Uplink Capacity Optimization by Power Allocation for Multimedia CDMA Networks with Imperfect Power Control

Transcription

1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL.**, NO.*, MONTH YEAR Uplink Capacity Optimization by Power Allocation for Multimedia CDMA Networks with Imperfect Power Control Tao Shu, Student Member, IEEE, and Zhisheng Niu, Senior Member, IEEE Abstract A closed-form capacity quasi-optimal power allocation scheme is presented for the uplink of multimedia CDMA systems with randomized received SIR resulted from the errors of power control. The optimality in capacity comes from that this scheme provides class-dependent SIR margins subject to the constraint of differentiated outage requirements. The statistics of signal under imperfect power control is modeled as log-normal random variable. The objective of capacity maximization is formulated as the minimization of total average received powers since the capacity of a CDMA system is interference limited. Under this model, we first derive the necessary conditions that a capacity-optimal power allocation should satisfy. By using conservative bounds, we provide a closed-form approximate solution to this optimization problem. This approximate solution provides nearly the same admissible region for multimedia traffic under imperfect power control as the accurate solution the optimal one) does. The closed-form quasi-optimal power allocation scheme proposed in this paper is just based on this approximate solution. By numerical example we verify our analysis and show that great capacity gain e.g. 9% as a maximum in the example) can be achieved by our scheme over its counterpart. Index Terms multimedia CDMA, capacity optimization, power control. I. INTRODUCTION THE next generation wireless networks are supposed to provide quality-of-service QoS) support for multimedia traffic based on Code Division Multiple Access CDMA) air interface. As one of the most important interferencereduction and capacity-enhancement techniques in CDMA system, power control or power allocation for multimedia traffic has received extensive studies in the literature[-[6. A basic requirement to any of these power control/allocation schemes, whether centralized or distributed, is that it should be computational effective and thus can be implemented real time. They are required to respond instantly to the change of user configuration and the change of channel status, and reallocate power among users whenever such changes happen. By this sense, a closed-form algorithm is the most desirable for such application. So far, the closed-form optimal power allocation which maximizes system capacity has been well studied under the assumption of perfect power control. By perfect power control, This work was partially presented in Globecom 00, Taipei, Nov. 00. This paper is jointly supported by the National Science Foundation of China 6070) and the Excellent Young Teachers Program of MOE. The authors are with the Department of Electronic Engineering, Tsinghua University, Beijing 00084, China shutao00@mails.tsinghua.edu.cn; niuzhs@tsinghua.edu.cn) we refer to the condition that the powers received at the base station can be kept at some reference levels constantly so that the received signal-to-interference ratio SIR) is a constant. Under this assumption, through simple linear optimization, it has been found that the proportional assignment of reference power level according to user s power index maximizes user capacity [-[4. However, the implementations of this closedform capacity-optimal power allocation scheme encounter difficulties because of the randomness of received SIR in a real system. In a practical CDMA system, two factors make the recieved SIR a random variable. The first factor is power control error due to the fading resulted from the movements of mobile users[0. This situation is generally referred to as imperfect power control. A direct consequence of imperfect power control is the randomness of received powers, which makes the received SIR of each user a random variable. Another factor contributing to the randomness of SIR is the burstiness of multimedia traffic, which makes the number of active users in the system a random variable. Clearly, user s QoS requirement under random SIR can be guaranteed by allocating power in such a way that each mobile has an extra margin of SIR, i.e., its SIR is somewhat above the minimum SIR threshold required for reception. The state that user s SIR goes below the predefined threshold is defined as outage. The determination of SIR margin is highly capacity-sensitive, e.g., increasing the margin of SIR reduces the probability of violating QoS requirement, but costs extra power and increases the interference, which turns to capacity loss. The traditional closed-form power allocation[-[4 did not take such factors into consideration thus leads to considerable capacity loss[5. From this sense, we also call that scheme as Limited Optimal Power Allocation LOPA) scheme in this paper since it is optimal only under perfect power control. To address this issue and utilize system resource more efficiently, recently there has been an continuously increased interest in the study of power allocations in the new context of randomized received SIR[7-[5. In the literature, the randomness of received powers and the randomness of number of active users are usually decoupled by considering only one of them while assuming the other one as a fixed parameter. Different mathematical models are applied to the analysis of these two factors: lognormal received signal model for imperfect power control and ON/OFF model for traffic burstiness. Due to the easier analysis under ON/OFF model, there have been some works considering the optimal power allocation under

