. The transmsson tme s the maxmum value n the set {T } for [, n] T M = max T = T,, T n. [,n] The transmsson tme T M s the tme used to transmt the nformaton bts. Based on ths defnton, the new problem s stated as follows. Problem 2. Mnmze the transmsson tme T M under the constrants 9 2. We now defne another mportant quantty called the average channel occupancy tme. Defnton 3. The average channel occupancy tme s the average of the transmsson perods T wth [, n] T S = n T = T,, T n. The average channel occupancy tme measures the average channel busy tme for a gven transmsson of nformaton bts. It s mportant to mnmze the transmsson tme for a gven energy effcency factor so that we free one or more channels as soon as possble for other users or other communcaton systems. Therefore, the new problem can be stated as follows. Problem 3. Mnmze the average channel occupancy tme T S under the constrants 9 2. In summary, Problems 2 and 3 fnd the specfc solutons of Problem that mnmze T M and T S, respectvely Transmsson tme mnmzaton Before provdng the soluton of Problem 2, we need the followng ntermedate result. Lemma. Under an energy effcency factor constrant and unform spectral effcences, the mnmzaton of the transmsson tme T M leads to the unform allocaton of the tme and unform allocaton of the nformaton bts over a gven subset of channels. See Appendx D. The mnmzaton of the transmsson tme under a unform spectral effcency constrant leads to fndng the channel subset whch mnmzes the tme T n 3. Wth n channels, there are 2 n possble subsets. The search of the subset of channels can be reduced to a number lower than n. In fact, let f : γ T 4 be the functon such that T = f γ satsfes 3. Ths functon s a decreasng functon. Then, the optmal subset of channels s defned by the hghest γ and at most n comparsons are needed as explaned n Algorthm. Algorthm. : Sort n decreasng order: γ π γ π2 γ πn 2: Set k = 3: Evaluate the cost functon f over the channel ndexes {π,, πk} 4: If f π,, πk s feasble,.e. T exsts, then k k + and go to step 3 whle k n 5: {γ π,, γ πk } s the optmal channel subset Algorthm can also be used to solve the water-fllng problem of Proposton. Let I be the optmal subset obtaned wth Algorthm. Then, the energy effcency factor s related to the spectral effcency α and the subset of channels I as reported by the followng corollary. Corollary. Under unform spectral effcency α, the mnmzaton of the transmsson tme to transmt Q nformaton bts Problem 2 leads to the energy effcency factor = α log e 2 2 α I I max j [,n] γj It s obtaned by usng 3 wth the sum over the subset I of channel and Lemma.. Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd. 7

8 The expresson of n Corollary can be compared to the one n Property 4, and the bound n Proposton 4 remans vald by summng over the channels n I nstead over [, n]. Corollary can also be compared to 8. The relaton between n 8 and α n Corollary s = α log e 2 2 α. 5 The unform spectral effcency constrant leads to a unform energy effcency factor over the channels. The total energy effcency factor s then equal to the energy effcency factor per channel weghted by the harmonc γ mean of the rato max j, wth I whereas j [, n]. The energy effcency factor depends on the subset I of channels used and t depends also on all the n channels through the maxmal normalzed SNR max j. However, snce the subset I s composed of the best channels, then max j s by defnton ncluded n ths subset. To compute the mnmum tme of transmsson Problem 2 for the transmsson of a gven quantty of nformaton bts Q, the followng lemma s also needed. Lemma 2. Under the energy effcency factor contrant and the nformaton bt allocaton constrant, the mnmzaton of the transmsson tme T M for the bt allocaton {Q } n over n parallel and ndependent channels leads to a unform channel