THE EMERGENCE of high-speed internetworks facilitates. Smoothing Variable-Bit-Rate Video in an Internetwork

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1 202 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 7, NO. 2, APRIL 1999 Smoothing Variable-Bit-Rate Video in an Internetwork Jennifer Rexford, Member, IEEE, Don Towsley, Fellow, IEEE Abstract The burstiness of compressed video complicates the provisioning of network resources for emerging multimedia services. For stored video applications, the server can smooth the variable-bit-rate stream by transmitting frames into the client playback buffer in advance of each burst. Drawing on prior knowledge of the frame lengths client buffer size, such bwidth-smoothing techniques can minimize the peak variability of the rate requirements while avoiding underflow overflow of the playback buffer. However, in an internetworking environment, a single service provider typically does not control the entire path from the stored-video server to the client buffer. This paper presents efficient techniques for transmitting variablebit-rate video across a portion of the route, from an ingress node to an egress node. We develop efficient techniques for minimizing the network bwidth requirements by characterizing how the peak transmission rate varies as a function of the playback delay the buffer allocation at the two nodes. Drawing on these results, we present an efficient algorithm for minimizing both the playback delay the buffer allocation, subject to a constraint on the peak transmission rate. We then describe how to compute an optimal transmission schedule for a sequence of nodes by solving a collection of independent single-link problems, show that the optimal resource allocation places all buffers at the ingress egress nodes. Experiments with motion-jpeg MPEG traces show the interplay between buffer space, playback delay, bwidth requirements for a collection of full-length video traces. Index Terms Bwidth-smoothing, internetwork, majorization, prefetching, variable-bit-rate video. I. INTRODUCTION THE EMERGENCE of high-speed internetworks facilitates a wide range of new multimedia applications, such as distance learning entertainment services, that rely on the efficient transfer of compressed video. However, constantquality compressed video typically exhibits significant burstiness on multiple time scales, due to the frame structure of the compression algorithm as well as natural variations within between scenes [1] [5]. The presence of this burstiness complicates the effort to allocate network resources to ensure continuous playback of the video at the client site. To avoid excessive loss delay in the network, service providers can provision link buffer resources to accommodate the Manuscript received July 29, 1997; revised September 16, 1998; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor S. K. Tripathi. J. Rexford is with the Networking Distributed Systems Research Center, AT&T Labs-Research, Florham Park, NJ USA ( jrex@research.att.com). D. Towsley is with the Department of Computer Science, University of Massachusetts, Amherst, MA USA ( towsley@cs.umass.edu). Publisher Item Identifier S (99) variable bwidth requirements. However, the high peak variability of the transmission rates substantially increases the network resources required to transfer the video. To reduce the end-to-end resource requirements, a storedvideo server can smooth the outgoing stream through workahead transmission into the client playback buffer. By initiating transmission early, the server can send large frames at a slower rate without disrupting the client application; the client system can then retrieve, decode, display frames at the stream frame rate. Drawing on a priori knowledge of the frame lengths client buffer size, bwidth-smoothing algorithms [6] [14] generate a server transmission schedule that consists of a sequence of constant-bit-rate runs that avoid underflow overflow of the playback buffer. With a modest client buffer size, effective smoothing algorithms can produce schedules with fairly small numbers of runs a significant reduction in both the peak variability of the transmission rates, as discussed in the overview in Section II. Smoothing substantially reduces network resource requirements under several different network service models, including peak-rate reservation, bwidth renegotiation, effective bwidth [9]. Previous work on bwidth smoothing has focused on video-on-dem servers that store the entire sequence of frames can directly coordinate access to the client playback buffer. However, in a realistic internetworking environment, a single service provider may not control the entire path from the stored video at the server through the communication network to the playback buffer at the client site. Instead, each network service provider has dominion over a portion of the route that may or may not include the server client locations. Still, bwidth smoothing can be extremely effective at reducing the overhead of transporting variable-bitrate video within a subset of the route through the network. To minimize resource requirements in an internetworking environment, this paper generalizes the bwidth-smoothing model to a sequence of nodes along part of the path from the stored-video server to the playback buffer [15]. These nodes can represent individual switches or routers in the domain of a single network service provider, or dedicated proxy servers that perform bwidth smoothing. For example, bwidth smoothing may be performed at network peering points or at the gateway to the client access network. Starting with a simple model of an ingress node that performs workahead transmission into an egress node, Section III describes how to compute the buffer allocation start up delay that allow bwidth smoothing to achieve the /99$ IEEE