2 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL.**, NO.*, MONTH YEAR the randomized SIR resulted by the burstiness of multimedia traffic[7-[9. Among them, [7 proposed a nonlinear iterativebased power assignment scheme which minimizes the total average received power subject to the average SIR and outage requirements. In [8, an iterative algorithm was also proposed to compute the capacity-optimal reference power levels for multi-class traffic according to the outage requirements and the channel conditions. [9 improved the work in [8 by proposing a more efficient iterative algorithm which converges much faster to the optimum reference values. On the other hand, although there have been some works study the power allocation under the imperfect power control[0-[5, only a little of them addresses the problem of optimization[5. [0- [ analyzed the uplink capacity and interference statistics in an imperfect power controlled CDMA system. [3 and [4 studied the satisfaction of differentiated outage requirements of different traffic class by power allocation, but they did not calculate the optimum power allocation in their analysis. [5 studied the optimal allocation of transmit powers under a Rayleigh fading environment, where power control is assumed only applied to the near-far effect and the large-scale shadowing while the small-scale fading is left untouched. The capacity optimization was formulated as a Geometric Programming problem and the complicated interior-point method was employed in its solution. In summary, nearly all existing optimal power allocation algorithms presented under randomized SIR are based on iterative computation or complicated numerical solutions. Under heavy traffic load, such algorithms become ineffective and will lead to large delay in power allocation due to the increased number of iterations and computation complexity it takes to achieve the optimum outcome. To the best knowledge of the authors, there is no literature addresses closed-form capacity-optimal power allocation under randomized SIR. The purpose of this paper is to address the closed-form capacity optimal power allocation by focusing on the first factor that results in randomized SIR, i.e., imperfect power control. We consider the application of power control against the fast fading and thus take a different signal model from [5. The objective of capacity maximization is formulated as the minimization of total average received powers since the capacity of a CDMA system is interference limited. The basis of our optimization is the the lognormal description of the statistics of the received power under imperfect power control, whose property has been maturely studied so far[0[[7[8. The most important contribution of this work lies in that we present a closed-form capacity quasi-optimal power allocation scheme by which class-dependent SIR margins are provided under the constraints of differentiated outage requirements. In addition, we offer a unified approach for analyzing power control under imperfect power control environment by analytically studying the relationship between outage probability and SIR margin. Based on the log-normal model of received signals, we prove the necessary conditions that an optimal power allocation should satisfy. A direct solution to this optimization problem face a group of transcendental equation sets, whose accurate closed-form solution is hard to be derived. By using conservative bounds instead, we provide the closed-form approximate Fig.. BS 6 BS 5 BS BS 0 cell of interest BS 4 BS BS 3 S,...S N intra-cell interference S K...S KNK inter-cell interference I...I fn I K...I KfNK BS 0 AWGN System model and received signals model at a reference base station solution i.e., the quasi-optimal scheme) to this optimization problem. The proposed quasi-optimal scheme converges to the optimal solution at the boundary of admissible region thus it provides the similar admissible region as the optimal one does. By numerical example we verify our analysis and show that great capacity gain e.g. 9% as a maximum in the example) is achieved by our scheme over its counterpart LOPA scheme. This closed-form scheme was employed in our subsequent work in [6, where we further considered the influence of bursty traffic from the link level and proposed a highefficiency call admission control scheme under randomized SIR environment. The rest of this paper is organized as follows. Section describes the models considered. In Section 3 we define system capacity by user outage probabilities and evaluate the capacity loss resulted by the LOPA scheme while employed under imperfect power control. In Section 4, we formulate the optimization problem and present the closed-form approximate solution a quasi-optimal scheme). Section 5 presents numerical examples and simulations, and Section 6 concludes our work. A. System Model II. SYSTEM MODEL AND SIGNAL MODEL We consider a multi-cell DS-CDMA system as shown in Fig., where cell 0 is the cell we focus on. In each cell there is a base station BS) which is responsible for the power control of the mobile users in the cell. There are K classes of users in the system and the number of active users in class i i =,..., K) in cell 0 is N i. The spread spectrum bandwidth is W Hz. The single-sided power spectrum density of the Additive White Gaussian Noise AWGN) is N 0 watt/hz. The QoS requirement of the user in class i is presented as a triple R i, α i, δ i ), where R i is the user s rate requirement, and Pr{E b /I 0 ) i < α i } δ i, in which E b /I 0 is the user s bitenergy-to-interference density ratio. Thus α i is the predefined SIR threshold and δ i is the largest outage probability that the user in class i can tolerate. In general, different classes of users require differentiated levels of outage probability. For example, the outage probability of the user with real time traffic such as voice or video can be as low as 0 or 0 3, while the outage probability of 0 may still be tolerable for non-real time data user such as WWW browsing or FTP.