tme allocaton. It s based on the proof of Lemma. See Appendx E. If one tme T s hgher than all other tmes, t s then possble to relocate energy from another channel to channel to reduce the tme T wthout any change n the allocaton of nformaton bts and wth the same total energy. The soluton of Problem 2 can now be establshed wth the followng theorem. Theorem Soluton of Problem 2. Under an energy effcency factor constrant, the mnmzaton of the transmsson tme of Q bts over n parallel and ndependent channels leads to Q = 0, λ, Q = B T M log 2 λ, λ >, wth λ such that Q = Q and the mnmal transmsson tme T M s the soluton of exp Q log e 2 B TM I = B log e I J 0 TM I wth I = { [, n] λ }. See Appendx F. B I I B +, B B I It s now possble to transmt Q nformaton bts under the energy effcency factor wth the mnmal tme of transmsson. The optmal subset I s obtaned usng Algorthm. The result n Theorem can be compared to Proposton. Corollary 2. The bt allocaton {Q } n whch mnmzes the transmsson tme Problem 2 under an energy effcency factor constrant s the bt allocaton whch mnmzes the energy under the mnmal transmsson tme. The Karush-Kuhn-Tucker KKT condtons of Proposton wth the soluton of Theorem are dentcal to the KKT condtons of Theorem wth the soluton of Proposton. Then, the solutons are also dentcal. Corollary 2 provdes another method to obtan the soluton of Theorem. The mnmum transmsson tme of Theorem can be obtaned usng Lemma 2, Proposton and an teratve algorthm, such as the bsecton or secant method Average channel occupancy tme mnmzaton We focus now on Problem 3. The goal now s to mnmze the mean of T, [, n] such that 9 2 are satsfed. Unfortunately, a closed form expresson can not be derved n the general case of unequal bandwdths B. If all the channel bandwdths B are equal to B, then the followng theorem provdes a smple soluton. 8 Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd.

9 Theorem 2 Soluton of Problem 3 wth B = B. Under an energy effcency factor constrant and wth B = B for all, the mnmzaton of the average channel occupancy tme to transmt Q nformaton bts over multple parallel and ndependent channels Problem 3 leads to transmt over the best channel. The mnmal average channel occupancy tme T S satsfes e Q log e 2 nb T S wth = arg max j [,n] γj. See Appendx G. Q log e 2 nb T S = 0, The best way to mnmze the channel occupancy, and therefore to mnmze the average channel occupancy tme, s to transmt the nformaton bts n the best channel. Ths result s vald only f all the channel bandwdths are equal. Numercal examples show that t s not the case when there are dfferent channel bandwdths. For example, let n = 2, Q = 0, = /2, {γ, γ 2} = {2, }. Wth B = B 2 =, and usng the Lambert functon W x, the mnmal average channel occupancy tme s TS Q ln 2 = = W e + Wth B 2 =, the confguraton {Q, Q 2} = {49, 5} wth {J, J 2} = {0.99 J 0, J 0 } leads to T S = 24 whch s lower than T S. Contrarly to Theorem, Theorem 2 can therefore not be appled wth unequal bandwdths Numercal examples To show some numercal results, we consder an orthogonal frequency dvson multplexng OFDM system wth 24 channels sub-carrers each havng dentcal bandwdth B and a total transmsson bandwdth of 20 MHz. Only n = 998 sub-carrers are used n the bandwdth [0.5;20] MHz. As an example of frequency selectve channel response, we assume the frequency response of a typcal power lne communcaton lnk as descrbed by the 5-path Zmmermann channel model [4]. The nose s assumed whte Gaussan wth a power spectral densty of dbm/hz. The transmsson of Q = Mbt s assumed. Wth all these assumptons, the asymptotc lmt of energy needed to transmt Mbt of nformaton s Proposton 3 J 0 = 6 log e 2 max 40 µj. 7 Now, n Fg. 3, we show the transmsson tme T M and the average channel occupancy tme T S as a functon of the energy effcency factor when