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threshold, the normalized amount of timeslots allocated can be quite different. In order to maintain outcome fair, the scheduler allocates more timeslots (i.e., exerts more effort) on a session with a very poor channel state. Thus, outcome fair is maintained at the expense of the system throughput. However, in order to avoid the pathological case that the poor session wastes too much of the system throughput (i.e., despite signi cant effort, the outcome is still not enough), a power factor is used to control the amount of effort exerted on such extremely unlucky sessions. In order to maintain outcome fair, the ELF approach cannot avoid wasting some bandwidth so as to achieve a fair distribution of realized throughput to the sessions with poor channel conditions. Proportional Fairness (PF). Designed for HDR (high-data-rate) services in CDMA systems, proportional fair [60] (PF) is considered to be a simple yet effective fairness notion. Speci cally, based on the utility based concept2 introduced by Kelly [66], HDR (downlink) scheduling is performed in a TDMA manner (i.e., only one user is selected for high-data-rate transmission in each burst session). At the scheduling time t, suppose that a session i has an average realized throughput Hi(t) over a past time window of length t (i.e., from time t - t to t), and the real throughput that can be achieved by session i at time t is li(t). A scheduler is said to achieve proportional fair if it selects for transmission the session with the largest value of l i (t ) H j (t ) (8.5)
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Furthermore, the notion of proportional fairness has the nice property that a proportional fair allocation cannot be replaced by any other arbitrary allocation that does not lead to a reduction in the aggregate fractional rate change. It should be noted that a proportional fair scheduler heuristically tries to balance the services of the sessions in terms of outcome, while implicitly maximizing the system throughput in a greedy manner. Obviously, the proportional fairness notion is a purely outcome fairness metric. Thus, while such a metric is simple to use, proportional fairness does not guarantee fairness in a strict sense. For example, consider the situation where a session has experienced a prolonged period of poor channel states and hence has a small value of Hi(t). It may not get service even though its channel condition improves [e.g., with a moderately large value of li(t)] if there is a dominant session that has a very good channel state [i.e., a very large value li(t)]. Furthermore,
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As illustrated in 6, proportional fair utility is de ned as a logarithmic function of the rate allocated to a user. Because of the convex nature of the logarithmic function, diminishing return is modeled.
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the delay experienced by sessions can also be very high in a proportional fair system. 8.4.4 Channel-Adaptive Fairness
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There is a relatively new notion of fairness called channel-adaptive fairness (CAF). Speci cally, a scheduler is channel-adaptive fair if in the short term the difference between the normalized throughput (normalized with respect to the channel capacity) of any two backlogged sessions i and j is bounded as follows Ti (t1, t 2 ) Tj (t1, t 2 ) <e ri f (F i ) rj f (F j ) (8.6)
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where Fi denotes the channel state (e.g., one of the ve classes A, B, C, D, and E), and f(Fi) = M(Fi)h, in which M(Fi) is the effective throughput factor (0 M(Fi) 1). The effective throughput factor is channel-state-dependent. For example, M(Fi) = 0.75 if Fi is channel state B. Here, h is a punish factor, which is a positive number. Thus, in our de nition of fairness, the throughput that a session receives will be proportional to its channel quality and outcome fair is maintained among all sessions in the long term. This fairness notion considers explicitly different channel states. Unlike the CIFQ algorithm that prevents the sessions without excellent channel state from transmitting, and unlike the ELF algorithm that distributes the normalized amount of service inversely proportional to their channel states, a CAF scheduler provides transmission opportunities to all sessions that do not suffer from the worst channel state in the short term, and at the same time, it punishes the sessions without good channel states to a different extent. Furthermore, unlike the proportional fair scheduler, using the CAF scheduler does not necessarily schedule the session with the best channel condition to transmit rst. With the channel-adaptive fairness, we can formalize a new fair queueing algorithm that is explained in detail in the following section. The punish factor h can help us decide between making use of bandwidth more ef ciently and treating every session more fairly. When a larger value of h is used, we punish the nonperfect channel state sessions that transmit packets more seriously and prevent them from wasting too much bandwidth. In effect, the bandwidth is used more ef ciently, the average delay of the total system is decreased, and the throughput is increased. However, if there is a session that is more unlucky than the others and has a higher probability of having a bad channel state, its average delay and throughput may be very bad because it is punished seriously and prevented from occupying the bandwidth. When a smaller punish factor is used, this kind of unlucky session will be punished only moderately and both the average delay and throughput of these sessions will be reduced. But as they have more chance to access the bandwidth and hence incur a larger wastage of bandwidth, the total throughput and
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