Multiuser Resource Allocation Maximizing the Perceived Quality
© A. Saul and G. Auer. 2009
Received: 1 August 2008
Accepted: 24 January 2009
Published: 3 March 2009
Multiuser resource allocation for time/frequency slotted wireless communication systems is addressed. A framework for application driven cross-layer optimization (CLO) between the application (APP) layer and medium access control (MAC) layer is developed. The objective is to maximize the user-perceived quality by jointly optimizing the rate of the information bit-stream served by the APP layer and the adaptive resource assignment on the MAC layer. Assuming adaptive transmission with long-term channel state information at the transmitter (CSIT), we present a novel CLO algorithm that substantially reduces the amount of parameters to be exchanged between optimizer and layers. The proposed CLO framework supports user priorities where premium users perceive a superior service quality and have a higher chance to be served than ordinary users.
With the high envisaged data rates of beyond 3rd generation (B3G) wireless communication systems [1, 2], multimedia broadband applications can be offered to mobile users. Multimedia applications are characterized by a multitude of data rate and quality of service (QoS) requirements. On the other hand, owing to the nature of the mobile radio channel, frequency selective fading, distance dependent path loss, and shadowing cause vast variations in the attainable spectral efficiency per user. The objective of multiuser resource allocation is to assign the available resources over the shared wireless medium to mobile users running different applications .
Orthogonal frequency division multiple access (OFDMA) provides orthogonal transmission slots in time and frequency, which may be flexibly assigned to the individual users [4, 5]. In B3G systems, this feature is exploited by the medium access control (MAC) layer to freely distribute the available bandwidth between users . Provided channel state information at the transmitter (CSIT) is available, the number of transmitted information bits per slot can be adjusted to the channel conditions of a particular user.
The application (APP) layer outputs encoded applications, for example, a video stream. For the scalable video coding (SVC) extension [7, 8] of the advanced video coding (AVC) standard H.264/MPEG-4 AVC the stream may be received with a variable information bit rate. Other kinds of video streams may be encoded or transcoded  with the desired data rate. In general, any application may be delivered with variable information bit rate, allowing to trade user-perceived quality with data rate.
The high level of flexibility and adaptability offered by emerging system architectures provides an opportunity for dynamic allocation of resources across users and applications, to increase the network resource usage and to enhance the user satisfaction. This effectively requires interaction between system layers, a paradigm known as cross-layer design [10–12]. For the multiuser resource allocation problem at hand, a global cross-layer optimization (CLO) problem is formulated: maximize the user-perceived quality by tuning the served data rate on the APP layer jointly with the adaptive resource assignment on the MAC layer. Application-driven CLO has been studied for systems supporting one single type of applications [11, 13, 14] as well as for various application classes .
Several publications [15–17] consider a logarithmic relation between utility metric and data rate, which may result in a concave optimization problem. A more realistic utility metric, measuring the user-perceived quality, is given by the concept of mean opinion score (MOS) . In , a framework is established that allows to mathematically formulate the MOS for multiple applications like voice, video streaming, and file download. The resulting nonconcave optimization problem may be approximated, for example, with a greedy algorithm that maximizes the sum of the MOSs for all users .
In this paper, the optimum multiuser resource allocation supporting multiple applications is derived in closed form for the case of adaptive transmission with long-term CSIT, assuming a logarithmic relation between utility metric and data rate. Interestingly, the cross-layer optimization problem is shown to become independent of the channel conditions but is entirely determined by the application characteristics, provided that the offered data rate at the APP layer is matched to the adaptive transmission parameters in the MAC layer. For the special case where all users share the same application class, it turns out that the overall perceived quality is maximized when all users are allocated the same bandwidth, which corresponds to equal resource sharing. This implies that users with good channel conditions transmit with higher rate and therefore enjoy better QoS, as adaptive transmission is more bandwidth efficient in this case. This is in a sharp contrast to conventional approaches for QoS provisioning that assume a fixed target rate per user [3–5], where users with poor channel conditions are allocated more bandwidth, so that all receivers perceive the same QoS.