3 SHU AND NIU: CAPACITY OPTIMIZATION FOR MULTIMEDIA CDMA NETWORKS WITH IMPERFECT POWER CONTROL 3 B. Signal and Intra-cell Interference Model Let S ij be the received signal power of the jth j =,..., N i ) user in class i i =,..., K) at the BS in cell 0. For the uplink of each user, we consider a similar receiver structure as that in [7: a RAKE receiver is used with L fingers to collect the power from L independent equal strength Rayleigh fading. Since our intent is to focus on the impact of short-term fading, we assume that open loop power control compensates fully for long-term variations in the path loss. The user s session is of long enough duration so that closed loop transmit power control can be performed to combat the short-term fading. Due to the limited power adjustment step and frequency, the power level received at base station suffers a deviation from the mean value. Insightful studies on the statistics of received power level under imperfect power control have been conducted in [0[[7[8. It has been found in these studies that under imperfect power control, S ij is very well approximated by a lognormal random variable with mean u ij and variance γ ij. The accuracy of this lognormal approximation was examined in [8 by comparing with the empirical received power probability density, and in [0 and [7 by comparing with the results of computer simulations, in both cases a very close match was presented. Let X ij be the decibel representation of S ij, i.e. X ij = 0 lg S ij. Then X ij is a Gaussian random variable, whose mean m ij and variance σij have the following relation with u ij and γ ij : u ij = e βm ij+ β σ ij, ) γ ij = e βm ij+β σ ij e β σ ij ), ) ln 0 where β = 0. The parameter σ ij reflects the imperfection degree of the power control and is determined by the fading rate and the receiver structure[7. For the purpose of tractability, we assume all users in cell 0 have similar mobility and the same receiver structure so that σij equals to σ for all possible i and j. This assumption justifies for the reason that the cell size of a multimedia CDMA system is small and the users mobility within a cell can be deemed to be homogeneous. One example is the case in highway, where nearly all mobile users own similarly high mobility and experience fast fading. Another example is the case in the downtown area, where nearly all mobile users have similar low mobility and experience slow fading. By this assumption, the users in the same class have the same values of u ij, γ ij, m ij, and σij, and we denote them as u i, γ i, m i, and σ. Therefore, for a particular i, S ij j =,..., N i ) are modeled as independently and identically distributed i.i.d.) lognormal random variables with mean u i and variance γ i, as described above. C. Inter-cell Interference Model As a common assumption in the study of multicell system, we assume the users in each class distribute uniformly throughout all neighboring cells. Another commonly assumed approximation is that the interference from class i users i = The readers interested in this topic are directed to these literature for details.,..., K) in the neighbor cells is equivalent as additional fn i class i users, namely equivalent users, in cell 0. The factor f is the mean ratio of inter-cell to intra-cell interference[9- [ of traffic class i, i.e. the ratio of the average total interference generated by class i users outside cell 0 to the average total interference generated by class i users in cell 0. Under the assumption of uniform user distribution, this ratio is completely determined by the propagation parameters of the channel[3, i.e. the exponent of path loss and the standard variation of shadowing, both of which are common parameters for all users in the system. Thus f is a system parameter and is independent from individual traffic class. The values of f corresponding to specific propagation parameters are derived in [3. For traffic class i, the inter-cell interference at BS0 contributed by any of its equivalent user in cell 0 is modeled as i.i.d. log-normal random variable with mean v i and variance ɛ i, where v i = u i but ɛ i γ i. The inequality between variance is due to the fact that the power of the mobile user outside cell 0 is not controlled by BS0[7. Accordingly, the decibel representation of this interference is modeled as Gaussian random variable with mean n i and variance τ, which have relations with v i and ɛ i in the same way as ) and ). The values of σ and τ are derived in [7 for various user speeds and receiver structures through simulation. Denote by I i the total inter-cell interference at BS0 contributed by all the users of class i outside cell 0. Since I i is the sum of the independent lognormal inter-cell interference generated by fn i equivalent users of class i, according to Wilkinson s method[ it can also be approximated tightly as a lognormal random variable with mean fn i v i and variance fn i ɛ i. Such approximation is much more accurate than Gaussian approximation and is also used in [9 and [. III. OUTAGE PROBABILITY AND CAPACITY ANALYSIS A. Derivation of Outage Probability Denote by I the total interference plus AWGN received at BS0, i.e., N N K I = S j S Kj + I j + 3 N 0W, 3) j= j= j= where 3/ is the inverse of the orthogonal factor of the chip with a rectangular shape in an asynchronous CDMA system[6. Since I is a sum of independent lognormal random variables, it can also be tightly approximated as a lognormal random variable[, whose mean and variance are given respectively by E[I = + f) N i u i + 3 N 0W, 4) D[I = N i γ i + fn i ɛ i. 5) The accuracy of this approximation is compared to that of Gaussian approximation in [5. It was shown that much better effect is achieved by lognormal approximation than Gaussian