2 REXFORD AND TOWSLEY: SMOOTHING VARIABLE-BIT-RATE VIDEO IN AN INTERNETWORK 203 maximum reduction in the peak transmission rate. By characterizing how the peak rate varies as a function of the buffer allocation start up delay, we show that a simple binary-search algorithm is sufficient for finding the optimal configuration. The proofs, presented in the Appendix, draw on the majorization smoothing algorithm introduced in [9] additional properties established in [16]. Section IV then describes how to minimize the buffer space start up latency subject to a constraint on the peak rate in the transmission schedule, drawing on earlier work [12], [17] on stored video. We show that it is possible to jointly minimize the buffer space start up delay using an algorithm with complexity that is linear in the number of frames in the video stream. For the common case where the ingress node receives the original unsmoothed stream, we show that the optimal buffer allocation divides the buffer space evenly between the ingress egress nodes. In Section V, we turn our attention to the tem-smoothing model with a sequence of nodes, describe how to efficiently compute a smooth transmission schedule for each link in the sequence. In concurrent work, the authors of [18] describe a similar tem-smoothing problem present heuristics for iteratively computing the transmission schedules. We show that optimal transmission schedules can be computed by solving independent instances of the singlelink problem. We then show that an optimal allocation of resources places all buffers at the ingress egress nodes, allowing us to apply the results from Sections III IV to the more general tem model. These results draw on the underlying properties of the majorization smoothing algorithm. In Section VI, we describe the results of experiments with full-length motion-jpeg MPEG video traces. These experiments highlight the performance of the smoothing algorithm provide guidelines for allocating buffer space start up latency. Section VII summarizes the paper discusses future research directions. II. SMOOTHING STORED VIDEO A multimedia server can substantially reduce the rate requirements for prerecorded video by transmitting frames into the client playback buffer in advance of each burst. Previous work on bwidth smoothing typically assumes an infinitesource single-link model, where the server stores the entire video stream generates a single transmission schedule based on the overflow underflow constraints on the client buffer. A. Smoothing Constraints As an example of constraints on bwidth smoothing, consider a server that stores an entire sequence of video frames with sizes for transmission to a client with an -byte playback buffer. For continuous playback of the video stream, the server must always transmit enough data to avoid buffer underflow, where indicates the amount of data consumed at the client by time, where. To prevent overflow of the playback buffer, the client should not receive more than bytes by time,as Fig. 1. The sample transmission plan S is a nondecreasing curve consisting of three linear runs that stay between the upper (U) lower (L) smoothing constraints. The second run increases the transmission rate to avoid an eventual buffer underflow, while the third run decreases the transmission rate. shown in Fig. 1. Any valid server transmission plan should stay between these vertically equidistant functions. Workahead transmission occurs whenever the schedule lies above the underflow curve. To permit smoothing at the beginning of the video, the model can introduce a playback delay by shifting the underflow overflow curves to the right by time units, resulting in for, where for. More generally, the vectors can represent a wide variety of constraints on the transmission of data from the server site. To formalize the bwidth-smoothing problem, suppose that we are given three vectors,,,, such that. We say that is feasible with respect to if only if,,. The vector represents the schedule as a sequence of transmission rates, where denotes the amount of data transferred during the time interval,. When there is no possibility of confusion, we write the schedule, as its rate vector as.if, then is said to be a change point in the vector. Moreover, it is said to be a convex change point if a concave change point if. The example in Fig. 1 has a convex change point, followed by a concave change point. Note that the run responsible for the peak transmission rate involves a bwidth increase (convex change point) followed by a bwidth decrease (concave change point). B. Majorization Schedule The constraints typically result in multiple feasible schedules, each with different performance properties. Several smoothing algorithms compute schedules that minimize the peak rate to reduce the bwidth requirements for transmitting the video stream. Minimizing other moments of the transmission rates can also reduce the video s empirical effective bwidth [9], [19], allowing the server the network to multiplex a larger number of streams. The theory of majorization [20] offers a useful way to compare the smoothness properties of different schedules. For,, let ( ) be the th largest component of ( ). Vector majorizes ( ), if for ; intuitively, the

3 204 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 7, NO. 2, APRIL 1999 requirement for transmitting stored video, as shown by the graphs in Fig. 2 taken from [21]. (a) (b) Fig. 2. The first graph plots a portion of the majorization schedule for a motion-jpeg encoding of the movie Jurassic Park, while the second graph plots the peak bwidth as function of the buffer size x for 20 full-length motion-jpeg traces. elements of vector are more evenly distributed than the elements of. The majorization relation implies that for any Schur-convex function :.In addition to common smoothness metrics such as the peak variance, this class of functions includes for all convex :. Applying the theory of majorization to the bwidthsmoothing problem, there exists an algorithm [9] which finds a vector such that for all. In constructing this majorization schedule, the algorithm computes linear trajectories that extend as far as possible between the lower upper constraints, ultimately encountering