Cognitive Access Algorithms For Multiple Access Channels

Transcription

1 203 IEEE 4th Workshop on Sgnal Processng Advances n Wreless Communcatons SPAWC) Cogntve Access Algorthms For Multple Access Channels Ychuan Hu and Alejandro Rbero, Department of Electrcal and Systems Engneerng, Unversty of Pennsylvana 200 South 33rd Street, Phladelpha, PA 904 Emal: ychuan, arbero}@seas.upenn.edu Abstract Ths paper consders a multple access fadng channel where each termnal has a dfferent belef about the channel states and adapts ts transmsson polcy to the belef. In ths settng, frequency dvson multple access FDMA) and channel aware random access RA) are two specal cases. To fnd solutons for general cases, we formulate the problem as a Bayesan game n whch each termnal maxmzes the expected utlty based on ts belef. We show that optmal solutons for both FDMA and RA are equlbrum ponts of the game. Therefore, the proposed game theoretc formulaton can be regarded as general framework for multple access channels. Furthermore, we develop a cogntve access algorthm that solves the problem approxmately. Numercal results show that the proposed algorthm acheves good performance and s adaptable to dfferent levels of channel belefs. I. INTRODUCTION Consder a multple access fadng channel n whch termnals contend for communcatng wth a central access pont. To explot favorable channel condtons, termnals adapt ther channel access as well as transmsson power to the random states of the fadng channel. The goal s to maxmze the expected value of the sum transmsson rates over all termnals subject to termnals average power constrants. Ths problem has been studed extensvely n the past and dependng on the avalablty of the channel state nformaton CSI) the optmal solutons are dfferent. When global CSI s avalable,.e., each termnal knows the channel states of others, they can cooperate wth each other to avod collson. Ths s known as frequency-dvson multple access FDMA) n whch the termnal wth the largest channel state gets the opportunty to transmt, see e.g.,, 2. However, global CSI s usually not avalable n many practcal scenaros and t s more reasonable to assume termnals only have access to local CSI. In ths case, termnals make transmsson decson and power allocaton based on ther local CSI wthout cooperatng wth each other. Ths s known as channel aware random access RA), and t has been show the optmal soluton s a threshold-based strategy,.e., transmsson s scheduled only when the local CSI exceeds certan threshold, see e.g., 3, 4. FDMA and RA can be regarded as two extreme cases for multple access channel where global CSI and local CSI are avalable for termnals. There are many other cases n between. For example, termnals may know ther local channels perfectly and have some mperfect nformaton about the channels of other termnals. In other words, termnals are cogntve n the sense that each termnal has a dfferent belef about the channel states. In ths settng, t s natural to formulate the problem as a Bayesan game n whch each termnal s a self-nterested but ratonal player that maxmzes the expected utlty based on ts belef. Bayesan game has been used to study random access channels n whch each termnal knows the pror dstrbutons of other channels 5, 6. However, these algorthms cannot be generalzed to the cogntve settng where belefs change over termnals and tme. Ths motvates us to develop cogntve access algorthms that determne an optmal allocaton of resources takng nto account the fact that dfferent termnals have dfferent belefs on the channel state. The rest of the paper s organzed as follows. Communcaton model s ntroduced n Secton II. In Secton III, we defne the Bayesan game and show that optmal solutons for FDMA and RA are Bayesan Nash Equlbrum BNE) ponts of the game. In Secton IV, we develop a cogntve access algorthm that fnds soluton for the Bayesan game approxmately. Numercal results and concluson are presented n Secton V and VI, respectvely. II. COMMUNICATION MODEL Consder a multple access channel wth n termnals contendng to communcate wth a common AP. The tme varyng channel h t) R + from user to the AP s modeled as block fadng and assumed ndependent for dfferent tmes t t 2 