Multimedia Systems (2010) 16:151–168 DOI 10.1007/s00530-010-0181-1
ORIGINAL RESEARCH PAPER
A relaxing bandwidth smoothing schedule for transmitting prerecorded VBR video in periodic network Zilei Wang • Hongsheng Xi • Guo Wei
Received: 7 January 2009 / Accepted: 13 January 2010 / Published online: 5 February 2010 Springer-Verlag 2010
Abstract Minimizing the number of bandwidth changes of individual stream and enabling the transmission schedule flexible can benefit increasing the number of concurrent multiplexed streams in periodic network without degrading quality of video. However, minimum change bandwidth allocation (MCBA) only provides a fixed transmission schedule for given video and client buffer although it achieves the minimum number of rate changes and the minimum peak rate. So the MCBA schedule cannot be adjusted when multiple streams are multiplexed over a limited bandwidth shared link. In this paper, we propose a relaxing transmission schedule that provides a range of rates for each interval and preserves the minimum number of rate changes without increasing any rate. As a result, the individual schedule can be adjusted by network service to try to satisfy bandwidth constraint when the total required bandwidth of multiplexed streams exceeds the available network capacity. First, we underline that the key of MCBA is to segment video with making each linear trajectory as long as possible. Then we use linear separability and support vector machine under the classification model to compute the longest interval and the required rate, respectively. Due to our proposed decoupled form between Communicated by Cormac Sreenan. Z. Wang (&) H. Xi Key Laboratory of Network Communication System and Control, Chinese Academy of Sciences, University of Science and Technology of China, 230027 Hefei, Anhui, China e-mail:
[email protected] G. Wei Electronic Engineering and Information Science Department, University of Science and Technology of China, 230027 Hefei, Anhui, China
segmentation and rate computation, finally, we construct the relaxing schedule bounded by the original MCBA rate and the minimum rate. The simulation using real MPEG4 and H.264 video traces confirms the philosophy and evaluates the relaxing performance of our proposed algorithm. Keywords Bandwidth smoothing Classification model Minimum changes bandwidth allocation (MCBA) Support vector machine (SVM) Statistical multiplexing Relaxing schedule
1 Introduction With the development of computer and network techniques and the standardization of compression algorithms, many interesting video services, such as video on demand, video conference, and IPTV, become very popular due to many valuable commercial systems. In these systems, media streaming server connected with large, fast disks retrieves the requested video data and transmits the consecutive packets to client through backbone network and access network [1]. Variable bit rate (VBR) encoder is widely employed in most video systems due to providing high quality of video at low bit-rate [2, 3]. However, VBR streams can result in lower network bandwidth utilization and increased price of reserved bandwidth because of such inherent burstiness [4]. By introducing playback buffer at the client site, bandwidth smoothing, which transmits the future big frames prior to their due playback time before each burst, can reduce the bandwidth requirement variation to alleviate this problem [5]. Such smoothness for individual stream can be measured through different metrics, such as the peak rate, the number of bandwidth changes, the variability of bandwidth requirements, and the buffer
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requirement. Various smoothing algorithms [3] can optimize one or multiple metrics, which is useful for different network services to multiplex streams [5]. Three different types of quality of service (QoS) may be provided by the underlying network for video transmission, including deterministic service, statistical service, and best effort service [6]. Deterministic service can offer the guaranteed delivery with deterministic bounds of delay, delay jitter, and loss rate. But it results in low network bandwidth utilization and high individual connection cost [7] because it needs reserve bandwidth resource, according to the peak rate. Statistical service can improve the resource utilization by statistically multiplexing streams over different intervals after partitioning video into small segments [8]. Finally, best effort service does not guarantee any QoS parameter, which transmits the needed packets only when the underlying network bandwidth is available [7]. Thus, best effort service is not suitable for transmitting high quality video. To improve bandwidth utilization, renegotiated constant bit rate (RCBR) [6], a statistical multiplexing network service with deterministic QoS guarantee, was proposed. This type of service periodically allocates bandwidth according to the locale maximum rate of individual stream in given interval [9]. Without specification, we assume RCBR with guaranteeing deterministic QoS is employed over the shared bandwidth link in this study. Both bandwidth smoothing and statistical multiplexing are effective multiplexing techniques to increase the number of concurrent streams [5]. For stored video with a prior knowledge of frame sizes and frame rate, many offline smoothing algorithms [3] have been proposed to achieve different optimization criteria. Minimum changes bandwidth allocation (MCBA) [10] results in the minimum number of rate changes, and minimum variability bandwidth allocation (MVBA) [11] results in the minimum variability of bandwidth requirements, along with the minimum peak rate. Piecewise constant rate transmission and transport (PCRTT) [12] and enhanced PCRTT (E-PCRTT) [13] connect the rate lines into a continuous transmission schedule after partitioning video duration into equal intervals. Rate-constrained bandwidth smoothing (RCBS) [14] results in the minimum buffer requirement under a constrained bandwidth by transmitting video as later as possible. These smoothing algorithms produce a piecewise linear schedule with a fixed rate during one interval such that they can be used in RCBR service. However, the above smoothing algorithms may not achieve high bandwidth utilization when many streams are multiplexed although they can achieve some optimal metric for individual stream. Statistical multiplexing can further improve bandwidth utilization of smoothed streams [5]. So many multiplexing algorithms for smoothed traffics
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[13, 15, 16] are also proposed. Hadar [13] multiplexes streams smoothed by E-PCRTT, and renegotiates the shared bandwidth at each boundary of intervals. Anastasiadis [15] provides the optimal lexicographic smoothing schedule under limited bandwidth and unlimited client buffer or unlimited bandwidth and limited client buffer. Furthermore, Stern [16] computes the optimal multiplexing schedule for equal intervals through linear programming. These multiplexing algorithms simultaneously allocate the required bandwidth of each ongoing stream, commonly resulting in high computational complexity. To decrease the complexity of admission control and guarantee QoS, Lai [17] proposed a monotonic-decreasing rate (MDR) schedule with the first rate maximal. It can decide whether new stream can be admitted only using the first required bandwidth. However, MDR may introduce a little waste of bandwidth for preserving the monotonicity property of summed stream. RCBR, different from deterministic service, may arise a competition failure when the total required bandwidth of multiplexed streams exceeds the available