- Research Article
- Open Access
QoS Differentiated and Fair Packet Scheduling in Broadband Wireless Access Networks
© Rong Yu et al. 2009
- Received: 12 February 2009
- Accepted: 7 September 2009
- Published: 27 October 2009
This paper studies the packet scheduling problem in Broadband Wireless Access (BWA) networks. The key difficulties of the BWA scheduling problem lie in the high variability of wireless channel capacity and the unknown model of packet arrival process. It is difficult for traditional heuristic scheduling algorithms to handle the situation and guarantee satisfying performance in BWA networks. In this paper, we introduce learning-based approach for a better solution. Specifically, we formulate the packet scheduling problem as an average cost Semi-Markov Decision Process (SMDP). Then, we solve the SMDP by using reinforcement learning. A feature-based linear approximation and the Temporal-Difference learning technique are employed to produce a near optimal solution of the corresponding SMDP problem. The proposed algorithm, called Reinforcement Learning Scheduling (RLS), has in-built capability of self-training. It is able to adaptively and timely regulate its scheduling policy according to the instantaneous network conditions. Simulation results indicate that RLS outperforms two classical scheduling algorithms and simultaneously considers: (i) effective QoS differentiation, (ii) high bandwidth utilization, and (iii) both short-term and long-term fairness.
- Schedule Problem
- Schedule Algorithm
- Mobile Host
- Bandwidth Utilization
- Packet Schedule
With the rapid growth of demands on wireless multimedia applications and wireless data services, it is expected that broadband wireless access (BWA) networks should be able to support high-speed, high-quality, realtime, and nonrealtime applications, such as video, image, voice, text, and data. Quality of Service (QoS) provisioning for heterogeneous traffic with diverse properties is a necessity in emerging BWA networks. Packet scheduling algorithms, which are in charge of the bandwidth allocation and multiplexing at the packet level, play key roles for resource sharing and QoS provisioning in BWA networks.
Different from traditional wired scheduling problems, packet scheduling in BWA networks will encounter two major difficulties: the high variability of wireless channel conditions and the unknown model of packet arrival process. The capacity of wireless channel usually suffers either moderate or drastic fluctuations due to the signal attenuation, inter-user interference, fading and user mobility, and so forth. As a result, the channel conditions of users may vary with time and location asynchronously at random, leading to the so-called time-dependent error and location-dependent error . In BWA networks, the process of packet arrivals is generally complicated and hard to accurately model in advance, since the packet arrivals are induced by the aggregation of multiple classes of traffic with different properties, such as constant and variable rates, and continuous and burst modes.
Besides the difficulties, packet scheduling algorithms for wireless multimedia networks are required to concurrently achieve three performance objectives: QoS differentiation and guarantee, wireless bandwidth utilization maximization, and fairness provisioning. Firstly, the transmission of heterogeneous classes of traffic with diverse QoS requirements naturally demands the scheduler to provide differential services. Secondly, the scheduler should be able to arrange the packet transmissions reasonably so that the precious wireless bandwidth can be utilized in an efficient way. Thirdly, the scheduler should also make sure that every traffic flow gets its fair share of the system resources. These three objectives, in most practical situations, are tightly coupled and complementary with each other. A good scheduling algorithm should provide satisfying performance tradeoff among them.
These special difficulties and performance objectives make the scheduling problem in BWA networks considerably challenging. The packet scheduler of a BWA network should operate across different sessions (i.e., connections or traffic flows) to guarantee that all sessions receive services with qualities according to the contract established when sessions are setup. At the beginning of each round of scheduling, the scheduler should decide one of the sessions for transmission. In order to make the right decision at each round of scheduling and hence accumulatively achieve the optimal long-term performance, the scheduler should be capable to predict the possible rewards (or costs) induced by different scheduling actions. However, the high variability of wireless channel conditions and the unknown characteristics of traffic flows make this long-term prediction difficult. Meanwhile, the multiple performance objectives introduce a set of interactional factors that should be considered together for prediction.