and dfferent termnals 2. The probablty densty functon pdf) of channel h s denoted as f ht)h) = f h h) and assumed to be tme nvarant and nonatomc. Ths latter condton s equvalent to havng contnuous cumulatve dstrbuton functons and holds true for practcal models ncludng Raylegh, Rcan, and Nakagam 7. For future reference defne the aggregate channel as ht) := h j t)} n j= and the channel complement of termnal as h t) := h j t)} n j=,j. We also assume a backlogged system where all termnals have packets to transmt all the tme. Upon observng ts own channel h t) node makes a decson on whether to transmt or not n the current tme slot and f t chooses to do so t selects a transmt power for the communcaton attempt. Transmsson decsons are based on the attempt functon q : R 0, } and the power allocaton functon p : R + 0, p nst, where p nst > 0 s a lmt on the nstantaneous power transmtted by termnal. Gven the channel value h t) termnal makes a transmsson attempt n tme slot t f and only f q h t)) = n whch case t does so wth power p h t)). The par of functons p := q, p ) s termed the transmsson strategy profle of termnal. The jont strategy s defned as the groupng p := p j } n j= of all ndvdual strategy profles and the complementary strategy of termnal as the groupng p := p j } n j=,j of all strateges except the one of. Observe that specfyng the jont strategy p s equvalent to specfyng the ndvdual strategy p and the complementary strategy p. A communcaton attempt wth power p h t)) when the channel s h t) proceeds at a rate Ch t), p h t)), where C : R + 0, p nst R + s a functon mappng channels and 20

2 203 IEEE 4th Workshop on Sgnal Processng Advances n Wreless Communcatons SPAWC) 2 powers to transmsson rates. If more than one termnal attempts transmsson n the same tme slot a collson occurs. Thus, user s able to reach the AP at tme t f and only f q h t)) = and q j h j t)) = 0 for all j. Therefore, the nstantaneous transmsson rate for termnal at tme t s r ht), pht))) = C h t), p h t)) ) n q h t)) q j h j t)), j=,j For jont transmsson strategy p, the average rate experenced by termnal s the expectaton E h r ht), pht))) wth respect to h of the nstantaneous rate n ). Adoptng the sum rate as our utlty functon, t then follows that the utlty assocated wth polcy p s gven by n Ūp) = E h Uph)) = E h r h, ph)). 2) where we dropped the tme ndex because the channel dstrbuton s assumed statonary and defned the nstantaneous utlty Uph)). Further note that each communcaton attempt, successful or not, ncurs a power cost p h ). Therefore, the average power consumpton of termnal s the expectaton E h q h )p h ) and n order to satsfy an average power budget p avg we must have E h q h )p h ) = p avg. 3) The goal of each termnal s to select the polcy p that maxmzes the average sum rate utlty Ūp) n 2) whle satsfyng the average power constrant n 3). However, termnal lacks the nformaton to do so. The rate r h, ph)) attaned by termnal s dependent upon the schedulng q h ) of all termnals cf. ). For termnal to be able to solve ths problem, t requres polces p and channels h of all other termnals. Snce nether p nor h s avalable to termnal, necessary assumptons need to be made n order to solve the problem. E.g., assume all channels have the same dstrbutons and all termnals employ the same polces,.e., f hj h) = f h h) and p j = p for all j, the problem can be formulated as optmal random access RA) 3, 4. In ths paper, we would lke to study more general cases n whch termnals have partal knowledge about others channels and apply dfferent polces. Assume termnal has access to channel estmatons of other termnals h := h j } n j=,j where h j s termnal s estmaton of termnal j s channel. We assume h j has the same dstrbuton as h j and the accuracy of h j s ndcated by condtonal probablty dstrbutons of h j gven h j,.e., h) see Remark ). When the channel estmaton s perfect,.e., hj = h j, t mples that h) = δh h j ). When h j does not reveal any nformaton about h j, the condtonal pdf h) s the pror dstrbuton of h j,.e. h) = f hj h). The condtonal pdf h) s called termnal s belef about termnal j s channel. We remark that the belef changes over tme. Remark The probablty dstrbuton h) depends on the channel estmaton method. A typcal way s to assume that h j s an outdated verson of h j modeled by an order- autoregressve AR) process. For example, suppose h j s complex Gaussan wth pdf CN 0, 2), then the estmaton can be modeled by h j = ρ h j + e j where ρ s the correlaton coeffcent between h j and h j and e j s complex Gaussan random nose wth pdf CN 0, ρ 2 ). In ths case, h) s a noncentral ch-square dstrbuton 8 ) h) = 2 ρ 2 ) exp h + ρ2 h j 2 ρ 2 I 0 ρ2 h h j ) ρ 2, ) where I 0 x) = =0 x2 /4) /!) 