capacity of shared link [6]. Many strategies have been proposed to cope with such congestion. Discarding some enhanced layer data for layer-coded video [18] or some unimportant B frames [19] and introducing a pause on the congested interval [19] can both reduce the bandwidth requirement. For generic case, the frame selection technique [1, 20] can be adopted, where each frame is scored according to its motion information. Frame selection aims to maximize the cumulative score without violating bandwidth constraint. The approaches, however, would decrease either frame rate or video resolution, degrading quality of perception (QoP) of viewers for high quality video services. The congestion commonly occurs within several intervals, where most streams require high bandwidth, if the multiplexed streams have been smoothed. Then adjusting the preset transmission schedule of individual stream is an effective strategy to cope with congestion without degrading QoS. In most time, client buffer has cached some future data or has extra space to cache more data, especially at the boundary of intervals. It is possible that we decrease the transmission rates of some intervals with guaranteeing QoS of the ongoing streams because the less transmitted data during the congested interval can be compensated through increasing the rates or prolonging the durations of adjacent intervals. Using E-PCRTT [13] on individual stream, Hadar [9] proposed a recursive renegotiation multiplexing algorithm called BFSM. It can decrease the required bandwidth of the maximum-rate stream by transferring some data into the adjacent intervals, whereby the overall multiplexed stream is still smooth. The algorithm, however, introduces another complicated problem how to choose good parameters, such
A relaxing bandwidth smoothing schedule
as the window length, which straightly determine renegotiation effectiveness of network service [9, 21]. In the existing smoothing algorithm, MVBA schedule [11] based on majorization complicates bandwidth management of periodic network because of its frequent rate changes if the smoothed streams are directly multiplexed [22]. On the contrary, MCBA [10] may decrease the probability of renegotiation failure because its minimum number of bandwidth changes requires less renegotiation times. However, the original search algorithm [10] of MCBA just greedily computes a serial of transmission rates, which, simultaneously, is coupled with finding inflection points (where the change of transmission rate occurs). Thus, MCBA under given client buffer results in a fixed schedule with fixed rates and fixed inflection points, not providing any optional schedule for network service. Consequently, RCBR cannot adjust the individual MCBA schedule especially when the ongoing streams compete for the limited shared bandwidth. In this paper, we propose a relaxing smoothing schedule corresponding to MCBA that not only preserves the minimum number of rate changes and the minimum peak rate but also provides a range of rates for each interval. Network service can select a suitable rate from the precomputed range to improve overall bandwidth utilization. Our main contributions focus on two aspects. First, we decoupled segmentation and rate computation of MCBA under our provided classification model. To minimize the number of rate changes, the key of MCBA is to modestly segment video by making each interval as long as possible. In this paper, we use the linear separability to examine for the bounded point of each interval (the reachable farthest point, after which the buffer constraint would be violated at any fixed rate). Then we use support vector machine (SVM) to fast work out each transmission rate and the critical points. Second, we proposed an efficient iterative algorithm to compute the minimum rate by connecting left overflow point and right underflow point. So we can construct a relaxing schedule bounded by the original MCBA rate and the minimum rate for each interval. The simulation using real-world MPEG4 and H.264 video traces shows the minimum rate transmission schedule, compared with the original MCBA schedule, can averagely reduce the bandwidth requirement by about 10% and use the decreased rate for 15–60% total duration under various buffer sizes. So we can consider that the relaxing schedule provides a large adjustment space for statistical multiplexing over a shared link. The rest of this paper is organized as follows. In Sect. 2, the related works of off-line bandwidth smoothing and MCBA are introduced. In Sect. 3, we formulate bandwidth smoothing into a binary classification problem, then decouple segmentation and rate computation of MCBA
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that are implemented through linear separability and SVM, respectively. In Sect. 4, we provide an iterative algorithm to compute the minimum rate and construct a range of rates for each interval. Section 5 presents some simulation results of the proposed algorithm on MPEG4 and H.264 prerecorded video traces. Finally, we summarize this paper with future works in Sect. 6. 2 Related works 2.1 Off-line bandwidth smoothing With a prior knowledge of frame sizes of stored video and client buffer size, off-line bandwidth smoothing can precompute a transmission schedule [3]. Media streaming server generally transmits video data at a series of fixed rates to make transmission curve as smooth as possible with guaranteeing QoS. Such smoothness can be measured by many metrics, which is represented by the number of bandwidth changes and the peak rate in this paper. For most transmission schedules, the change of bit rate occurs only when the buffer constraint would be violated at the last rate so that the reserved buffer resource can be fully utilized. In this respect, bandwidth smoothing becomes a trade-off technique between network bandwidth and client buffer size [21]. Before going into details, we summarize the used major notations in Table 1. Given an N-frame video with frame size si (i = 1,…, N) bytes and frame slot DT = 1/F (F is the frame rate, typiP cally 25 or 30), the total size of frames is C = Ni=1si, and the average rate is ravg = C/N, where time is measured in units of frame slots. Without loss of generality, assume client starts to playback at zero time. For video display, the client decoder processes one frame data every time, i.e., video data in the client playback buffer is taken out as one entire frame. Thus, the cumulative amount of consumed data by client for i frames can be formulated into a staircase function as [3] i X LðiÞ ¼ sj : j¼1
Suppose a0 bytes have been cached in the client buffer before playback. Let ai denote the amount of data transmitted during the ith frame slot. Then the cumulative amount of data transmitted by frame i is AðiÞ ¼
i X
aj þ a0 :
j¼1
Let B denote the client buffer size. Define the overflow curve [3] as UðiÞ ¼ min fLði 1Þ þ B; Cg:
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Besides the overflow and underflow curves, any feasible schedule is also constrained by the start line (0, B) on the cumulative bytes coordinate and the finish line A(i) = C. The above four curves form a closed constraint region [7], and any feasible transmission curve must stay inside the region, as shown in Fig. 1. The transmission schedule is commonly over at the time t0 less than N due to cached data in client buffer, which is defined as the minimum time by which the cumulative amount of transmitted bytes has reached the total size of video C
Table 1 Notations used in the paper Symbol Meaning F
The frame rate of video
N
The number of frames in the video
si
The frame size of video (1 B i B N)
ti
The time when the ith frame is taken by decoder P The total size of video, C ¼ Ni¼1 si