both the curves. Whenever a bwidth increase (decrease) is necessary to avoid crossing the ( ) curve, the algorithm starts a new trajectory as early as possible to limit the size of the change. Hence, change points always reside on the or curves, with at convex points (rate increases) at concave points (rate decreases) [9], as in the example in Fig. 1. The majorization algorithm produces a sequence of constantbit-rate runs that substantially reduces the peak bwidth C. Domination Refinement Results The majorization algorithm has several properties that facilitate comparisons between different instances of the bwidthsmoothing problem; in particular, the following domination refinement lemmas [16] play an important role in analyzing the models introduced in Sections V III-B. The domination result compares the constraints of two smoothing configurations: Lemma 2.1: Let be such that. Then. Moreover, if, is a concave change point in,itisa concave change point in. Similarly, if, is a convex change point in, then it is a convex change point in. Proof: This was established in [16]. The proof of is by contradiction. If the schedule dips below at some time, then the schedule must eventually have a rate increase. This rate increase would occur at a convex change point, where hits. But, which violates the assumption that lies below. Hence,. If has a convex change point at time (hitting ), then implies that also has a convex change point at time (hitting ). Similarly, if has a concave change point at time (hitting ), then implies that also has a concave change point at time (hitting ). Focusing on one set of constraints, the domination lemma implies that raising the curve causes concave change points to gradually disappear in the corresponding majorization schedules; similarly, lowering the curve causes convex change points to gradually disappear. These change points disappear at a series of critical buffer sizes. Drawing on this result, the refinement property characterizes how the majorization schedule changes with the size of the playback buffer. With a slight rephrasing of the result in [16], we have: Lemma 2.2: Let be such that, where is a vector whose components are equal to.if, then the change points in are a superset of the change points in. Proof: This was established in [16] by invoking Lemma 2.1 twice. Applying the lemma to the original pair of smoothing constraints shows the result for concave change points; applying the lemma a second time, after raising the curves by, implies the result for convex change points. Capitalizing on the majorization algorithm its properties, the next section analyzes a more general model where the ingress node receives the video stream performs workahead transmission into the buffer at the egress node. III. MINIMIZING THE PEAK RATE We begin by considering bwidth smoothing over the route between an ingress node an egress node,

4 REXFORD AND TOWSLEY: SMOOTHING VARIABLE-BIT-RATE VIDEO IN AN INTERNETWORK 205 Fig. 3. This figure shows a single-link system with an x-bit egress buffer an (M 0 x)-bit ingress buffer, with an arrival vector A, a playout vector D, the optimal transmission vector S 3. (a) (b) (c) Fig. 4. As a function of the buffer allocation x, the peak-rate curve p(x) has (a) shrinking growing phases; (b) shrinking phase only; (c) growing phase only. determine how to compute the buffer allocation start up delay that allow the maximum reduction in the peak transmission rate. By characterizing the shape of the peakrate curve, we show that a simple binary-search procedure is sufficient to determine the optimal allocation of buffer space between the ingress egress nodes. A similar result holds for the selection of the start up latency. The section ends with a brief discussion of practical issues, including propagation delay, jitter, limited knowledge of frame sizes. A. Single-Link Model The single-link model consists of an ingress node that performs workahead transmission of the arriving video stream into the egress node, as shown in Fig. 3. The incoming stream is characterized by an arrival vector,,, where corresponds to the amount of data which has arrived at the ingress node by time where,. The stream has a playout vector where corresponds to the amount of data that must be removed from the -bit egress buffer by time, where 0,. For a valid playout vector for the arriving stream, we assume that for with at the end of the transfer. At time a total of bits are distributed between the -bit egress buffer the -bit ingress buffer, requiring for. Any valid transmission schedule must avoid underflow overflow of the egress buffer underflow overflow of the ingress buffer Let denote the majorization schedule, subject to these constraints. When there is no ambiguity, we refer to the majorization schedule as. The schedule has a peak rate of. In the next subsection, we determine how to compute the value of that minimizes by characterizing the shape of the curve. Then, we follow a similar approach to compute the start up delay that minimizes the peak rate, where a playout delay of shifts to the right by time units. B. Buffer Allocation We first consider how to determine the optimal division of the -bit buffer budget between the ingress egress nodes, with the goal of minimizing the peak bwidth of the smooth transmission schedule. To characterize how, we parameterize the smoothing constraints varies with Recall that is the majorization schedule associated with,,,,. Note that as more buffer space is allocated to the egress node, the pair, rises relative to the pair,, which remains stationary. By characterizing how the schedule changes with, we can show that: Lemma 3.1: The peak rate of the single-link system is a piecewise-linear, convex function of. The proof, which extends the results in Section II-C, is presented in the Appendix. The main consequence of the lemma is that the peak-rate curve has one of the shapes shown in Fig. 4, where the slope of the curve increases in. Consequently, a binary search is sufficient to determine a value of that minimizes