2 s the zeroth order modfed Bessel functon of the frst knd. Ths partcular form for the condtonal pdf h) s used to provde numercal results n Secton V. The rest of the development n the paper holds ndependently of the partcular form of ths pdf. III. BAYESIAN NASH EQUILIBRIUM Snce termnals are not allowed to cooperate wth each other, each termnal has to make transmsson decsons ndependently. On the other hand, the system utlty depends on the actons of all termnals cf. 2). Ths stuaton can be modeled as a game wth ncomplete nformaton where each termnal makes decsons on ts own channel and belefs about channels of others whle receves a global utlty. From termnal s perspectve, the utlty Up) s a functon of ts own acton p and actons of other termnals p. As a result, we can wrte Up) as Up, p ). In each tme slot, termnal observes ts own channel h and estmatons about channels of other termnals h. Based on h, termnal has a belef about h whch s represented by the condtonal pdf h). Suppose other termnals polces p are gven, the best response termnal can take s to maxmze the expected value of the sum rate utlty based on ts belef subject to the average power constrant p ) := argmax E h, h E h h Up, p ) s.t. E h, h q h )p h ) p avg, p P where P = p q h ) 0, }, p h ) 0, p nst } s the set of values that p can take. Note that the double expectaton E h, h E h h Up, p ) n 5) s the same as the expected utlty n 2): the outer expectaton n 5) s taken over termnal s observatons h and h whle the nner expectaton n 5) s wth respect to termnal s belef about channels of other termnals,.e., h gven h. In ths game, each termnal solves an optmzaton problem lke 5). The equlbrum pont of the game s defned by the followng: Defnton p BNE all the followng holds true 4) 5) s Bayesan Nash Equlbrum BNE) f for p BNE = argmax E h, h E h h Up, p BNE ) s.t. E h, h q h )p h ) p avg, p P At BNE, every termnal plays best response to the polces of other termnals. A natural queston arses s how to obtan solutons for BNE. As we wll see next, two conventonal formulatons of optmal wreless access that follow from 2) and 3), namely optmal FDMA and optmal RA, are BNE for two specal cases when termnals have perfectly correlated belefs h) = δh h j ) for all j ) and uncorrelated belefs h) = f hj h) for all j ). 6) 2

3 203 IEEE 4th Workshop on Sgnal Processng Advances n Wreless Communcatons SPAWC) 3 A. Perfectly correlated belefs When termnals belef about others s perfect, ths mples that each termnal has access to the global channel state. By usng global CSI, termnals can cooperate wth each other so that collson can be avoded. To do so, we ntroduce a constrant n = q h ) whch allows only at most one termnal to transmt n each tme slot. Snce collson s avoded, the nstantaneous rate r can be rewrtten as q h )Ch, p h )). As a result, each termnal can solve the followng global optmzaton problem locally = argmax E h q h )Ch, p h )) 7) = s.t. E h q h )p h ) p avg, p P q h ) = Notce that problem 7) s the optmal FDMA wth sngle frequency 2. It can be shown that problems lke 7) have null dualty gap 9, 0, and the optmal soluton s unquely determned by the optmal soluton for ts dual problem. Let λ FDMA be the optmal dual varable for the dual problem of 7), the optmal soluton for 7) s then gven by h ) = max Ch, p ) λ FDMA p, 8) p P g FDMA h ) > max max j gfdma j h j ), 0 q FDMA h ) = H where H ) s the Heavsde functon and g FDMA h ) s defned by g FDMA 9) h ) = Ch, h )) λ FDMA h ). 