In this paper, we observe that the problem of packet scheduling in BWA networks could be described as a semi-Markov Decision Process (SMDP). There are two major advantages to solve the scheduling problem by the methodology of SMDP. Firstly, by modelling the scheduling problem as an SMDP, the entire scheduling process is divided into a series of stages. Decisions are made stage by stage. In other words, the original complicated problem is divided into multiple relatively simple subproblems, which are much more tractable than the original one. Secondly, there exist various of advanced mathematical techniques to solve the formulated SMDP problems, especially the learning-based approach. With the aid of learning-based approach, the scheduler has underlying ability to capture the main characteristics of the system and accurately predict the long-term rewards (or costs), even though there is no full knowledge of the system model.
The remainder of this paper is organized as follows. A brief introduction of existing packet scheduling algorithms is given in Section 2. The SMDP formulation is presented in Section 3. After introducing the system model, we cast the packet scheduling problem into the framework of SMDP. The construction of cost function is explained in details. Section 4 discusses the solution of the formulated SMDP problem. The methodology of reinforcement learning  is employed. A feature-based linear approximating architecture is adopted to approach the optimal average cost. The Temporal Difference (TD) technique  is used to continuously regulate the tunable parameter variables. Section 5 reports the results of simulation experiments, where the proposed algorithm is compared with two existing algorithms. Conclusion is presented in Section 6.
In literature, the problem of packet scheduling is extensively studied in both wired and wireless environments. We broadly categorize the algorithms into three classes and present a brief introduction of them. For comprehensive overview of existing scheduling algorithms, readers are referred to .
2.1. Algorithms in Wired Environment
Algorithms of First-Come-First-Served (FCFS) and Round Robin (RR) are first developed for wired networks. Their original versions and improved versions (e.g., Weighted Round Robin and Deficit Round Robin ) are underlying schemes for wireless networks. Another well-known algorithm is Generalized Processor Sharing (GPS) , which is ideal and unimplementable in packet-based networks. Many other scheduling algorithms try to approximate GPS as far as possible. The merit of these wired scheduling algorithms is their simplicity and ease of implementation. But if directly used in BWA networks, these algorithms cannot achieve high bandwidth utilization and promise fairness guarantee.
2.2. Algorithms in Wireless Environment with GE-Model
The classical two-state Gilbert-Eilliot (GE) model [6, 7] is firstly used to model the wireless link variation. The channel is simply described to be either in "good" or in "bad" state in this model. A survey of this class of scheduling algorithms can be found in . Work in  devised Idealized Weight Fair Queuing (IWFQ) algorithm. In , Channel-condition Independent Fair Queueing (CIF-Q) algorithm was put forward. Both algorithms of IWFQ and CIF-Q need to simulate a virtual error-free fair queueing system and try to schedule packets in the same order as the ideal reference system does. IWFQ and CIF-Q differ in the way they compensate the lagging flows. Wong et al.  presented Token Bank Fair Queueing (TBFQ) algorithm, which uses token pools and token bank to keep track of the service status of each flow and dynamically regulate the priorities of the flows to occupy the channel resource. All these three algorithms achieve good performance tradeoff among the three performance issues aforementioned. But they only work under the GE channel model, which is too coarse to characterize the practical variability of the wireless channel conditions.
2.3. Algorithms in Wireless Environment with Multistate Model
Some researchers considered variable-rate transmission in scheduling algorithms [11–15]. The Modified Largest Weighted Delay First (M-LWDF) algorithm  schedules packets according to a simple but effective constructed metric. The scheme was proved analytically to be throughput optimal. But it does not explicitly guarantee any fairness. The algorithm of Jalali et al.  was proposed to satisfy the so-called proportional fairness; the algorithm of Liu et al.  tried to maximize the long-term average total throughput for a given time fraction; the algorithm of Borst and Whiting  was made to be Pareto-optimal. The algorithm of Park et al.  selects user for transmission based on the cumulative distribution function (CDF) of the user rates.
Besides the above three classes of scheduling algorithms, researchers also consider standard-compatible schemes recently. In , a scheduling algorithm which is practical and compatible to the IEEE 802.16 standard is proposed. It provides QoS support in terms of bandwidth and delay bounds for all types of traffic classes as defined by the standard.