C B
The client buffer size, B [ max1BiBN si
ai
The transmitted data during the ith frame slot
rk L(i)
The transmission rate over the kth segment P The cumulative transmitted bytes, AðiÞ ¼ ij¼0 aj Pi The cumulative consumed bytes, LðiÞ ¼ j¼1 sj
U(i) bk
The ith overflow point The intersection of (k - 1) and kth SVM trajectories
b0k
The intersection of (k - 1) SVM and kth altered trajectories
b00k
The intersection of (k - 1) altered and kth SVM trajectories
bvk
The intersection of (k - 1) and kth altered trajectories
mk
The bounded point of kth trajectory
xk
The first intersection of kth trajectory with L or U
cuk,1
The first critical point on U over the kth segment
cuk,l
The last critical point on U over the kth segment except mk
clk,1 clk,l
The first critical point on L over the kth segment
A(i)
t0 ¼ min fti jAðiÞ Cg: 0 ti N
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The last critical point on L over the kth segment except mk
To prevent client buffer from underflowing and overflowing, any valid transmission schedule must stay between the overflow curve U and the underflow curve L [7], as shown in Fig. 1 LðiÞ AðiÞ UðiÞ; 0 i N: ð1Þ Figure 1 shows a transmission schedule that consists of three rate lines, which changes the bit rate from r1 to r2 at t1, then switches to r3 at t2. Finish line Cumulative Bytes
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In bandwidth smoothing algorithms, MCBA results in the minimum number of bandwidth changes and the minimum peak rate through pursuing the longest trajectory for each run. So it can be used in both deterministic and statistical network services. MCBA [10] is first proposed as an extension of CBA to compute the optimal schedule in terms of the number of rate changes, including both increasing and decreasing. The original MCBA algorithm iteratively computes next rate trajectory by searching the inflection point on frontier of the current trajectory so that new trajectory can reach the farthest point without violating the buffer constraint. Using the strict definition and proofs, Junbiao [7] further describes details of MCBA via hub and split, and explains a smoothing relationship between connect costs and QoS of video, including the initial delay. MCBA has a worst-case computational complexity of O(N2log C) [3] because of the binary search on frontiers. Every trajectory of MCBA schedule is tightly clamped by the underflow curve L and the overflow curve U, whose tangent points are called critical points in [10] or constraint points in [7]. Without causing ambiguity, we call such points as critical points, denoted by clk or cuk . Now, we recall the computation procedure of MCBA and some deduced important properties to direct the design of our algorithm. Figure 2 shows an example of the first trajectory, which reaches the farthest point t3 by searching the starting point on the frontier (0, B). As a result, a0 bytes as the initial buffer occupancy are cached before playback that represents the startup delay of schedule. The transmission line R1 lying between L and U is tangent to the overflow and underflow curves at t1 and t2, respectively. Then the frontier (t2, t3) for next run is formed that is bounded by the last intersection on U and the last critical point on the opposite side L. The frontier splits the constraint region into the left and right regions. The above procedure is repeated on new frontier until the transmission curve reaches the finish line. The final schedule, a continuous transmission curve between U and L, is formed that
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U
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i Fig. 2 MCBA finds the longest trajectory R1 between U and L by searching on the frontier (0, B), forming the new frontier (t2, t3) for next run, where ti is the tangent point or the intersection between R1 and L or U
is composed of the starting points and the last ending point together with the corresponding transmission rates. We can see that the transmission rate is worked out by searching the inflection point on the latest frontier [10] in this algorithm. The segmentation of MCBA is coupled with the rate computation. The resulting MCBA schedule is compact so that it cannot be adjusted on multiplexing since every trajectory is tightly clamped by U and L [7].
3 MCBA under classification model Smoothing algorithms aim to construct a feasible path from the starting line to the finish line under the river model in Sect. 2.1, and make the required bandwidths as smooth as possible or decrease the required client buffer size [3]. Actually, the transmission schedule is feasible as long as it does not violate the buffer constraint at every discrete temporal point in Fig. 3 [7]. Thus, the transmission curve is also a classification curve of the overflow and underflow points. We can formulate bandwidth smoothing into a binary classification problem (for details please refer to our earlier paper [21]). As for MCBA, the schedule introduces a new segment only when the buffer constraint has to be violated whichever point of frontier the current rate line starts from. The transmission line would intersect with either U or L at the farthest point, called upperbounded or lowerbounded, respectively [7]. In this paper, we decouple segmentation and rate computation of MCBA through the bounded point that is the reachable farthest point of one fixed-rate line. We first segment video by examining for the bounded point of each interval under the classification model. Then we apply some classification algorithm to calculate the transmission rate of any provided interval (from the starting point to the bounded point). As a result, the MCBA schedule can be worked out through two independent steps:
Fig. 3 Classification model of off-line bandwidth smoothing, consisting of the underflow points (dots) on L and the overflow points (multiplication symbol) on U, which are determined by video frame sizes and client buffer size B
finding the bounded point and calculating the transmission rate, which offers a chance to compute some relaxing schedule with different rates. 3.1 Segmentation In this study, we first use the classification model in Fig. 3 to examine for the bounded point. Let Li = (i, li), i = 1, 2,…, N denote the underflow points on L (‘‘’’ in Fig. 3) and Uj = (j, uj) = (j, lj ? B - sj), j = 1, 2,…, M be the overflow points on U (‘‘9’’ in Fig. 3). The bounded point the kth segment is denoted by mk, before which we can transmit data at the latest rate rk, while not for the next point mk ? 1 where it would violate the buffer constraint. Thus we can denote by (ck, mk) the frontier on the kth rate line, where ck = clk,l for the upperbounded case and ck = cuk,l for the lowerbounded case. Now, we use the classification approach to sequentially obtain all bounded points rather than checking the rate line in the original search algorithm. Under the classification model, we can show that finding the MCBA bounded point of one fixed-rate line is equivalent to constructing the maximum linear separable set of overflow and underflow points. Theorem 1 Given the frame sizes of video si, i = 1, 2,…, N and the client buffer size B. Then finding the MCBA bounded point mk of the longest trajectory Rk from the frontier (ck-1, mk-1) is equivalent to examining for the farthest separable point nk, to which the overflow and underflow points from ck-1, mk-1 are linear separable, but not for the next point nk ? 1, i.e., mk = nk. Proof Without loss of generality, we only consider the case that the previous rate line Rk-1 is lowerbounded, i.e., the critical point ck-1 lies on the overflow curve, and the last bounded point mk-1 lies on the underflow curve.