the resource allocaton algorthms assocated to Lemma and Theorems and 2 are appled. In partcular, Lemma : ths algorthm leads to the mnmum transmsson tme T M under unform channel spectral effcences. The Algorthm s used to calculate the optmal resource allocaton; Theorem : ths algorthm provdes the mnmum transmsson tme wthout any spectral effcency constrant; Theorem 2: ths algorthm mnmzes the average channel occupancy tme. As expected, the mnmum T S s obtaned when Theorem 2 s appled. However, f we apply ths resource allocaton the transmsson tme T M wll be ncreased. Theorem mnmzes the transmsson tme as expected but wth the nconvenence that the average channel occupancy tme T S s not mnmzed. The soluton obtaned wth Lemma s an ntermedate soluton that provdes a transmsson tme close to the mnmum transmsson tme. When goes to, T M = nt S and both T M and T S go to nfnty. All resource allocaton solutons become dentcal snce a sngle subcarrer s used. We now compare these results to a conventonal best effort communcaton where the objectve s to maxmze the btrate. In ths case, the btrate s maxmzed under a power spectrum densty contrant of 50 dbm/hz, whch s a peak power constrant. The soluton leads to T M best effort = T Sbest effort = 9.4 ms to transmt Q = Mbt. Consequently, the energy effcency factor can be computed and t becomes equal to best effort = 56 db. Therefore, f we compare ths result wth the one presented n Fg. 3, we notce that to transmt Mbt of nformaton the conventonal rate maxmzaton soluton requres about tmes less tme but 53 db of more energy compared to the energy effcent algorthms that target an effcency of = db. Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd. 9

10 Fgure 3. Tme n second, T M and T S, needed for the transmsson of 6 nformaton bts for 3 transmsson strateges versus energy effcency factor. 6. MULTI-OBJECTIVE OPTIMIZATION The transmsson tme T M and the average channel occupancy tme T S have been mnmzed under an energy effcency factor constrant. If the mnmzaton of T M s not more mportant than T S, t wll be nterestng to know f there exsts a Pareto fronter for the transmsson tme and average channel occupancy tme optmzaton problem. Ths Pareto fronter s an mportant fronter because t defnes equvalent confguratons for the multobjectve optmzaton problem. The varables T M and T S are defned n R + but they can not take all values n R 2 + for a gven energy effcency factor. A qualtatve example of feasble set of varables {T M, T S} s gven n Fg. 4 where three partcular ponts, A, B, and C, and two half-lnes, d and d 2 are hghlghted. Prevous results have shown that T M T M B wth T M B beng the mnmal transmsson tme gven by Theorem, and T S T SC wth T SC beng the mnmal average channel occupancy tme gven by Theorem 2. Note that Theorem 2 s vald only wth equal bandwdths, whch s the case treated hereafter. Other bounds on T M and T S as dscussed below can also be specfed. Property 8. n tmes the average channel occupancy tme nt S s lower bounded by the transmsson tme,.e. T max T. Property 9. The average channel occupancy tme T S s upper bounded by the transmsson tme,.e. T max T. These two propertes allow us to draw the half-lnes d and d 2 n Fg. 4. All couples {T M, T S} are above d and below d 2. The half-lne d s defned by nt S = T M wth T M T M C and the half-lne d 2 n by T S = T M wth T M T M B. Another bound s gven by the lne segment AB where T M A = T M B, T SA = T M B and nt SB = I T M B wth I gven by Theorem. An other bound can also be defned n the regon of feasble solutons. It corresponds to the curve segment BC n Fg. 4. It s defned as follows. Defnton 4. For every pont {T M, T S} n the curve segment BC and for every transmsson tme T M [T M B, T M C], the average channel occupancy tme T S Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd.