4 4 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL.**, NO.*, MONTH YEAR approximation, especially when the number of users is not large enough. Let Y = 0 lg I. Y is also a Gaussian random variable. By Wilkinson s method[, its mean p and variance η are given respectively by p = β [ ln E[I β ln + D[I E[I) η = β ln i [ + D[I E[I), 6). 7) For any class i user in cell 0 with received power S i, its received E b /I 0 at BS0 is given by ) Eb = 3W S i. 8) I 0 R i I S i Then the outage probability of class i user in cell 0 is given by { } { } 3W S i I P i = Pr < α i = Pr S i < R i I S i + 3. W R iα i 9) Let g i = and A + 3W i = 0 lg g i. By using the decibel R i α i representations of S i and I, 9) is further rewritten as P i = Pr{X i < Y + A i }. 0) Since X i and Y are independent Gaussian random variables with mean m i and p, and variance σ and η, respectively, the outage probability is then given by ) m i A i p P i = Q, ) σ + η where Qx) = π x B. System Capacity e t dt. The system s admissible region S is the set of N,..., N K ) Z+ K, which satisfies the outage probability constraints for all traffic classes under a specified power allocation scheme Γ, i.e. S = {N,..., N K ) Q m i A i p σ + η ) δ i, m,..., m K ) = ΓN,..., N K ), i K}.) The vector N,..., N K ) is defined as user configuration in this paper and the element of S is called feasible user configuration. The capacity of the system is defined as the boundary of S. In addition, we also use the term capacity and the term admissible region interchangeably when there is no confusion. It is indicated by ) that the outage probability of class i user is closely related to the power allocation scheme. By power allocation scheme, we mean to the determination of the average received signal power vector u,..., u K ) Strictly speaking, Y should be dependent on X i. However, because Y is composed of a large amount of X i s, its dependency on a particular X i can be negligible. In this sense, Y is approximately independent of X i. or equivalently m,..., m K ) for a given user configuration N,..., N K ). When a user configuration is beyond the boundary of S, no feasible power vector is available under the power allocation scheme employed. Thus the admissible region S is dependent on and determined by the power allocation schemes. For the given traffic classes, the maximum admissible region S max is selected from all the possible admissible regions so that every possible admissible region is a subset of S max. What we are interested in is the power allocation scheme which provides S max. C. Capacity Loss Resulted by the LOPA Scheme In [3 and [4, the LOPA power allocation scheme which minimizes the sum of total received signal powers or equivalently provides the maximum capacity for multimedia CDMA users is derived under the assumption of perfect power control. By this scheme, for a user configuration N,..., N K ) and the QoS requirements {R i, α i ), i =,..., K}, the allocated received power for class i user is given by u i = g i g Σ 3 N 0W, 3) where g Σ = K N ig i. It is easy to examine the optimal nature of 3) in the perfect power control environment. However, this scheme will lead to great capacity loss under imperfect power control because of the ignorance of differentiated outage probabilities. According to 3), the power allocated to class i user is proportional to g i, or the power index of class i. Adopting 3) under imperfect power control, all u i are scaled up by some scalar M to achieve sufficient margin of SIR in order to satisfy the desired outage probability. By ), this scheme leads to the mean received signal power of class i in db as m i = A i + 0 lg 3 N 0 W + 0 lg M g Σ βσ. 4) Substituting 4) into ), the outage probability of class i user is given by ) 0 lg 3 N 0 W g P i = Q Σ + 0 lg M pm) βσ, 5) σ + η M) where pm) and η M) indicate the influence of M on the total interference. Eq.5) shows that the LOPA scheme results in a unified outage probability for all classes of users regardless of their individual QoS requirement. Thus only δ min = minδ,..., δ K ) will be the practical constraint in the increase of admissible users and capacity are lost by inefficient power allocation. Under imperfect power control, a power allocation scheme which provides differentiated outage probabilities will provides better capacity performance than 3) does. IV. FORMULATION OF CAPACITY OPTIMIZATION AND A QUASI-OPTIMAL SOLUTION Under multi-class environment, system capacity is a set of vectors. In this paper, we define the capacity maximization as seeking S max so that every possible admissible region is one of

5 SHU AND NIU: CAPACITY OPTIMIZATION FOR MULTIMEDIA CDMA NETWORKS WITH IMPERFECT POWER CONTROL 5 its subsets. CDMA system is characterized as an interference limited system and any reduction of received interference will lead to an increase of capacity. In this context, the objective of maximizing the capacity is consistent with the minimization of received total powers, i.e., K min u,...,u K ) N iu i s.t. Q mi A i p ) δ σ i, i =,..., K). +η 6) Concerning the optimization presented by 6), we have the following proposition. Proposition : If 6) owns the optimal solution, all outage probability constraints are met with equality at the optimal solution. Proof : see Appendix I. By Proposition, the optimal power allocation scheme which minimizes the sum of total received power or equivalently provides the largest admissible region while providing QoS guarantees to multimedia users must satisfy the following equation set of m i : m i A i p σ + η = Q δ i ), i =,..., K) 7) where p and η are functions of m i i =,..., K) and are given by )-) and 4)-7). Eq.7) is a group of transcendental equation sets and its accurate closed-form solution is hard to be derived. Instead of trying to solve it by numerical methods, we seek the approximate solution, which leads to a closed-form quasioptimal power allocation scheme for multimedia users. As mentioned in Section, by quasi-optimal, we mean that the performance of this scheme converges to the performance of the optimal one only at the boundary of the admissible region. Thus it can provide nearly the same maximum capacity provided by the optimal one but its power sum does not reach the minimum for the user configuration inside the boundary. The approximate solution to 7) needs the following two propositions. Proposition : When a user configuration is at the boundary of the admissible region provided by 7), i.e., reaches the largest possible capacity, then the power allocation u i i =,..., K) tends to be infinite. Proof : see Appendix II. In fact, Proposition indicates such a fact that the capacity loss caused by the AWGN can be compensated by scaling up the received powers of all users. Proposition 3: When all elements of received signal power vector u,..., u K ) have infinite values, the ratio among u i uniquely determines the outage probability of each traffic class. Proof : see Appendix III. In the following, we seek the closed-form approximate solution to 7). Two steps are applied: firstly, we explore the power allocation at the admissible region boundary provided by 7); and then, we extend this allocation to the inside of the admissible region. By 7), the received signal power required for class i user is given by m i = A i + p + σ + η Q δ i ). 