The theoretical analysis serves as a basis for a novel CLO algorithm that allows for a more realistic utility function that is based on the MOS. The proposed algorithm for the underlying nonconcave optimization problem is easy to implement and exhibits significantly lower complexity than the generic solutions in [19, 20]. Moreover, priority classes can be supported in the way that premium users perceive superior service quality and are more likely to be served, even under poor channel conditions. The proposed framework also allows to cater for additional constraints, such as a guaranteed minimum perceived quality for all users.
The developed CLO framework for application driven multiuser resource allocation is evaluated by mathematical and numerical analysis. We elaborate for which application classes CLO attains the most significant gains, and the origin of these gains is identified. Furthermore, the computational cost and the overhead due to exchange of CLO related parameters between layers is studied. It is demonstrated that the overhead of the proposed CLO framework grows only linearly with the number of users and available slots, which compares to an exponentially growing overhead for conventional techniques [11, 12, 21, 22]. This is particularly relevant to B3G systems with their high degree of freedom for resource allocation, due to the large number of served users and available slots.
The remainder of this paper is structured as follows. Section2 provides an overview of the considered multiuser downlink with focus on MAC and APP layers. Section 3 introduces the CLO framework and the flow of exchanged parameters between layers and optimizer. In Section 4, the optimum multiuser resource allocation strategy is derived, assuming idealized application characteristics. The proposed CLO framework for the more realistic nonconcave optimization problem is established in Section 5, and its performance is evaluated by computer simulations in Section 6.
2. System Overview
A wireless downlink shared by users is considered. An application server is transferring multimedia applications via core network and base station to mobile users. There are applications, which, without loss of generality, generate bit-streams, associated to different users.
2.1. Link and Physical Layer
2.2. Application Layer
The objective MOS is recommended as utility metric for voice transmission by the ITU-T  as a measure for the user satisfaction. Practically, the MOS may take values between 1 (not acceptable) and 4.5 (very satisfied). In , the MOS is extended to other services like video streaming, file download, and web browsing. The obtained mathematical model of the user-perceived quality can be used as universal utility metric for CLO, allowing for joint optimization of different application classes.
The application characteristic is mainly influenced by data rate and packet losses, described by the applications' rate-loss distortion . In this paper, the perceived quality is exclusively expressed as a function of the data rate , while packet losses are not considered as an explicit parameter. While this conveniently simplifies the analysis, this choice requires some further motivation, since certain kinds of source encoded bit-streams are sensitive to packet losses .
Packet losses may be caused by transmission errors over the mobile radio channel or by system overload. Regarding the wireless channel the link layer may compensate for packet losses by means of adaptive modulation and channel coding in combination with automatic repeat request (ARQ). While link adaptation ensures that transmission errors occur with low probability, low latency retransmissions of erroneous packets within the link layer  maintain reliable delivery of packets, at the expense of a certain rate reduction.
In an overloaded scenario, the offered load by the APP layer exceeds the capacity of the wireless link. Such an overload scenario can be effectively avoided by a fine grained adjustment of the offered data rate at the APP layer so as to match the capacity of the wireless link.
For instance, in case of video streaming, transcoding  or using the SVC extension of H.264/MPEG-4 AVC [7, 8] allows to vary the data rate in a rather fine granularity. As packets can be dropped at either the application server or the base station, a low latency rate adaption mechanism is feasible, at the same physical location as the scheduler in the MAC layer, effectively allowing to express perceived quality by data rate.
Moreover, the possibility to selectively drop packets offers one further opportunity to adjust the data rate. Likewise, for file downloads the data rate can also be adjusted in arbitrarily small steps. Hence, it is reasonable to assume that the application data rates can be adjusted continuously.
2.2.1. Video Streaming
The parameters , , and characterize a specific video stream or sequence, which is source encoded with rate . These parameters may be determined by matching the distortion-rate model to the measured bit stream of a video.
3. Application-Driven Cross-Layer Optimization
3.1. Layer Model
A major challenge in cross-layer design is the abstraction of parameters exchanged as control information. In order to limit the amount of control information, we introduce a layer model at the optimizer that emulates the relevant characteristics of the layer. The parameters of the layer model are determined at the corresponding layer, and only these parameters are sent as control information to the optimizer. The optimizer then tunes the model so as to identify the operating modes that maximize the chosen utility, which are then fed back to the system layers.