for a given value of. Optimizing requires, at most, log invocations of the majorization smoothing algorithm, resulting in a worstcase complexity of to determine an optimal value an optimal transmission schedule. To motivate the theoretical result, consider the simpler case of stored video, where for, the ingress buffer is large enough to store the entire video (i.e., bits). The peak bwidth decreases as the egress buffer size increases, with diminishing returns for larger buffer sizes, as shown for several sample video traces in Fig. 2(b). Typically, the peak bwidth in a transmission plan is determined by a region of large frames in a small portion of the stream (e.g., the second run in Fig. 1). The upper constraint curve rises as the egress buffer size increases, causing a gradual decrease in the slope of the peak-rate run. In fact, the peak rate decreases in proportion to the increase in, resulting in a linear segment in the peak-rate curve. These linear segments are clearly visible in Fig. 2(b). Appealing to the domination result in Lemma 2.1, the buffer size eventually grows large enough for the convex change point to disappear, causing the peak rate run to occur in a different part of the video. This begins a new linear segment in the peak-rate curve. The diminishing returns account for the convexity of the peak-rate curve. The proof of Lemma 3.1 in the Appendix formalizes generalizes this argument. With buffer constraints at both the ingress egress nodes,

5 206 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 7, NO. 2, APRIL 1999 the peak-rate curves typically have a cup shape, as shown in Fig. 4(a). C. Start Up Delay In addition to allocating buffer space to smoothing, a video service can minimize the bwidth requirements of the variable-bit-rate stream by introducing a start up latency of at the egress node. Delaying playout by time units effectively shifts the playout curve at the egress node, as discussed in Section II-A. Hence, we define the vectors to be Given a buffer allocation is constrained to be less than., the startup latency otherwise some windows of frames would overflow the combined buffer space. Let denote the peak rate as a function of startup latency for given arrival playout vectors buffer allocation. Define to be startup latency that minimizes the peak rate To characterize how varies with, we parameterize the smoothing constraints As the start up delay increases,, rise relative to the pair,, which remains stationary. Recall that is the majorization schedule associated with,,,,. As in Section III-B, we can show that: Lemma 3.2: The peak rate,, of the single link system is a piecewise-linear convex function of. The arguments in the proof of Lemma 3.1 in the Appendix are easily modified to apply here. Whereas the pair of vectors in the buffer-allocation problem rise with respect to the other pair of vectors as increases, move to the right, with respect to the pair of vectors. The points on the majorization schedule exhibit similar behavior under a latency change as under a buffer change. Hence, the peak-rate curve has one of the shapes shown in Fig. 4. In the stored video case, increasing always decreases the peak rate, since a longer start up delay provides more opportunity for smoothing, though there are diminishing returns for longer delays. The diminishing returns account for the convexity of the peak-rate curve. When there is finite buffer space at both the ingress egress nodes, the peak-rate curve typically has a cup shape, as in Fig. 4(a). If the start up delay is too short, the ingress node does not have sufficient time to perform workahead transmission into the egress buffer. On the other h, if the start up delay is too long, the ingress node must transmit aggressively to avoid overflow of the ingress buffer. The primary consequence of Lemma 3.2 is that a binary search can be used to determine the optimal startup latency for a given buffer allocation. Given a maximum feasible start up latency of, the optimization procedure has a worst-case complexity of. D. Practical Constraints The smoothing model presumes prior knowledge of the arrival playout vectors, does not account for propagation delay jitter. The basic model can be extended to incorporate these practical constraints. To account for a constant propagation delay of time units, the playout curve should be shifted to the left by units, to ensure that each frame arrives at the egress node by its deadline. Delay jitter can be hled by making the smoothing constraints more conservative, based on the minimum maximum number of bits that may arrive at any time [9]. For example, the overflow constraints can assume the maximum number of arriving bits at each time unit, while the underflow constraints can assume the minimum number of arriving bits at each unit of time. In addition, the ingress node may not have full knowledge of the arrival vector, particularly in the case of live video. Experiments with motion-jpeg MPEG traces show that having information about a small window of future frames is sufficient to approximate the performance of a scheme that has complete knowledge of frame sizes [22]. Our experiments show that relatively simple frame-size prediction schemes are effective [22]. To focus the discussion on the basic properties of the smoothing model, the remainder of the paper assumes that there is no propagation delay or jitter, that the arrival vector is known in advance. IV. RATE CONSTRAINTS The previous section described how to allocate buffer space start up delay to minimize the peak bwidth in the transmission schedule. However, in many cases the network may impose a specific rate constraint, such as modem bwidth, that drives the selection of the other smoothing parameters. This section presents efficient techniques for minimizing both the start up delay the buffer allocation, subject to an upper bound on the transmission rate between the ingress egress nodes. Our approach is to consider greedy transmission schedules that send as late or as early as possible, subject to the rate constraint the arrival playout vectors. These greedy schedules can be used to determine the minimum sizes for the egress buffer, ingress buffer, start up delay. A. Joint Minimization of Start Up Delay Egress/Ingress Buffer Size Before investigating the general problem of rate-constrained smoothing on a single link, we discuss the important special case of stored video, where the ingress buffer stores the