8) When the user configuration reaches the maximum capacity, m i reaches infinite thus the AWGN item in 3) can be ignored according to Proposition. η is simplified as η = β ln [ + C K N ie βmi + f) K N ie βmi ), 9) where C = e β σ + fe β τ f is a measure of control errors and inter-cell interference. Let u i = g i φ, where φ tends to be infinite. According to Proposition 3, g i i =,..., K) completely determines P i i =,..., K) and thus the capacity. From 8), the g i which achieve the maximum capacity should satisfy ) 0 lg g i g j = m i m j = 0 lg gi g j ) + σ + η [ Q δ i ) Q δ j ) 0) i, j =,..., K). It is a conservative approximation to substitute η with its lower bound ηmin in 0). By conservative approximation, we mean that this substitution tends to reduce the difference between the outage probability of the users with δ min and the outage probability of other class of users when the maximum capacity is reached. Thus the resulted capacity will be no less than that caused by the LOPA scheme. From 9), the lower bound of η subject to m i is [ ηmin = β ln C + + f) K N. ) i Substituting ) into 0), we get g i g j g i0 g j 0 σ +η min 0 Q δ i ) σ +η min 0 Q δ j ). ) Because g i is the scalar of u i, we let g i σ +η min = g i 0 0 Q δ i), i =,..., K). 3) Then we derive the approximate solution to 7) at the boundary of its admissible region as σ +η min u i = g i 0 0 Q δ i ) φ, φ ). 4) Eq.4) is also a power allocation scheme which provides the similar admissible region as the optimal solution of 7) does. At the boundary of the admissible region, 4) converges to the accurate solution of 7) with an error introduced by approximating η with ηmin in 0). In the numerical example, we will show that great capacity gain is still achieved in spite of this approximation. When the user configuration is inside the boundary of the admissible region, 4) provides smaller

6 6 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL.**, NO.*, MONTH YEAR P i for each class of users than δ i because of the infinity of φ. This indicates that φ can be lowered as much as possible only if every P i is maintained not more than δ i. We keep the relations among u i given by 3) to seek the minimum φ which maintains every P i not more than δ i for the user configuration inside the admissible region. This leads to our following quasi-optimal power allocation scheme. Substituting 4) into ), we have m i = βσ + 0 lg g i + 0 lg φ. 5) Substituting 5) to the constraints of 6), we have TABLE I STANDARD DEVIATION DETERMINED FROM LOGNORMAL FITTING OF SIGNAL POWER AFTER POWER CONTROL, DATA EXTRACTED FROM [7 V km/h) L = L = 3 L = 4 σ db) τ db) σ db) τ db) σ db) τ db) lg φ p σ + η Q δ i ) βσ 0 lg g i + 0 lg g i, i =,..., K). 6) In order to satisfy 6), it is sufficient to satisfy the following inequality 0 lg φ p σ + ηmaxq δ i ) βσ 0 lg g i+0 lg g i, 7) where ηmax is the maximum value of η subject to φ, which is a mono-increasing function of φ, and is given by η max = η φ) φ = β ln [ + C K N ig i, 8) g Σ where g Σ = + f) K N ig i. The parameter p in 7) is given by p = 0 lg g Σ + 3 ) [ N 0W 0 lg + Cφ K N ig i g Σ φ + 3 N. 0W ) Substituting 8) and 9) into 7), we have [ φ g Σφ + 3 ) N K 0W + Cφ N i g i B i 9) g Σφ + 3 N 0W ) 4, 30) βσ + σ +η max where B i δ i ) = 0 σ +η min )Q δ i ) 0 is a constant corresponding to class i. Noting that φ 0 and 30) holds for all i, it is easy to show that the minimum of φ satisfying 30) is φ min = { 3 N 0 W F max g Σ F max 3 N 0W B max, C = 0; g Σ B, C = 0, 3) max +4CB max C and B max = B i δ min ). where F max = By our quasi-optimal power allocation scheme, for a given user configuration N,..., N K ), the average received power allocated to class i user is u i = g i φ min, where g i and φ min are explicitly given by 3) and 3), respectively. Therefore, this scheme is of closed-form and can determine the power allocation among users in real time. In our scheme, g i can be seen as a generalized power index GPI) in the imperfect power control environment, as compared to g i, since g i account for not only the rate and SIR threshold but also the error of power control and outage constraint as well. Power allocated to class i user is exactly proportional to its GPI. In addition, in order to ensure that the average received power is positive, it is necessary that { g Σ F max 0 C 0; g Σ B 3) max 0 C = 0. Inequation 3) can be used as the criterion in the system call admission control CAC). Our closed-form scheme takes the traffic characteristics, user QoS parameters and power control errors into consideration and thus it is more practical than LOPA scheme. It is easy to examine the consistency of this closed-form scheme with the optimal LOPA scheme in the perfect power control environment. In the perfect power control environment, σ = τ = 0, thus g i = g i, C = 0 and B max =. This makes the power allocated for class i user to be u i = N 0W, which is exactly 3). Thus the LOPA scheme can be included as a special case of our quasi-optimal power allocation scheme in the environment of perfect power control. gi 3 g Σ V. NUMERICAL EXAMPLES AND DISCUSSIONS We consider a multi-cell CDMA system with only two classes of users, which have QoS representation of 3Kb/s, 4, 0 ) and 64Kb/s, 6, 0 ), respectively. The spread spectrum bandwidth is W =5MHz and the single-sided power density of AWGN is N 0 = 0 9 watt/hz. The inter-cell to intra-cell interference ratio is assumed to be f =0.55. Our Monte Carlo simulation is directly based on the lognormal model of received powers under imperfect power control. The values of σ and τ under different user mobility and Rake receiver structures are from [7. For the purpose of integrity, we extract those values we employed in our simulation from [7 and present them in Tab. I, where L is the number of the fingers of RAKE receiver and V is the speed of user. Admissible region comparison between the LOPA scheme and the quasi-optimal scheme are plotted in Fig. and Fig.3 for user mobility of 5km/h and 40km/h, respectively. Computer simulation results are also shown in the same figures for comparison. It shows that much larger admissible region is achieved by using the quasi-optimal power allocation scheme. The superiority in capacity comes from that the new scheme provides class-dependent SIR margins subject to differentiated outage constraints. In contrast, LOPA scheme only provides a

