Joint Throughput Maximization and Fair Uplink Transmission Scheduling in CDMA Systems
© S. Papavassiliou and C. Li. 2009
Received: 9 July 2008
Accepted: 20 February 2009
Published: 24 March 2009
We study the fundamental problem of optimal transmission scheduling in a code-division multiple-access wireless system in order to maximize the uplink system throughput, while satisfying the users quality-of-service (QoS) requirements and maintaining fairness among them. The corresponding problem is expressed as a weighted throughput maximization problem, under certain power and QoS constraints, where the weights are the control parameters reflecting the fairness constraints. With the introduction of the power index capacity, it is shown that this optimization problem can be converted into a binary knapsack problem, where all the corresponding constraints are replaced by the power index capacities at some certain system power index. A two-step approach is followed to obtain the optimal solution. First, a simple method is proposed to find the optimal set of users to receive service for a given fixed target system load, and then the optimal solution is obtained as a global search within a certain range. Furthermore, a stochastic approximation method is presented to effectively identify the required control parameters. The performance evaluation reveals the advantages of our proposed policy over other existing ones and confirms that it achieves very high throughput while maintains fairness among the users, under different channel conditions and requirements.
The continuous growth in traffic volume and the emergence of new services have begun to change the structure and requirements of wireless networks. Future mobile communication systems will be characterized by high throughput, integration of services, and flexibility [1–5]. With the demand for high data rate and support of multiple quality of service (QoS), the transmission scheduling plays a key role in the efficient resource allocation process in wireless systems. The transmission scheduling determines the time instances that a mobile user may receive service, as well as the resources that should be allocated to support the requested service, in order to make the resource distribution fair and efficient.
The fundamental problem of scheduling the users transmission and allocating the available resources in a realistic uplink code-division multiple-access (CDMA) wireless system that supports multirate multimedia services, with efficiency and fairness, is investigated and analyzed in this paper. A transmission scheduling method which achieves the maximum system throughput under the constraints of satisfying certain users QoS requirements and maintaining throughput fairness among them is provided and evaluated.
1.1. Related Work and Motivation
A class of scheduling schemes, namely, the opportunistic scheduling schemes, has been proven to be an effective approach to improve the system throughput by utilizing the multiuser diversity effect [6, 7] in wireless communications. Specifically, for a system with many users that have independent varying channels, with high probability there is a user with channel much stronger than its average SNR requirement. Therefore, the system throughput may be maximized by choosing the user with "relatively best" channel for transmission at a given slot. However, some fairness constraints must be imposed on the scheduling policies to ensure the fair resource allocation.
It has been shown in  that scheduling users one-by-one can result in higher system throughput for high data rate traffic in the CDMA downlink. However, this work does not exploit the time-varying channel conditions. In [7, 9], a high-speed data rate scheme (HDR) is introduced, where the base station schedules the downlink transmission of a single user at a given time slot with the data rates and slot lengths varying according to the specific channel condition. In [10–12], a transmission scheduling scheme for multiple users, which considers both the channel condition and queueing delay/length, is proposed and shown to be throughput optimal if it is feasible. However, the fairness issue is not explicitly addressed in that work. In [13–15], a framework for opportunistic scheduling that maximizes the system performance by exploiting the time-varying channel conditions of wireless networks is presented. Three categories of scheduling problems—the temporal fairness, utilitarian fairness, and minimum-performance guarantee scheduling—are studied and optimal solutions are given.
Although the downlink transmission assignment is important for several applications, the efficient uplink transmission scheduling plays an important role as well, especially with the prevailing of multimedia communications and applications. It has been argued that the downlink scheduling method is not suitable to be applied to the uplink transmission scheduling, where simultaneous transmissions may result in higher throughput [16, 17]. The uplink transmission scheduling problem is more complicated and requires further consideration of additional elements to make the corresponding scheduling policies feasible . The achievable throughput in such a case depends not only on the service access time, but also on the transmission powers and the corresponding users interference. In addition, multiple users can be scheduled simultaneously to transmit in the same time slot, which is a major difference from the wireline and TDMA-like scheduling schemes, making the respective scheduling processes either inapplicable or inefficient in the CDMA environment. The simple temporal fairness scheduling, where the main resource to be shared is the time, fails to provide rational fairness in this case. As a result, the throughput optimal and fair uplink transmission scheduling problem needs to jointly consider multiple factors such as access time, transmission power, channel conditions, and number of users to be scheduled at the same time. Heuristic approaches to address the problem of short-term fairness and demonstrate the tradeoff between fairness and throughput under some special cases have been introduced in [19–21].