11 T S s d A B C Fgure 4. Feasble set of varable {T M, T S }. wth T [0, T M ], T SBC = mn T, n Q = Q and J = J 0. By defnton BC s a Pareto front. The followng theorem specfes the values of the ponts wthn ths front. Theorem 3. T M For every pont {T M, T S} n the Pareto fronter BC, T {0, T, T M } such that the sum of T s mnmal and wth no more than only one T j equal to T where T s the soluton of B T M log 2 α + B j T log2 α = Q, I j I j B T M α + B j T α = J0, and where j s the ndex of the poorest channel n I. See Appendx H. As n prevous cases, we are left wth the determnaton of the subset I for whch T 0. A modfed verson of Algorthm can be used, where the optmal subset s the one whch provdes the mnmal average channel d occupancy tme, and, as n Theorem, Q = B T log 2 α, J = B T α for all I. The cardnalty of I vares from the value gven by Theorem, n the pont B, to, n the pont C. T vares from 0 to T M. Thus, we have found the Pareto fronter BC. All ponts n ths fronter are equvalent from the mult-objectve optmzaton pont of vew. To show some numercal results, we consder the example of Secton 5.4, wth = 0.5. The mnmal transmsson tme T M s gven by Theorem and t s equal to 2.0 s, see Fg. 3. Ths confguraton leads to T S = 64.5 ms, see Fg. 3, and the correspondng pont n Fg. 4 s B. The mnmal channel occupancy tme s gven by Theorem 2 and t s equal to 28.3 ms. The correspondng pont n Fg. 4 s C wth T M C = 28.3 s. Then, n ths Pareto fronter, T M vares from 2.0 s to 28.3 s and T S vares from 28.2 ms to 64.5 ms. In ths example, the range of varaton of T M s hgher than the range of varaton of T S. In pont C, T M C = nt SC, wth n = 998, because only one channel s used. In pont B, only 3.2 % of the channels are used then T M B = 3.2 % T SB. Note that n pont A, all the channels are used and T SA = T M A = T M B. 7. CONCLUSION Wthout any transmsson tme constrant, the mnmzaton of the energy needed to transmt an amount of nformaton leads to an nfnte tme of transmsson. Conventonal energy effcency formulatons deal wth the mnmzaton of the transmt energy and lead to the use of an nfnte transmsson tme or bandwdth. In ths paper, we have nstead formulated resource allocaton problems wth a gven energy constrant near to the asymptotc energy lmt that s achevable wth a fnte bandwdth and tme of transmsson. To solve such problems, we have defned the energy effcency factor as the rato between the energy requred n the asymptotc regme and the energy requred to transmt a certan amount of nformaton wth lmted tme, bandwdth resources. Wth ths new energy Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd.

12 metrc defnton, we have studed the resource allocaton problem n parallel ndependent Gaussan channels from an nformaton theoretc pont of vew. We have nvestgated the mnmzaton of the transmsson tme and of the average channel occupancy tme under an energy effcency factor constrant. Fnally, the Pareto front for the jont transmsson tme and average channel occupancy tme optmzaton problem has been derved. The proposed resource allocaton algorthms provde a method to mplement energy effcent communcaton systems. A. PROOF OF PROPOSITION B. PROOF OF PROPOSITION 3 The problem s to fnd the nformaton bt allocaton that mnmzes the transmtted energy,.e. J 0 = mn { Q =Q, Q 0 } J,0. 2 Assume there exsts one j such that > j, then J,0 = log e 2 Q + j loge 2 log e 2 Q 22 s mnmal f and only f Q = 0 for all j. If two, or more, channels are maxmal,.e. = 2 > for all / {j, j 2}, then the total nformaton Q can be splt between these channels j and j 2 wthout any restrcton. Otherwse, only one channel carres all the nformaton Q and there s one sngle soluton. The problem s to fnd the nformaton bt allocaton that mnmzes 4,.e. {Q } = arg mn 2 Q B T BT. 8 under the constrants n Q = Q and Q 0, The functon Q J Q s convex. The problem s then a convex optmzaton problem and the Lagrangan s L = J Q + λ Q Q µ Q. 9 The bt allocaton {Q } n that satsfes the KKT condtons Q = Q, µ 0, [, n], Q 0, [, n], 20 µ Q = 0, [, n], Q log e 2 B 2 T λ µ = 0, s optmal. Let λ = λ log e 2. If µ = 0 then Q = B T log 2 λ 0 and λ. If µ 0 then Q = 0 and λ. C. PROOF OF PROPOSITION 4 Under a unform channel transmsson perod allocaton and a unform nformaton bt allocaton, the mnmal needed energy s obtaned wth nfnte transmsson tme, as t s stated by Proposton 2, lm T B2 n Q nt B = T Q log e 2 n. 23 The transmsson wth energy effcency fector s feasble f and only f the mnmal needed energy 23 s lower than J 0,.e., Q log e 2 Q log e n max In ths case, the transmsson s then feasble otherwse the target energy effcency factor s not achevable. Note that the transmsson s feasble n fnte tme n the case of strct nequalty. 2 Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd.