7 SHU AND NIU: CAPACITY OPTIMIZATION FOR MULTIMEDIA CDMA NETWORKS WITH IMPERFECT POWER CONTROL Class user number LOPA scheme theory) LOPA scheme simulation) quasi-optimal scheme theory) quasi-optimal scheme simulation) user mobility: 5km/h Rayleigh paths Capacity gain Rayleigh paths 3 Rayleigh paths 4 Rayleigh paths Class user number User mobility km/h) Fig.. fading Capacity under LOPA scheme and quasi-optimal scheme in slow Fig. 4. Capacity gain vs. user mobility and receiver structure Class user number LOPA scheme theory) LOPA scheme simulation) quasi-optimal scheme theory) quasi-optimal scheme simulation) user mobility: 40km/h Rayleigh paths Outage probability E-3 E-4 E-5 P and P by LOPA scheme P by quasi-optimal scheme P by quasi-optimal scheme user mobility: 5km/h Rayleigh paths theoretical simulation Fig. 3. fading Class user number Capacity under LOPA scheme and quasi-optimal scheme in fast uniform margin for all users, thus capacity loss is resulted since the user with the most stringent outage constraint will determine the needed margin. In Fig.4 We show the capacity gain achieved by the quasioptimal scheme over the LOPA scheme. Here, the capacity gain is defined as the ratio of the extra area of the admissible region achieved by the quasi-optimal scheme to the area of the admissible region achieved by the LOPA scheme. In general, with the increase of user mobility larger capacity gain is achieved by our quasi-optimal scheme over the LOPA scheme. In addition, the simpler Rake receiver structure is e.g. the less resolvable Rayleigh fading paths) the larger the capacity gain is. These advantages are due to the fact that under higher user mobility and simpler receiver structure, capacity is degraded more seriously by a not-so-good power allocation. In this case, a more reasonable power allocation will be more effective in the relief of capacity degradation. For example, in the case of resolvable Rayleigh fading paths and user mobility of 40km/h, the capacity gain reaches 9%. The outage probabilities under both power allocation schemes in slow and fast fading are plotted in Figs.5 and 6, respectively. We fix the number of class users to be and increase the number of class users gradually to explore Fig. 5. E Class user number Outage probabilities vs. class user number in slow fading the extra admissible region. The differentiation effect of the quasi-optimal scheme is clearly demonstrated in these figures: by the quasi-optimal scheme class users always experience larger outage probability than class users do. On the contrary, by the LOPA scheme, both classes of users experience the same outage probability in spite of the difference in their QoS constraints. This differentiation effect of the quasi-optimal scheme leads to a more efficient utilization of power and lead to the capacity gain demonstrated in the Figs.-4. From Figs.5 and 6 we can also examine the effect of the approximation introduced in our previous theoretical analysis. According to Proposition, the outage probabilities resulted by the accurate solution to 7) should be exactly 0 for class users and 0 for class users inside the admissible region. However, because of the approximation introduced in our previous analysis, both the outage probabilities of class and class users under the quasi-optimal scheme are monoincreasing function of class user number. However, if we extend the network user configuration to be N, N ) R+, it is seen that the outage probabilities converges to 0 for class user and 0 for class user nearly at the same user configurations approximately 8.8 in Fig.5 and 4.3 in Fig.6), which are the respective boundaries of the admissible region in Figs.5 and 6. The fact that the outage probabilities of

8 8 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL.**, NO.*, MONTH YEAR Outage probability Fig. 6. Average received power watt) Fig E-3 P by quasi-optimal scheme P and P by LOPA scheme P by quasi-optimal scheme user mobility: 40km/h Rayleigh paths theoretical simulation E E-3 Class user number Outage probabilities vs. type user number in fast fading total received average MAI class user class user user mobility: 5km/h user mobility: 40km/h E Class user number Power allocation and total MAI power at BS vs. user number both classes of users converge to their respective constraints nearly at the same time just indicates that the approximation introduced in our analysis only has a trivial influence on the final capacity derived, i.e., the capacity provided by the approximate solution is nearly the same as the capacity provided by the accurate solution. The major difference between the approximate solution and the accurate one lies in the power allocated inside the boundary of admissible region, where more power is allocated by approximate solution. The power allocations for both types of users and the total received MAI powers at the BS in slow and fast fading are plotted in Fig.7, where class user number is fixed to. It shows that with the increase of class user number, larger average received powers are needed for both types of users to meet the outage probabilities requirements. For the same user configuration, users experiencing faster fading will need larger average powers to meet the outage probability requirements. power control environment. Based on the lognormal model of received powers, we have derived the necessary conditions that an optimal power allocation scheme should satisfy in order to minimize the total received interference at the BS or equivalently provide the