0) g FDMA h ) can be regarded as a local utlty functon that only depends on termnal s local CSI h. In each tme slot, termnals compute ther local utltes g FDMA h ) and the one wth the largest nonnegatve utlty gets the opportunty to transmt. We show ths soluton has the followng property. Proposton Suppose termnals have perfect belefs about other termnals,.e., h) = δh h j ), then obtaned by 8) and 9) that solves problem 7) s BNE of the game as s defned by 6). Proof: When termnal has perfect belefs about other termnals, we have h = h. As a result, we can wrte the maxmzaton problem n 5) as p ) = argmax E h Up, p ) ) s.t. E h q h )p h ) p avg, p P. To show s BNE, we need to show that gven p = the optmal soluton for ) s ) =. We prove ths by contradcton. Suppose gven p = optmal soluton for ) s ) E h Up BR the. Ths mples ), ) > E h U, ). 2) Moreover, the constrant n ) mples that ) s feasble for the FDMA problem. Ths contradcts wth the fact that s the global maxmzer for the FDMA problem 7). Therefore, t must be ) =. B. Uncorrelated belefs In ths case, termnals only have pror knowledge about channels of others and solves the followng optmzaton problem p RA = argmax E h Up) 3) s.t. E h q h )p h ) p avg, p P Ths s known as the optmal random access problem n whch termnals operate ndependently wthout cooperatng wth each other and the optmal soluton can be found by 3, 4. We show that p RA has the followng property: Proposton 2 Suppose termnals have uncorrelated belefs about other termnals,.e., h) = f hj h), then p RA that solves problem 3) s BNE of the game as s defned by 6). Proof: When termnal has uncorrelated belefs about other termnals, we have f h h h ) = f h h ). As a result, we can rewrte the maxmzaton problem n 5) as p ) = argmax E h Up, p ) 4) s.t. E h q h )p h ) p avg, p P. To show p RA the soluton for 4) s s BNE, we need to show that gven p = p RA p RA ) = pra. To see ths s true, note that gven p = p RA problem 3) and 4) are dentcal. In ths case, optmal soluton for 3) s the optmal soluton for 4),.e., p RA ) = pra In summary, when termnals have perfectly correlated and uncorrelated belefs about other termnals t s possble to formulate the problem as FDMA or RA and fnd correspondng optmal solutons. Interestngly, optmal solutons for these two dfferent problems both concde wth the BNE defned by 6). In other words, BNE can be used as a unfed framework to model multple access channels. Indeed, from an ndvdual termnal s pont of vew the only dfference between FDMA and RA s the knowledge about other channels whch s captured by the belef n BNE. However, for ntermedate cases where belefs are nether perfectly correlated nor uncorrelated fndng the BNE soluton s not an easy task because the objectve n 6) evolves polces of other termnals whch s unknown to termnal. Next, we develop algorthms that solve 6) approxmately.. IV. COGNITIVE ACCESS ALGORITHMS The key n desgnng an algorthm for solvng 6) s to model the actons of other termnals. Let q j ) be the modelng of termnal j s acton. The easest way of modelng termnal j s to assume that h j s the true channel gan and termnal j makes decson based on t,.e., q j h j ). The ntuton behnd ths modelng s that each termnal fnds a strategy that s optmal when ther belef about other termnals are perfect as n the case of optmal FDMA 7). As a result, termnal solves the followng problem locally p CA, p CA } = max E h q h )Ch, p h )) 5) + q E hj j h j )C h j, p j h j )) j=,j s.t. E h q h )p h ) p avg, p P q E hj j h j ) p j h j ) p avg j, p j P j j q h ) + q j h j ). j=,j 22

4 203 IEEE 4th Workshop on Sgnal Processng Advances n Wreless Communcatons SPAWC) 4 Note that problem 5) s the same as the one for FDMA 7) except that p s replaced by p. Suppose the optmal dual varable for the dual problem of 5) s λ CA, then the optmal soluton for 5) s gven by p CA h ) = max Ch, p ) λ CA p, 6) p P q CA h ) = H g CA h ) > max max j gca j h j ), 0, 7) where g CA h ) s gven by g CA h ) = Ch, p CA h )) λ CA p CA h ) 8) In practce, each termnal solves 5) offlne locally to fnd the optmal multpler λ CA. Based on λ CA, termnal makes transmsson and power allocaton decsons accordng to 6)- 8). When termnal s scheduled for transmsson, t assumes other termnals are slent,.e., q j = 0 for all j, and the attaned nstantaneous transmsson rate s Ch, p CA h )). However, ths may not be true snce the modeled acton q j may be dfferent from the real acton q j of termnal j