13 D. PROOF OF LEMMA E. PROOF OF LEMMA 2 Lemma s proven n the general case wthout any constrant on the nformaton quantty. The varables {J } n are consdered nstead of {T } n. The Lagrangan of the convex optmzaton problem s wth n L = max T + λ J0 λ + µ J, 25 In Appendx D t has been proven that for all {Q } n and wth unform channel capactes, the optmal transmsson perods {T } n that mnmze the transmsson tme are ndependent of. Ths result can be extended wthout channel capacty constrant. Usng 3, let f be the functon such that f T = J. Ths functon s monotonc and convex. The Lagrangan of the problem s gven by 25 replacng T by f J nstead of 26. The soluton T s then T = J B 2 α. 26 The nfnte norm s not dfferentable but t s the lmt of the p-norm for p. We then use ths p-norm and the roots of the dervatve of the Lagrangan are T = J0 2 j I and T = T M for all I. Q j B j T j Bj 3 p p J p j= B j2 α = p λ + µ. 27 p B 2 α As n Appendx A, the KKT condtons are used and the soluton s ndependant of p J = J0 j I B 28 B j for all n I = { [, n] µ = 0}. Usng 26 and 28, t follows T = J0 j I 2 α B j and T s ndependent of. Wth B = B for all, Q = αbt = J0 and Q s then ndependent of. α 2 α j I, 29 γ j 30 These results have been obtaned wthout any constrant over α and {Q }. It then remans vald wth the constrant Q = Q. Note that the unform tme allocaton s also vald even f all B are not equal. F. PROOF OF THEOREM Let f be the functon ft, {Q } n = 2 Q B T BT = J0. 32 Ths functon s convex and decreasng along the T axs. The nverse functon s then convex and f f T, {Q } n, {Q} n = T. 33 The Lagrangan of the problem s and L =f J0, {Q}n + λ Q L Q = Q µ Q 34 j= B j 2 Q j B j T log e 2 2 Q B T Qj log e 2 B jt + λ µ. 35 Solvng the KKT condtons wth I beng the ndex subset such that f I then µ = 0, the optmal tme T s the Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd. 3

14 soluton of exp Q log e 2 T B I B log e I 36 J 0 B I I B = T +. B B I I To reduce the complexty of the root fndng, we prove that the root of the functon exsts and s unque. To do ths, let gx be the functon correspondng to the prevous equaton wth T x = and gx = e ax b cx d. 37 The study of ths functon shows that t s convex and lm fx = +. The nequalty between the weghted x arthmetc mean and the weghted geometrc mean leads to g0 0 wth equalty f and only f = for all and j: the soluton T of 36 exsts and s unque. Usng 35, the optmal Q s Q = B T log 2 λ 38 for I and wth λ such that I Q = Q. G. PROOF OF THEOREM 2 Let = arg max and T S be the average channel j occupancy tme. If all the nformaton bts Q are transmtted through the channel, then J 0 = 2 Q nbts nbt S, 39 whch proves the last part of the theorem. Let j be another channel, wth Q j = Q Q and T j = nt S T. Usng the Taylor seres expanson of the exponental functon, we obtan J + J j J0 = p= log p e 2 p!b p Q + Qjp T + T j p Q p T p + γ Q p j T p j. 40 But Q p T p + Qp j T p j Q + Qjp 0 4 T + T j p for all p N, {Q, Q j, T, T j} R 4 + and wth equalty when Q T j = Q jt. Then, J + J j J wth equalty f and only f = and Q jt = Q T j. Ths means that the target energy J 0 can not be reached f the channels that transmt nformaton are not the ones wth the hghest. Consequently, the average channel occupancy tme s mnmzed f all the nformaton bts Q are transmtted through the best channel. H. PROOF OF THEOREM 3 The optmzaton problem s derved from Defnton 4. Wth the new constrant T T M,, the Lagrangan of ths problem s L = T + o T M T ν T ξ Q n k J 0 k + λ Q Q + µ J. 43 The KKT condtons are B 2 Q Q ln 2 B T + B T 2 Q B T BT Q = Q = J0 ν T = o T M T = ξ Q = 0 2 Q B T + + λ + ξ = 0 µ ν o = 0 µ 44 If ν = o = ξ = 0 then only one pont satsfes the KKT condtons. Let T be ths pont, j the correspondng channel ndex, I the ndex subset such that I = { Q 4 Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd.