By summarizing the above-mentioned scheduling algorithms, we notice that most of the algorithms are designed based on heuristic methods. Either introducing the virtual/ideal service models as reference systems [8, 9] or defining some key metrics as criteria [11–14], many existing algorithms are built on the observation, judgement, intuition, and experience of the designers. Although heuristic algorithms have the merit of simplicity, their underlying drawbacks are twofold. Firstly, heuristic algorithms are generally proposed with respect to some given special outside conditions, which means that they are less of adaptivity. When the practical conditions do not match the provided ones, it is hard for heuristic algorithms to maintain their scheduling performance. Secondly, designing a heuristic scheduling algorithm is easy when the problem is relatively simple but could be rather tough when the scheduling environments and objectives become complicated. For packet scheduling problem in BWA networks, the scheduler is required to consider three main objectives simultaneously and, meanwhile, combat highly variable channels and unknown traffic models. Such a complicated scheduling problem is far out of the ability of a heuristic algorithm to handle.
To overcome the disadvantages of heuristic algorithms, we propose in this paper a learning-based algorithm for the packet scheduling problem in BWA networks. The proposed algorithm has in-built capacity of studying from outside environment and adapting system parameters according to the performance requirements. This attractive feature makes the proposed algorithm appropriate for the scheduling problem in BWA networks. More specifically, when this learning-based algorithm is applied in BWA networks, we only need to translate the scheduling objectives to the system reward (or cost) and then connect the scheduling problem with the ingredients of the solution framework. Little attention should be paid on the details of the entire scheduling process such as how to dynamically adapt the scheduler for the optimal performance. Because the scheduler has potential ability of self-training, it could automatically tuning itself towards an optimal system reward (or cost) according to the actual environment. Actually, the works in [13, 14] have realized that fixed and unadaptable scheduling schemes are incapable in complicated scheduling problem. Undetermined constants are integrated into the key metrics at the system initialization phase so that the scheduler could adjust its strategy at some extent. Nonetheless, the learning abilities of these algorithms are very limited, and they should still be looked on as heuristic algorithms.
In this section, we cast the packet scheduling problem into the framework of Semi-Markov Decision Process (SMDP). As the base of our discussion, the wireless network model and channel model are firstly introduced. After that, we connect the packet scheduling problem with the ingredients of SMDP. The construction of cost function is comprehensively discussed.
3.1. System Model
3.1.1. Wireless Network Model
3.1.2. Wireless Channel Model
3.2. SMDP Ingredients
In a general shared-channel wireless network, the scheduler should make scheduling decisions based on the knowledge of the system state, including the queueing delay of all packets, the channel conditions of all sessions, and the practical service rates of all sessions. Suppose that there are sessions in the system, and each session has a queueing buffer that can store packets at the most. (Without loss of generalization, we assume that all queues can store same number of packets. A more practical assumption may be that the buffer length of the th session is in packets.) At time , the queueing delay of the th packet in the th session is denoted by . For the th session, let denote the maximum available channel rate at time , the practical average service rate at time , and the nominal service rate (negotiated when the session is setup). We now connect the packet scheduling problem with the main ingredients of the Semi-Markov Decision Process (SMDP) as follows.
The scheduling process is naturally divided into a series of stages by events , which represent packet arrivals and departures (we say that a packet "departs" if its service completes). Stages are indexed by integers Let denote the time that event happens. The time duration of the th stage is represented by .
A state consists of the queueing status of the whole scheduling system, including the current queueing delay of all packets, the wireless link conditions of all sessions, and the actual service rates of all sessions. In particular, the state at the beginning of stage is defined by (for ease of presentation, we also call the state of stage in this paper)
where , , and . For the computation of the actual service rate , we improve the approach provided in . We define a time window which is sufficiently long. If session is assigned for transmission, the actual service rate of session at the end of the th stage is updated by ; otherwise, . The set of all possible states is called state space, denoted by , which is a multiple dimension space with huge size.
At stage , with the occurrence of event , the scheduler should make a decision according to the state . The decision is defined by the session number of the next transmission, that is, , . The decision space is denoted by . At stage , if event is packet departure, since the wireless channel is idle, the decision subset contains all sessions that can transmit. (A session that "can transmit" is a session with nonempty queue and nonzero channel rate.) If event is packet arrival, in the case that the channel is busy (being used for transmission), is a singleton; in the case that the channel is idle (as no session can transmit, e.g., an empty system), the decision subset contains all sessions that can transmit.
Policy is a sequence of decision functions, where decision function is a mapping from state space to the decision space . A policy is said to be stationary if for all , , which means that the decision function does not change with stages. The space of stationary policy is denoted by . We only consider stationary policies as candidate policies in this paper, and without confusion, we simply use as the denotation of stationary policy. Note that, under policy , the decision at stage is represented by .
