In this paper, we investigate the network utility maximization problem in FDMA systems. We summarize with a suite of criteria on designing utility functions so as to achieve the global optimization convex. After proposing the general form of the utility functions, we present examples of commonly used utility function forms that are consistent with the criteria proposed in this paper, which include the well-known proportional fairness function and the sigmoidal-like functions. In the second part of this paper, we use numerical results to demonstrate a case study based on the criteria mentioned above, which deals with the subcarrier scheduling problem with dynamic rate allocation in FDMA system.
As the downlink access data rate increases, future wireless networks are expected to support various services with different quality of service (QoS) demands [1]. We can classify high speed services into two classes based on their delay tolerance. The QoS services are delay and rate sensitive, and require a certain access data rate. This type of application includes many high speed downlink data services that are widely studied over the last decades, such as Video on Demand (VOD) and packet-switched voice services. The other class corresponds to the best-effort services conducting more elastic applications such as file transfer and e-mail. This kind of services can adjust their data rate gradually and is often delay tolerant [2]. It is commonly believed that the concept of utility function, which maps the access data rate to the level of user satisfaction or QoS, is appropriate to characterize the elasticity of services.
In the past few years, utility-based radio resource management problems such as rate control and power allocation have been widely studied, and most of them dealt with the situations where the utility functions are either convex ones for which efficient theories and algorithms such as the Karush-Kuhn-Tucker (KKT) conditions exist, or specific nonconvex ones including the well-known sigmoidal-like function of which the dual problems are explored and solved by centralized or distributive algorithms. However, some research shows that the convex utility functions are appropriate only to model elastic services and do not capture the properties of services with strict QoS demands [2]. And the nonconvex utility maximization problem is significantly hard to be analyzed and solved, even by centralized computational methods. Particularly, nonconvexity makes a local optimum may not be a global optimum and there exists strictly positive dual gap. The standard distributive algorithms solving the dual problem may produce infeasible or suboptimal rate allocation, and the global maximization of nonconcave functions is an intrinsically difficult problem [3].
Due to these issues, in this paper, we would like to study the utility maximization problem in another way, by not specifying any particular forms of utility functions, even not assuming their convexity, but proposing sufficient and necessary conditions under which a utility function can guarantee leading the objective function convex. Although the content in this paper deals mainly with the network utility maximization problem in FDMA systems, the idea of exploring utility properties that achieves the global problem convex can be further studied in other scenarios, such as CDMA systems.
In the numerical analysis, we will use the proposed criteria to demonstrate and solve an FDMA subcarrier scheduling problem with dynamic rate allocation. Various service types are considered, and we use different utility functions based on the criteria mentioned above to characterize the QoS demand properties of different services. For instance, for the VoIP and video streaming services, sigmoidal-like functions are appropriate to model their utilities, since decreasing the transmission data rate below a certain threshold would result in a significant drop in the QoS. And the utility of the best-effort service, which does not need a constant data rate support, would be more likely modeled using convex functions such as the logarithm function, since the more bits transmitted to the user, the more satisfied the user would be. However, it should be noticed that the sigmoidal-like function and the logarithm function are only two of many function forms that consistent with the criteria proposed in this paper. The main contribution of this paper is that we try to present a way which the researchers may follow to choose or design appropriate utility functions that could nicely model the characteristics of both real-time and best-effort services while guaranteeing that the optimization problem will always be convex.
The rest of the paper is organized as follows. In Section II, we present the system description and formulate the optimization problem. In Section III, we propose the criteria on utility designing and present an optimal power allocation algorithm. Based on the criteria mentioned, in Section IV we will use numerical results to demonstrate an FDMA subcarrier scheduling problem with dynamic rate allocation. And Section VI gives a summary and concludes the work.
In this section, we briefly describe the studied FDMA network scenario and formulate the general global network utility maximization problem. Consider a single-cell downlink Orthogonal FDMA system with N users. Inter-cell interference is not taken into consideration. The total system bandwidth W is divided into K subcarriers with bandwidth / f W K Δ =
. Let ik H denote the channel frequency response at subcarrier k with user i, so the SNR of user i at this subcarrier is expressed as
, where is the transmi
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