For instance, applied to a video stream the local utility could be the PSNR or MOS, and according to (2) the parameters might represent source coding parameters such as the chosen codec, the frame rate, and the data rate . As a second example, applied to the PHY layer the local utility might be the sum throughput of all users, and are parameters such as the channel coefficients or the velocity of the mobile terminal.
Following the conventional idea of parameter exchange, an intralayer optimization might deliver the subset of operating modes that maximize the utility function , called efficient set in , also known as Pareto frontier. These operating modes are the crosses being located on the curve in Figure 4. A subset of operating modes is selected as operating points (circles). These are provided to the optimizer, which performs CLO by choosing the overall best operating point.
The proposed layer model is the curve in Figure 4, which represents an approximation of the utility metric as a continuous function. As demonstrated in the following the proposed parameter abstraction by a layer model exhibits a significant advantage for multiuser resource allocation, due to the potentially large number of available slots.
3.1.1. Link Layer Model
For conventional CLO the parameters that are provided to the optimizer are the set of possible data rates for all users in (1). Considering an OFDMA-based B3G air interface with a large number of available slots, a prohibitive set of possible data rates is obtained. Instead of offering a set of discrete values to the optimizer, the link layer model defines the shares of the available resources per users, in (1), as continuous functions. The factors allow the optimizer to tune the link layer model. Then, according to (1) an arbitrary number of data rate combinations can be emulated at the optimizer. The only required parameters at the optimizer are the set of parameters . Hence, the link layer model for the optimizer is fully determined by (1). Once the optimizer has found an optimum set of coefficients , these are fed back to the link layer.
3.1.2. Application Layer Model
Each user's application characteristic can be parametrized by only two parameters, , or alternatively . The optimizer then tunes the model by maximizing the user-perceived quality and returns the set of optimum user data rates to the APP layer.
3.2. Parameter Exchange
3.2.1. System Description
The parameter describes the application characteristic for user , which is and for the APP layer model from Section 3.1.2. In addition more detailed information about the applications in a real system may also be contained in .
containing the MOS of all users, which according to Figure 3 is delivered to the optimizer.
3.2.2. Required Overhead
Reviewing the exchanged parameters, we notice that the vectors and contain only long-term information. No instantaneous CSIT, power allocation, modulation, or schedules have to be exchanged between PHY/MAC layer and the optimizer. Likewise the APP layer model specified in Section 3.1.2 is determined by only two parameters that are slowly time varying. This has the advantage that the system is less sensitive against delays caused by parameter exchange between layers and the optimizer. Robustness against delays is of importance for CLO as base station and application server are most likely located at different physical locations so that control information is to be exchanged over the core network.
data rate values have to be transmitted to the optimizer and one value is fed back as the chosen schedule.
4. Optimum Resource Assignment
Based on the model-based CLO framework the optimum resource allocation assuming an idealized utility is derived in closed form in this section. The mathematical analysis is the basis of an optimization algorithm presented in Section 5, which maximizes a more realistic utility.
4.1. Problem Statement
As the above optimization problem is neither convex nor concave, we first define an idealized utility that produces a concave optimization problem.
4.2. Unbounded Application Characteristic
Note that and, hence, the solution of the unbounded optimization problem is independent on the physical radio channel, characterized by , and only depends on , which is determined by the ratio between and .
For finding a closed form solution of the optimum resource assignment in (19), in the following we prove the concavity of the optimization problem, derive the optimum share of resources between two users, and find a solution for the absolute resource share of a user.
as the final solution of the unbounded optimization problem (19).
Interestingly, given that all users use the same application, the optimum resource allocation for the unbounded problem results in an equal resource scheduler where all users are assigned the same number of slots. This implies that users experiencing a good channel receive higher data rates and therefore enjoy better QoS, as adaptive transmission is more bandwidth efficient in this case.
