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7.3 Monotonic Decreasing Rate Scheduler
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As discussed in Section 7.2, the fundamental limitation of existing smoothing algorithms is the need for upward rate adjustments, of which correct operation depends on the successful and timely completion of network resource reservations. To remove this limitation, we can use only downward rate adjustments in the transmission schedule. In other words, the initial transmission rate will be the highest, with each subsequent rate lower than the previous one. We call this algorithm the monotonic decreasing rate (MDR) scheduler for obvious reasons. In this chapter, we focus on prerecorded videos. The MDR schedule for a video is computed of ine and is stored with the video for use during video streaming. We present an algorithm to compute the MDR schedule in the next section and derive several properties of the algorithm in Section 7.3.2 to Section 7.3.4.
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7.3.1 Computing the MDR Schedules
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Compared to existing smoothing algorithms, the MDR property introduces an additional constraint only downward rate adjustments can be used. Note that although this reduces the set of possible schedules within the feasible region, it still does not uniquely determine the transmission schedule for a given video. In fact, there are still an in nite number of possible MDR schedules within the feasible region. To select a MDR schedule, we need to consider the resultant resource requirements. The choice of the MDR schedule can affect the peak transmission rate and the client buffer requirement, both of which should be minimized. Interestingly, it turns out that we can always compute a MDR schedule that has minimum peak rate and minimum client buffer requirement, among all possible MDR schedules. We de ne a MDR transmission schedule with the set of rate-time tuples: {ri , Ti |i = 1, 2, . . . , n}, where ri and Ti are the transmission rate and commencing time for the ith segment in the transmission schedule as depicted in Figure 7.2; and n is the total number of segments in the MDR transmission schedule. For a MDR transmission schedule, the rates will be monotonic decreasing, i.e., ri > r j , for all i, j where n j > i 1. To compute the schedule, we begin from the origin as depicted in Figure 7.2, and assign the rst segment with the highest transmission rate, i.e., r1 = max A(t) A(0) t > 0 t (7.5)
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and mark the time, denoted by T1 , at which the rate is maximized. The tuple {r1 , T1 } then represents the rst segment of the MDR transmission schedule. Next, we repeat this process with T1 as the starting point to obtain {r2 , T2 }, and so on. In general, the transmission rate for the next segment can be computed from ri+1 = max A(t) A(Ti ) t > Ti t Ti (7.6)
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ri: transmission rate for segment i r3 Accumulated Data r2 MDR schedule S(t) Second bit-rate reduction point: T2 First bit-rate reduction point: T1
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Figure 7.2 A monotonic decreasing rate schedule generated by the MDR scheduler
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until it reaches the end of the video. It can be shown (see Appendix) that the above procedure guarantees that the generated transmission schedules are monotonic decreasing. The transmission schedule is then de ned from the resultant rate-time tuples {ri , Ti }:
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S(t) =
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s( )d , where s( ) = ri , for Ti < Ti+1
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(7.7)
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The rate-time tuples are then stored together with the video data. The video server will simply schedule the transmission of the video according to this MDR transmission schedule. Its monotonicity property ensures that once a stream is admitted, there will always be suf cient system bandwidth for the whole duration of the video stream, even if there are other random traf cs such as web, le transfer, etc., in the system. This MDR scheduler has several additional desirable properties, namely modest admission complexity, minimum peak rate, and minimum client buffer requirement. These are discussed in the following sections.
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