6 REXFORD AND TOWSLEY: SMOOTHING VARIABLE-BIT-RATE VIDEO IN AN INTERNETWORK 207 frames as late as possible, this process jointly minimizes both, since it is not possible to reduce either the egress buffer utilization or start up latency by sending data any earlier than necessary. A similar approach can be used to jointly minimize the ingress buffer size the start up latency without any constraint on the egress buffer size by initially generating a schedule that transmits frames as early as possible, subject to the rate constraint the arrival curve, with. In particular Fig. 5. Smoothing schedule that transmits data as late as possible, subject to a rate constraint r. No schedule that obeys the rate constraint r; the underflow constraint D can enter the region between S late D. entire video in advance. For stored video with an ingress buffer of size, the smoothing constraints simplify to,asin the example in Section II-A Fig. 1. In this context, it is possible to minimize both the start up delay the egress buffer size by generating a schedule that transmits frames as late as possible, subject to the rate constraint [12], [17]. Drawing on the playout curve this algorithm, of complexity starts from the last frame sequences backward to the beginning of the video; when necessary, the algorithm transmits frames at rate, as shown in Fig. 5. That is As a result, closely follows the playout curve except in regions with large frames, which require workahead transmission to avoid exceeding the rate constraint. The minimum buffer size corresponds to the maximum distance that this schedule lies above the lower constraint curve, while the minimum start up latency depends on the amount of workahead transmission required for the first frame in the video [12]. The schedule that transmits as late as possible serves as a building block for minimizing start up latency buffer sizes in the general case, where for all. We begin by minimizing without constraining the ingress buffer size. This results in the constraint vectors. As in the case of stored video, the minimum buffer size can be computed from the schedule that transmits frames as late as possible, based on the playout curve ; that is However, the start up latency now depends on the maximum amount of workahead transmission necessary at any point in the schedule, relative to the arrival vector. Based on the schedule that transmits data as late as possible, the minimum start up latency is which can be determined in time by sequencing through the vectors, independent of. By transmitting. Based on the schedule, the optimum values of are We observe that the optimizations based on the optimization based on must achieve the same minimum start up delay. In the next subsection, we show how to minimize the ingress (egress) buffer, subject to jointly minimizing the egress (ingress) buffer the start up latency. Both approaches generate a feasible schedule in time, subject to the rate constraint, with the minimum start up latency a minimum size for at least one of the buffers ( or ). Both approaches also minimize the total buffer allocation. Also, in the special case of, it is possible to jointly minimize both, with. After determining the values for,,, the majorization algorithm can compute the optimally smooth transmission schedule. B. Joint Minimization of Start Up Delay Total Buffer Allocation After jointly minimizing the start up latency the egress (resp. ingress) buffer size, it is possible to minimize the ingress (resp. egress) buffer size based on the schedule (resp. ). More formally, let be the smallest ingress buffer size, given a start up latency the minimum egress buffer size ; similarly, let be the smallest egress buffer size, given a start up latency the minimum ingress buffer size. We focus on the derivation of after determining. While the egress buffer size was determined from the schedule subject to, the ingress buffer size can be found by generating a schedule that sends as early as possible subject to. Consequently as shown in Fig. 6(a). Similarly, can be determined from the schedule that transmits as late as possible subject to,, resulting in.

7 208 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 7, NO. 2, APRIL 1999 (a) (b) Fig. 6. (a) How to compute the minimum ingress buffer size y 0 subject to the minimum egress buffer size x 3. (b) Increasing the egress buffer size does not decrease the minimum total buffer requirement x + y. Based on these derivations, we can show that for all. The proof of this result draws on properties of the greedy schedules that transmit as late, or as early, as possible: Lemma 4.1: Given a rate constraint, let be the schedules that send as late as possible, subject to constraints, respectively. The schedule that sends as late as possible, subject to, satisfies,. Similarly, the schedule that sends as late as possible subject to, satisfies,, where are the schedules that send as early as possible, subject to constraints, respectively. Proof: First, we observe that, is a feasible rate-constrained schedule subject to the lower constraint, since imply that,. Next, we show by contradiction that no feasible schedule can send frames later than,. Assume that the schedule lies below, at some time. Without loss of generality, assume that dominates at time (i.e., ). This implies that. But this violates the definition of as the schedule that transmits as late as possible subject to. Hence,,. A similar result holds for. In effect, each define an inadmissible region above respectively, that cannot support any schedule with rate constraint ; a similar result holds for. Lemma 4.1 shows that, when there are