In summary, the optimum resource allocation for the unbounded optimization problem (32) is independent of the channel conditions; the number of assigned slots (the allocated bandwidth) is exclusively determined by the application characteristics; users with a good channel enjoy higher data rates. On the other hand, all users are given a fair share of the available resources. This is in a sharp contrast to a maximum throughput scheduler, which exclusively serves good users while users experiencing a poor channel starve for resources. The significance of this finding is that the maximized utility in (19) is an idealized measure of user-perceived quality.
4.3. Subset of Users
For solving the bounded optimization problem (18), it is useful to solve the unbounded problem only for a subset of "variable" users . The remaining users have fixed optimization coefficients and are not subject to optimization. Here, the notation denotes the relative complement of set in set .
5. Optimization Algorithm Maximizing the User-Perceived Quality
Based on the analytical solution for the unbounded problem in Section 4, an optimization algorithm for the bounded problem (18) is presented in this section. In an intermediate step a solution for the upper bounded problem is derived, where the application characteristic is upper bounded at an MOS of . Then the solution of the bounded problem is developed, and its computational complexity is assessed. Finally, the proposed CLO algorithm is extended to support different priority classes.
5.1. Upper Bounded Problem
Let denote the optimum rate allocation of user of the unbounded problem (32). In case , the rate for user may be reduced to without sacrificing service quality, and the retained resources can be given to users with , . A solution of this concave problem is found by the iterative algorithm:
Solve unbounded problem (36).
If any user has been moved in Step 3, continue with Step 2, otherwise stop.
If any of the application characteristics deviates from (4), Step 2 can be replaced by a conventional algorithm that solves the unbounded problem. Alternatively, appropriate values for and can be chosen to approximate the real application characteristic, giving rise to a certain deviation to the exact solution. Optionally, this approximation could be used as a starting point for an applicable conventional algorithm.
5.2. Bounded Problem
We approach the bounded optimization problem (18) by dividing it into two subproblems: first, a subset of users is determined who cannot be served and therefore get no resources, ; second, for the remaining users the upper bounded optimization problem from Section 5.1 is solved. In case dropped users are selected appropriately in the first step, the remaining served users will always achieve data rates so that the solution for the bounded problem is optimum.
The following iterative algorithm for the solution of the bounded problem is formulated as follows.
Initially, all users are served.
Drop users as detailed in Steps 2.1–2.4.
Solve upper bounded problem for the served users as described in Section5.1.
Continue with Step 2.1.
Solve upper bounded problem for the served users as described in Section 5.1 and stop.
In this algorithm the determines how many users are served. When the objective is to maximize the sum of all users' MOS, referred to as "increase sum MOS", an appropriate strategy is to continue dropping users until this does not further improve the sum MOS.
This condition checks whether the MOS that would be achieved with the allocated resources exceeds a certain minimum . Setting ensures that only a minimum of users are dropped, while no resources are wasted to users that would anyhow experience unacceptable service quality of . On the other hand, higher values of enforce a certain minimum perceived quality. This variant of the algorithm is therefore termed "reduce outage".
As the above discussion touches upon the issue of admission control, other criteria that determine which users are admitted to the system might be introduced. For example, in a cellular system it might be desirable to give priority to users that hand over from a neighboring cell rather than to serve a user who wishes to enter the network.
5.3. Computational Complexity
An appealing feature is that the proposed optimization algorithm deterministically terminates after a certain time. To prove this the worst case run time is calculated in the following. Since in each iteration at least one user is dropped, there are at most iterations in the outer loop. The inner loop computes the solution of the upper bounded problem. In the worst case, one user is moved from to so that the number of iterations at most equals the number of served users. The total number of iterations is therefore upper bounded by .
An observation from the simulation results in Section 6 is that typically most users can transmit. Hence, the number of iterations for the outer loop is likely to be significantly smaller than . Likewise, trials suggest that for the inner loop it is rather unlikely that more than two iterations are required. Since the essential calculation within the inner loop is given by the closed form expression (36), the total complexity of the optimization algorithm is low.
5.4. Priority Classes
In order to support different priority classes, the utility function is adjusted in the following.
