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SCHEDULING MODEL
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FFQ assumes that the channel is error-free, or at the very least, that errors are not location-dependent (i.e., all backlogged ows have the ability to transmit at a given time, or none of the ows can; here, a ow is called backlogged if it has packets pending to be transmitted). Speci cally, FFQ is neither fair nor able to provide minimum throughput bounds in the presence of locationdependent channel errors. In order to capture the behavior of ows in a wireless environment while bearing the constraints of the channel in mind, the error-free service of a ow is used as the service that it would have received at the same time instant if all channels had been error-free, given identical offered load. A ow is then said to be leading if it has received channel allocation in excess of its errorfree service. On the other hand, a ow is said to be lagging if it has received channel allocation less than its error-free service. A ow that is neither leading nor lagging is said to be in sync. Ideally, an ef cient packet scheduling algorithm should support a fair service model with the following properties: 1. Short-term fairness among in-sync backlogged ows that perceive an error-free channel 2. Short-term throughtput bounds for ows with an error-free channel 3. Channel-conditioned delay bounds for packets 4. Long-term fairness among backlogged ows with bounded channel errors 5. Long-term throughtput bounds for all ows with bounded channel errors 6. Support for both delay-sensitive and error-sensitive data ows 7. (Optional) Optimization of the schedulable region by decoupling the delay and bandwidth requirements of ows 8.2.3 A Motivating Example
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In FFQ, each ow i is given a weight ri (the negotiated service share), and for any time interval [t1, t2] during which there is no change in the set of backlogged ows, the channel capacity granted to each ow i, denoted by Wi(t1, t2), satis es the following property: Wi (t1, t 2 ) Wj (t1, t 2 ) =0 ri rj (8.1)
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Now, consider three backlogged ows during the time interval [0, 2] with r1 = r2 = r3. Flow 1 and ow 2 have error-free channels, while ow 3 perceives a channel error during the time interval [0,1). If the scheduler is aware of the channel state of the ows, then it will not consider ow 3 during [0,1). Thus, by applying Equation (8.1) over the time periods [0,1) and [1,2], the following allocations result: W1[0,1) = W2[0,1) = 0.5, W1[1,2] = W2[1,2] = W3[1,2] = 1 . 3
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CHANNEL-ADAPTIVE WIRELESS FAIR SCHEDULING
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However, over the time interval [0,2], the allocations are W1[0,2] = W2[0,2] 5 = 6 , while W3[0,2] = 1 . Obviously, the overall allocation is unfair to ow 3. 3 This example illustrates that it is dif cult to de ne fairness and design a suitable scheduler. The problem is that wireless fair queueing must distinguish between a nonbacklogged ow (for which no compensation is provided in fair queueing) from a backlogged ow that perceives an error channel. However, compensating for the latter will void the separation property of the fair queueing.
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In general, the behavior of a wireless packet scheduler can be structured as consisting of the following components: (1) error-free service model, (2) lead lag model, (3) compensation model, and (4) slot and packet queues. They are elaborated in the following text. 8.3.1 Error-Free Service Model
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The error-free service provides a reference for how much service a ow should receive in an ideal error-free channel environment. Typically, the error-free service is some packetized approximation of FFQ and a commonly used algorithm is the weighted fair queueing (WFQ). In WFQ, each ow i in a set of ows is allocated a weight ri. The kth packet pik of ow i is assigned a starttime tag S(pik ) and a nish-time tag F(pik ) according to the following algorithm:
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Lk -1 i k S( pik ) = max V ( A( pik )), S( pik -1 ) + , where Li is the length of the kth ri packet of the ow i, A(pik ) is the arrival time of the packet, and V(t) is the virtual time at time instant t. Lk i F ( pik ) = S( pik ) + . ri dV C = . dt i ri At each time, the packet with the minimum nish tag (i.e., the packet whose last bit would complete transmission rst among all backlogged packets in the uid model) is transmitted. Lead Lag Model
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There are two approaches in this model: 1. The lag of a ow is computed as the difference between the error-free service and real service received by the ow. In this case, a ow that falls
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