Power-and Rate-Adaptation Improves the Effective Capacity of C-RAN for Nakagami-m Fading Channels

Transcription

1 Power-and Rate-Adaptation Improves the Effective Capacity of C-RAN for Nakagami-m Fading Channels Hong Ren, Nan Li, Cnha Pan, Maged Elkashlan, Armgam Nallanathan, Fellow, IEEE, Xiaoh Yo, Fellow, IEEE and Lajos Hanzo, Fellow, IEEE Abstract We propose a power-and rate-adaptation scheme for clod radio access networks C-RANs), where each radio remote head RRH) is connected to the baseband nit BBU) pool throgh optical links. The RRHs jointly spport the sers by efficiently exploiting the enhanced spatial degrees of freedom. Or proposed scheme aims for maximizing the effective capacity EC) of the ser sbject to both per-rrh average-and -power constraints, where the EC is defined as the maximm arrival rate that can be spported by the C-RAN nder the statistical delay reqirement. We first transform the EC maximization problem into an eqivalent convex optimization problem. By sing the Lagrange dal decomposition method and solving the Karsh- Khn-Tcker KKT) eqations, the optimal transmission power of each RRH can be obtained in closed-form. Frthermore, an online tracking method is provided for approximating the average power of each RRH. For the special case of two RRHs, the expression of the average power of each RRH can be calclated in explicit form. Hence, the Lagrange dal variables can be compted in advance in this special case. Frthermore, we derive the power allocation for two important extreme cases: ) no delay constraint; ) extremely stringent delay-reqirements. Or simlation reslts show that the proposed scheme significantly otperforms the conventional algorithm withot considering the delay reqirements. Frthermore, when appropriately tning the vale of the delay exponent, or proposed algorithm is capable of garanteeing a delay otage probability below 9 when the maximm tolerable delay is ms. This is sitable for the ftre ltra-reliable low latency commnications URLLC). Index Terms URLLC, Effective capacity, Delay constraints, Cross-layer design, C-RAN. I. INTRODUCTION The fifth-generation 5G) wireless system to be deployed by is expected to offer a sbstantially increased capacity. To achieve this ambitios goal, the C-RAN concept has been regarded as one of the most promising soltions. In particlar, C-RAN is composed of three key components: ) a pool of BBUs centrally located at a clod data center; ) low-cost, low-power distribted RRHs deployed in the network; 3) highbandwidth low latency fronthal links that connect the RRHs H. Ren, N. Li and X. Yo are with the Sotheast University, Nanjing, China. {renhong, nanli, xhy}@se.ed.cn). C. Pan, M.Elkashlan and Armgam Nallanathan are with the Qeen Mary University of London, London E 4NS, U.K. {c.pan, maged.elkashlan, a.nallanathan}@qml.ac.k). L. Hanzo is with the School of Electronics and Compter Science, University of Sothampton, Sothampton, SO7 BJ, U.K. lh@ecs.soton.ac.k). This work is partially spported by the National Natral Science Fondation of China nder Grants 6573 and 656, the National 863 Program 4AAA7). L.Hanzo wold like to acknowledge the financial spport of the Eropean Research Concil, Advanced Fellow grant Beam-Me-Up. to the BBU pool. Under the C-RAN architectre, most of the baseband signal processing of conventional base stations has been shifted to the BBU pool and the RRHs are only responsible for simple transmission/reception fnctions, some of the hitherto centralized signal processing operations can be relegated to the BBU pool. Hence, the network capacity can be significantly improved. Recently, the performance of C-RAN has been extensively stdied 5, albeit these papers have been focsed on the physical layer isses, which giving no cognizance to the delay of the pper layer s. However, most of the mltimedia services, sch as video conferencing and mobile TV have stringent delay reqirements. De to the time-varying characteristics of fading channels, it is impossible to impose a deterministic delay-bond garantee for wireless commnications. For the sake of analyzing the statistical QoS performance, W et al. 6 introdced the notion of effective capacity EC), which can be interpreted as the maximm constant packet arrival rate that can be spported by the system, whilst satisfying a maximm bffer-violation probability constraint. De to the complex expression of EC, most of the existing papers focs on the EC maximization problem for the simple scenario, where there is only a single transmitter 7. Specifically, a QoS-driven power-and rate-adaptation scheme was proposed for single-inpt single-otpt SISO) systems commnicating over flat-fading channels in 7, with the objective of maximizing EC sbject to both delay-qos and average power constraints, which is characterized by the QoS exponent θ. A smaller θ corresponds to a looser QoS garantee, while a higher vale of θ represents a more stringent QoS reqirement. The reslts of 7 showed that in the extreme case of θ, the power allocation redces to the conventional water-filling soltion. By contrast, when θ, the optimal power allocation becomes the channel inversion scheme, where the system operates nder a fixed transmission rate. The EC maximization problem was stdied in 8 in the context of cognitive radio networks, where the power constraints of 7 were replaced by the maximm tolerable interference-power at the primary ser. Closed-form expressions of both the power allocation and the EC were derived for the secondary ser. The power minimization problem sbject to EC constraints was considered in 9 for three different scenarios. Significant power savings can be achieved by sing the power allocation scheme of 9. To a frther advance, both sbcarrier and power allocation were investigated in for a one-way relay

2 network wherein the optimal sbcarrier and power allocation was derived by adopting the Lagrangian dal decomposition method. Most recently, Wenchi et al. considered both the average-and -power constraints when maximizing the EC, and provided the specific conditions, when the power constraints can be removed. From the information-theoretical point of view, it is difficlt to solve the ergodic capacity maximization problem sbject to both the average-and power constraints, even for the basic Gassian white noise channel. In Shannon s landmark paper, only the asymptotically low and high SNR for the bandlimited continos time Gassian channel was stdied. This problem was later stdied by Smith 3 and showed that the capacity achieving distribtion is discrete. In 4, the athors considered the more general case of the qadratre additive Gassian channel and derived the lower and pper bonds of this channel s capacity. Khojastepor et al. 5 stdied the capacity of the fading channel nder both types of power constraints nder the assmption of the perfect channel state information available at both the transmitter and receiver side. However, the above papers 4 neither consider the pper layer delay nor the mltiple transmission points. To avoid any traffic congestion and attain a good C-RAN performance, the adaptive power allocation scheme shold take the diverse QoS reqirements into accont to garantee the satisfaction of sers. In this paper, we aim for jointly optimizing the power allocation of each RRH in order to maximize the EC of a ser of the C-RAN, where both the average power and power constraints of each RRH are considered. This ser is jointly served by all RRHs of the C- RAN de to the powerfl comptational capability of the BBU pool. Unfortnately, the power allocation schemes developed in the aforementioned papers for single-transmitter scenarios 7 cannot be directly applied to C-RAN s relying on mltiple RRHs for serving the ser and to simltaneosly exploit the spatial degrees of freedom. The reason can be explained as follows. In these papers, there is only a single transmitter and only a sm-power constraint is imposed. The Lagrange method can be sed to find the optimal power allocation, which is in the form of a water-filling-like soltion in general. However, for the C-RAN, all RRHs have their individal power constraints and the power cannot be shared among the RRHs. Y et al. 6 provided a detailed reason as to why the classic Lagrange method cannot be readily applied in C-RAN. Hence, new methods have to be developed. Specifically, the contribtions of this paper can be smmarized as follows: ) In this paper, we derive the optimal power allocation for each RRH by resorting to the Lagrangian dal decomposition and the Karsh-Khn-Tcker KKT) conditions. The power allocation soltions depend both on the ser s QoS reqirements and on the joint channel conditions of the RRHs. ) The Lagrangian dal decomposition reqires s to calclate the sbgradient, where the average power of each iteration shold be obtained. However, it is nmerically challenging to derive the expression of average power for each RRH. To tackle this isse, we provide an online training method for tracking the average power of each RRH. For the special case I I T f m α i θ EC θ) μ D max B P avg i i TABLE I THE LIST OF NOTATIONS The nmber of RRHs The set of RRHs The time frame length Nakagami parameter Channel-power-to-noise ratio CPNR) QoS exponent The effective capacity specified by θ Constant arrival data rate Maximm delay bond System bandwidth Average power constraint of RRH i Peak power constraint of RRH i of a single RRH, the closed-form expression of average power can be obtained. For the more complex case of two RRHs, we also provide the expression of the average power for each RRH in explicit form, which can be nmerically evalated. 3) We provide the closed-form power allocation soltions for two extremely important cases: ) When the QoS exponent θ approaches zero, which corresponds to the conventional ergodic capacity maximization problem; ) When the QoS exponent θ tends to infinity, which represents the strictly delaylimited case. We also extend or work to the more general mltiser case, where the optimal transmit power for each ser is derived. 4) Or simlation reslts will show that the proposed algorithms significantly otperforms the existing algorithms. Additionally, by appropriately choosing the QoS exponent, or proposed delay-aware algorithm can garantee a delayotage probability below 9, when the maximm tolerable end-to-end EE) delay is ms, which satisfies the strict delay reqirements of the ftre ltra-reliable low latency commnications URLLC). The most closely related paper of or work is, where single transmitter is considered. The optimal power allocation can be readily derived in closed form by the aid of the Lagrangian dal decomposition, which is in the water-fillinglike form. However, or considered C-RAN involves mltiple independent transmitters, the power allocation derivations are mch more involved, and the power allocation soltion is not the same as the conventional water-filling form. In, only one dal variable needs to be optimized, which can be obtained by sing the bisection search method, while mltiple dal variables are involved in the C-RAN scenario and the sbgradient method is adopted to pdate the dal variables. Frthermore, in both papers, the average power expression is reqired in the pdating of dal variables. However, this isse is not stdied in, while or paper provides more details abot the analysis of this isse. The rest of this paper is organized as follows. In Section II, the C-RAN system is introdced along with the concept of EC and or problem formlation. In Section III, we provide the optimal power allocation for the general case of any nmber of RRHs. In Section IV, we derive the integral expressions s closed-form soltion concerning the average transmit power of each RRH for the sake of pdating the Lagrangian dal variables. In Section V, we obtain the power allocation for two extreme cases, namely for the delay-tolerant scenario and