multiple smoothing constraints, the inadmissible region is the union of the regions defined by each individual constraint. Apply these results to, we see that in regions where is determined by. Similarly, in regions where is determined by. Drawing on this observation, we can show that: Theorem 4.1: Given a rate constraint, an arrival vector, a playout vector, a startup latency, the total buffer size is minimized by allocating an egress buffer of size an ingress buffer of size. Similarly, is minimized by allocating an egress buffer of size an ingress buffer of size. Selecting results in the minimum total buffer. Proof: Consider the buffer allocation with at the egress ingress, respectively, where is derived from is derived from, as shown in Fig. 6(a). To see that no other allocation can decrease the total size of the two buffers, consider the effect of increasing the egress buffer size in Fig. 6(b). Suppose that is determined by the value of at time. Drawing on Lemma 4.1, we know that is determined by the schedule that sends as early as possible subject to either or.if is determined by, then an increase in raises by the same amount; as a result, the increase in is exactly offset by the corresponding decrease in.if is determined by, then increasing does not decrease, the total buffer allocation increases. Finally, if increasing moves the selection of to a different part of the video, then the decrease in does not offset the increase in, again the total buffer allocation increases. Hence, it is not possible to reduce the total buffer allocation below. A similar result holds for, ensuring that both allocations minimize the total buffer requirement for a given value of. Since are nondecreasing in, the choice of results in the minimum total allocation of. One consequence of Theorem 4.1 is that the total buffer allocation can be minimized through the calculation of either or. More generally, any buffer allocation can satisfy the rate constraint, so long as : Corollary 4.1: Under start up latency, any buffer allocation with such that results in a transmission schedule with a minimum peak rate of. Proof: As shown in Lemma 3.1, the peak rate has a decreasing region, followed by a constant region, ending with an increasing region. Since allocations of both produce the same peak rate, the function cannot have a larger value in the interval. Also, note that an egress buffer with cannot result in a smaller peak rate under the same value of, since it is not possible to reduce the rate without increasing the total buffer allocation. Hence, any allocation of can satisfy the rate constraint with the minimum total buffer allocation. A stronger version of Theorem 4.1 holds for the important special case of. As before, the joint minimization of stems from the schedule that transmits frames

8 REXFORD AND TOWSLEY: SMOOTHING VARIABLE-BIT-RATE VIDEO IN AN INTERNETWORK 209 Fig. 8. The tem model consists of nodes i =0; 1; 111;nwith buffer space b i. The video stream has an arrival vector A at node 0 a playout vector D at node n. Given these constraints, a solution to the tem-smoothing problem produces link transmission schedules S i for i =1; 2; 111;n. optimal allocation of resources places all buffers at the ingress egress nodes, allowing us to apply the results from Sections III IV to the tem model. Fig. 7. Playback curve D(w 3 ) the upper smoothing constraints for the special case of A D. A single segment of the video stream determines both the egress buffer size x 3 the start up latency w 3 under the rate constraint r, where x 3 = rw 3. as late as possible subject to the playout curve the rate. The schedule also minimizes the egress buffer size : Theorem 4.2: Given a rate constraint,. Proof: When, a single value of determines both, as shown in Fig. 7. In fact, the two values are directly related by the rate constraint, with. To determine, the schedule is computed from the two upper constraints. These two constraints have exactly the same shape except for the constant time space offsets (, respectively). Hence, the schedules that send as early as possible are, respectively. Since the two schedules obey the rate constraint, we have for all. That is, the upper constraint always dominates, by Lemma 4.1. Increasing above does not influence the constraint, hence, does not allow us to decrease. To show that, note that the segments in are parallel to. In fact, the ingress buffer size is determined by the same run that determines the egress buffer size. Since it is not possible to decrease by increasing, we also have, resulting in. The theorem shows that when the ingress node receives an unsmoothed version of the video stream, the optimal buffer allocation divides the resources evenly between the ingress egress nodes. V. OPTIMAL TANDEM SMOOTHING The previous sections have focused on the single-link system, where the buffer space resides solely at the ingress egress nodes. In this section, we generalize the results to a tem of nodes, show how to efficiently compute a smooth transmission schedule for each link in the sequence. After introducing the tem model, we derive an efficient technique for computing an optimal schedule for a given allocation of buffers in the tem system. In particular, we show that it is possible to compute an optimal tem schedule by applying the majorization algorithm on a collection of independent single-link problems. Then, we show that an A. Tem Model The tem-smoothing model consists of nodes with node feeding node, as shown in Fig. 8. Each node includes a buffer of size. Consider a stream of data which arrives at node 0 destined for node. The stream has an arrival vector where corresponds to the amount of data which has arrived at node 0 by time. It is assumed that. Similarly, the stream has a playout vector where corresponds to the amount of data that must be removed from node by time. It is assumed that. For a valid playout vector for the arriving stream, we assume that for with at the end of the transfer. To ensure that the smoothing buffers can store any outsting data in the tem system, we also require that for. Each node has a smoothing schedule where denotes the cumulative amount of data that node has sent to node by time. As in Section II,. Each schedule has a corresponding rate vector where. In deriving the smoothing constraints on the tem system, we focus on the schedule as shown in Fig. 8. To avoid buffer underflow overflow at node,a feasible schedule for node must satisfy the inequality A similar inequality holds for the schedule. Consequently, the set of link schedules in the tem system must satisfy the constraints for, where. Let denote the set of feasible schedules for the buffer allocation. For the case of, this system reduces to the single-link smoothing problem with as discussed in Section III. (1)