whch s obtaned by termnal j solvng another maxmzaton problem lke 5). Therefore, collson may happen when all termnals operate by followng 6)-8). Before proceedng to show the performance of the proposed algorthm, we frst prove the next propertes. Proposton 3 Let λ CA and λ FDMA be optmal multplers assocated wth termnal s average power constrants n 5) and 7), then λ CA = λ FDMA. Let p CA h ) and h ) be the optmal power allocatons for termnal that solve problems 5) and 7), then p CA h ) = h ). Proof: Snce h j and h j have the same pdf, the dual problems of 5) and 7) are the same. Therefore, the optmal dual varables for ther dual problems are the same,.e., λ CA = λ FDMA. To see p CA h ) = h ), observe that p CA h ) and h ) are functons of λ CA and λ FDMA, respectvely cf. 6) and 8). Snce we have shown λ CA = λ FDMA, t follows that h ) = h ). Ths completes the proof. p CA Let the expected utltes acheved by p CA and are Ū CA and Ū FDMA, respectvely. We show that the performance of the proposed algorthm for the followng two cases. Proposton 4 If termnals have perfectly correlated belef,.e., h) = δh h j ), the expected utlty acheved by p CA s the same as that of FDMA,.e., Ū CA = Ū FDMA. Proof: When termnals have perfectly correlated belef, h j = h j. Ths mples that the problem 5) solved by the proposed algorthm s the same as the problem 7) solved by FDMA. Therefore, the performance acheved by both algorthms are dentcal,.e., Ū CA = Ū FDMA. Proposton 5 If termnals have uncorrelated belef,.e., h) = f hj h), the expected utlty acheved by p CA s a fracton of that of FDMA,.e., Ū CA = βū FDMA, where β 0, s a constant. In partcular, when channels are symmetrc, β = /e as the number of termnals goes to nfnty. Proof: In both FDMA and the proposed algorthm, transmsson decson for each termnal s made by comparng ts local utlty wth maxmum of the utltes of others cf. 9) and 7). Therefore, gven local channel h termnal transmts wth certan probablty under both polces. Let α FDMA h ) and α CA h ) be the probablty that termnal transmt n the proposed algorthm and FDMA, respectvely. Accordng to 9) and 7), α FDMA h ) and α CA h ) are gven by α FDMA h ) = Pr g FDMA h ) > max max j gfdma j h j ), 0 α CA h ) = Pr g CA h ) > max 9) max j gca j h j ), 0. 20) and g CA j By defnton, gj FDMA are functons of the local channel h, correspondng optmal multplers and optmal power allocatons cf. 0) and 8). Snce λ CA = λ FDMA and p CA h ) = h ) by Proposton 3, for the same channel we have gj CA h j ) = gj FDMA h j ). Moreover, snce h j, h j have the same dstrbuton, we conclude that α CA h ) = α FDMA h ) = α h ). As a result, the expected utlty acheved by s Ū FDMA = and the expected utlty acheved by p CA Ū CA = E h Ch ), h ))α h ), 2) E h Ch ), p CA h ))α h ) j Ehj α j h j ), s 22) where the product j Ehj α j h j ) n 22) represents the probablty that all termnals other than termnal are slent. Defne β := j Ehj α j h j ) and rewrte 22) as Ū CA = β E h Ch ), p CA h ))α h ). 23) Let β mn = mn β and β max = max β, then there must exst β β mn, β max such that Ū CA = β = β E h Ch ), p CA h ))α h ) E h Ch ), h ))α h ), 24) where the second equalty follows from the fact that p CA h ) = h ). Substtute 2) nto 24) yelds Ū CA = βū FDMA. 25) When the channels are symmetrc, t can be show that β = /n) n. Snce lm n /n) n = /e, Ū CA goes to /e Ū FDMA as n goes to nfnty. V. NUMERICAL RESULTS Numercal tests are conducted to evaluate performance of the proposed algorthm. We assume local channel h follows a complex Gaussan dstrbuton CN 0, 2) and the mperfect channel estmaton h j s modeled by 4). Assume capacty achevng codes are used for transmsson and the capacty functon takes the form of Ch, p h )) = log + h p h )/N 0 ) where N 0 s normalzed nose power. Wthout loss of generalty, we assume N 0 =. The average power budget s for all termnals,.e. p avg = for all. We conducted smulatons for dfferent total number of termnals n 0, 20, 30, 40, 50} and dfferent correlaton coeffcent ρ 0, 0., 0.2,, }. In the smulaton, 23