3 3 for extremely stringent delay constraints. Nmerical reslts are also provided in Section VII. Finally, or conclsions are drawn in Section VIII. The other notations are smmarized in Table. I. II. SYSTEM MODEL Consider a downlink C-RAN consisting of I RRHs and a single ser, where each RRH and the ser have a single antenna, as depicted in Fig.. The set of RRHs is denoted as I = {,,,I}. All the RRHs are assmed to be connected to the BBU pool throgh the fronthal links relying on highspeed fiber-optic cables. In Fig., the pper layer packets are first bffered in first-in-first-ot FIFO) qee, which will be transmitted to the physical layer of the RRHs. Since the RRHs are only responsible for simple transmission/reception, we do not consider the storage fnction at the RRHs. At the datalink layer, the pper-layer packets are partitioned into frames and then each frame will be mapped into bit-streams at the physical layer. The channel is assmed to obey the stationary block fading model, implying that they are fixed dring each time frame of length of T f, while it is switched independently over different time frames. To elaborate, we consider a Nakagami-m block-fading channel, which is very general and incldes most of the practical wireless commnication channels as special cases 7. The parameter m represents the severeness of the channel, where the fading channel flctations are redced with m. The probability density fnction PDF) of the Nakagami-m channel spanning from the i-th RRH to the ser is given by: ) m ) fα i )= αm i m exp mᾱi α i, α i, ) Γm) ᾱ i where Γm) = w m e w dw is the Gamma fnction, α i denotes the instantaneos channel-power-to-noise ratio CPN- R) from the ith RRH to the ser, and ᾱ i is the average received / CPNR at the ser from the ith RRH, denoted as PL i σ, where PL i is the large-scale fading channel gain spanning from the ith RRH to the ser that incldes the path loss and shadowing effect, and σ is the noise power. Let s define α =α,α,,α I T. Since α,,α I are independent, the joint PDF of α is given by fα) =fα )fα ) fα I ). ) A. Effective Capacity EC) This paper considers the EE delay reqirement for each packet. In C-RANs, the EE delay, denoted as D max, incldes the plink UL) and downlink DL) transmission delays of D T, qeeing delay at the bffer of the BBU pool of D q, and the fronthal delay of D F. Since the fronthal links are sally deployed with high-speed fiber, the fronthal delay is mch less than ms 8. In addition, since the packet size in URLLC is very short, the UL and DL transmission can be finished within a very short time. Hence, the EE delay is mainly dominated by the qeeing delay at the bffer of the BBU pool, which is given by D q = D max D F D T.In The method developed in this paper can also be applied to the mltiser scenario, where all sers apply the classical orthogonal freqency division mltiplexing access OFDMA) techniqe to remove the mlti-ser interference. Fig.. Fig.. Topology of a C-RAN with I RRHs. Cross-layer transmission model. the following, we mainly focs on the stdy of the qeeing delay, which is characterized by the EC. The EC is defined as the maximm constant frame arrival rate that a given service process can spport, while obeying the delay reqirement indicated by the QoS exponent θ that will be detailed later. Let the seqence Rk,k =,, represent the data service-rate, which follows a discrete-time stationary and ergodic stochastic process. The parameter k is the time frame index. Let s denote by St) t k= Rk the partial sm of the service process over the time seqence spanning from k =to k = t. Let s frthermore assme that the Gartner-Ellis limit of St, which is denoted by Λ C θ) = lim t /t) loge{e θst }), is a convex differentiable fnction for all real-vale of θ 7. Then, the EC of the service process specified by θ is ECθ) = Λ C θ) = θ loge{e θrk }) 3) θ where E{ } denotes the expectation operator. Let s assme that a data sorce enters a qee of infinite bffer size at a constant data rate μ. The probability that the delay exceeds a maximm delay bond of D q satisfies 6 Pdelay ot =Pr{Delay D q } εe θμdq, 4) where fx) gx) indicates that lim fx)/gx)=, and x ε is the probability that the bffer is non-empty, which can be calclated by sing the method in 6. The parameter θ can be fond by letting the constant arrival rate to be eqal to the EC, i.e. setting it to μ =ECθ). Hence, if the delay-bond violation probability is reqired to be below Pdelay ot, one shold limit its incoming data rate to a maximm of μ =ECθ). It is seen from 4) that θ is an important parameter, representing the decay rate of the delay violation probability. A smaller θ corresponds to a slower decay rate, which indicates that the delay reqirement is loose, while a larger θ corresponds to a faster decay rate, which implies that the system

4 4 is capable of spporting a more stringent delay reqirement. In other words, when θ, an arbitrarily long delay can be tolerated by the system, which corresponds to the capacity stdied in Shannon information theory. On the other hand, when θ, this implies that no delay is allowed by the system, which corresponds to the very stringent statistical delay-bond QoS constraint of allowing no delay at all. B. Problem formlation For the ser, we assme that the non-coherent joint transmission is adopted as the BBU pool 9. Then, the instantaneos service rate of a single frame, denoted by Rν), can be expressed as follows, 3: Rν) =T f B log + p iν)α i ), 5) i I where B is the system s bandwidth, ν =α,θ) Δ represents the network condition that incldes both the channel s power gains and the EC exponent reqirement, and p i ν) represents the power allocation for RRH i that depends on the network s condition ν. In this paper, we aim for optimizing the transmit power in order to maximize the EC for the ser nder two different types of power limitations for each RRH: nder an average power constraint and a power constraint. The first one is related to the long-term power bdget, while the second garantees that the instantaneos transmit power is below the linear range of practical power amplifiers. Mathematically, this optimization problem can be formlated as max {p iν),i I} θ log E α e θt f B log + piν)αi)) i I 6a) s.t. E α p i ν) P avg i, i I, 6b) p i ν) i, i I, 6c) where E α { } denotes the expectation over α, while P avg i and denote the ith RRH s maximm average transmit power i constraint and transmit power constraint, respectively. III. OPTIMAL POWER ALLOCATION METHOD By exploiting the fact that log ) is a monotonically increasing fnction, Problem 6) can be eqivalently simplified as min E α + ) εθ) p i ν) α i 7a) {p iν), i I} i I s.t. E α p i ν) P avg i, i I, 7b) p i ν) i, i I, 7c) where we have ε θ)=θt f B/ln. In Appendix A of or longer version paper 4, we prove that Problem 7) is a convex optimization problem. Hence, the Lagrangian dality method can be sed to solve Problem 7) with zero optimality gap. Note that in Problem 7), only the average power constraints are ergodic, while the others are instantaneos power constraints. Similar to 5, we shold first introdce the dal variables associated with the average transmit power constraints. Then, the original problem can be decomposed into several independent sbproblems, where each one corresponds to one fading state. In addition, the instantaneos power constraints are enforced for each fading state. Let λ =λ,,λ I T represent the nonnegative dal variables associated with the average power constraints. The Lagrangian fnction of Problem 7) can be written as LP ν), λ) = E α + p iν)α i ) εθ) i I + i I λ ie α p i ν) P avg i ), 8) where P ν) =p ν),,p I ν) T. Let s now define P = {P ν) 7c)}. The Lagrange dal fnction is then given by g λ) = min Pν) P LP ν), λ). 9) The dal problem is defined as g λ). ) max λ i, i As proven in Appendix A of 4, Problem 7) is a convex optimization problem, which implies that there is no dality gap between the dal problem and the original problem. Ths, solving the dal problem is eqivalent to solving the original problem. To solve the dal problem in ), we shold solve the problem in 9) for a fixed λ, then pdate the Lagrangian dal variables λ by solving the dal problem in ). Iterate the above two steps ntil convergence is reached. ) Solving the dal fnction in 9): Foragivenλ, we shold find the dal fnction gλ), which can be rewritten as gλ) =E g λ) λ ip avg i I i, ) where we have: g λ)= min + ) εθ)+ p i ν) α i λ ip i ν). ) Pν) P i I i I Note that g λ) can be decopled into mltiple independent sbproblems, each corresponding to a specific fading state. Those sbproblems have the same strctre for each fading state. Hence, to simplify the derivation, ν is omitted in the following. Each sbproblem can be expressed as: min P s.t. + ) εθ) p iα i + ip i i I i I 3a) p i i, i I. 3b) The above problem is convex. In the following, we obtain the closed-form soltion by solving the KKT conditions of Problem 3). First, we introdce the nonnegative dal variables of μ i, i, and δ i, i for the associated constraints in 3b). The KKT conditions for Problem 3) can be expressed as εθ) + ) εθ) i I p i α i αi +λ i + μ i δi =, i 4a) μ i p i i )=, i 4b) δi p i =, i 4c) p i i, i 4d) p i, i, 4e) with p i, δi, and μ i, i. Then, we have the following lemma. Lemma : For any two arbitrary RRHs i and j, ifp i > and p j =, then the following relationship mst hold λ i λ j. 5) α i α j Proof : Please see Appendix B of 4.

