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where Y denotes a set of sessions that are backlogged, and for a nonleading session chosen to transmit packets in the real system, Ni is updated as follows Ni Ni + li ri f (F i ) (8.12)
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and Li is de ned in a similar way. Here, Li keeps track of the normalized amount of services received by session i that is proportional to its channel state function when it is leading. When a session i becomes leading and no longer suffers from the worst channel state, Li will get initialized in a way analogous to that expressed in Equation (8.11). In the real system, selection is made among the nonleading sessions rst. The session with the minimum Ni will be selected, and the packet at the head of the waiting queue of this session will be transmitted and Ni will be updated accordingly. If there is no such kind of session that is nonleading and backlogged, the system will select from the leading ones in the increasing order of the sessions Li, and then Li will be updated accordingly. If all sessions are not backlogged (a very unlikely situation in a mobile computing system with a reasonable number of active users), dummy packets will
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be sent. If the session j selected in the real system is not the one chosen in the error-free one and it is i that is selected in the error-free system, the D of i and j will both be updated: Di Di + li, Dj Dj - lj; otherwise, D will not be changed. When a session with a comparatively bad channel state transmits a packet, either Ni or Li will increase more rapidly than will a session with a better channel state.As the punish factor changes, we can decide how serious we should punish a session that does not have a perfect channel and transmits packets.The larger the punish factor is, the more serious we punish the unlucky sessions. Let us consider a simple example. Suppose that session i has channel state 0.75 (i.e., class B), session j has channel state 0.25 (i.e., class D), and both are nonleading and have the same service rate, and all packets are of the same length l. If the punish factor is 1, then after both of them transmit one packet, l l Ni increases by 1.33 and Nj increases by 4 .The reason why Nj increases ri rj much more than Ni is that j has a much poorer channel state than does i while both of them get one packet transmitted. After that, j has less chance of transmitting because of the large Nj value, so what the system does is give j the chance to transmit but punishes it because it wastes bandwidth. If we change l the punish factor to 0.5, then Ni increases by 1.15 after i transmits one ri l packet and Nj increases by 2 after that. So, as expected, j is punished only rj moderately as the punish factor decreases. Long-Term Fairness. Nonetheless, there is still one issue to be considered. Although the sessions that do not have excellent channel states but get packets transmitted are punished, they are given some chance to transmit, and part of the bandwidth of the system is wasted and can never be compensated. Because the scheduler will not schedule a leading session to transmit if there is a lagging one that is backlogged and is not in the worst channel state (i.e., state E), the scheduler will not conserve the effort of the system as most of the other scheduling algorithms do. Thus, we assign a service share to a virtual compensation session to help in the long term. This preallocated service share is used to help the lagging ones with perfect channel state because only when a session has a perfect channel state, can it get compensation most ef ciently. When a lagging session exits from nonperfect channel states, its session ID will be queued in the virtual compensation session. Sessions that are queued in the virtual compensation session are in the decreasing order of their D. So we give bonus service to the lagging sessions if it has a perfect channel state, and the session that lags the most will get it rst so that it can be assisted in catching up and thus, longterm outcome fair can be maintained. The design of the virtual compensation session is quite different from that in SBFA. In SBFA, only sessions with perfect channel states will be scheduled while sessions with bad channel states will get their transmission opportuni-
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