Furthermore, how to maximize the throughput of uplink CDMA system was first analyzed in . The sole purpose of uplink throughput maximization can be achieved by choosing the "best" users in terms of their received power, when they transmit at their maximum power. However, such throughput maximization does not consider fairness, that is, the equal opportunity for all users to receiving service despite their channel conditions. Therefore, among the objectives of our approach in this paper is to identify the actual "best" users that should transmit in order to maximize the throughput, when the fairness constraints are introduced and respected.
In , several scenarios of scheduling uplink CDMA transmission with voice and data services are analyzed. With the number of voice users and their corresponding transmission rates fixed, that work attempted to maximize the throughput of data service. It was shown that when the synchronization overhead is reasonable, a smaller number of simultaneous transmission users achieve higher system throughput and at the same time lower the average transmission power. However, in this case the achievable throughput is affected by the "weakest link." Therefore, this approach can be regarded only as a static analysis that considers the relationship between the performance and the number of users chosen for transmission. The problem of identifying the actual set of users to transmit based on their channel conditions, which may reduce the impact of the "weakest link", has not yet been investigated and is one of the main objectives of our paper.
In addition, the problem of uplink CDMA scheduling is further complicated by the fact that the conventional concept of capacity used in the wireline networks, for example, total bandwidth of the physical media, is not directly applicable in the CDMA systems. In this case, the actual system capacity is not fixed and known in advance, since it is a function of several parameters such as the number of users, the channel conditions, and the transmission powers.
Therefore, in summary the main contributions of this paper are as follows. (1) Jointly consider the factors of channel capacity, number of users and their interference, transmit power, and fairness requirements. (2) Formulate an optimization problem that stresses the fairness requirement under time-varying wireless environment and proves the existence of an optimal solution based on all constraints. (3) Exploit the power index concept and power index capacity, as a novel and effective way, to treat the fairness issue in the transmission scheduling policy under the considered uncertain and dynamic environment. (4) Devise a scheduling policy that achieves throughput fairness among the users and optimal system throughput under certain constraints.
1.2. Paper Outline
The rest of the paper is organized as follows. In Section 2, the system model that is used throughout our analysis is described, and the problem of the uplink scheduling in CDMA systems is rigorously formulated as a multiconstraint optimization problem. It is demonstrated that this problem can be expressed as a weighted throughput maximization problem, under certain power and QoS constraints, where the weights are the control parameters that reflect fairness constraints. Based on the concept of power index capacity, this optimization problem is converted into a simpler linear knapsack problem in Section 3.1, where all the corresponding constraints are replaced by the users power index capacities at some certain system power index. The optimal solution of the latter problem is identified in Sections 3.2 and 3.3, while in Section 3.4, a stochastic approximation method is presented in order to effectively identify the required control parameters. Section 4 contains the performance evaluation of the proposed method, along with some numerical results and discussion, and finally Section 5 concludes the paper.
2. System Model and Problem Formulation
where is the one-sided power spectral density of additive white Gaussian noise (AWGN), and determines the proportion of the interference from other users received power. Without loss of generality in the following, we assume . Obviously, to meet the SINR requirement, the received SINR has to be larger than the corresponding threshold , that is, . In the following, we assume perfect power control in the system under consideration, while users are scheduled to transmit at the beginning of every fixed-length slot. The objective of the optimal scheduling policy is to find the optimal number of allowable users and their transmission rates, which achieves the maximum system throughput while maintaining the fairness property.
2.1. Problem Formulation
The fairness constraint, that is, , is represented by the choice of . By adjusting the value of , the user will get more or less opportunities to transmit data, and hence the corresponding normalized throughput is balanced. As we discuss later in this paper, the value of can be approximated by a stochastic approximation algorithm, which has already found its application in [14, 15] under similar situations. Note that since we assume perfect power control in the CDMA system under consideration, only the equality case of (7) is considered here.