9 210 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 7, NO. 2, APRIL 1999 B. Optimal Transmission Schedules To relate the tem model to the single-link problem, observe that the link entering node has upstream buffer space from nodes downstream buffer space from nodes as shown in Fig. 8. Let be the majorization schedule associated with a single-link system with a source buffer of capacity a receiver buffer of capacity. That is, is the majorization schedule associated with the constraints This is accomplished by establishing the following four inequalities for (3) (4) (5) (6) for. Note that these constraints are typically looser than those imposed on the link schedules in the tem system, since each link schedule in the tem system also depends on the schedule chosen for the upstream downstream links. The main result in this subsection is to show that, despite having tighter constraints, the tem system has the same optimal transmission schedule for each link. To establish this result, we first observe that the optimal singlelink schedules are also a feasible solution to the tem system: Lemma 5.1: The schedule satisfies the following constraints for : Proof: It suffices to establish the following: for. Observe that are nonincreasing in. Moreover,. Hence, the inequalities in (2) follow from an application of Lemma 2.1. Then, we show that these optimal single-link solutions are also the optimal link transmission schedules for the tem system: Theorem 5.1: The majorization schedules associated with the finite-source, single-link problems with playout function client buffers satisfy the following relations: feasible Proof: We begin by showing that any feasible solution for the tem model is a feasible solution for the set of single-link models; that is (2) Inequality (3) is satisfied by because it is defined as ; for, inequality (3) follows through repeated application of inequality (1). Similarly, as, repeated application of (1) yields (4). Inequalities (5) (6) are established in the same manner. Lemma 5.1 can now be applied to establish the theorem. Henceforth, we shall let denote the majorization schedules associated with. The schedules can be computed in time by solving the independent single-link problems. The resulting rate vectors minimize any function of the form,, where is a nondecreasing function the are Schur-convex functions. C. Optimal Buffer Allocation Next, we consider how to optimally allocate buffer resources across the nodes in the tem system. We initially focus on an important special case of the tem-smoothing model, where the ingress node has sufficient buffer space to store the entire video; for example, this scenario arises when the ingress node is a server that stores a prerecorded video stream. When, the buffer allocation problem reduces to allocating bytes among nodes,. Let, ; be the set of feasible buffer allocations. Then: Lemma 5.2: If the ingress node has sufficient buffer to store the entire video, then the optimal allocation of units of buffer to the remaining nodes is in the sense that Proof: The proof follows from Theorem 5.1. As a consequence of this majorization relationship, the optimal buffer allocation minimizes cost functions of the form where is a nondecreasing function the are (possibly different) Schur-convex functions. A similar result holds when the egress node has an arbitrarily large buffer (i.e., ) bytes are allocated among the remaining nodes. A similar, albeit slightly weaker, result applies to the more general problem of allocating bytes among all nodes. A feasible smoothing schedule requires to ensure that the tem of nodes can accommodate the outsting data at any time. For the set of ;, ; of feasible buffer allocations, we have:

10 REXFORD AND TOWSLEY: SMOOTHING VARIABLE-BIT-RATE VIDEO IN AN INTERNETWORK 211 Lemma 5.3: For any buffer allocation exists a buffer allocation, such that there, with whenever is nondecreasing is Schur-convex. Proof: Let. Clearly However, this can be achieved by simply allocating units of buffer to the ingress node (node 0) units of buffer to the egress node (node ). Note that Lemma 5.3 shows that the buffer allocation with, minimizes nondecreasing cost functions that apply the same Schur-convex function to each link, in contrast to the stronger result in Lemma 5.2. Still, for a wide set of performance metrics, such as the peak the variability of the transmission rates, Lemma 5.3 shows that we need only consider allocations with. Also, we know from Theorem 5.1 that we need only consider majorization schedules associated with particular buffer allocations. Once the tem model is reduced to a single-link system, the results for the single-link model in Sections III IV can be applied. For example, suppose that each link in the tem model imposes a peak rate constraint,. Given these rate constraints, we attempt to minimize the total buffer allocation the start up latency. Drawing on Lemma 5.3, we can reduce this problem to a single-link system with buffers at the ingress egress nodes. In particular, applying Lemma 5.3 with, shows that it suffices to consider buffer allocations. Hence, the optimal allocation for the tem system derives from analyzing a single-link system with a constraint on the transmission rates. VI. PERFORMANCE EVALUATION Service providers can