3.2.5. Cost Function
As mentioned above, the definition of system cost of scheduling problem in BWA networks should take into account three objectives: (i) wireless bandwidth utilization maximization, (ii) QoS differentiation and guarantee, and (iii) fairness provisioning. Since cost function has major influence to the scheduler's performance, we place the construction of cost function in a single subsection which will be presented shortly. In the th stage, the cost with state and decision is represented by . Under policy , the average cost of the system is provided by
3.2.6. Bellman Equation
Generally, the optimal policy of the SMDP problem could be obtained by solving the Bellman optimality equation. In the average cost SMDP problem, Bellman optimality equation takes the following form :
Here, stands for expectation; represents the duration of current stage; denotes the decision set at current time; is the state at next stage. The value is the optimal average cost, and is called the optimal differential cost, which has the following interpretation: when we operate the system under an optimal policy, then equals to the expectation of the difference of the total cost (over the infinite horizon) for a system with initial state , compared with a system with initial state . State is a recurrent state with zero differential cost (e.g., the state in which all queues are empty).
As shown in (5), the optimal policy is composed by a series of optimal decisions at all stages, which maximizes the value of the righthand side of (5).
The desire for the exact solution of in (4) is unrealistic because of the huge size of state space and the overwhelming computational requirement. Hence, we employ the methodology of Reinforcement Learning (RL)  to produce a near optimal solution. The central idea is to approach by a parameterized approximating function . Online learning is carried out to continuously regulate the parameter vector and make more and more close to . The details of the RL solution are presented in the next section. Before that, we explain the construction of cost function in a separated subsection.
3.3. Construction of Cost Function
Simplicity. A simple form of cost function can significantly reduce the computation complexity of the whole scheduling algorithm.
Effectiveness. Since the wireless packet scheduler is expected to concurrently address performance issues of QoS, bandwidth utility and fairness, the definition of cost function should take into account all these three aspects.
Scalability. In different practical services and applications, the requirements on every single performance objective may be quite different. Therefore, the structure of the cost function may have some tunable coefficients which allow the network supervisor to regulate according to actual situations.
Our discussion of the cost function starts from a draining problem, where given a number of packets in the system, and assuming no more packet arrivals, the scheduler should try to arrange the order of packet transmissions so that the total cost is minimized. Draining problem is indeed a simplified version of the original scheduling problem. By assuming that there is no new packet arrival, draining problem is much more analyzable than the original scheduling problem. Here, we consider a draining problem with the following setup. There are sessions in the system. The initial queue length of session is in packets, . At the beginning, the queueing delay of all the packets is supposed to be zero. We separately define three different forms of cost function below. Each definition actually involves one performance objective.
3.3.1. Bandwidth Utilization
where is the head-of-line (HOL) delay of session at the beginning of the th stage; is the indicator function, satisfying that when is true, and when is false. By this definition, the total cost is equal to sum of the delay of all packets. To minimize the total cost, the scheduler will always assign the session with the highest channel rate to transmit. Obviously, this scheduling policy will result in bandwidth utilization maximization.
3.3.2. QoS Differentiation
where is the weight with session . The existence of weights makes the scheduling rule different from the one under definition (6). To be concrete, consider two sessions and with weights and , respectively, and . Suppose at a certain stage that , and both are the highest channel rates of all sessions. The scheduler will choose to assign for transmission first, because for a same period of waiting time, the cost of is larger than . We can see that different weights give the session different priority levels. Following the definition of cost in (7), the scheduler can achieve QoS differentiation.
where and are, respectively, the nominal and actual average service rate of session . Under this definition, the scheduler is insensitive to the bandwidth utilization or QoS differentiation. What concerns the scheduler is whether every traffic gets its ordered service. The following theorem explains how the scheduler provides fair services for multiple traffic.
Considering a scheduling system with cost function of (8) and working under the Fluid limit model , if all sessions have the same channel rate , the proportion of service that session receives from the scheduler is given by , where the sum of service proportion is .
See the appendix.