The following Proposition 1 states that the optimal solution is achieved when a user either transmits at full power or does not transmit at all.
Since is positive number, obviously (12) is nonnegative, while the objective function is a convex function of . Hence, the optimal solution of this problem is that the transmit power obtains the value of its boundary, that is, either 0 or .
In Section 3, the corresponding optimization problem is transformed to an equivalent problem of a simpler form, which facilitates the identification of the optimal solution. However, in the following we first introduce the concept of power index capacity which is used to represent the corresponding constraints, under the problem transformation.
2.2. Power Index Capacity
Let us regard as the actual system load, which is the sum of power indices assigned to all backlogged users, while we assume that there is a target system load . It should be noted that here is not fixed but has value . The meaning and motivation for the definition of the target system load are that the system will attempt to provide the appropriate scheduling in order to make the actual system load reach the target load (however, it serves as an upper bound and cannot be exceeded). For an arbitrarily selected in the range of , there exist two possible cases concerning the relationship between the actual system load and the target system load. When considering small values for the target system load , the system can easily make the actual system load reach the target load under consideration, that is, . On the other hand, when is large, especially when it approaches to 1, it may be impossible for the actual achievable system load to reach due to the limitation imposed by (13)). Let us assume that in time slot the maximum system load this system can achieve based on all users channel states and all possible user schedulings is . We will now consider the two cases mentioned above, that is, and .
2.2.1. Target Load Is Less than or Equal to Maximum System Load
2.2.2. Target Load Is Larger than Maximum System Load
If the target load is larger than the maximum system load, that is, , it means there will be no feasible transmission power solution in (7) and (8) to achieve this target load and therefore the relationship in (15) does not hold any more. In this case, we simply apply the power index restriction of (16) to each user. The consequence is that the final achieved system load becomes since .
In fact, unless all possible transmission user sets are searched, it is unknown in advance whether or not the actual system load can reach the chosen . Therefore, applying (16) to the case unifies the definition of the power index range, within which a user can be assigned a feasible power index without knowing the value of . One key principle and rule regarding the algorithm proposed in this paper is to assign to an individual user a power index that is less than or equal to its power index capacity. In the power index assignment algorithm described in Section 3.2, the situation where may occur. However, it should be noted here that as proven by Theorem 1 later in the paper, the global optimal solution must be the one satisfying . The target load range where is then not possible to be the optimal solution. The intentionally introduced restriction of (16) in the case of allows the algorithm to rule out such values of due to the fact that in this case.
2.2.3. Definition of Power Index Capacity
In a CDMA system with backlogged users at time slot , given the target system power index , the maximum power index that does not violate (13) for a single user whose channel gain is is defined as the power index capacity (PIC) of this user.
Note that in (17) the power index capacity is limited by the target system power index. This is reasonable since a power index capacity that is greater than will have no practical meaning and application. Furthermore, since our focus in this paper is to find an optimal scheduling policy as well as the optimal system load , the value of in (17) is not determined in advance.
Intuitively, the power index represents the relationship between the transmission power and the corresponding interference that is caused to other users. If we considered that the total system power index is fixed to , larger power index for user indicates that it has relatively higher signal-to-interference ratio compared to the other users with smaller power index, while at the same time it causes more interference to them. Accordingly, users with high-power indices may lower their transmission power to reduce the interference they may cause, which in turn means that they will have smaller power index to limit the intracell interference of the system, and therefore satisfy (13) that guarantees the existence of a feasible transmission power solution.
3. Problem Transformation and Optimal Solution
3.1. Problem Transformation
In the following, let denote the set that contains all the power and rate vectors that satisfy constraints (7) and (8) and . The elements and represent the transmit power and rate of the th user in the th vector. Similarly, we define another set containing the power and rate vectors that satisfy constraints (19), (20), and (21). By definition, it is obvious that any power and rate vector is feasible. However, since in constraint (21), may take a value, that is, close to 1 , the required transmit power could also accordingly become larger than maximum allowable transmit power if we simply look at the result from (15). The following proposition states that if perfect power control is assumed, for any rate (or power index) vector that satisfies constraints (19), (20), and (21), there always exists a feasible transmit power vector.
which still satisfies the above inequality and proves this proposition. The power control of the CDMA system will choose the minimal transmit power, that meets the required SINR.