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Assume the kth rate line is Rk, and its bounded point is mk. It is obvious that Rk can classify the overflow points in [ck-1, mk] and the underflow points in (mk-1, mk]. If we move mk onto next point ðmk mk þ 1Þ; the buffer constraint has to be violated because the MCBA trajectory Rk is longest. Therefore, mk limits the biggest linear separable set of the overflow points from ck-1 and the underflow points from mk-1, which actually is the last point nk of the kth interval. On the other hand, assume that the linear separable set of the overflow points from ck-1 and the underflow points from mk-1 can reach the farthest point nk. Then there exists a line that can correctly classify the overflow points in [ck-1, nk] and the underflow points in [mk-1, nk] but not for the point (nk ? 1). So the MCBA rate line Rk reaches nk at most since Rk also belongs to the classification lines. Due to the MCBA’s greediness, mk = nk. h Theorem 1 indicates that we can find out the bounded point mk using any classification algorithm that can check the linear separability if the starting points are given. In particular, the convex envelopes can fast check the linear separability through sequentially adding a pair of overflow and underflow points Li and Ui at the same temporal point, which was proposed in our earlier paper as the envelop algorithm [23]. In this study, we use the algorithm to examine for the bounded points. 3.2 Rate computation The following step of segmentation is to calculate the transmission rate rk of the provided interval that can reach the bounded point mk. In MCBA, the frontier with boundaries of the critical point ck and the bounded point mk need to be used for the rate computation of next run. Thus the rate-computation algorithm is demanded not only to calculate the rate but also to obtain the current critical point ck. In the previous paper [21], we have proven that SVM [24] can achieve the minimum buffer requirement for constant rate transmission and transport (CRTT) [12]. The SVM rate rsv k can be worked out using the shifted overflow and underflow points [21]. Thus we introduce the SVM classification algorithm in this study to calculate the MCBA rate that can simultaneously obtain rsk and the critical point ck. Support vector machine is a supervised machine learning technique based on the statistic learning theory. Consider a binary classification problem where samples ðxi ; yi Þ; xi 2 RN ; yi 2 f1; 1g; i ¼ 1; 2; . . .; l are linearly separable. The goal of SVM is to find the optimal classification hyperplane of samples, which maximizes the minimum distance between the hyperplane and all samples (called margin) [24] as
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min kx xi k : x 2 RN ; ðw xÞ þ b ¼ 0
max w;b
yi ððw xi Þ þ bÞ [ 0; i ¼ 1; . . .; l;
s.t.
ð3Þ
where w is the weighting vector and b is the offset, which together determine the hyperplane. The resulting margin equals 2=kwk; which means that maximizing the margin is equivalent to minimizing the magnitude of the weighting vector w. By introducing Lagrange multipliers ai and relaxing variables fi for the soft margin, the problem can be transformed into a dual quadratic programming problem under the Karush-KuhnTucher (KKT) conditions. P minðkwk2 =2 þ C li¼1 fi Þ ð4Þ s.t.yi ðhw; xi þ bÞ 1 fi ; i ¼ 1; 2; . . .; l fi 0; i ¼ 1; 2; . . .; l: When fi = 0, the problem is a hard classification that does not permit any misclassified sample and requires the samples be linearly separable. The KKT conditions and the obtained vector of the above optimization problem are ai ½yi ððxi wÞ þ bÞ 1 ¼ 0; w¼
l X
y i ai xi :
ð5Þ
i¼1
We can see that w is determined only by the samples (xi, yi) having nonzero Lagrange multiplier ai, called support vectors. These samples lie on the boundaries of the decision region in geometry and determine the best hyperplane. The distances between the hyperplane and two class of support vectors are equal for symmetric samples. In MCBA, assume that we have obtained the SVM classification line from (5) for the given interval that is optimal to classify the overflow points and the underflow points X ai yi hx; xi i þ b ¼ 0; ð6Þ i2sv
where xi = Li or Ui, i [ sv denotes the overflow and underflow support vectors, hx1, x2i denotes the inner product of the vectors x1 and x2, and yi [ {-1, 1} indicates whether the point xi belongs to overflow points (?1) or underflow points (-1). Without loss of generality, we uniformly use (ti, vi) to represent xi in the following discussion. Then the SVM transmission rate rsv k and the corresponding initial buffer occupancy a0 for the rate computation can be calculated as P ai yi ti sv rk ¼ P i2SV ; ð7Þ i2SV ai yi vi X a0 ¼ b= ai yi vi : ð8Þ i2SV
Obviously, they are also determined by all support vectors. The final rate line of the kth interval in Eq. (6) is formulated into
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v ¼ rksv t þ a0 ;
ð9Þ
and t is the time index within the kth interval represented by the frame number. As an example, Fig. 4 shows how to use SVM in offsv line bandwidth smoothing, where Usv i and Lj denote the support vectors on U and L, respectively. The resulting SVM rate line Rk has equal margin d with the overflow and underflow points, i.e., the maximized minimum distance between the classification line Rk and the samples [24]. In addition, we can directly mark the critical point ck from the support vectors owning to such equality. The MCBA rate line can achieve the longest interval and is tightly clamped by U and L. Thus the SVM rate line is that of MCBA, i.e., s rsv k = rk, since the previous segmentation is greedy like the original search algorithm. Lemma 1 After obtaining the kth bounded point mk for the frontier (ck-1, mk-1), MCBA can transmit video data at the SVM rate rsv k within the current interval, and the critical point ck is the last support vector on the opposite side of mk (ck and mk lie on different curves: ck on L and mk on U, or ck on U and mk on L), which form the new frontier (ck, mk) for next run. Proof Without loss of generality, we assume that the trajectories Rk-1 and Rk are both upperbounded, i.e., mk-1, mk lie on U, and ck-1, ck on L. Let rsv k denote the SVM rate and rsk is the original MCBA rate. The required buffer size for rsk is B. For the current interval with the overflow range (mk-1, mk) and the underflow range (ck-1, mk), rsv k has the buffer requirement Bsv. Then Bsv B B because the SVM rate has the minimum buffer requirement. Thus rsv k is also a feasible transmission rate under a larger buffer B. As defined, the critical points of MCBA are the tangent points between the rate line and the buffer constraint curves. Then they must also be support vectors of SVM that has the minimum distance from the rate line Rk. Therefore, the needed critical point ck is the last support U U 2sv
U1sv
d
Rk L
d Lsv1
Fig. 4 SVM rate line A classifies the overflow points on U and the sv underflow points on L with overflow support vectors {Usv 1 , U2 } and }, where the margin is d for both L and underflow support vector {Lsv 1 U
vector on the opposite side of mk, which form the new frontier (ck, mk). h According to our previous paper [21], the SVM rate can keep invariable if the projection of the maximum overflow support vector is always greater than that of the minimum underflow support vector on the SVM rate line for all buffer sizes. So we can compute the SVM rate line under a larger buffer that makes the overflow points and the underflow points more distant. Consequently, the consumed time can be greatly shortened since it strongly depends on the distribution of two classes of points [21]. Furthermore, It is very important that the critical point ck can simultaneously be obtained from the support vectors according to Lemma 2 even under the larger buffer. 