5 5 Lemma shows that the RRHs associated with a smaller λ i /α i shold be assigned a non-zero power, while the RRHs having a larger vale shold remain silent. Let s hence introdce π as a permtation over I, so that we have λ πi) α πi) λ πj) α πj), when i<j,i,j I. Let I Ibe the set of RRHs that transmit at a non-zero power. Then, according to Lemma, it can be readily verified that we have I = {π),,π I )}. Lemma : There is at most one RRH associated with < p i <P i, and the RRH index is i = π I ). Proof : Please see Appendix C of 4. Next, we derive the optimal power allocation of Problem 3) shown as in Theorem. Theorem : The optimal soltion of Problem 3) is shown as follows, pπa) = where T πa) is given by T πa) = α π I ) p = πa), a < I ; { } min π I ),T π a), a = I ;, a > I ; λπ I ) εθ)α π I ) ) I εθ)+ 6) πa) α πa) a= with I being the largest vale of integer x, so that we have x εθ) λ πx) < εθ)α πb) α πb) +. 7) πx) b= Proof : Please refer to Appendix D of 4. It can be seen from 6) that the RRHs with high vale of λ i /α i shold be switched off, and the ones with larger vales can transmit with power. The dal variable λ i can be regarded as the price, and larger vale of λ i will incr higher cost when the RRH is active. Corollary : When the nmber of RRHs is eqal to one, i.e. I =, the optimal transmit power is given by, if α < λ εθ) ; where F is given by F = F, if α λ εθ), if α λ εθ) and F<P ; and F P. λ /εθ)) /+εθ) α εθ)/+εθ) α. 8) Proof : It is can be readily derived sing Theorem. Note that the above reslt is consistent with the point-topoint reslt obtained in. ) Solving the dal problem ): To solve the dal problem ), we invoke the sbgradient method, which is a simple method of optimizing non-differentiable objective fnction 6. The sbgradient is reqired by the sbgradient of g ) at λ k) =λ k),,λk) I T in the kth iteration. Theorem : The sbgradient of g ) at λ k) iteration is given by d k) = E α P in the kth λ ν) P avg, 9) k) T wherep ν)= p λ k),λ k) ) ν),,p I,λ k) ) ν) is the opti- According to 6, a vector d is a sbgradient of gλ) at λ k), if for all λ, gλ) gλ k) )+d T λ λ k) ) holds. mal soltion of Problem 9) when we have λ = λ k), and P avg =P avg,,p avg I T. Proof : Please see Appendix E of 4. Based on Theorem, the Lagrangian dal variables can be pdated as λ k+) = λ k) + ζ k) d k), ) where ζ k) is the step in the k th iteration. The sbgradient method is garanteed to converge if ζ k) satisfies lim k ζ k) =and k= ζk) = 7. The step size in the simlation section is set as ζ k) = a/k, where a is the constant step-size parameter. In smmary, the soltion of Problem 6) is given in Algorithm. To execte Algorithm, there is another isse Algorithm Solving Problem 6) Initialize: Iteration nmber k =, λ ) =λ ),,λ I ) ; Repeat. Compte P k) with fixed λ k) sing 6);. Compte the sbgradient d k), sing 9); 3. Update λ k+) sing ), increase k by ; Until convergence that has to be tackled, namely how to calclate the average power for each RRH in order to obtain the sbgradient d k). Given the dal variables λ k), the expression of optimal power at each RRH depends on the generations of channel gains α. Different generations of α will lead to different orders of λ i /α i,i =,,I, and ths different power expressions. Even for a fixed problem-order the power allocation expressions reqire mltiple integrations, which imposes a high comptational complexity. As a reslt, it is a challenge to obtain the expression of average power for each RRH in closed-form for any given λ k). In fact, even for the simple case of two RRHs, the average powers generally do not have simple closed-form soltions, as shown in the next section. To resolve the above isse, we propose an online calclation method for tracking the average power reqired for each RRH. The main idea is to replace the expectation operator by averaging the power allocations for all samples of channel generations dring the fading process. Specifically, let s define P k ) i as P k ) i = k k j= p i,λ j) ) ) α j),θ,i=,,i, ) where p i,λ j) ) α j),θ ) denotes the optimal power allocation of the ith RRH for the jth channel generation, and λ j) represents the corresponding dal variables. Then, the expectation over p ν) can be approximated as i,λ k) ) E α p i,λ k) ) ν) P k) i = p i,λ k) ) α k),θ ) k ) +k ) P i, i I, ) k which can be recrsively obtained based on the previos P k ) i, i. It is worth noting that this algorithm can be readily applied to the case when the fading statistics are nknown. Fortnately, for the case of a single RRH 3, the average 3 Note that did not provide the closed-form expression for the average power for the case of one RRH.

6 6 Fig. 3. The properties of the fnction hx) =+ax) b cx. power can be obtained in closed-form, which is given in Appendix F of 4. The average power expression for the case of I =is even more complex, which is extensively stdied in the following section. IV. SPECIAL CASE: I = To show the difficlty of obtaining the closed-form soltion for each RRH, we only consider the simplest scenario of two RRHs, i.e., I =. To this end, we first introdce the following lemma, which will be sed in the ensing derivations. Lemma 3: Let s define the fnction hx) =+ax) b cx, where a>,c >,b>. Then, when x, there are only three possible crves for the fnction hx), which are shown in Fig. 3. The conditions for each case are given as follows: ) Case : ab c ; ) Case : ab c< and c /b ) ab) b)+b ; 3) Case 3: ab c< and c /b ) ab) b)+b<; Proof : Please see Appendix G of 4. Based on Lemma 3, we commence by deriving the expression of the average transmit power. According to Theorem, there are three different possible cases for I = : ) I =;) I =;3) I =. Obviosly, the average power contribted by the first case is zero, hence we only consider the latter two cases, which are discssed in the next two sbsections. Withot loss of generality, we assme that λ λ 3) α α holds. The case for λ α > λ α can be similarly derived. A. Scenario : There is only one RRH that transmits at a non-zero power, i.e., I = Since we have assmed that the ineqality 3) holds, only RRH is transmitting at a non-zero power and remains silent in this case. According to Theorem, the conditions for I =are given by λ εθ), α +) 4) ε θ) α λ <, 5) ε θ) α and only RRH transmits non-zero power, which is given by { ) } p =min λ εθ)+,. 6) α ε θ) α It may be readily seen that, there are two possible vales of p, and the conditions for each vale will be discssed as follows. ) p = : In this case, the following condition shold be satisfied: ) λ εθ)+, 7) α ε θ) α which is eqivalent to + α ) +εθ) ε θ) λ α. 8) Notice that the left hand side of 8) is in the form of hx) defined in Lemma 3, with x = α, a, b, c given by a =,b=+ε θ),c= ε θ)/λ. 9) To garantee that there exists a positive α, the fnction hx) shold be reminiscent of Fig. 3-c), and the conditions for Case 3 in Lemma 3 shold be satisfied. Otherwise, p is not eqal to. In the following, we assme that the conditions are satisfied. Let s denote the soltions of hα )= as α l and α, where α l < α. These soltions can be readily obtained by sing the classic bisection method. Then, condition 8) is eqivalent to α l α α. 3) By combining 3) and 4), we obtain the feasible region of α in the form of: λ λ ) ) +εθ) α min α, + α λ ε θ) λ ) +εθ), = + α 3) ε θ) where the last eqality holds by sing 8). Similarly, by combining 5) and 3), we obtain the feasible region of α as: ) max α, l λ α α. 3) ε θ) In this context, we prove that λ /ε θ) <α l in Appendix H of 4. Hence, the feasible region of α is given by α l α α. Based on the above discssions, we obtain the conditions for p =,p =as follows: c ) /b ) C :ab c<, b)+b<, 33) ab where a, b, c are given in 9). If Condition C in 33) is satisfied, the average power assigned to RRH in this case is given by α TRRH C Δ = α l λ εθ)+ α ) +εθ) fα )f α ) dα dα. 34) By sbstitting the PDF of α in ) into 34), 34) can be simplified to TRRH C = Γm) α α l fα )γ m, ) +εθ) mλ ᾱ εθ) + α )dα. 35) Unfortnately, the closed-form expression of TRRH C cannot be obtained even for the special case of m =. However, the vale of TRRH C can be obtained at a good accracy by sing the nmerical integration fnction of Matlab. ) ) p = λ εθ)+ α εθ)α : In this case, the following condition shold be satisfied: ) λ εθ)+ <, 36) α ε θ) α which leads to hα ) > with a, b, c defined in 9). As seen from Fig. 3, when the first two conditions in Lemma 3 are

