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Using the tness values, a selection process can be applied in that after eliminating all the zero- tness strings from the population, new strings are generated by duplicating the remaining strings in a proportionate manner the ttest string gets the largest share of duplication, and so on. A new generation can begin using the new population. The time complexity of the genetic algorithm is O(nNp) because each evaluation of the tness requires O(n) time. With carefully crafted implementation, the genetic algorithm can be quite ef cient in practice. The merit of a genetic approach is that it can generate near-optimal solutions [35]. 7.6.3 Approaches that Maintain Fairness
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The fairness concept has its roots in the design of the packet service disciplines at a router in which different sessions of packets are contending for a single outgoing link [149].Assuming that the packets are in nitesimally divisible (i.e., the uid model), a fair policy can be implemented by using the generalized processor sharing (GPS) approach [108]; in a round-robin manner, each packet session gets an in nitesimally small share of transmission time on the outgoing link. The shares may be weighted by the sessions requested rates, which are determined when they are admitted in the system. However, because in practice packets are not in nitesimally divisible and are of variable sizes, the GPS approach cannot be used in actual implementation. Thus, a variety of different approaches is devised to approximate the behavior of the GPS approach [149]. These approaches differ in time complexity and analytical performance in terms of delay. For wireless networks, there have also been a number of more recent attempts in designing fair allocation policies. The major difference between the rate allocation problem for wireline networks and that for wireless networks is that in a wireless network, the channel quality is time-varying and location-dependent. Thus, it is possible that the channel quality of a user can be so poor that data cannot be successfully transmitted. Thus, the wireline policies cannot be directly applied in a wireless setting. The general approach of tackling the problem is that a wireline policy is used as a reference system in which the channel is assumed to be error-free. The allocation for each user is then computed in this ideal system. In the real system, for each user we try to allocate the computed rate share to the user as far as possible subject to the constraint that the user s channel condition can support such an allocation. If a user s channel condition happens to be poor for one or more rounds of allocations, the user will then be a lagging user because a smaller amount of rate share will be obtained as compared to the one computed in the reference system. On the other hand, the surplus rate allocations (because the users with poor channel quality cannot transmit) are shared proportionately by the users with better channel conditions. Thus, these users get more rate shares as compared to those computed in the reference system, and are called leading users.
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The existing algorithms for fair wireless rate allocation are based mostly on this general principle. However, because these algorithms are designed using a very simpli ed channel model a two-state model (either good or bad), they are not suitable for use in our model, in which the channel is accurately simulated, taking into account both fast fading and shadowing effects. In our study, we devise a channel-adaptive fair rate allocation policy using a priority metric Qi that is simple to implement: Qi = g i e- bD i (7.12)
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Here, gi is the SIR (i.e., the channel state) of user i, Di is the leading amount in terms of the actual number of information bits transmitted, and b is a scaling factor for balancing the effects of gi and Di. We use the STFQ (start-time fair queueing) [149], which is an ef cient variant of the GPS approach, as the reference allocation algorithm. Thus, the allocation policy works by using the following lling procedure: 1. Sort the requests in the descending order of Qi. 2. Allocate one SCH to the rst request and check the constraints. If the constraints are satis ed, repeat this step with the next request; otherwise, undo the allocation and stop. 3. Update the Qi of all requests. Go back to the rst step. The time complexity of this above channel-adaptive fair policy is O(Mn log n), where M is the largest possible value of mj . As mentioned before, in practice the value of M is quite small (less than 10) and thus, the channel-adaptive fair policy is quite fast. 7.6.4 User-Oriented Heuristics
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The schemes described above are based on a more global view of the system: maximizing system throughput, maximizing aggregate utility, and maintaining systemwide user fairness. We also consider two heuristics that are more useroriented. The rst one is called longest queue rst (LQF), which gives a higher priority to a user that has a larger backlog of packets to be sent. This heuristic is useful for mobile devices with only a limited amount of buffer space. Indeed, variants of LQF were found to be highly effective in one study [63]. The second one is largest delay rst (LDF), which gives a higher priority to a user that has its head of line packets delayed by the longest amount of time. However, it is important to note that unlike the four approaches described earlier, these user-oriented heuristics do not make use of the channel adaptation mechanism in that users are not selected on the basis of their relative channel conditions. Thus, although a channel-adaptive physical layer (e.g., the VTAOC scheme) is still incorporated in the model, there is no synergy between the rate allocation policy and the physical layer.
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