3.3 SVM transmission schedule Using the above proposed algorithms for single interval, including the segmentation and the rate computation, we can construct an entire continuous transmission schedule. We first explain how to choose the starting points of next run so that the procedure can be repeated and the adjacent rate lines are connected between U and L. Consider two possible case in Fig. 5 where Rk is upperbounded and lowerbounded, repectively. In Fig. 5a, Rk intersects the overflow curve U at mk. It lies between the (i - 1)th frame slot and the ith frame slot, i.e., mk [ [ti-1, ti), and Rk has the critical point ck on L, which together with mk forms the frontier (ck, mk). In order to guarantee Rk is connected with Rk?1 between U and L (rk?1 \ rk) under the classification model, the overflow point ck and the underflow point ti in Fig. 5a need to be included into the next classification set. On one hand, Rk and Rk?1 can be connected between U and L only if Rk?1 can classify ck and ti. On the other hand, any feasible classification line Rk?1 of next run must correctly classify ti and ck. Therefore, the starting points of next overflow and underflow classification points are floor(mk ? 1) and ck, respectively, where floor(x) represents the maximum integer not greater than x. For the lowerbounded case in Fig. 5b, the bounded point mk is identical with some frame slot ti. Similarly, the critical point ck on U and the bounded point mk (ti) are the starting points of next run. So far we have decoupled the computation of MCBA schedule, in which we use the linear separability to examine for each bounded point and apply SVM to calculate the transmission rate. The SVM transmission schedule can be constructed by connecting all rate lines. However, our algorithm is still greedy to pursue the longest trajectory, just like the original search algorithm [10]. As a result, the schedule cannot be adjusted for multiplexing streams yet. To tackle this problem, we propose a relaxing schedule in the next section with preserving the minimum
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the durations of the adjacent intervals that have higher rates would be prolonged, and the current duration with the lower rate is shortened. Otherwise the low-rate duration can be increased if a larger allocated bandwidth is provided. Thus, network service may achieve some graceful compromise between the allocated bandwidth and its duration for each stream if the provided schedule can be adjusted. Compared with the original MCBA schedule produced by the greedy search, some different schedule may result in lower required bandwidth on some intervals while simultaneously preserving the minimum number of rate changes as MCBA. Consider a virtual example in Fig. 6, where the longest trajectory Rk from the frontier (cuk-1, mk-1) is obtained, and the new frontier (cuk , mk) for next run Rk?1 is produced. From Fig. 6, it is obvious that Rk does not achieve the minimum rate since the rate line R0k has a lower required bandwidth that still connects Rk-1 and Rk?1 at b0k and b00kþ1 , respectively.
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Fig. 5 The longest trajectory Rk reaches the farthest intersection mk under the discrete time model on a the overflow curve U, and b the underflow curve L, forming new frontier (ck, mk)
number of rate changes and the minimum peak rate. Making use of the decoupled form of our proposed algorithm, we try to decrease the required bandwidth so that network service can adjust downwards the rate of individual stream during the congested interval and satisfy the limited bandwidth constraint as much as possible.
Utilizing our proposed decoupled form of MCBA in which the bounded and critical points can be obtained in advance, we can explore the minimum rate of the given segment. Intuitively, the schedule need cache enough data in the client buffer at the starting point and consume almost all data at the ending point in order to achieve the minimum rate. Thus, the minimum rate line can be constructed by connecting the left overflow point and the right underflow point in the classification model, i.e., the minimum rate line R0k has the overflow critical point cuk and the underflow critical point clk with cuk \ clk that are the tangent points or the intersections between the rate line and U (L). In this Cumulative Bytes
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Network service needs to reduce the bandwidth requirement of some streams when the shared link becomes congested [9]. As for MCBA, it is expected that transmission rates can be decreased without increasing the number of rate changes and the peak rate, and that the new rate lines can form a continuous feasible schedule inside the constraint region. In this paper, we propose a relaxing schedule that is implemented through calculating the minimum rate of each interval. Network service can allocate individual stream, an appropriate bandwidth between the minimum rate and the original MCBA rate, according to the rate–duration relationship of each stream during the congestion interval. When network service allocates a smaller bandwidth to some stream on congested interval,
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paper, we propose an algorithm to find out two such points on the overflow and underflow curves. Before digging into details of our algorithm, we first introduce some useful notations from Ref. [7] that denotes a rate line with respect to different relationship with its adjacent rate lines: • • • •
[inc, inc]: the previous rate is smaller than the current rate, and the next rate has an increase; [dec, inc]: both the previous rate and the next rate are greater than the current rate; [inc, dec]: both the previous rate and the next rate are smaller than the current rate; [dec, dec]: the previous rate is greater than current rate, and the next rate has a decrease.
The current rate line Rk is lowerbounded with rk?1 [ rk for both [inc, inc] and [dec, inc] cases, as shown in Fig. 6. Rk has the underflow critical points clk and the overflow critical point cuk with clk \ cuk . Thus, the SVM rate rsv k can be further decreased because its underflow tangent point is before the overflow one. Let xk?1 denote the intersection between the next line Rk?1 and L. Then xk?1 is not greater than mk. We can calculate the minimum rate rm k within the overflow range (cuk-1, xk?1) and the underflow range (mk-1, xk?1), from which we choose two critical points to form the rate line R0k . On the other hand, rk?1 \ rk for both [inc, dec] and [dec, dec] cases in Fig. 7. Rk orderly has the critical points cuk and clk with cuk \ clk. Thus, Rk has already achieved the minimum rate. In the following discussion, we focus on the first case that rk?1 [ rk, and such Rk that has the lowerbounded point mk is called the rate-increase segment in this paper. We can show the new rate lines, whichever the original SVM rate line Rk or the shifted rate line R0k with lower rate,
can form a continuous feasible schedule between U and L, i.e., R0k can connect its adjacent rate lines Rk-1 (R0k1 ) and Rk?1 (R0kþ1 ) without violating the buffer constraint. First, we verify that Rk can connect the SVM rate lines Rk-1 and Rk?1. Without loss of generality, we consider Rk in Fig. 6 where Rk-1 is also lowerbounded. The current rate line is bounded by the frontier (cuk-1, mk-1) on Rk-1 and the next rate line Rk?1. The shifted line R0k has the intersections b0k and b00kþ1 with Rk-1 and Rk?1, respectively. The condition R0k \ rk decides b0k \ bk and b00kþ1 \ bk?1. In addition, the inflection points b0k and b00kþ1 are limited by the boundary points, i.e., b0k [ cuk-1 and b00kþ1 [ xk?1. So R0k can connect Rk-1 and Rk?1 between U and L. However, the adjacent rate lines may be shifted to R0k1 and R0kþ1 with lower required bandwidth, as shown in Fig. 8. For the previous rate line in Fig. 8a, R0k1 and Rk intersect at the point b00k . Let bvk denote the intersection between R0k1 and R0k . We can see R0k1 must be under Rk-1 from the critical point cuk-1. Consequently, cuk-1 \ bvk \ b0k ,