Theorem 1 indicates that under the definition of (8), each traffic flow will get its fair share of service being proportional to the square root of its nominal service rate. A more general form of cost function that relates to fairness issue is provided by extending the definition of (8) as follows:
Although none of the above three cost functions can simultaneously relate to all the three performance issues, they lead us to construct an appropriate one. Actually, by simply combining the definitions given by (6), (7), and (8), we obtain the cost function as follows:
Here is a constant associated with session that relates to fairness. The specific values of 's and 's are left to be determined according to the practical situations by the network supervisor. Regarding the form of the cost function in (10), it is worth to mention that a cost function of similar form is introduced in . The authors assembled the factors of HOL delay, channel rate, and constant weight in a multiplication manner to produce an effective cost function. It is easy to see that the form of cost function definition by (10) has three important properties as mentioned at the beginning of this subsection. Firstly, it is simple and easy to compute in each stage. Secondly, as a combination of (6), (7), and (8), it effectively takes into account the three performance issues: bandwidth utilization, QoS guarantee, and fairness. Thirdly, the scalable coefficients 's and 's provide flexibility for the network supervisor to regulate the scheduler according to the practical requirements.
4.1. Function Approximation Architecture
The function approximation architectures could be broadly categorized into two classes: the nonlinear and the linear structures. One typical nonlinear architecture is the multilayer perceptron or feedforward neural network with a single hidden layer (see, e.g., ). Under this architecture, the output approximating function is the nonlinear combination of input feature vector with internal tunable weight vector . This architecture is powerful because it can approximate arbitrary functions of . The drawback, however, is that the dependence on is nonlinear, and tuning can be time-consuming and unreliable.
Alternatively, a linear feature-based approximation architecture  is much simpler and easier to implement, in which the output approximating function is the linear combination of the input feature , given by
Since the linear combination of feature vector should approximate the optimal differential average cost, the definition of feature vector explicitly relates to the definition of cost function. As (10) has shown, the cost at each stage is determined by the weighted queueing delay of the packet being assigned to transmit. Thus, the components of the feature vector should be defined with respect to all packets in the system, each component corresponding to one packet. Let denote the packet size of session . At stage , the remaining waiting time for the th packet in session could be estimated by , where is the current actual service rate of session . Thus, the feature component with respect to the th packet of session is defined by
Here, the item could be viewed as the estimation of the queueing delay for the th packet in session . Note that if provides an accurate prediction of the service rate of session , (13) coincides well with (10) and the loss of optimality of the linear architecture could be sufficiently small. Extending (12) yields
4.2. Online Parameter Tuning
After establishing the approximating architecture, we employ the technique of Temporal-Difference (TD) learning  to perform online parameter tuning. The algorithm starts with an arbitrary initial parameter vector and improves the approximation as more and more state transitions are observed. At each stage, a greedy decision is made based on current and . Specifically, the decision should be made by
where is the estimation of state at the next stage. The computation of is trivial if we simply assume that no packet arrives during the current stage. More concretely, let denote the number of packets in session . At the beginning of a stage, given the state
if no more packet arrives before the current transmission completes, then the state at the end of current stage (i.e., the beginning of next stage) is estimated by
In this section, we evaluate the performance of RLS by network simulation. Two simulation experiments are carried out. The first experiment is to investigate the convergence performance of the parameter tuning procedure. The second one is to compare the performance of RLS with two existing packet scheduling algorithms: CSDP-WRR  and CIF-Q .
5.1. Simulation Setup
Properties of the 5 sessions in the simulation.
Generally, audio and video traffic has higher priority than data traffic, due to the requirement of realtime transmission. Thus, we set the priority factors of the audio and video traffic flows as , that is, , and the priority factors of data traffic flows as , that is, . In the aspect of providing fair service, data traffic is usually more sensitive than audio and video traffic. This is because data traffic flows, such as FTP and e-mail, are generally served in the mode of best effort (BE). There is no guarantee how much service the system has to provide for these traffic flows. Without consideration of fair service, multiple data traffic flows in a same system may receive fairly different amounts of service, due to the different wireless channel conditions. Hence, we set larger fairness factors for the data traffic flows. The fairness factors of audio and video traffic are set as , that is, , and the fairness factors of data traffic are set as , that is, .