7 7 satisfied, the ineqality 36) holds for any α. When the third condition in Lemma 3 is satisfied, the ineqality 36) holds when α α l and α α, where α l and α α l <α ) are the soltions of hα )=with a, b, c defined in 9). Now, we first assme that the first two conditions in Lemma 3 are satisfied c b C: ab c, or ab c<and b)+b 37) ab) where a, b, c are given in 9). Then, the condition in 36) can be neglected. According to 3) and 4), the feasible region of is given by λ λ ) ) +εθ) α min α, + α = λ α, 38) λ ε θ) λ where 36) is sed in the last eqality. From 5), the feasible region of α is given by α λ εθ). As a reslt, the average power of RRH contribted by the case when condition C is satisfied is given by T C RRH = λ εθ) = Γm) λ λ α λ εθ) p f α ) f α ) dα dα, p γ m, mλ ) α f α ) dα. 39) ᾱ λ Fortnately, when m is an integer, the closed-form expression of TRRH C can be obtained. Let s define U = Δ λ εθ),v = Δ +εθ),w = Δ mλ ᾱ λ, 4) Z = Δ m/ᾱ)m m )!,Y = Δ mλ ᾱ λ + m ᾱ. If m =, the Nakagami-m channel redces to the Rayleigh channel, and the average power of RRH in 39) can be simplified to: ) TRRH C = ᾱv ΓV,U/ᾱ ) + U V ᾱ Ei Ūα 4) ΓV,UY ) ᾱ Y V U V ᾱ Ei UY). If m is an integer that is larger than one, i.e., m, wecan obtain the closed-form expression of TRRH C : TRRH C = J + J J 3 J 4, 4) where J, J, J 3 and J 4 are respectively given by J 3 = Z U V J 4 = J = ᾱv J = Z m m ) Γ V +m, mu ᾱ, m )!m V U V l= l= m ᾱ Γ m )! W l l!y l+m Γl + m,yu), W l l!y l+v +m Γl + V + m,yu), m, mu ᾱ ). 43) 44) The details of derivations can be fond in Appendix I of 4. Next, we consider the case, where Condition 3) in Lemma 3 is satisfied. According to the condition in 36), the feasible region of α is α α l and α α. Additionally, from 5), α U shold hold. According to Appendix H of 4, U α l always holds. Hence, the overall feasible region is α l α U and α α. Frthermore, the feasible region of α is the same as in 38). As a reslt, the expression of the average power for RRH contribted in this case can be similarly obtained as in 39), except for the different integration intervals for α. The expression for the special case, when m is an integer can be similarly obtained. Let s now define the following fnction F x) = α ) V γ m, W α ) fα )dα. Γm) x α U Then, TRRH C given in 39) is eqal to TRRH C = F U). If the following condition is satisfied: c ) /b ) C3 : ab c<and b)+b<, 45) ab with a, b, c given in 9), the average power for RRH contribted nder this condition is TRRH C3 = F U) F α) l + F α ). 46) Note that Condition C3 is the same as Condition C, bt the power allocation for RRH is different. B. Scenario : Both RRHs are transmitting at a non-zero power, i.e., I = In this case, RRH will transmit with fll power, i.e., p =, and will transmit at a non-zero power. According to Theorem, the following condition shold be satisfied: λ ) εθ) < + α 47) ε θ) α and the transmit power of is given by { ) } p =min λ +εθ), P α. 48) α ε θ) α The conditions for each vale of p will be discssed as follows. ) p = : In this case, the following condition shold be satisfied: ) +εθ) P α. 49) α λ ε θ) α Combining conditions 3), 47) and 49), we can obtain the feasible region of α as follows ε λ α α < ) θ) α +εθ) P λ α A. 5) λ To garantee that we have a nonempty set of α, the following condition shold be satisfied: + α + λ λ α ) +εθ) < ε θ) λ α, 5) which is in the form of fnction hx) defined in Lemma 3 with x = α, and a, b, c given by a = + λ,b=+εθ),c= ε θ). 5) λ λ To garantee the existence of a positive α, the graphical crve of the fnction hα ) shold be similar to that in Fig. 3-c, and the conditions are given by c ) /b ) C4 :ab c<, and b)+b<, 53) ab

8 8 where a, b, c are given in 5). Let s denote the soltions of hα )=as α l and α, where we have α l <α. Then, the feasible region of α is given by α l <α <α. Under Condition C4, the average powers of RRH and are respectively calclated as TRRH C4 = α Γm) T C4 α l fα ) γ = T C4 RRH m, ma ) γ m, mλ ) α dα, 54) ᾱ ᾱ λ, 55) where A is defined in 5). ) ) p = λ +εθ) α εθ)α α : In this case, the following condition shold be satisfied: > ) λ +εθ) P α. 56) α ε θ) α By combining 3), 47) and 56), the feasible region of α can be obtained as follows: ε B θ) α λ ) +εθ) >α max { λ } α,a λ, 57) where A is defined in 5). Note that B > A always holds. Hence, to garantee that there exists a feasible α, the following condition shold be satisfied: + λ ) +εθ) α ε θ) α. 58) λ λ Again, the right hand side of 58) is in the form of the fnction hx) defined in Lemma 3, with x=α, and a, b, c are given by a = λ,b=+εθ),c= ε θ). 59) λ λ To garantee that there exists a positive α, the third condition in Lemma 3 shold be satisfied: c ) /b ) C X :ab c<and b)+b<, 6) ab where a, b, c are given in 59). When a, b, c are given in 59), we can denote the soltions of h α ) = as α l and α with α l < α. Hence, the feasible region of α is given by α l <α < α. The remaining task is to determine the lower bond of α as shown in 57). Case I: If the following condition is satisfied: λ α A, 6) λ the feasible region of α is given by B>α λ α, 6) λ where B is defined in 57). The ineqality 6) can be rewritten as ) + α + λ +εθ) α ε θ) α, 63) λ λ which is eqivalent to hα ) with a,b and c given in 5). As seen from Fig. 3, when the first two conditions in Lemma 3 are satisfied, the ineqality 63) holds for any α >. Combining this with the condition 58), the feasible region of α is given by α l <α < α, where α l and α are the soltions of hα )=with a, b, c defined in 59). Define the following condition with a, b, c defined in 5) 4 : C5 : C X and ab c, or C X,ab c< and ) c /b ) 64) ab b)+b, and C X as ε C X ) θ) α +εθ). 65) α λ Under Condition C5, the average power of RRH and RRH are respectively given by TRRH C5 = Γm) α α l γ m, mb ) γ m, mλ ) α fα )dα, 66) ᾱ ᾱ λ T C5 = J J 67) with J = Γm) J = ᾱp mγm) α α l C γ α α l )) m, mb ᾱ ) γ m, mλ ᾱ λ α f α ) dα α E G) f α )dα, where E and G are respectively given by E = γ m +, mb ),G= γ m +, mλ ) α ᾱ ᾱ λ with B defined in 57), α l and α are the soltions of hα )= with a, b, c defined in 59). On the other hand, when the third condition in Lemma 3 is satisfied, ineqality 63) holds for α α l or α α, where α l and α are the soltions of hα )=with a, b, c defined in 5). Additionally, it is easy to verify that the crve of hα ) associated with a, b, c defined in 5) is above the crve of hα ) with a, b, c defined in 59). Hence, the following relations hold: α l <α l <α < α. 68) Combining 68) with condition 58), the feasible region of α is given by α l <α <α,α l <α < α. 69) As a reslt, when the following conditions are satisfied with a, b, c defined in 5): c ) /b ) C6 : C X,ab c<and b)+b<, 7) ab the average power reqired for RRH and, denoted as TRRH C6 and TRRH C6 respectively, is similar to the expressions of TRRH C5 and TRRH C5 except that the integration interval of α becomes 69). Case II: If the following condition is satisfied: λ α <A, 7) λ the feasible region of α is B>α >A, where B is defined in 57). The above condition is eqivalent to hα ) < with a, b, c defined in 5). To garantee the existence of a positive α, the third condition in Lemma 3 shold be satisfied. 4 Under Condition C5, a, b, c are defined in 59).