Under these conditions, the conclusions from Section 4.2 apply: the utility function that supports priority classes (40) is strictly concave downwards, and the underlying optimization problem is solved by substituting with in (31), (32), and (36).
which is similar to the approach described in . This function is applied for obtaining the numerical results presented in Section 6.5.
There are several possibilities how to further incorporate priority classes, for example, by adjusting the upper bound of the upper bounded optimization problem, the stop criterion or by using an alternative criterion for dropping users.
6. Performance Evaluation
Link layer parameters.
Number of subcarriers
Cyclic prefix duration
BPSK, 4-, 16-, 64-QAM
WINNER urban macro-cell 
Duplex ratio DL/UL
Noise spectrum density
Maximum Doppler speed
Number of users
Number of available slots
6.1. Simulation Setup
We consider an OFDMA downlink that occupies a bandwidth of . Due to the inherent orthogonality of orthogonal frequency division multiplexing (OFDM), each subcarrier in each OFDM symbol may be assigned to a different user without causing interference, so that users can be scheduled independently in time and frequency. Adjacent subcarriers and OFDM symbols are correlated and, therefore, experience a similar channel gain. In order to limit the signaling overhead symbols are grouped to form one slot.
The WINNER typical urban macrocell channel (model C2 ) is used, which models channel attenuation due to frequency selective fading, distance dependent path loss and log-normal shadowing . Instantaneous channel variations due to velocities of mobile users are generated using Jakes' model . The channel model is implemented such that the average SNR always allows transmission with the lowest supported modulation and coding scheme. This is motivated by the fact that users with lower SNR would not be able to establish a connection to the base station and, hence, cannot request to be served. While the average SNR always exceeds the given limit, the instantaneous SNR may be significantly lower due to frequency selective fading.
Mobile velocities up to are assumed, which implies that instantaneous CSIT may not be available. It is assumed that the average SNR over all simultaneously transmitted slots is available for link adaptation. Hence, the same modulation and coding scheme is applied to all subcarriers of one user during one slot duration. However, slots assigned to different users will typically use a different modulation and coding scheme.
Simulations are executed as follows: every 100 milliseconds independent shapshots of path loss and shadowing realizations are generated for each user according to a uniform user distribution within the cell area. Then is estimated and passed to the optimizer. CLO is performed to determine the optimum share of resources , which is subsequently fed back to the PGPS scheduler at the MAC layer.
After the 100-millisecond snapshot, the actually achieved average data rates are determined. The actually achieved data rates may deviate from the optimizer's estimate . Each user's MOS is determined based on the user's application and the achieved data rate. Then, the CDF of the MOS averaged over all users is calculated.
6.2. Performance of Different Optimization Algorithms
As seen in Figure 8, both variants outperform equal resource allocation and achieve a comparable average MOS as greedy resource allocation. Compared to equal resource allocation, any performance improvement of the considered optimization algorithms is due to the bounds in the MOS trajectory, since users with perceive the same QoS as if they were served with the reduced rate . Likewise, users with perceive the same QoS as a user who is not served at all. The "reduce outage" variant serves practically all users, although some perceive a poor service quality. In contrast, the "increase sum MOS" variant tends to drop users with poor quality and assigns the freed resources to served users. This is due the objective, which aims to maximize the sum MOS of all users: a user will be dropped, if the increase in MOS of the served users outweighs the decrease in MOS of dropping a certain user.
6.3. Deviation Due to Application Model Abstraction
In order to assess the effect of rate variations of the video stream over time, for each 100-millisecond PHY channel snapshot a new (random) snippet of the respective video stream is used. For the proposed optimization algorithm from Section5 the parameters and are estimated by the application server for each video snippet and provided to the CLO. Because the optimization algorithm is based on the bounded logarithmic relationship (4), which deviates from the actually used video model (3), the decided resource distribution will be suboptimum. For comparison CLO with greedy optimization using the exact video model (3) is also simulated.
6.4. Guaranteed Service Quality
It may be desirable to support the demand for minimum QoS. This may be accomplished by tuning the parameter of the stop criterion in the "reduce outage" variant of the proposed optimization algorithm. As the stop criterion controls which users are dropped from the list of active users (see Section5.2), setting to a value in the range ensures that all served users achieve at least a minimum perceived quality of .