5 203 IEEE 4th Workshop on Sgnal Processng Advances n Wreless Communcatons SPAWC) )*#& 56 +0)*#& '% #&:+-)"* 0.9 4!" # 546 # 576./0%,% '2* -#%!3! *#&)/%0 %2%,%0 '3* -#% !"#$$%& %')*#)+$,+--%&#)+$! Fg.. Comparson of the expected sum rate utlty acheved by the optmal FDMA ρ = ), the optmal RA ρ = 0) and the proposed algorthm ρ 0, 0., 0.2,,. The total number of termnals s n = !"#$$%& %')*#)+$,+--%&#)+$! Fg. 2. The expected sum rate utlty acheved by the proposed algorthm normalzed by that acheved by the optmal FDMA for n = 0 and n = 50. The horzontal lne s /e stochastc subgradent descent algorthm 0 s used to teratvely compute the prmal and dual varables. Optmal solutons for FDMA when ρ = ) and RA when ρ = 0) are also computed. Fg. compares the expected sum rate acheved by optmal FDMA when ρ = ), optmal RA when ρ = 0) and the proposed algorthm when ρ 0, 0., 0.2,, for n = 0. When ρ = the expected utlty acheved by the proposed algorthm s.87 whch s equal to that acheved by the optmal FDMA. Ths corroborates the results n Proposton 4. As the correlaton ρ decreases, the performance of the proposed algorthm degrades gracefully and acheves an expected utlty of 0.72 when ρ = 0. Ths s very close to the expected utlty acheved by the optmal RA 0.78). In Proposton 5, t s shown for symmetrc channel the expected utlty acheved by the proposed algorthm for ρ = 0 s about /e of the utlty acheved by optmal FDMA as n goes to nfnty. To show ths s true, we normalzed the expected utlty acheved by the proposed algorthm by the utlty acheved by the optmal FDMA. Fg. 2 shows the normalzed expected utlty acheved by the proposed algorthm for n = 0 and n = 50. The horzontal lne s /e Indeed, for n = 50 the normalzed utlty converges to /e when ρ = 0. Moreover, notce that the normalzed utlty decreases as n ncreases. Ths s because when n ncreases the mperfect channel estmaton s more lkely to cause collsons. VI. CONCLUSION We consdered algorthms that adapts transmsson polcy to the random channel states n multple access fadng channels where each termnal has a dfferent belef about the channel states. In ths settng, we formulated the problem as a Bayesan game n whch each termnal maxmzes the expected utlty based on ts belef subject to an average power constrant. We showed that optmal solutons for both FDMA and RA are Bayesan Nash Equlbrum BNE) ponts of the formulated game. Therefore, the proposed game theoretc formulaton can be regarded as general framework for multple access channels. Moreover, a cogntve algorthm s developed to solve the problem approxmately. Numercal results show that the proposed algorthm acheves performance equal to as the optmal FDMA when the channel estmaton s perfectly correlated and performance very close to the optmal RA when channel estmaton s uncorrelated.. REFERENCES H. G. Myung, J. Lm, and D. J. Goodman, Sngle carrer fdma for uplnk wreless transmsson, Vehcular Technology Magazne, IEEE, vol., no. 3, pp , Sep A. Rbero and G. Gannaks, Optmal fdma over wreless fadng moble adhoc networks, n ICASSP Acoustcs, Speech and Sgnal Processng, IEEE Internatonal Conference on, Apr. 2008, pp Y. Yu and G. B. Gannaks, Opportunstc medum access for wreless networkng adapted to decentralzed cs, IEEE Trans. Wreless Commun., vol. 5, no. 6, pp , Jun Y. Hu and A. Rbero, Adaptve dstrbuted algorthms for optmal random access channels, IEEE Trans. Wreless Commun., vol. 0, no. 8, pp , Aug Y. Cho, C.-S. Hwang, and F. Tobag, Desgn of robust random access protocols for wreless networks usng game theoretc models, n INFOCOM The 27th Conference on Computer Communcatons. IEEE, Apr. 2008, pp B. Yang, Y. Shen, M. Johansson, C. Chen, and X. Guan, Opportunstc multchannel access wth decentralzed channel state nformaton, Wreless Communcatons and Moble Computng, 203. Onlne. Avalable: 7 A. Goldsmth, Wreless Communcatons. Cambrdge, UK: Cambrdge Unversty Press, J. Proaks and M. Saleh, Dgtal communcatons, 5th ed. McGraw-Hll, A. Rbero and G. B. Gannaks, Separaton prncples of wreless networkng, IEEE Trans. Inf. Theory, vol. 56, no. 9, pp , Sep A. Rbero, Optmal resource allocaton n wreless communcaton and networkng, EURASIP Journal on Wreless Communcatons and Networkng, vol. 202, no., p. 272, 202. Onlne. Avalable: Work n ths paper s supported by the Army Research Offce grant P NS and the Natonal Scence Foundaton CAREER award CCF Emal at ychuan, arbero}@seas.upenn.edu 24