9 9 TABLE II MAIN RESULTS WHEN I =, AND λ λ α α Conditions Average Power I = I = C C C3 C4 C5 C6 C7 T C RRH T C RRH T C3 RRH T C4 RRH,T C4 T C5 RRH,T C5 T C6 RRH,T C6 T C7 RRH,T C7 Assming that this condition is satisfied, the feasible region of α is given by α l <α <α, where α l and α are the soltions of hα )=with a, b, c defined in 5). Again, by sing the relations in 68) and the condition 58), the feasible region of α is given by α l <α <α. Hence, if the following conditions are satisfied with a, b, c defined in 5): c ) /b ) C7 : C X,ab c<and b)+b<, 7) ab that the average powers reqired for RRH and are respectively given by TRRH C7 = α Γm) α l J = Γm) J = γ α α l m, mb ) γ m, ma ) fα )dα, 73) ᾱ ᾱ TRRH C7 = J J, 74) where J and J are given by ) )) C γ m, mb ᾱ γ m, ma ᾱ f α ) dα, ᾱp mγm) α α l α H L) f α )dα. where H and L are respectively given by H = γ m +, mb ),L= γ ᾱ m +, ma ᾱ ). C. Discssion of the reslts In this sbsection, we smmarize the reslts discssed in the above two sbsections in Table II. The average power reqired for each RRH contribted nder condition 3) is given by P λ α λ α RRH = 7 i= T Ci RRH εci),p λ α λ α = 7 i=4 T Ci εci), 75) where ε ) is an indicator fnction, defined as {, if Condition Ci holds, ε Ci) = 76), otherwise. Then, the average power reqired for each RRH is given by P RRH =P λ α λ λ α α > λ RRH +P α λ α > λ α RRH RRH,P RRH =P λ α > λ α λ α λ λ α α > λ +P α α, 77) where P and P denotes the average transmit power nder condition λ α > λ α for RRH and, which can be calclated as the condition of λ λ α. To provide more insights concerning the joint power allocation reslts for the case of I =, in Fig. 4 we plot the regions corresponding to different cases of the dal variables. For clarity, we only consider the case of λ α > λ α. Fig. 4-a) corresponds to the case of λ = λ =. From this figre, we can see that or proposed joint power allocation algorithm divides the region of λ α > λ α into two exclsive regions by the solid lines. If α,α ) falls into the region T, both RRHs remain silent. On the other hand, if α,α ) falls into T, Condition C is satisfied and only RRH will transmit with positive power, bt less than the power. Fig. 4-b) corresponds to the case of λ =/5,λ =. In this case, the region of λ α > λ α is divided into five exclsive regions. Similarly, in region T, none of the RRHs transmit. In region T and T 4, Condition C3 is satisfied and only RRH is assigned non-zero power for data transmission, bt less than the power. If α,α ) falls into region T 3, Condition C is satisfied, and RRH will transmit at power and RRH still remains silent. However, if α,α ) falls into the region T 5, Condition C5 holds, RRH will transmit at power and is allocated positive power that is lower than the power. Fig. 4-c) corresponds to the case of λ =λ =/. In this case, the region is partitioned into as many as eight regions. Regions T -T 4 are the same as the case in Fig. 4- b). If α,α ) falls into regions T 6 and T 8, Conditions C6 is satisfied, hence RRH will transmit at its power and is assigned non-zero power for its transmission, bt its power is less than the power. If α,α ) falls into region T 7, Condition C7 is satisfied. Similarly, RRH transmits at its maximm power and transmits at a non-zero power below its power. On the other hand, if α,α ) falls into region T 7, Condition C4 is satisfied and both RRHs will transmit at their power. It is interesting to find that with the redction of the dal variables, more RRHs will transmit at non-zero power or even the power. This can be explained as follows. The dal variables can be regarded as a pricing factor, where a lower dal variable will encorage the RRHs to be involved in transmission de to the low cost. V. POWER ALLOCATION IN EXTREME CASES In this section, we discss the power allocation for two extreme cases: ) when θ, representing no delay reqirement; ) when θ and i, i I, representing the extremely strict zero delay reqirement. A. θ No delay reqirements) When θ, the system is delay-tolerant. The EC maximization problem is eqivalent to the ergodic capacity maximization problem, which is given by max E α log + i I p ) i α) α i {p iα), i} s.t. E α p i α) P avg i, i 78) p i α) i, i. By sing a similar method as in Section III, the optimal transmit power at each RAU is given by πa), a < I ; p πa) = { min P π I ), T } πa),a= I ; 79), a > I ; where T πa) is given by T πa) = α π I ) ln απ I ) ) I λ π I ) a= πa) α πa)

10 Fig. 4. The partitions of the region of λ > λ nder different dal variables: a) λ α α = λ =; b) λ =/5,λ =; c) λ = λ =/. The system parameters are set as : Time frame of length T f =. ms, the system bandwidth B = KHz, delay QoS exponent θ = ln)/ so that εθ) =, power constraint i =W, i =,. with I being the largest vale of x sch that In απx) > + x λ πx) a= πa) α πa) 8) λ and π is a permtation over I so that πi) α πi) λ πj) α πj), when i<j,i,j I. When there is only one RRH, the optimal power allocation redces to the conventional water-filling soltion that is given by p = λ ln P, 8) α where x y represents that the vale is zero if x<, x if <x<y, and y if x>y. B. θ and i, i IExtremely strict delay reqirement) For this case, the system cannot tolerate any delay, and the EC is eqal to the zero-otage capacity. Hence, the optimal power allocation for each RRH redces to the channel inversion associated with a fixed data rate 8 given by p i = β i, i I, 8) α i where β i is specifically selected to satisfy the average power constraints. For the Nakagami-m channel defined in ), we have { } { m E αi = m )ᾱ i,m> 83) α i, m. Then, β i can be calclated as { m )ᾱip avg i m β i =,m>, m. The EC can be obtained as { Tf Blog EC = + i I β i),m>, m. 84) 85) Note that when m =, the EC is eqal to zero, which is consistent with the reslt in 8, namely that the zero-otage capacity for Rayleigh fading is eqal to zero. However, when m>, the EC increases with m. This coincides with the intition that with the increase of m, the channel variations redce and the channel becomes more stable. As a reslt, the qee-length variations are redced and hence higher delay reqirements can be satisfied. VI. EXTENSION TO THE MULTIUSER CASE The above sections focs on the single ser case. However, in C-RAN, de to the powerfl comptational capability of the BBU pool, C-RAN is designed with the goal of serving mltiple sers. To simply the analysis, we assme that the RRHs transmit the signals to different sers in orthogonal channels to avoid mltiser interference. Similar assmptions have been made in, 3, 9, 3. Let s denote the total nmber of sers by K and K = {,,K}. The QoS exponent for ser k is denoted as θ k, and the set of QoS exponents is denoted as θ = {θ,,θ K }. The channel s power gain from RRH i to ser k is denoted as α i,k. The set of channel gains from all RRHs to ser k is denoted as α k = {α,k,,α I,k }. The set of all channel gains in the network is denoted as α = {α,, α K }.For simplicity, the power constraints are not considered. We aim for optimizing the power allocation to max max {p i,k ν), i,k} k K log T f Bθ k s.t. E α k K p i,k ν) E α e θ kr k ν) ) 86a) P avg i, i I, 86b) p i,k ν), i I,k K, 86c) where ν = {α, θ} represents the network condition that incldes both the channel s power gains and the QoS exponent reqirements, R k ν) is the data rate achieved by ser k that is given by R k ν) =T f Blog + i I p i,kν)α i,k ). Note that in the mltiser case, each ser s power allocation depends both on its own channel condition and on its own delay reqirement as well as those of other sers. Frthermore, the power allocation soltion of each ser is copled with the per-rrh average power constraints, which complicates the analysis. Unlike in the single ser case, Problem 86) cannot be decomposed into several independent sbproblems for each channel fading state. The reason for this is that the objective fnction in Problem 86) cannot be transformed into the format of Problem 7). However, since the EC is a concave fnction 3 and the constraints are linear, Problem 86) still remains a convex optimization problem. Hence, the Lagrangian dality method can still be sed to solve this problem. Specifically, we introdce the dal variables of λ i, i, and δ i,k, i, k for the corresponding constraints in Problem 86). The KKT conditions for the optimal soltions of Problem 86)