Suppose that , and therefore it satisfies constraints (7), (8). It is apparent that . Since, as shown earlier, constraints (7), and (8) can also be represented by (13) , also satisfies (13). Using function (22), we can convert the rate vector into the corresponding power index vector . Let . For a feasible power and rate vector, with known ( ), we can find each user power index capacity . Since satisfies (13), based on Proposition 2 and the definition of power index capacity, we conclude that . That means that the assigned powers and rates in also satisfy the constraints (19), (20), and (21). Therefore, .
The above proposition shows that the optimal solution can also be obtained with the new constraints since they define the same solution set. Please note that, as mentioned before, the fairness constraints in the original problem are replaced by parameters . The choice of the proper values of that maintain fairness is discussed in detail later in this paper.
Among the new constraints, the right-hand sides of inequalities (19) and (20) are not fixed values, but are functions of the selected target system load . Hence, whether or not the final solution is feasible also depends on the choice of . For any value of , there could be many feasible solutions among which one will be the optimal. Moreover, there must exist an optimal system load that can achieve the overall best solution. It is natural to regard the objective as the function of system load , , and thus is the local optimal result at some specific . The maximum is achieved when . The ultimate objective of the proposed method is to find this optimal and the optimal power index assignment vector under it.
In Sections 3.2 and 3.3, we propose a two-step approach to solve the optimization problem (17)–(20). More specifically, in the first step (Section 3.2), we assume a fixed and then given that fixed parameter we propose a simple method (greedy algorithm) trying to find the optimal set of users to receive service. However, this optimality is not a global optimality. In general, as mentioned before, could get any value within the interval . The global optimal solution can be obtained when parameter is chosen to be the optimal one . The actual objective of the second step of our approach (Section 3.3) is to find this optimal , by which the global optimal set of users that will be scheduled to receive service can be identified.
3.2. Greedy Algorithm for a Given System Load
Note that (26) is a nonlinear continuous knapsack problem with the taking continuous values between 0 and 1. In general, solving this type of problem is proven to be difficult or even impossible in some cases . However, Proposition 1 limits the transmit power of a user , to either or for the optimal solution. This condition provides a possible method to solve the above nonlinear knapsack problem. Without loss of generality, we suppose that the optimal solution is when the first users transmit at their maximum power, , . The optimal system load is . The following theorem states that the power index of an individual user is equal to its power index capacity under , that is, .
Let the optimal solution allow users to transmit at their maximum power and the system achieves the system load . The power index that an individual user received in this case is equal to its power index capacity, that is, .
With reference to the optimal solution of problem (26), we can prove the following theorem.
It has been shown in  that in worst case . Let represent the result that corresponds to the integer solution of (32) when is assigned a value from , and be the result when . From the definition of , we know that is the maximum value among all , that is, . Based on Proposition 1 and the analysis in the previous subsection, it is easy to find that . Therefore, when the optimal system power index is chosen, . Since and the equality holds only when , and the optimal solution can be obtained.
3.3. Optimal System Load
As we discussed in the last subsection the optimal solution of problem (26) depends on the selected system load . Relation (17) shows that the power index capacity increases as decreases. At the first point when , the power index capacity reaches its largest value and then it decreases linearly following the value of . Although a smaller value of may increase the single user power index capacity at some range, the finally achieved objective function could be low due to the small system load . On the other hand, setting large reduces the individual user power index capacity as (17) indicates. The consequence of smaller power index capacity is that more users are required to share , and probably a small objective function should be used due to the concavity of function that converts the power index to throughput. Therefore, whether or not the objective function reaches its maximum value depends not only on the value of the system load , but also on how it is shared among the candidate users. There must exist an optimal value of system load that can achieve the maximum weighted rate.
Problem (37) is a simple unconstrained maximization problem that searches for the maximum within the interval . The disadvantage of (37) is that it does not have an explicit expression. Hence, algorithms that rely on the first- or second-order derivatives will not be applicable in this case. Therefore, the searching process depends on the result of (34). Note that every time when a new value of is chosen, the order of may be different from that of previous .