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which implies R0k1 and R0k can be connected between U and L. Similarly, we can assert R0k and R0kþ1 can also be connected between U and L, as shown in Fig. 8b. Finally, the lower rate lines R0k , k = 1, 2,…, M can form a continuous feasible schedule, and keep the number of rate changes invariable. Therefore, we can decrease the rate so long as it does not violate the buffer constraint. Now, we show how to calculate the minimum rate rm k so that the relaxing schedule can transmit video at any rate in s (rm k , rk). Without loss of generality, we explain the details through the rate increase case [dec, inc] in the following discussion. A direct approach is to examine the connection rate of any pair of overflow and underflow points within the given ranges, then to find the minimum one. But it has the computation complexity of O(N2) because of the exhaustive search. In practice, we can speed up the procedure. 4.1.1 Search range We first improve the computation complexity by narrowing the search ranges that can decrease the sample number N of the complexity. In the proposed decoupled algorithm, the rate rsk is worked out by SVM, which simultaneously produces two border lines with equal margins that are represented by all support vectors. According to such feature, we can significantly narrow the search ranges to hasten the computation of the minimum rate. Figure 9 shows two different cases of Rk with rk-1 [ rk and rk?1 [ rk, where the border lines of SVM are denoted by the dashdot line. In Fig. 9, the overflow search range of Rk is (mk-1, l xk?1), and the underflow one is (ck-1,l , xk?1). Let cuk,1 represent the first overflow support vector. Due to the border lines of SVM, any lower rate line R0k cannot overflow after cuk,1 if it can satisfy the buffer constraint at cuk,1. So we can choose (mk-1, cuk,1) as the overflow search range for calculating the minimum rate. Now, we narrow the underflow search range according to the non-decrease property of rate lines. We first consider the case in Fig. 9a, where there is no any underflow support vector after cuk,1, i.e., clk,j \ cuk,1 for any j. Let ek represent the contour point on L where its cumulative bytes are equal to that of the bottom overflow point mk-1, i.e., L(ek) = U(mk-1). Since the transmission rate must be positive, we choose (ek, xk?1) as the underflow search range if ek B xk?1. Otherwise ek [ xk?1, as shown in Fig. 9a. Such case likely occurs if the current segment Rk is short and its adjacent rates rk-1 and rk?1 is much greater than rk. It is common in our following simulation. Then two contour line Ruk and Rlk through ek and xk?1 are formed within the current interval. Therefore, it is permitted that the schedule does not transmit any data at all during the kth interval, i.e., the schedule can extremely select any zerorate line between Ruk and Rlk, just accompanied with the
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shift of the inflection points b0k and b00kþ1 . As for the case in l Fig. 9b, where the last underflow vector support ck,l before u l xk?1 is larger than ck,1, (ck,l, xk?1) is chosen as the underl flow search range since the underflow points before ck,l cannot violate the buffer constraint. Concluding, we can narrow the underflow search range to (max{ek, clk,l}, xk?1) for calculating the minimum rate. 4.1.2 Calculating the minimum rate In this paper, we design a fast algorithm to calculate the minimum rate rm k if the narrowed search ranges are given. The algorithm iteratively computes the critical points cuk and clk that can form the minimum rate line, instead of the exhaustive search. In particular, we explain the principle of our iterative algorithm through the first several loops in Fig. 10.
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For convenience, let plk,j (puk,j) denote the jth intermediate critical point on L (U) for calculating rm k , i.e., the tangent point between the intermediate rate line and L (U) by fixing the starting point on U (L), as shown in Fig. 10. Here, we start the iterative computation from the last point in the overflow search range (mk-1, cuk,1), i.e., cuk,1 is the initial fixed point. Then the first intermediate critical point plk,1 in the underflow search range is obtained at which the intermediate rate line ! cuk;1 ; plk;1 and L are tangent without violating the underflow ! buffer constraint. However, the line cuk;1 ; plk;1 may cause the client buffer overflow before cuk,1. Thus, we need continue to compute the overflow critical point puk,1 in the overflow search range (mk-1, cuk,1) by fixing the starting point plk,1. The above procedure, including to find the overflow critical point by fixing the latest underflow critical point and to find the underflow critical point with fixing the latest overflow critical point (which is called to compute the critical point on the opposite side in the following discussion), is repeated until the final connection line does not violate both buffer constraints. For example, the iteration in Fig. 10 is over through three times of computation, and two final critical points cuk = puk,1 and clk = plk,2 construct the minimum rate line R0k . Actually, the following theorem can guarantee the convergence of the iterative algorithm. Theorem 2 Given the overflow and underflow search ranges, if we iteratively compute the critical point on the opposite side according to the above method, which is started from the overflow support vector cuk,1, then the minimum rate rm k without violating the buffer constraint can be achieved in a finite loops. Proof
As shown in Fig. 10, we focus on the first inter ! ! mediate line cuk;1 ; plk;1 and the second one plk;1 ; puk;1 : Without loss of generality, we assume that both of them violate the
buffer constraints so that the iteration has to go on. Then ! cuk;1 ; plk;1 achieving the rate rk,1 may overflow only before cuk,1. To satisfy the overflow buffer constraint, the following ! line plk;1 ; puk;1 needs to increase the rate with rk,2 [ rk,1. Similarly, the new rate needs to be increased if the current rate line violates the underflow buffer constraint, such as ! puk;1 ; plk;2 : Therefore, the latest rate keeps increased if the intermediate rate line violates whichever the overflow constraint or the underflow constraint. Next, we show that the proposed iterative procedure can finish in a finite loop using this feature. We first check the second underflow critical point plk,2 that is found with fixing puk,1, and the new rate rk,3 ! ([ rk,2 [ rk,1) is achieved. The line puk;1 ; plk;2 with rk,3 can ! not introduce underflow after plk,1 since cuk;1 ; plk;1 with the smaller rate rk,1 does not underflow after plk,1, i.e., plk,2 B plk,1. For generic situation, it is easily verified that plk;jþ1 plk;j puk;jþ1 puk;j ; where the equality is satisfied only if the buffer constraints are not violated, then the iteration is over. Therefore, the overflow critical point cuk keeps moving to the left and the underflow one clk to the right. Consider the triangle in Fig. 10 with the vertices cuk,1, plk,1, and puk,1. The new underflow critical point plk,2 only stays inside the triangle according the increasing rate. Consequently, the underflow search range is significantly narrowed, so is the overflow search range. On the other hand, the initial overflow and underflow search ranges are both composed of the finite number of discrete points. Thus, our iterative algorithm can finish in a finite loops since it keeps narrowing the ranges. The resulting line achieves the minimum rate by connecting two final critical point. h Our iterative algorithm is efficient in practice. The minimum rate can be worked out within several loops (typically 1–2 times) in our following simulation. For an extreme case, connecting the first overflow support vector cuk,1 and the last underflow support vector clk,l can sometimes result in the final minimum rate without underflowing and overflowing. 4.2 Schedule So far we have been able to work out a transmission schedule with the minimum rate for each interval, along with the minimum number of rate changes and the minimum peak rate. Then a relaxing schedule can be constructed, whose rate is bounded by the minimum rate rm k
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and the original MCBA rate rsk for any segment Rk. We can transmit video data at any rate rk between rsk and rm k. In this paper, we propose a simple strategy to determine the transmission line for achieving any middle rate s m rk [ (rm k , rk). Let xk represent the intersection between the original rate line Rk and the minimum rate line R0k . We demand that any rate line must run through that point, i.e., we adjust the rate of given segment through rotating the rate line with fixing the point xm k . The s resulting rate line with rk is feasible since rm B r k k B r k. After determining the transmission rate with the fixed point xm k , the inflection points can be worked out by computing the intersections between adjacent rate lines. Thus, an appropriate schedule derived from the relaxing schedule can be determined by network service for multiplexing streams, which may require a lower bandwidth on some intervals with the cost of a little shift of inflection points.