maximum admissible region under imperfect power control. By using conservative bounds, we have suggested a closed-form approximate solution to the optimization problem. This approximate solution leads to a closedform quasi-optimal power allocation scheme for multimedia users under imperfect power control, by which class-dependent SIR margins are provided subject to the differentiated outage requirements. This new scheme provides nearly the same capacity as provided by the accurate solution but more average received powers are needed. By numerical examples, we have verified the correctness of our analysis and showed that great capacity gains are achieved by this new scheme over the traditional LOPA scheme. Our results could be of guidance meaning in the operation of real-time power allocation in practical CDMA systems. APPENDIX I PROOF OF PROPOSITION Proof: The class i user s outage probability in ) is a mono-decreasing function of u i and a mono-increasing function of u j j =,..., K, j i), i.e., P i u i < 0, P i u j > 0 for i, j =,..., K and j i. 33) Thus at least one constraint in 6) is met with equality at the optimal solution. For a given user configuration N,..., N K ), let the optimal solution be u o,..., u o K ). We use counterevidence method to prove that all equalities hold at the optimal solution. Assume that at least one inequality in 6) strictly holds at the optimal solution. Without loss of generality, let it be the kth constraint, i.e. P k < δ k. By the continuous nature of the function of P k, there must be a u o k > 0 by which uo k is reduced to uo k i.e., uk o = uk o uo k ) and the resulted outage probability of class k user increases but the inequality P k < δ k still holds strictly. The outage probability of the other classes i =,..., K, i k) of users decrease with the decrease of class k user s mean received power from u o k to uo k. Thus the adjustment of uk o to u o k makes all constraints still be satisfied. By this, we find a feasible solution u, o..., u k o,..., u o K ) to 6), which makes all constraints be met but owns smaller sum than the original optimal one u o,..., u o K ). This conflicts with the given condition that u o,..., u o K ) owns the minimum sum for the network user configuration N,..., N K ). Thus our assumption of P k < δ k is invalidated and all constraints in 6) must be met with equality at the optimal solution. VI. CONCLUSIONS We have considered the problem of closed-form capacityoptimal power allocation for the uplink of multimedia CDMA networks with imperfect power control. By theoretical analysis, we have pointed out the capacity-inefficiency of the traditional LOPA scheme when it is applied in the imperfect APPENDIX II PROOF OF PROPOSITION Proof: Extend user configuration to positive real space, i.e., N,..., N K ) R K +. Accordingly, we mathematically define that N i where N i R + ) users of class i in cell 0 will present lognormal interference with mean N i u i and variance N i γ i at

9 SHU AND NIU: CAPACITY OPTIMIZATION FOR MULTIMEDIA CDMA NETWORKS WITH IMPERFECT POWER CONTROL 9 BS0. For a given u,..., u K ), the user s outage probability is a mono-increasing function of the user numbers, i.e., P i N j > 0 for i, j =,..., K. 34) Let N,..., N K ) be a user configuration at the boundary of the admissible region provided by 7). Let the corresponding average received power vector be u,..., u K ). The outage probability of class i users is given by ) m i A i p P i = Q, 35) σ + η where p = β ln [ + f) β ln + η = β ln + m i = βσ + β ln u i, 36) N i u i + 3 N 0W K N iγ i + K fn iɛ i + f) K N iu i + 3 N 0W K N iγ i + K fn iɛ i + f) K N iu i + 3 N 0W ) ), 37). 38) Since N,..., N K ) is at the boundary of the admissible region of 7), according to Proposition, there must be P i = δ i i =,..., K). 39) We use the counterevidence method to prove that u i must tend to be infinite for all i i =,..., K). Assume u i is finite, then we can scale up u i by a scalar of M > for all i, i.e. u i = Mu i. The resulted outage probability of the class i user will be ) P i m i = Q A i p, 40) σ + η where p = β ln [ + f) β ln + η = β ln + m i = βσ + β ln u i + ln M, 4) β N i u i + 3 N 0 W M K N iγ i + K fn iɛ i + f) K N iu i + 3 N 0 W M ) K N iγ i + K fn iɛ i + f) K N iu i + 3 N 0 W M 4) + ln M, β ). 43) Compare 40) with 35), it is clear that the adjustment of scaling u i s up by M times is just equivalent to scaling the power of the AWGN down by M times while maintaining u i s constant. Because u i s are of limited values by our assumption, the outage probability of all classes of users is reduced by this adjustment, i.e., By 39) and 44), we have P i < P i, i =,..., K). 44) P i < δ i, i =,..., K). 45) Considering the continuous property of the outage probability as a function of N i, there must be a N > 0, by which the increase on class users N + N, N,..., N K ) will still make 45) hold. Then we find a feasible user configuration N + N, N,..., N K ), which is beyond N,..., N K ), and whose corresponding power vector is Mu,..., Mu K ). That N + N, N,..., N K ) is a feasible user configuration conflicts with the given condition that N,..., N K ) is the boundary of the largest admissible region. Thus the assumption that u i s are of limited values is invalidated and then Proposition follows. APPENDIX III PROOF OF PROPOSITION 3 Proof: Let the network user configuration be N,..., N K ) Z+ K and its received signal power vector be u,..., u K ). Let the ratio among u i be u :... : u K = g :... : g K, i.e., u i = g i φ, where φ is a constant. From ) and 4)-7), the m i, p and η in ) can be written as p = β ln [ + f) β ln + η = β ln + m i = βσ + β ln g i + ln φ, 46) β N i g i + 3 N 0 W φ C K N ig i + f) K N ig i + 3 N 0W φ C K N ig i + f) K N ig i + 3 ) N 0 W φ 47) + ln φ, β ). 48) When u i s own infinite values, φ is infinite. Then the items of 3 N 0 W φ in above equations equal to 0. The outage probability of P i in ) is fully determined by g i and is independent of φ. Then Proposition 3 follows. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments and constructive suggestions which have definitely improved the quality of this paper.