11 are given by εθ k)α i,k Z k ν) εθ k) fα)+λ i fα) δ i,k =, i 87a) κ k ) λ i E α k K p i,kν) P avg i =, i 87b) δi,kp i,kν) =, i, k 87c) p i,kν), i, k 87d) where fα) represents the joint PDF of α, Z k ν) = + i I p i,k ν)α i,k and εθ k )=θ k T f B/ln, κ k is given by κ k = T f Bθ k E α Z k ν) εθ k). 88) To satisfy the above KKT conditions, we first find the optimal power allocation associated with the fixed dal variables λ i, i, then pdate the dal variables by sing the sb-gradient method. Given λ i, i, we then have the following theorem. Theorem 3: There is at most one RRH serving each ser and the index of this RRH is given by i λ i =argmin, 89) i I α i,k while the corresponding power allocation is given by p εθk ) i,k = κ k λ i ) +εθ k ) α εθ k) +εθ k ) i,k α i,k +, 9) where x + =max{,x}. Proof: The proof is similar to that of the single ser scenario of Section III, which is omitted for simplicity. Althogh each ser is served by only one RRH for each channel state, they still benefit from the mlti-rrh diversity, as seen in 89). Similarly, the online calclation method sed for the single ser case can be adopted to approximate the vale of κ k. VII. SIMULATIONS In this section, we characterize the performance of or proposed algorithm. We consider a C-RAN network covering a sqare area of km km. The ser is located at the center of this region and the RRHs are independently and niformly allocated in this area. Unless otherwise stated, we adopt the same simlation parameters as in : Time frame of length T f =. ms; system bandwidth of B = KHz; average power constraint of P avg i =.5 W, i; power constraint of i =W, i; QoS exponent of θ =.5; Nakagami fading parameter m =. The path-loss is modeled as PL i = log d i db) 3, where d i is the distance between the ith RRH and the ser measred in km. The channel also incldes the log-normal shadowing fading with zero mean and 8 db standard derivation. The noise density power is -74 dbm/hz 3. We compare or algorithm to the following algorithms: ) Nearest RRH serving algorithm: In this algorithm, the ser is only served by its nearest RRH, and the optimization method proposed in for point-to-point systems is adopted to solve the power allocation problem in this setting. This algorithm is proposed to show the benefits of cooperative transmission in C-RAN. Y coordinate m) 5-5 RRH User X coordinate m) Fig. 5. Random network topology with two RRHs. The ser is located at the center of the C-RAN. The coordinates of RRH and are given by 6, 8 and 9, 946, respectively. The CPNRs are ᾱ =3.89 and ᾱ =.43. ) Constant power allocation algorithm: In this algorithm, the transmit power is set to be eqal to the average power constraint for any time slots. Since the power limit is higher than the average power limit, both the average power constraints and power constraints can be satisfied. This approach was assmed to illstrate the benefits of dynamic power allocation proposed in this paper. 3) Independent power allocation algorithm: In this algorithm, each RRH i independently optimizes its power allocation withot considering the joint channel conditions of the other RRHs. The optimization problem is given by max p iν) s.t. θ log E α e θt f B log +p iν)α i) ) 9a) E α p i ν) P avg i, 9b) p i ν) i. 9c) The optimal power allocation soltion is given by 8). This algorithm is invoked for showing the benefits of jointly optimizing the power allocation according to the joint channel conditions of all RRHs. 4) Ergodic capacity maximization algorithm: This algorithm aims for maximizing the ergodic capacity withot considering the delay reqirement. The optimal power allocation soltion is given by 79). 5) Channel inversion algorithm: This algorithm imposes an extremely strict delay reqirement. The power allocation is the channel inversion given by 8). In the following, we first consider the case of I =, where the average power reqired for each RRH can be nmerically obtained according to the reslts of Section IV. Then, we stdy the more general case associated with more than two RRHs. Fig. 5 shows a randomly generated network topology relying on two RRHs. The average CPNRs related to RRH and are then given by ᾱ =3.89 and ᾱ =.43. Figs. 6-8 are based on this network topology. In this setting, the ser is only served by RRH de to its shorter distance compared to, when sing the nearest RRH algorithm. In Fig. 6, we stdy the convergence behavior of the proposed algorithm nder different delay exponents and stepsize parameters. The left three sbplots correspond to the QoS exponent θ =.5, while the right three correspond to θ =.5. It can be observed from this figre that the

12 Average Power W) θ=.5,a=.4 RRH θ=.5,a=.6 RRH θ=.5,a= RRH 5 5 Nmber of iterations Average Power W) θ=.5,a=.6 RRH 5 5 θ=.5,a= RRH θ=.5,a=.6 RRH 5 5 Nmber of iterations Fig. 6. Convergence behavior of the proposed algorithm for the special case of two RRHs nder different step parameters. Left three sbplots correspond to the QoS exponent θ =.5, and the right three correspond to θ =.5. Normalized EC bit/s/hz) Proposed allo. alg. Constant power allo. alg. Nearest RRH serving alg. Independent power allo. alg. Ergodic capacity maxi. Channel inversion - - QoS Exponent θ Fig. 7. Normalized EC for varios algorithms vs QoS exponent θ. convergence speed mainly depends on step-size parameter a. For both cases of θ, a smaller step-size parameter leads to slower convergence and larger one leads to faster convergence speed. For example, for the case of θ =.5, nearly 7 iterations are reqired for convergence when a =.4, while only iterations are needed when a =. However, since the sbgradient method is not an ascent method, a larger vale of a may lead to a flctation of the power vale. A small vale of a yields a smooth bt slow convergence. Hence, the stepsize parameter shold be careflly chosen to strike a tradeoff between the convergence speed and the smoothness of the power crve. It shold be emphasized that the main advantage for the case of two RRHs is that the Lagrange dal variables can be calclated in an off-line manner by sing the reslts of Section IV, and stored in the memory. Let s now stdy the impact of the delay reqirements on the EC performance for varios algorithms in Fig. 7. The normalized EC performance which is defined as the P ot Delay -5 - Proposed alg., D max =3 ms Proposed alg., D max = ms Proposed alg., D max = ms - - QoS Exponent θ Fig. 8. Delay-otage probability verss QoS exponent θ for varios vales of D max for or proposed algorithm. EC divided by B and T f with the nit of bit/s/hz ) is considered. As expected, when the delay-qos reqirement becomes more stringent, i.e., the vale of QoS exponent θ increases, the normalized EC achieved by all algorithms is redced. By employing two RRHs for jointly serving the ser to exploit higher spatial degrees of freedom, the proposed algorithm significantly otperforms the nearest RRH based algorithm, where only one RRH is invoked for transmission. For example, for θ =. the normalized EC provided by the former algorithm is roghly 5% higher than that of the latter algorithm. It is observed that the performance of the latter algorithm is even inferior to that of the naive constant power allocation algorithm. Since or proposed algorithm adapts its power allocation to delay-qos reqirement, whilst additionally taking into accont the channel conditions, it achieves a higher normalized EC than the constant power allocation algorithm, and the performance gain attained is abot % when θ =.. By optimizing the power allocation based on the joint channel conditions, the proposed algorithm performs better than the independent one, where the joint relationship of the channel conditions is ignored. The performance gain is more prominent when the delay-qos reqirement is very stringent, which can be p to 8.6% in this example. The proposed algorithm has similar performance to that of the ergodic capacity maximization algorithm for small θ, while the performance gain provided by the proposed algorithm becomes higher for a large θ. For a high θ, the normalized EC achieved by the ergodic capacity maximization algorithm is even lower than that of the constant power allocation algorithm. It is seen from Fig.7 that the performance of the channel inversion based power allocation method is mch worse than that of or proposed algorithm, since it aims for an extremely strict delay bdget rather than adaptively adjsting the power allocation according to the delay reqirements. Fig. 8 shows the delay-otage probability verss QoS exponent θ for varios vales of D max for or proposed algorithm. In this example, the UL and DL transmission dration and the fronthal delay are set as D T = D F = T f =. ms 33. Then, the corresponding qeeing delay is given by D q = D max D T D F = D max. ms. The maximm incoming data rate is set to be eqal to the EC, i.e. to μ =ECθ). The method in 6 is sed for calclating the delay-otage probability. As expected, the delay otage probability increases with the redction of D max. By appropriately choosing the QoS exponent θ, the delay otage probability can be redced below 8. For example, for the case of D max = ms, the delay otage probability is as low as 4 when θ =.585, which satisfies the stringent delay reqirement of the ltra-reliable low latency commnications URLLC) in 5G. For a given target Pdelay ot, the corresponding vale of θ obtained from Fig. 8 can be sed in Fig. 7 to find the achievable EC. For example, when the delay otage probability reqirement is Pdelay ot = 6 for the case of D max =ms, the corresponding θ is given by.. The resltant normalized EC becomes.78 bit/s/hz according to Fig. 7. We now consider a more general case in Fig. 9, where five RRHs are randomly located in a sqare. The average CPNRs received from these RRHs are then ᾱ =64.3, ᾱ =5.3,