The time of calculating the best result for a newly chosen , including the time of reordering the users (if needed), is easily obtained as if is assumed to be large enough. Moreover, there are many possible local maximum points within the range . The final optimal must be a global best value. Although in  many searching algorithms on how to locate the minimum/maximum solution within a range are described, to make these algorithms effective there must be only one extreme point in the specified range. However, in general it is not possible to know the range which contains only the global optimal value. Thus, an exhaustive search within would be needed. However, the following proposition provides a lower bound with respect to the searching range instead of 0 in order to restrict the corresponding feasible searching range.
With the decrease of the target system load , the individual power index, provided by (14)), will keep increasing till reaches the point for user , that is . With respect to user , if its power index . is given by , which varies with different users since their are not likely the same. Let be the minimum among all 's. Once all backlogged users will have the same power index capacities . Define a small increment and let . Apparently, for all users their power indices will all have small increment such that . Maintaining the previous power index assignment and giving to any backlogged user will help increase the objective function (18). We hence can keep adding to till it reaches , which proves this proposition.
Since the optimal can reside between and , we need to calculate a series of sample values after every interval . Apparently, the smaller the the more samples we get and thereby the more accurate is the obtained result. On the other hand, it also increases the required computational time and power.
Therefore, in practice we only use reasonably small in order to reduce the corresponding computational power and complexity, while still obtain a good approximation of the optimal solution. It should be noted though that in theory when becomes infinitely small the above methodology can be used to find the optimal solution. Specifically, there exists an algorithm with complexity of that guarantees the finding of the optimal solution, however its high complexity limits its applicability for real-time computations and can be used only for benchmarking purposes. Let us assume that the order in (34) is known and fixed. Under this condition, there are only possible results satisfying the optimal condition in Proposition 1, that is, try the maximum transmission power in the fixed order with number of users from 1 to . The maximum result is the optimal one. For any two users in the possible system load range from , their order of will change at most three times. Therefore, there are totally order changes for users. Every order change requires first the sorting operation and then the comparison operation that have complexity of and respectively, which makes the overall complexity of this method .
Compare the result of round ( ) to that of round . If the result in round ( ) is less than round , then stop the calculation. In that case, the result of round is the best result in this order between and .
The largest result obtained in step (4) is the global optimal solution.
Once the order is fixed in the range at step (2), the method provided in Section 3.2 that finds the best local solution can be applied here, which will provide the largest , , users with this fixed order. The only difference is that the target system load is not provided directly by a specific known value , but lies within a specific range. Based on Proposition 1, according to which the users allowed to transmit will use their maximum transmission power, we perform rounds of calculation in step (3) and compare the results to find the optimal users.
3.4. Fairness Conditions
where , . We can simply let . In most situations, the value of may not be directly available, but instead the , where is the observation noise. In this case, the above approximation approach still applies, with the observed value replaced by . The convergence of to the root requires .
where is the smooth factor which determines how the estimated follows the change of actual achieved system throughput. In the remaining of the paper, throughout the performance evaluation of our approach, the value is chosen. The numerical results presented in Sections 4.2.2 and 4.2.3, with respect to the convergence of 's and the achievable fairness, demonstrate that such a method is very effective in approximating the optimal values of and therefore controlling and maintaining fairness.
4. Performance Evaluation
In this section, we evaluate the performance of the proposed method in terms of the achievable fairness and throughput, via modeling and simulation. Furthermore, to better understand the performance of the proposed scheduling algorithm-in the following we refer to as throughput maximization and fair scheduling (MAX-FAIR)—we compare it with the maximum throughput (MAX) scheme , which achieves the maximum total uplink throughput by allowing only the best users in terms of their received power to transmit, and with the HDR algorithm [7, 9], which is a single user scheduling algorithm. The principles and operation of HDR basically refer to a proportional fair scheduling scheme, which can be used in the uplink scheduling to demonstrate the one-at-a-time proportional fair scheduling. Following the HDR principles the transmission of a single user at a given time slot is scheduled, with the data rates and slot lengths varying according to the specific channel condition. In the MAX scheme parameter, is determined by iteratively comparing the throughput of best users, , where is the total number of users. The throughput achieved by MAX scheme is regarded as the upper bound throughput in the uplink CDMA scheduling. On the other hand, since HDR achieves temporal fairness, we consider it here to mainly observe the difference between temporal fairness and throughput fairness and their corresponding advantages in specific cases.