6.5. Traffic Priority Classes
Figure 12 shows the CDF of the sum MOS. Premium users exhibit a significantly better MOS than ordinary users and are more likely to be served.
6.6. Application Characteristic
As seen from Figure 13, the attainable gains of CLO maximizing the sum MOS (solid lines) over equal resource allocation (dashed lines) are dependent on both and the ratio . For low data rate requirements the CLO gain diminishes, as there is an excess of available resources to serve all users with excellent quality . For increasing data rate requirements the CLO gain depends on the ratio , in the way that the CLO gain increases with decreasing . This is explained by the fact that for an increasing ratio the MOS characteristic as a function of the data rate, in (4), approaches the unbounded problem addressed in Section 4.2, for which according to (33) equal resource allocation is optimum. In other words, the attainable CLO gains over equal resource allocation with are due to users whose rates are outside the logarithmic range of . As the logarithmic range is specified by the ratio , the lower the higher the gains to be achieved by optimization.
6.7. Mixed Service Classes
Figure 15 shows the CLO gain in sum MOS relative to equal resource allocation against the number of users in each group. Results are plotted for different values of and , for a total number of users and . Interestingly, in some cases the overall MOS gain for scenarios with mixed service classes exceeds the case when all users are within either of the service classes. This is due to the freed resources by replacing a high-rate user by a less demanding low-rate user, which allows the remaining users to fetch some of the freed resources.
The relationship between low- and high-rate users is further investigated in Figure 16, which shows the CDF of the sum MOS for both user groups. Corresponding to the maximum in Figure 15, there are 4 users with and 12 users with . An appealing observation is that both user groups gain from CLO. While the average gain is , low- and high-rate users gain and in overall perceived quality, respectively.
6.8. System Performance
Resource allocation with QoS constraints where multiple users share a wireless downlink is one key challenge in the design of future wireless systems. The MOS is chosen as a universal utility metric for the user-perceived quality for CLO between link and APP layer.
Adaptive transmission based on long-term CSIT over a time and frequency selective fading channel is considered, including distance dependent path loss and log-normal shadowing. Applications are described by a rate-distortion characteristic, expressed by the MOS. With these settings a model-based CLO framework is devised, which emulates the functionalities of the system layers within the optimizer. Compared to known CLO approaches significantly less parameters need to be exchanged. Simulations of a video streaming scenario confirm that model mismatch, where the APP layer model at the optimizer is not perfectly matched to the actual application, only results in modest performance degradation.
As a metric for the user satisfaction we chose to maximize the sum MOS, which resulted in a nonconcave optimization problem. Given an idealized utility metric with an unbounded logarithmic relation between perceived quality and data rate, a concave problem is retained, so that the optimum resource allocation is derived in closed form. One noteworthy result of the analysis is that the optimum solution is independent of the physical channels and is solely described by the application characteristics.
The theoretical findings are the basis for a low complexity and easy to implement CLO algorithm for the more realistic nonconcave optimization problem. The proposed iterative optimization algorithm is significantly less complex than known optimization algorithms and has the appealing feature to deterministically terminate.
The proposed algorithm offers an additional degree of freedom to the network operator to configure its own policies, such as enhancing user satisfaction, ensuring a minimum perceived quality to all users, or to operate the wireless system with higher load so as to maximize revenue. Furthermore, different priority classes can be supported.
The attainable gains of CLO strongly depend on the application characteristics. The higher the sensitivity of the perceived quality to changes of the data rate, the more considerable the gains that can be achieved. Dependent on the application more than , additional users can be served without sacrificing user satisfaction. If multiple service classes with different application characteristic are running simultaneously, all users can be expected to benefit from CLO. In some cases additional CLO gains that exploit a certain mix of service classes are observed.
This paper was presented in part at the IEEE Int. Conf. on Communications (ICC'2007), Glasgow, UK, at the IEEE Int. Symp. on Wireless Communication Systems (ISWCS'2007), Trondheim, Norway, and at the IEEE Vehicular Technology Conference (VTC'2008 Spring), Singapore.
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