13 3 Y coordinate m) 5-5 RRH User RRH 4 RRH 3 Y coordinate m) 5-5 RRH 4 User User RRH RRH 3 Fig. 9. Fig.. RRH X coordinate m) Random network topology with five RRHs. Average PowerW) θ=.5,a=4 RRH RRH 3 RRH 4 RRH θ=.,a=5 RRH RRH 3 RRH 4 RRH θ=.5,a= RRH RRH 3 RRH 4 RRH Nmber of iterations Convergence behavior of the online tracking method. ᾱ 3 =63., ᾱ 4 =3.8 and ᾱ 5 =5.. Based on Fig. 9, the convergence behavior of the online tracking method is shown in Fig., where different QoS reqirements are tested. As shown in Fig., the online tracking method converges promptly for all the vales of θ considered which is achieved by properly choosing the step parameter a. These observations demonstrate the efficiency of the online tracking method. In Fig., we plot the effective capacity performance for varios algorithms for this more general scenario. Similar observations can be fond from this figre. For example, the normalized EC achieved by all algorithms decrease with the QoS exponent θ, and the proposed algorithm significantly otperform the other algorithms. Finally, we stdy the EC performance of the mltiser scenario. The simlation setp is given in Fig., where there are two sers located at, and,, respectively, while the locations of the for RRHs are given by 65, 65, 65, 65, 65, 65, and 65, 65. We shold em- Normalized EC bits/s/hz) Proposed allo. alg. Constant power allo. alg. Nearest RRH serving alg. Independent power allo. alg. Ergodic capacity maxi. Channel inversion - - QoS Exponent θ Fig.. Normalized EC for varios algorithms vs QoS exponent θ for the more general case of I =5. Fig X coordinate m) Simlated network topology for two sers with for RRHs. Normalized EC bit/s/hz) Proposed allo. alg. Ergodic capacity maxi. - - QoS Exponent θ Fig. 3. The sm normalized EC for varios algorithms vs QoS delay exponent θ. phasize that or algorithm sed for the mltiser scenario is applicable for any network setp. In Fig. 3, we compare or proposed EC maximization algorithm of Sbsection VI to the mltiser ergodic capacity maximization algorithm in terms of the sm EC of the two sers. Note that a similar performance trend has been observed to that in Fig. 7. For example, both algorithms have almost the same performance in the low QoS exponent regime, while or proposed EC-oriented algorithm performs mch better than the ergodic capacityoriented algorithm for high exponents. When θ =.4, the performance gain is p to.3 bit/s/hz. VIII. CONCLUSIONS We considered joint power allocation for the EC maximization of C-RAN, where the ser has to garantee a specific delay-qos reqirement to declare sccessfl transmission. Both the per-rrh average and power constraints were considered. We first showed that the EC maximization problem can be eqivalently transformed into a convex optimization problem, which was solved by sing the Lagrange dal decomposition method and by stdying the KKT conditions. The online tracking method was proposed for calclating the average power for each RRH. For the special case of two RRHs, the expression of average power for each RRH can be obtained in closed-form. We also provided the power allocation soltions for the extreme cases of θ and θ. The simlation reslts showed that or proposed algorithm converges promptly and performs mch better than the existing algorithms, inclding the conventional ergodic capacity maximization method operating withot delay reqirements. A delay otage probability of 9 can be achieved at D max =

14 4 ms by or proposed algorithm throgh controlling the vale of θ, which is very promising for the applications in ftre URLLC in 5G. In this paper, we only extend the single-ser case to the mltiple-ser interference-free case. For the more general case where each ser sffers from the mlti-ser interference, the formlated optimization problem is non-convex, where the Lagrangian dality method is not applicable. How to deal with this problem will be left for ftre work. In addition, the fronthal capacity is generally limited de to the fast oversampled real-time I/Q digital data streams transmitting over the fronthal links. How to design the transmission scheme by taking into accont this constraint will also be considered in the ftre work. REFERENCES J. G. Andrews, S. Bzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, What will 5G be? IEEE J. Sel. Areas Commn., vol. 3, no. 6, pp. 65 8, 4. M. Peng, S. Yan, and H. V. Poor, Ergodic capacity analysis of remote radio head associations in clod radio access networks, IEEE Wireless Commn. Lett., vol. 3, no. 4, pp , 4. 3 Y. Shi, J. Zhang, and K. B. Letaief, Grop sparse beamforming for green clod-ran, IEEE Trans. Wireless Commn., vol. 3, no. 5, pp , 4. 4 B. Dai and W. Y, Energy efficiency of downlink transmission strategies for clod radio access networks, IEEE J. Sel. Areas Commn., vol. 34, no. 4, pp. 37 5, 6. 5 C. Pan, H. Zh, N. J. Gomes, and J. Wang, Joint precoding and RRH selection for ser-centric green MIMO C-RAN, IEEE Trans. Wireless Commn., vol. 6, no. 5, pp , 7. 6 D. W and R. Negi, Effective capacity: a wireless link model for spport of qality of service, IEEE Trans. Wireless Commn., vol., no. 4, pp , 3. 7 J. Tang and X. Zhang, Qality-of-service driven power and rate adaptation over wireless links, IEEE Trans. Wireless Commn., vol. 6, no. 8, pp , 7. 8 L. Msavian and S. Aissa, Effective capacity of delay-constrained cognitive radio in Nakagami fading channels, IEEE Trans. Wireless Commn., vol. 9, no. 3, pp. 54 6,. 9 X. Zhang and J. Tang, Power-delay tradeoff over wireless networks, IEEE Trans. Commn., vol. 6, no. 9, pp , 3. Y. Li, L. Li, H. Li, J. Zhang, and Y. Yi, Resorce allocation for delay-sensitive traffic over LTE-advanced relay networks, IEEE Trans. Wireless Commn., vol. 4, no. 8, pp , 5. W. Cheng, X. Zhang, and H. Zhang, Statistical-QoS driven energyefficiency optimization over green 5G mobile wireless networks, IEEE J. Sel. Areas Commn., vol. 34, no., pp , 6. C. E. Shannon, A mathematical theory of commnication, ACM SIGMOBILE Mobile Compting and Commnications Review, vol. 5, no., pp. 3 55,. 3 J. G. Smith, The information capacity of amplitde-and varianceconstrained sclar gassian channels, Information and Control, vol. 8, no. 3, pp. 3 9, S. Shamai and I. Bar-David, The capacity of average and power-limited qadratre gassian channels, IEEE Transactions on Information Theory, vol. 4, no. 4, pp. 6 7, Jl M. A. Khojastepor and B. Aazhang, The capacity of average and power constrained fading channels with channel side information, in 4 IEEE Wireless Commnications and Networking Conference IEEE Cat. No.4TH8733), vol., March 4, pp Q. Y. Y, Y. T. Li, W. Xiang, W. X. Meng, and W. Y. Tang, Power allocation for distribted antenna systems in freqency-selective fading channels, IEEE Trans. Commn., vol. 64, no., pp., 6. 7 M. K. Simon and M.-S. Aloini, Digital commnication over fading channels. John Wiley & Sons, 5, vol G. Zhang, T. Q. S. Qek, A. Hang, M. Kontoris, and H. Shan, Delay modeling for heterogeneos backhal technologies, in 5 IEEE 8nd Vehiclar Technology Conference VTC5-Fall), Sept 5, pp X. Chen, X. X, X. Tao, and H. Tian, Dal decomposition based power allocation for downlink ofdm non-coherent cooperative transmission system, in IEEE 75th Vehiclar Technology Conference VTC Spring), May, pp. 5. X. Chen, X. X, J. Li, X. Tao, T. Svensson, and H. Tian, Optimal and efficient power allocation for ofdm non-coherent cooperative transmission, in IEEE Wireless Commnications and Networking Conference WCNC), April, pp T. V. Chien, E. Bj?rnson, and E. G. Larsson, Joint power allocation and ser association optimization for massive mimo systems, IEEE Transactions on Wireless Commnications, vol. 5, no. 9, pp , Sept 6. L. Li, S. Bi, and R. Zhang, Joint power control and fronthal rate allocation for throghpt maximization in ofdma-based clod radio access network, IEEE Transactions on Commnications, vol. 63, no., pp , Nov 5. 3 C. He, B. Sheng, P. Zh, X. Yo, and G. Y. Li, Energy- and spectralefficiency tradeoff for distribted antenna systems with proportional fairness, IEEE J. Sel. Areas Commn., vol. 3, no. 5, pp , May 3. 4 H. Ren, N. Li, C. Pan, M. Elkashlan, A. Nallanathan, X. Yo, and L. Hanzo, Power-and rate-adaptation improves the effective capacity of C-RAN for Nakagami-m fading channels, arxiv preprint arxiv:74.94, 7. 5 R. Zhang, On verss average interference power constraints for protecting primary sers in cognitive radio networks, IEEE Trans. Wireless Commn., vol. 8, no. 4, pp., 9. 6 W. Y and R. Li, Dal methods for nonconvex spectrm optimization of mlticarrier systems, IEEE Trans. Commn., vol. 54, no. 7, pp. 3 3, 6. 7 S. Boyd and L. Vandenberghe, Convex optimization. Cambridge niversity press, 4. 8 A. J. Goldsmith and P. P. Varaiya, Capacity of fading channels with channel side information, IEEE Transactions on Information Theory, vol. 43, no. 6, pp , Nov R. G. Stephen and R. Zhang, Joint millimeter-wave fronthal and ofdma resorce allocation in ltra-dense cran, IEEE Transactions on Commnications, vol. 65, no. 3, pp. 4 43, March 7. 3 Y. Li, M. Sheng, Y. Zhang, X. Wang, and J. Wen, Energy-efficient antenna selection and power allocation in downlink distribted antenna systems: A stochastic optimization approach, in 4 IEEE International Conference on Commnications ICC), Jne 4, pp A. Helmy, L. Msavian, and T. Le-Ngoc, Energy-efficient power adaptation over a freqency-selective fading channel with delay and power constraints, IEEE Trans. Wireless Commn., vol., no. 9, pp , 3. 3 E. U. T. R. Access, Frther advancements for E-UTRA physical layer aspects, 3GPP TR 36.84, Tech. Rep.,. 33 C. She, C. Yang, and T. Q. S. Qek, Cross-layer optimization for ltrareliable and low-latency radio access networks, IEEE Transactions on Wireless Commnications, vol. 7, no., pp. 7 4, Jan 8. Hong Ren received the B.S. degree from Sothwest Jiaotong University, in Electrical Engineering in, Chengd, China, and the M.S. and Ph.D. degrees from Sotheast University, Nanjing, China, in Electrical Engineering in 4 and 8, respectively. From October 6 to Jan 8, she was a visiting stdent in the School of Electronics and Compter Science, University of Sothampton, UK. She is crrently a postdoctoral scholar with the School of Electronic Engineering and Compter Science, Qeen Mary University of London, UK. Her research interests lie in the areas of commnication and signal processing, inclding green commnication systems, cooperative transmission and cross layer transmission optimization.