4.1. Model and Assumptions
Throughout our numerical study, we consider a single cell DS-CDMA multirate system with multiple active users. All active users are continuously backlogged during the simulation and generate packets with average size of 320 bytes. The maximum transmission power is assumed the same for all users, that is, , while the system chip rate is as defined in IS-95 and the required SINR is dB for data service, the same for all users. The transmission time is divided into 1 millisecond equal length slots, which is the algorithm scheduling interval, while the simulation lasts for slots.
In most of the numerical results presented in the next subsection, unless otherwise is explicitly indicated, all users are assumed to have the same weight. Such a scenario allows us to better understand and compare the achievable performances of the various scheduling schemes, when users have different channel conditions. However, the operation and effectiveness of the proposed MAX-FAIR policy is also demonstrated in an environment, where users present different weights.
4.2. Numerical Results and Discussion
The numerical results presented in Sections 4.2.1 and 4.2.2 refer mainly to the impact of some of the parameters associated with the proposed MAX-FAIR algorithm on its operation and achievable performance and allow us to obtain a better understanding of its operational characteristics and properties. Then in Sections 4.2.3 and 4.2.4, comparative results about the achievable throughput and fairness of the MAX-FAIR, MAX and HDR algorithms are presented.
4.2.1. Finite System Power Index Samples
Specifically, in Figure 2, we present the probabilities of the number of simultaneously served users in each scheduling cycle. For this experiment, we consider 40 backlogged users in the system and perform 200 trials. In each trial, users are randomly assigned the SNRs in the designated SNR range, following the 8-state model  described in Section 4.1. We observe that when there are users having high SNR values, for example, in the cases of [−3,3] dB and [2,4] dB, only a small number of users (at most 2 in this experiment), are served concurrently. However, in the case that all users have small SNR values, for example, in the case of [−4,−2] dB, the number of simultaneously served users increases significantly (it is distributed between 4 and 17 in our case as can be seen by Figure 2). Such user distribution indicates that in the case that a single user cannot consume all the system resources (e.g., the case where users have low SNR values), more users will be scheduled simultaneously in order to achieve a more efficient resource utilization and as a result increase the total system throughput. This also demonstrates the advantage of our proposed scheduling algorithm over the one-by-one scheduling algorithms that have been proposed in literature. As a result, with respect to Figure 1, for the case of [0,1] dB, multiple users are scheduled to reach the maximal throughput. Increasing the number of active users enables the system to schedule more available candidates to achieve higher throughput, and therefore the achievable result is less sensitive to the number of samples. However, for the case [−3,3] dB at most only 1 or 2 users are scheduled for simultaneous transmission. In the following experiments and numerical results, we adopt the accuracy of 100 samples, which is sufficient to reach 95% of the optimal-weighted throughput.
4.2.2. Parameter Convergence by Stochastic Approximation
4.2.3. Throughput and Fairness Performance
4.2.4. Number of Users
In this paper, the CDMA uplink throughput maximization problem, while maintaining throughput fairness among the various users, was considered. It was shown that such a problem can be expressed as a weighted throughput maximization problem, under certain power and QoS requirements, where the weights are the control parameters that reflect the fairness constraints. A stochastic approximation method was presented in order to effectively identify the required control parameters. The numerical results presented in the paper, with respect to the convergence of the control parameters and the achievable fairness, demonstrated that this method is very effective in approximating the optimal values and therefore controlling and maintaining fairness. Furthermore, the concept of power index capacity was used to represent all the corresponding constraints by the users power index capacities at some certain system power index. Based on this, the optimization problem under consideration was converted into a binary knapsack problem, where the optimal solution can be obtained through a global search within a specific range.
The performance of the proposed policy in terms of the achievable fairness and throughput was obtained via modeling and simulation and was compared with the performances of other scheduling algorithms. The corresponding results revealed the advantages of the proposed policy over other existing scheduling schemes and demonstrated that it achieves very high throughput, while satisfies the QoS requirements and maintains fairness among the users, under different channel conditions and requirements.
This work has been partially supported by EC EFIPSANS Project (INFSO-ICT-215549).
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