## 4 Generalized Weighted Fairness (GWF)  
From the discussion so far, we have described mechanisms on how to achieve throughput fair allocation by applying various modulation and coding schemes (MCSs) to calculate the fair share (as a  
Figure 3: Fairness Definitions. (a) Slot Fair Allocation and Byte Fair Allocation; (b) Weighted Fair Allocation [11].  
Figure 2: Steps in a Simple Max-Min Fairness Algorithm [10].  
resource allocation unit). However, in wireless networks, e.g., wireless local area networks, or WLANs, there are two ways to define fairness: _temporal fairness and throughput fairness_[14].  
The former approach is to allocate equal number of service time units to each user. Again, in IEEE 802.16e Mobile WiMAX Networks, the resource allocation is in terms of the number of slots [1,2]. Therefore, here the unit used to achieve the temporal fairness is _slot_. In other words, the base station will allocate equal number of slots to each user. The users, who are near the base station, and therefore have a very good link, will be able to use good MCSs (multiple bits per symbol), and consequently receive good throughput in bytes per second. This is also called slot fairness or temporal fairness.  
On the other hand, the other approach is to allocate equal throughput (or bytes) to each user. The users near the base station will need fewer slots than those who are far away to transmit the same number of bytes. This is called byte fairness or throughput fairness. Figure 3a shows formal descriptions for both slot and byte fairness schemes.  
When setting either slot fairness (temporal fairness) or byte fairness (throughput fairness) for each user, it can be argued which definition is better. For example, the users want to run certain applications, such as VoIP (Voice over Internet Protocol), which require a certain bit rate, and so the users would prefer a guaranteed bit rate for their applications or a fair share of the available bit rate in case of best effort service. This is the argument in favor of byte fairness or throughput fairness.  
The argument in favor of temporal fairness or slot fairness goes as follows: The service provider has a fixed number of slots, and if the user happens to choose a bad location, e.g., the basement of a building on the edge of the cell, the provider will have to allocate a significant number of slots to provide the same quality of service as to a user who is near the base station. Since the providers have no controls over the user locations, they can argue that they will provide the same resources to all users, and the throughput observed by the user will depend upon their locations.  
Some service providers will prefer throughput fairness while the others will favor the slot fairness. The equipment manufacturers should allow both possibilities. One way to do this is to implement the weighted fairness introduced in this paper [11]. The formal definition of the weighted fairness is shown in Figure 3b.  
In general, setting the weight, \(w\), to 0 results in byte fairness, and setting it to 1 results in slot fairness. A carrier can select the weight to be either of the two values or any values between 0 and 1. Note that the GWF formulation as defined in Figure 3b uses equal weight for all mobile stations. It is possible to generalize this formulation further by allowing different weights for different mobile users based on their priorities or the price paid.