5 Experimental results In this paper, we assume that RCBR network service is employed over the shared bandwidth link. For off-line bandwidth smoothing, the transmission schedule is carried out in a constant bandwidth channel with deterministic delay. Thus, we evaluate the performance of different algorithms directly through their smoothing results on some video traces. Our used metrics include the required buffer size (in units of MB), the bandwidth requirement (in units of KB/[Frame Slot] or the corresponding abbreviation KBpf), and the number of rate changes. MCBA would result in different schedule with different rates and different inflection points for different video. To clearly demonstrate the effectiveness of our proposed algorithm, some real video traces encoded by the typical MPEG4 and H.264 are used, including Star Wars IV and NBC News, whose clips are specified in Sect. 5.1 (their related information can be downloaded from http://trace.eas.asu.edu/ h264/index.html) [25]. In the simulation, we first compare the minimum rate schedule and the original MCBA schedule under two fixed buffer sizes so that the philosophy of our proposed algorithm can be clearly shown. Then we smooth the video clips under various buffer sizes, and evaluate the relaxing degree of our schedule. It is measured through the rate decrease coefficient and the total rate-decreased temporal duration of the minimum rate rm k , compared with the original MCBA rate rsk, which are defined in Sect. 5.3. The relaxing degree intuitionally represents the adjustable space for network service to multiplex streams. In the following results, our proposed algorithm and the original MCBA algorithm are represented by Min and Search, respectively.
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Without any specification, we always consider the proposed algorithm results in the minimum rate schedule. 5.1 Video trace MPEG4 and H.264 codec have been widely applied in many commercial VoD and IPTV services due to providing high-quality video at low bit rate. We use real MPEG4 video Star Wars IV and H.264 video NBC News [25] to examine the smoothing performance of our proposed algorithms. Table 2 lists some statistics of Star Wars IV and NBC News. Star Wars IV is a science fiction movie with frequent and sharp scene changes. NBC News is a news-like video that has relatively steady scenes. In Table 2, the mean frame size of Star Wars IV is smaller than that of NBC News, but the variability of Star Wars IV is greater. So Star Wars IV would result in larger bandwidth requirements if any bandwidth smoothing algorithm were not applied. Figure 11 exhibits the distribution of first 5,000 frame sizes of two video traces. Star Wars IV in Fig. 11a has many short-term burstiness, and the maximum ratio of different frame sizes is greater than 10. For an example, the frame size suddenly becomes much larger around the 100th frame slot. NBC News in Fig. 11b, however, is relatively stable during the entire video duration so that it is able to be smoothed under a smaller client buffer size. We use the first 5,000 frames of each video trace in the following experiments. Each video clip is treated as a full movie, i.e., the constraint region has a finish line of C. 5.2 Performance analysis We smooth two different video clips using Min and Search algorithms, and show the philosophy of our proposed algorithm through comparing their smoothing results under the same buffer size. Min, compared with Search, can decrease the required bandwidth on some intervals. Then we use the decrease degree of required bandwidth to measure the effectiveness of our algorithm that is represented by the rate decrease coefficient as Table 2 Some statistics of Star Wars IV and NBC News Encoder
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the first six segments, where the last two rows are the rate decrease coefficient and the iteration number for calculating the minimum rate, respectively. We can see that Min can significantly decrease the required bandwidth on the first, third, and fifth segments as the following rates are increased. On the contrary, Min and Search result in the equal required bandwidth on the second and fourth segments. So our proposed algorithm can only reduce the
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Table 3 Smoothed bandwidth requirements of Search and Min for the first 5,000 frames of Star Wars IV Index
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bandwidth requirement of the rate-increase segments, while not for the rate-decrease case. In particular, Star Wars IV has 21 (of 44) rate increase segments with the total temporal duration 2,632 (of 5,000) if Search is used. While the total temporal duration is decreased to 2,242 (of 5,000) if Min is applied, which is the compensation of the rate decrease. In addition, Min extremely results in the zero rate on the first and third segments. Then network service can introduce a pause to the zero-rate streams without affecting their consecutive display if the shared link is congested during such intervals. Such case may occur only for [dec, inc] segments, where the cached data can maintain consecutive video display due to a big buffer occupancy at the starting point of interval. Even for some long rate-increase segments, like the fifth segment, Min can significantly decrease the required rate. Therefore, our proposed schedule provides a big relaxing space for network service to improve bandwidth utilization. Table 4 provides the smoothing results of NBC News, where the client buffer size 1.66 mb is used according to its larger mean frame size. Both Min and Search result in 25 segments. Our proposed algorithm has a similar effectiveness like for Star Wars IV except the smaller rate variation since NBC News has the steady frame sizes. From Tables 3 and 4, Star Wars IV requires lower transmission rates than NBC News because the rate is mainly determined by the mean frame size if the short-term burstiness has been eliminated by smoothing algorithms. The average rate-decrease coefficient of two clips are about 11.95 and 11.62% on the entire video duration. They are nearly equal since Min is almost independent on the fluctuation sharp of different video. In addition, the iteration number for two clips is not less than 2, which shows that our algorithm can fast work out the minimum rate in practice.