15 5 Nan Li received the B.Eng. degree in electrical engineering from Beijing University of Posts and Telecommnications, Beijing, P. R. China in, and the Ph.D. degree in electrical and compter engineering from University of Maryland, College Park, MD in 7. From 7-8, she was a postdoctoral scholar in the Wireless Systems Lab, Department of Electrical Engineering, Stanford U- niversity. In 9, she became a professor in the National Mobile Commnications Research Laboratory, School of Information Science and Engineering in Sotheast University, Nanjing, China. Her research interests are in network information theory for wireless networks, SON algorithms for next generation celllar networks and energy-efficient commnications. Cnha Pan M 6) received his B.S. and Ph.D. degrees in school of Information Science and Engineering, Sotheast University, Nanjing, China, in and 5, respectively. From 5 to 6, he worked as a research associate in University of Kent, UK. He is crrently a postdoc with Qeen Mary University of London, UK. His research interests mainly inclde Ultra-dense C-RAN, UAV, internet of things IoT), NOMA, and mobile edge compting. He serves as the Stdent Travel Grant Chair for ICC 9, and a TPC Member for many conferences, sch as ICC and Globecom. Xiaoh Yo F ) was born in Agst 5, 96. He received his Master and Ph.D. Degrees from Sotheast University, Nanjing, China, in Electrical Engineering in 985 and 988, respectively. Since 99, he has been working with National Mobile Commnications Research Laboratory at Sotheast University, where he held the rank of director and professor. His research interests inclde mobile commnication systems, signal processing and its applications. He has contribted over IEEE jornal papers and books in the areas of adaptive signal processing, neral networks and their applications to commnication systems. From 999 to, he was the Principal Expert of the C3G Project, responsible for organizing Chinas 3G Mobile Commnications R&D Activities. From -6, he was the Principal Expert of the China National 863 Beyond 3G FTURE Project. Since 3, he has been the Principal Investigator of China National 863 5G Project. Professor Yo was the general chairs of IEEE WCNC 3 and IEEE VTC 6. Now he is Secretary General of the FTURE Form, vice Chair of China IMT- Promotion Grop, vice Chair of China National Mega Project on New Generation Mobile Network. He was the recipient of the National st Class Invention Prize in, and he was selected as IEEE Fellow in same year de to his contribtions to development of mobile commnications in China. Maged Elkashlan M 6) received the Ph.D. degree in Electrical Engineering from the University of British Colmbia, Canada, 6. From 7 to, he was with the Commonwealth Scientific and Indstrial Research Organization CSIRO), Astralia. Dring this time, he held visiting appointments at University of New Soth Wales and University of Technology Sydney. In, he joined the School of Electronic Engineering and Compter Science at Qeen Mary University of London, UK. His research interests fall into the broad areas of commnication theory and statistical signal processing. Dr. Elkashlan crrently serves as Editor of IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. He received the Best Paper Awards at the IEEE International Conference on Commnications ICC) in 6 and 4, the International Conference on Commnications and Networking in China CHINACOM) in 4, and the IEEE Vehiclar Technology Conference VTC-Spring) in 3. Armgam Nallanathan S 97-M -SM 5-F7) was an Assistant Professor with the Department of Electrical and Compter Engineering, National University of Singapore, from to 7. He was with the Department of Informatics, Kings College London, from 7 to 7, where he was a Professor of wireless commnications from 3 to 7. He has been a Professor of wireless commnications and Head of Commnication Systems Research CSR) Grop with the School of Electronic Engineering and Compter Science, Qeen Mary University of London, since 7. He has athored nearly 4 technical papers in scientific jornals and international conferences. His research interests inclde 5G wireless systems, Internet of Things, and moleclar commnications. He was a co-recipient of the Best Paper Award presented at the IEEE Global Commnications Conference 7 GLOBECOM 7) and IEEE International Conference on Commnications 6 ICC 6). He is an editor for IEEE Transactions on Commnications. He has been selected as a Web of Science ISI) Highly Cited Researcher in 6. He is an IEEE Distingished Lectrer. Lajos Hanzo M 9-SM 9-F 4) received the D.Sc. degree in electronics in 976 and the Ph.D. degree in 983. Dring his 4-year career in t- elecommnications he has held varios research and academic posts in Hngary, Germany, and the U.K. Since 986, he has been with the School of Electronics and Compter Science, University of Sothampton, U.K. He has sccessflly spervised Ph.D. stdents. In 6, he was admitted to the Hngarian Academy of Science. He is crrently directing a 5-strong academic research team, where he is involved in a range of research projects in the field of wireless mltimedia commnications sponsored by indstry, the Engineering and Physical Sciences Research Concil, U.K., the Eropean Research Concils Advanced Fellow Grant, and the Royal Societys Wolfson Research Merit Award. He is an enthsiastic spporter of indstrial and academic liaison and he offers a range of indstrial corses. He has co-athored 8 John Wiley/IEEE Press books on mobile radio commnications totaling in excess of pages, and pblished over 7+ research contribtions on IEEE Xplore. He was FREng, FIET, and a fellow of the EURASIP. In 9, he received an honorary doctorate from the Technical University of Bdapest and in 5 from the University of Edinbrgh. He has served as the TPC chair and the general chair of varios IEEE conferences, presented keynote lectres and has received a nmber of distinctions. He is also a Governor of the IEEE VTS. From 8 to, he was Editor-in-Chief of the IEEE Press and a Chaired Professor with Tsingha University, Beijing. He is the Chair in telecommnications with the University of Sothampton. He has 33,+ citations and an h-index of 73.