5.2. Convergence of Parameter Tuning
The first experiment investigates the convergence of RLS in the procedure of parameter tuning. Since the optimal average cost is the most important metric of the system performance, the variation of reflects the convergence of the learning algorithm. We run the simulation and record the value of . The step size of the learning algorithm should be set before the simulation. As mentioned in Section 4, the setting of step size parameters and should satisfy the conditions of (19). One efficient way to determine these parameters is to set the value of step size roughly inversely proportional to the stage number (or running time) . Let represent the step size with respect to feature component at stage . We set
It is observed that RLS converges very fast in the procedure of parameter tuning. The approximate optimal average cost has reached a relatively stable value after 100 seconds. Although keeps varying as time goes by, the fluctuation is still in an acceptable range. It is clear that the learning algorithm has converged. For comparison, we also choose another group of step size parameters. We set
We observe in Figure 6 that RLS converges at a smaller speed than in the situation of Figure 5. It takes about 300 seconds for the scheduler to regulate the parameters and reach a stable approximate optimal average cost . But the variation of the value of in Figure 6 is much more moderate than that in Figure 5. These results indicate that the smaller the step size parameters, the slower the convergence speed, but the stabler the system performance. The selection of step size parameters 's and 's depends on the actual network conditions. The scheduler should not only provide a sufficiently stable performance but also be fast reactive to follow the variability of outside environment.
5.3. Scheduling Performance
We know from Figures 7(a) and 7(b) that both the average and maximum delays of audio and video sessions in RLS are clearly smaller than in either of the other two algorithms. In other words, RLS is able to provide better QoS guarantee for realtime traffic flows. It should be noticed that the satisfying QoS guarantee for audio and video sessions in RLS is not obtained by sacrificing the throughput of the other sessions. This is validated by Figure 8.
Figure 8 consists of three subfigures. Figures 8(a) and 8(b), respectively, report the throughput of data traffic flows in the first 400 seconds and the entire 1200 seconds. Figure 8(c) reports the total throughput of all the three data traffic flows. In Figure 8(a), compared to the other two algorithms, RLS significantly improves the throughput of data-1 and data-2, except that the throughput of data-3 in CIF-Q is a little more than that in RLS. But we know from Figure 8(c) that the total throughput of the three data traffic flows in RLS is much more than that in the other two algorithms. More specifically, in the first 400 seconds, the total throughput of data traffic is 1.62 Mbps, which is higher than that in CSDP-WRR (1.19 Mbps) and higher than that in CIF-Q (1.22 Mbps). In the entire 1200 seconds of simulation, the overall throughput of data traffic in RLS is 1.77 Mbps, which is higher than that in CSDP-WRR (1.57 Mbps) and higher than that in CIF-Q (1.63 Mbps). Results of Figure 8 indicate that RLS has a considerable improvement in bandwidth utilization.
To demonstrate the short-term and long-term fairness properties of RLS, we record the service received by the data sessions over the whole simulation time in Figure 9. It is observed that the curves diverge moderately in the error-prone period and then converge gradually in the error-free period. We give interpretation as follows. To promise short-term fairness, RLS does not force the leading sessions to give up all their leads but just makes them degrade gracefully. So data-1 session receives relatively more service in the first 400 seconds. However, when the system becomes error-free, the lagging sessions will gradually get back their lags, and finally all data sessions obtain same throughput. This observation validates that RLS offers both short-term and long-term fairness guarantee in the sense that it provides short-term fairness for error-free sessions and long-term fairness for error-prone sessions. This exactly coincides with the spirit of CIF properties introduced in .
In this paper, we address the problem of packet scheduling in BWA networks. We cast the problem into the framework of semi-Markov Decision Process and solve it by the method of reinforcement learning. The proposed scheduling algorithm RLS simultaneously considers three performance issues in BWA networks: (i) QoS differentiation and guarantee, (ii) bandwidth utilization, and (iii) fair service. Being different from traditional scheduling algorithms, RLS is learning-based. It has the potential to learn from outside environment and adaptively regulate its scheduling policy. Simulation experiments are carried out to evaluate the performance of RLS. The numeric results indicate that RLS outperforms two existing scheduling algorithms in the sense that it provides better QoS for realtime traffic flows and meanwhile significantly improves the throughput of nonrealtime ones. Moreover, RLS achieves both long-term and short-term fairness.
Proof of Theorem 1
Suppose that the total bandwidth of the system is . Let denote the proportion of bandwidth allocated by the scheduler to session under the Fluid limit, that is, . The scheduling problem could be formulated as a linear programming:
The work in this paper is supported by programs of NSFC under Grant nos. 60903170, U0835003, and U0635001.
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