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We show the example of Star Wars IV in Fig. 12, including the original MCBA schedule and the minimum rate schedule, to exhibit the philosophy of our algorithm. Here, the first three segments are plotted under two client buffer sizes (818.3 and 1,727.6 kb) so that the difference between Search and Min can be highlighted. Now, we show why and how our algorithm can decrease the required bandwidth for the rate-increase segments. In Fig. 12, the MCBA schedule on the rate-decrease segment (the second segment) has a left overflow critical point and a right underflow critical point so that it has achieved the minimum rate. So Min and Search have the identical rate line on those segments except for the inflection points. Let us focus on the second segment in Fig. 12a. Search ends this interval around 281, while Min prolongs its duration to 291 with the same rate. Then the minimum rate schedule transmits more data during the second interval and an idle duration is following (the idle is eliminated under a grater buffer size in Fig. 12b which is replaced by a low rate segment). Thus, the minimum rate schedule pre-transmits some video data during the second interval that should be transmitted by the original MCBA schedule during the third interval. As a result, the minimum rate schedule partly compensates some video of the idle segment through prolonging the duration of previous high-rate segment. Similarly, if the next rate is higher, the minimum rate schedule transmits data first at rm k , then at the m higher rate rm k?1 ([rk ) so that the total transmitted data can reach that of rsk, i.e., the minimum rate schedule needs to start next high-rate segment in advance, compared with the original MCBA schedule, such as the first interval in Fig. 12. Therefore, Min shortens the duration of rate-increase segment. The inflection points are shifted towards the lower rate segment. From another viewpoint, Min makes full use of free buffer space or cached data around the inflection points by adjusting the transmission rate. 5.3 Performance comparison We have shown that Min and Search produce the equal number of intervals in theory. Now, we compare Search and Min under various client buffer sizes by simulation. In particular, we define two metrics to evaluate our algorithm, including the rate-decrease degree and the rate-decreased duration. They together express the overall relaxing performance of our proposed schedule. The rate-decrease degree is represented by the average rate-decrease coefficient pd, which is defined as PM i s PM i m s pi;k k¼1 ðri;k ri;k Þ=ri;k ¼ k¼1 ; pdi ¼ Mi Mi where Mi represents the achieved number of segments for the ith buffer size, and rsi,k, rm i,k, k = 1,…, Mi denote the kth rate of the MCBA schedule and the minimum rate
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Fig. 12 Smoothed transmission curves achieved by Search and Min for the first three segments of Star Wars IV under two buffer sizes
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The rate-decreased duration tdi is the total temporal duration of Min decreased rate, compared with the original MCBA schedule, which is defined as
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Fig. 13 The average ratedecrease coefficient and the rate-decreased temporal ratio versus the buffer size for two video clips
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So tdi represents the temporal effectiveness of Min for the client buffer Bi, which, together with pdi, represents the adjustment space of our relaxing schedule. In following results, however, we use the ratio between tdi and the total number of frames N = 5,000 to replace tdi.
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In simulation, we choose different initial buffer sizes for two clips according to their maximum frame size, then add the buffer size by 4 kb every time until the total added amount reaches 1 mb. As a result, 256 different buffer sizes are produced. Figure 13 shows the smoothing metrics versus client buffer size relationship for both video clips, including the average rate-decrease coefficient and the ratedecreased temporal ratio. The average rate-decrease coefficient is irregularly fluctuant for two video clips. As we have known, both Search and Min maximize greedily segment video that maximize the current trajectory regardless of the rest video data. The bounded points are shifted backwards as the buffer size is increased while determining whether the transmission rates can be decreased is not considered at all. Thus, the average rate-decrease coefficient depends on both the distribution of frame sizes and the client buffer size, resulting in an irregular curve. In Fig. 13, the average ratedecrease coefficient of Min varies around 10% for both Star Wars IV and NBC News, i.e., the minimum rate schedule can averagely decrease the required bandwidth by 10% although it only works on the rate-increase segments. In addition, the coefficient of Star Wars IV has sharper fluctuation than that of NBC News because its frame sizes are more bursty. Therefore, our proposed algorithm has relatively stable bandwidth effectiveness to different video and different buffer sizes although the average ratedecrease coefficient depends on both of them. However, the rate-decreased temporal ratio demonstrates different fluctuation for two video clips. Star Wars IV in Fig. 13a is irregular (40–60%), and averagely achieves near 50%. So the minimum rate schedule can use the decreased rate for almost half of video duration. But NBC News in Fig. 13b results in shorter temporal ratio with a wider fluctuation range (15–40%). Furthermore, the ratedecreased duration has a tendency to become lower as the buffer size is increased. Due to the steady frame sizes, NBC News can achieve a good smoothing schedule even under a smaller buffer. So NBC News can almost be fully smoothed when the provided buffer is big enough, and has no longer obvious effect if the buffer size is further increased. Thus, our relaxing schedule would result in lower rate-decreased temporal ratio for larger buffers. From the results of Star Wars IV and NBC News, the relaxing schedule is more effective to the bursty movies.
6 Conclusion By decoupling the segmentation and the rate-computation of MCBA under the binary classification model, we propose a relaxing transmission schedule in this paper that can preserve the minimum number of rate changes and the
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minimum peak rate. This enables network service to adjust the bandwidth requirement of individual stream when the shared bandwidth link is congested. First, the MCBA schedule can be worked out by linear separability and SVM, then we proposed an iterative algorithm to calculate the minimum rate that can finish in several loops in practice. Finally, the relaxing schedule can transmit video at any rate between the minimum rate and the original MCBA rate by rotating the rate lines of rate-increase segments. Through the simulation using two real video traces, the proposed schedule provides a large adjustable space for network service to multiplex streams with QoS guarantee so that the bandwidth utilization can be improved. One main contribution of our algorithm is to independently determine the longest segment and its rate. So we, utilizing this feature, can design some on-line smoothing algorithm to adapt variable network bandwidth and achieve a smaller number of rate changes for RCBR. In addition, network service should multiplex rate-increase segments of one stream with rate-decrease segments of other stream. This type of multiplexing can likely adjust the summed required bandwidth of different streams when bandwidth congest occurs. Thus, designing some adaptive algorithm and some effective strategy of network service with MCBA may become our future works, where our proposed algorithm in this paper is their benchmark. Acknowledgments The authors would like to thank G. V. der Auwera, P. T. David and M. Reisslein for supplying the MPEG4 and H.264 video traces at http://trace.eas.asu.edu/h264/index.htm. In addition, we appreciate the reviewers for their provided valuable suggestions in revising the paper. This work was supported by National High-Tech Research and Development Program of China (863 Program) under grant no. 2008AA01A317 and National Natural Science Foundation of China under grant no. 60774038.
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