10 0 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL.**, NO.*, MONTH YEAR REFERENCES [ K. S. Gilhousen, I. M. Jacobs, et al, On the capacity of a cellular CDMA system, IEEE Trans. Veh. Technol., vol.40, pp.303-3, May 99. [ S. Yao and E. Geraniotis, Optimal power control law for multimedia multirate CDMA systems, in Proc. Vehicular Technology Conf. VTC 96), pp [3 L. C. Yun and D. G. Messerschmitt, Variable quality of service in CDMA systems by statistical power control, in Proc. Int. Conf. Communications ICC 95), June 995, pp [4 A. Sampath, P. S. Kumar and J. M. Holtzman, Power control and resource management for a multimedia CDMA wireless system, in Proc. IEEE Int. Symp. Personal, Indoor and Mobile Radio Communications PIMRC 95), vol., pp.-5, 995. [5 A. Sampath, N. B. Mandayam and J. M. Holtzman, Erlang capacity of a power controlled integrated voice and data CDMA system, in Proc. Vehicular Technology Conf. VTC 97), pp [6 D. K. Kim and F. Adachi, Theoretical analysis of reverse link capacity for an SIR-based power controlled cellular CDMA system in a multipath fading environment, IEEE Trans. Veh. Technol., vol.50, no., pp , Mar. 00. [7 P. R. Larijani, J. W. Chinneck and R. H. Hafez, Nonlinear power assignment in multimedia CDMA wireless networks, IEEE Commun. Lett., vol., no.9, Sept. 998, pp [8 S. J. Lee and D. K. Sung, Capacity evaluation for DS-CDMA systems with multi-class on/off traffic, IEEE Commun. Lett., vol., no.6, Jun. 998, pp [9 M. K. Park and S. K. Oh, Comments on Capacity evaluation for DS- CDMA systems with multi-class on/off traffic, IEEE Commun. Lett., vol.4, no., Dec. 000, pp [0 B. Hashem and E. S. Sousa, Reverse link capacity and interference statistics of a fixed-step power-controlled DS/CDMA system under slow multipath fading, IEEE Trans. Commun., Vol.47, no., pp.905-9, Dec [ E. Kudoh, On the capacity of DS/CDMA cellular mobile radios under imperfect transmitter power control, IEEE Trans. Commun., vol.4, pp , Aug [ R. Prasad, M. Jansen and A. Kegel, Capacity analysis of a cellular direct sequence code division multiple access system with imperfect power control, IEICE Trans. Commun., vol. E76-B, no.8, pp , Aug [3 D. Kanade and V. K. Bhargava, Power assignment strategy for the reverse link of multimedia DS-CDMA system, in Proc. IEEE ICPWC 99, vol., Feb.999, pp [4 M. K. Park and S. K. Oh, Power allocation and capacity analysis on multi-class DS-CDMA systems with power control error, in Proc. Int. Conf. Communications ICC 00), 00, pp [5 S. Kandukuri and S. Boyd, Optimal power control in interferencelimited fading wireless channels with outage-probability specifications, IEEE Trans. on Wireless Commun., vol., no., Jan. 00, pp [6 T. Shu and Z. Niu, Call admission control using differentiated outage probabilities in multimedia DS-CDMA networks with imperfect power control, IEICE Trans. Commun., vol.e86-b, No., Jan. 003, pp.6-4. [7 B. Hashem and E. Sousa, Increasing the DS/CDMA system reverse link capacity by equalizing the performance of different velocity users, in Proc. Int. Conf. Communications ICC 98), pp , 998. [8 A. M. Viterbi and A. J. Viterbi, Erlang capacity of a power controlled cellular CDMA system, IEEE J. Select. Areas Commun., vol., no.6, pp , Aug [9 J. M. R. Jerez, M. R. Garcia and A. D. Estrella, Effects of multipath fading on BER statistics in cellular CDMA networks with fast power control, IEEE Commun. Lett., vol.4, no., pp , Nov [0 B. Hashem and E. Sousa, Performance evaluation of DS/CDMA systems employing adaptive transmission rate under imperfect power control, in Proc. IEEE Int. Symp. Personal, Indoor and Mobile Radio Communications PIMRC 98), pp , 998. [ J. M. R. Jerez, M. R. Garcia and A. D. Estrella, Effects of power control errors and multipath fading on BER in a cellular CDMA system, in Proc. Vehicular Technology Conf. VTC 000), pp , 000. [ N. C. Beaulieu, A. A. A. Dayya and P. J. Mclane, Estimating the distribution of a sum of independent lognormal random variables, IEEE Trans. Commun., Vol.43, no., pp , Dec [3 A. M. Viterbi, A. J. Viterbi and E. Zehavi, Other cell interference in cellular power-controlled CDMA, IEEE Trans. Commun., vol.4, no./3/4, pp , Feb./Mar./Apr PLACE PHOTO HERE PLACE PHOTO HERE Tao Shu received his B.S. and M.S. degrees in electronic engineering from South China University of Technology, Guangzhou, P.R. China in 996 and 999 respectively. He is currently pursuing his Ph.D. degree at the Department of Electronic Engineering, Tsinghua University, Beijing, P.R. China. His research interests include resource allocation in wireless networks, wireless personal communications, and queueing theory. He is a student member of IEEE. Zhisheng Niu graduated from Northern Jiaotong University, Beijing, China, in 985, and got his M.E. and D.E. degrees from Toyohashi University of Technology, Toyohashi, Japan, in 989 and 99, respectively. In 994, he joined with Tsinghua University, Beijing, China, where he is now a full professor at the Department of Electronic Engineering. He is also an adjunction professor of Northern Jiaotong University and the Institute of Automation of China Academy of Science. From 99 to 994, he was with Fujitsu Laboratories Ltd., Kawasaki, Japan. From October 995 to February 996, he was a visiting research fellow of the Communications Research Laboratory of the Ministry of Posts and Telecommunications of Japan. From February 997 to February 998, he was a visiting senior researcher of Central Research Laboratory, Hitachi Ltd. From January 00 to February 00he was a visiting professor of Venture Business Laboratory at Saga University, Japan. He also visited Polytechnic University, NY, in Feb. 00. He received the PAACS Friendship Award from the Institute of Electronics, Information, and Communication Engineers IEICE) of Japan in 99 and the Best Paper Award st prize) from the 6th Chinese Youth Conference on Communication Technology in 999. His current research interests include teletraffic theory, performance evaluation of high-speed broadband integrated networks, and wireless communication networks. Dr. Niu is a senior member of the IEEE and the Chinese Institute of Electronics CIE) and a member the IEICE. He is now serving as the Chair of Membership Development Committee of Asia-Pacific Board of IEEE Communication Society, the Chair of APCC Steering Committee ASC), and the Chair of IEEE Communication Society Beijing Chapter.