15 0} and α = λ µ, then and B T M log 2 α + B j T log2 α = Q, I I B T M α + B j T α = J0 45 J T M B α I T = B jα Q T M B log 2 α I = B j log 2 α. 46 Only one α satsfes the prevous constrants and t s the soluton of 46. Thus, T j = T and T = T M for all I wth j, otherwse T = 0. Unfortunately, nether the ndex nor the ndex j s provded by the equatons. But the average channel occupancy tme s mnmzed, as the transmsson tme, f the channels are fully exploted. Then, the best channels among I are such that T = T M and the poorest one s such that T j = T. The case where ν 0 or ξ 0 leads to T = 0 or Q = 0 and the ndex s outsde the subset I. The case o 0 leads to T = T M whch s taken nto account n the defnton of the subset I. REFERENCES. G. L, Z. Xu, C. Xong, C. Yang, S. Zhang, Y. Chen, and S. Xu, Energy-effcent wreless communcatons: tutoral, survey, and open ssues, IEEE Wreless Communcatons, vol. 8, no. 6, pp , Dec G. Auer, V. Gannn, C. Desset, I. Godor, P. Skllermark, M. Olsson, M. Imran, D. Sabella, M. Gonzalez, O. Blume, and A. Fehske, How much energy s needed to run a wreless network? IEEE Wreless Communcatons, vol. 8, no. 5, pp , Oct C. Shannon, Communcaton n the presence of nose, Proceedngs of the I.R.E. Insttute of Rado Ingneers, vol. 37, pp. 2, Jan S. Verdú, Spectral effcency n the wdeband regme, IEEE Transactons on Informaton Theory, vol. 48, no. 6, pp , Jun T. Chen, H. Km, and Y. Yang, Energy effcency metrcs for green wreless communcatons, n Internatonal Conference on Wreless Communcatons and Sgnal Processng, Suzhou, Chna, Oct. 20, pp E. Belmega, S. Lasaulce, and M. Debbah, A survey on energy-effcent communcatons, n IEEE Personal, Indoor and Moble Rado Communcatons Symposum, Istanbul, Turkey, Sep. 20, pp K. Fall, A delay-tolerant network archtecture for challenged nternets, n Conference on Applcatons, Technologes, Archtectures, and Protocols for Computer Communcaton. Karlsruhe, Germany: ACM SIGCOMM, Aug M. Ashraf and S. Sohab, Energy-effcent delay tolerant space tme codes for asynchronous cooperatve communcatons, Transactons on Emergng Telecommuncatons Technologes, vol. 204, pp. 7, Jan M. Lopez-Bentez, F. Casadevall, A. Umbert, J. Perez-Romero, R. Hacheman, J. Palcot, and C. Moy, Spectral occupaton measurements and blnd standard recognton sensor for cogntve rado networks, n Internatonal Conference on Cogntve Rado Orented Wreless Networks and Communcatons, Jun. 2009, pp. 9.. A. Hamn, J.-Y. Baudas, and J.-F. Hélard, Best effort communcatons wth green metrcs, n IEEE Wreless Communcatons, Networks Conference, Cancun, Mexco, Mar. 20, pp , Green resource allocaton for powerlne communcatons, n IEEE Internatonal Symposum on Power Lne Communcatons and Its Applcatons, Udne, Itale, Apr. 20, pp Y. Polyansky, H. Poor, and S. Verdú, Mnmum energy to send k bts through the gaussan channel wth and wthout feedback, IEEE Transactons on Informaton Theory, vol. 57, no. 8, pp , Aug , Channel codng rate n the fnte blocklength regme, IEEE Transactons on Informaton Theory, vol. 56, no. 5, pp , May M. Zmmermann and K. Dostert, A multpath model for the powerlne channel, IEEE Transactons on Communcatons, vol. 50, no. 4, pp , Apr. Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd. 5

16 Trans. Emergng Tel. Tech. 0000; 00: John Wley & Sons, Ltd.