minimize the bwidth requirements for transferring variable-bit-rate video through careful provisioning of their buffer resources the start up delay. Using full-length motion-jpeg MPEG video traces, simulation experiments evaluate a network with a -byte egress buffer a -byte ingress buffer. The experiments assume that the ingress node receives an unsmoothed sequence of video frames ( ), which are removed from the egress buffer time units later. The simulation experiments investigate how,, affect the peak the variability of the transmission rates in the majorization schedule. The empirical results motivate simple heuristics for selecting these parameters to minimize the bwidth requirements of the video stream. A. Buffer Allocation To investigate the impact of the buffer allocation policy, Fig. 9 plots the peak the coefficient of variation (stard (a) (b) Fig. 9. The graphs show bwidth requirements for a motion-jpeg encoding of Beauty the Beast after smoothing with a (M0x)-byte ingress buffer x-byte egress buffer using a window w =20frames (for a minimum of W =590kb). From upper left to lower right, the curves correspond to M = W; 1:25W;1:5W; 2W; 3W. deviation divided by the mean) of the transmission rates for a motion-jpeg encoding of the movie Beauty the Beast with a 20-frame start up delay. Each curve corresponds to a different total buffer size, where the -axis varies the fraction allocated to the egress buffer. To avoid overflow, the combined ingress egress buffers must hold at least bytes; for this video trace, a sliding window of frames has a maximum of bytes. From top to bottom, the curves in Fig. 9 correspond to increasing values of as expected, larger buffer configurations reduce both the peak variability of the bwidth requirements. Plots for other video traces other values of show the same basic trends. Placing Mbytes of buffer space at gateway routers or network proxies for each video stream is quite reasonable given the relatively low cost of memory. In exchange, smoothing offers a substantial reduction in network bwidth requirements. Consistent with Lemma 3.1, the peak-rate curves are piecewise-linear convex in the egress buffer size. Unbalanced buffer allocations limit the amount of workahead transmission (when is small) or require excessive workahead transmission (when is small) during regions of large frame sizes. As a result, each curve reaches its maximum at the end points, since no smoothing is

11 212 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 7, NO. 2, APRIL 1999 possible when the network has only ingress or egress buffering; in these cases, the peak rate stems from the maximum frame size ( bytes), while the coefficient of variation arises from the variability in the frame sizes (a stard deviation of 3580 bytes a mean of bytes, resulting in a ratio of 0.28). Allocating some buffer space at both the ingress egress of the network has an immediate effect on both metrics. This effect is especially dramatic for MPEG videos (not shown), since a small buffer can remove the significant amount of burstiness that occurs within a group-of-pictures. Following the left-h side of the curves in Fig. 9(a) shows how the minimum egress buffer size changes as a function of the rate for each value of. Observe that, in the decreasing region, all of the curves share the same minimum buffer size for each peak rate. Consistent with the discussion in Section IV-A, the minimum buffer size does not depend on the total buffer size, as long as the combined ingress egress buffers are large enough to satisfy the rate constraint. Similarly, each curve reaches the same peak rate at, when the ingress buffer has size. As a result, the peak-rate curves are each symmetric with for. As a consequence of Theorem 4.2, each peak-rate curve has a minimum at. Smaller values of offer limited opportunities for smoothing, resulting in a nearly flat peakbwidth function when, as seen in the top curve in Fig. 9(a). Once is large enough to permit a moderate amount of workahead transmission, the benefits of smoothing become more sensitive to the distribution of buffer space between the ingress egress points. For a given value of, a more balanced distribution of buffer space gives greater latitude for smoothing throughout the entire video, even in regions with relatively small frame sizes compared to the peak rate. As grows from to, the constant region of the peak-rate curve gradually shrinks, while remaining symmetric around. The curves for have the same minimum peak rate of bytes/frame-slot, with larger values of resulting in wider regions of optimal buffer allocations in Fig. 9(a). In these flat portions of the peak-rate curve, the ingress egress buffers are both larger than bytes, ensuring that each buffer can hold any sliding window of frames; this results in smoothing constraints that do not depend on the exact distribution of the buffer space. Even when the peak-rate curve is flat, moving closer to a symmetric buffer allocation of offers further reductions in the bwidth variability, as shown in Fig. 9(b). Although the bwidth variability curves appear to have the symmetry properties of the peak-rate curve, the other smoothness metrics are not always perfectly symmetric. Having the same peak rate does not necessarily imply that two schedules match in other parts of the smoothed video. Still, the bwidth variability metrics are effectively symmetric across a range of traces buffer sizes. B. Start Up Latency To study the sensitivity of smoothing to the start up delay, Fig. 10 plots the peak bwidth as a function of (a) (b) Fig. 10. These graphs plot the peak bwidth requirements for two videos after smoothing with a start up delay of w frames. Each curve corresponds to a different buffer allocation of x = M=2, with seven evenly spaced values of M; increasing from the upper leftmost curve to the lower rightmost curve. for a motion-jpeg encoding of E.T. The Extra Terrestrial an MPEG encoding of Star Wars [2]. Each peak-rate curve corresponds to a different value of with an optimal buffer distribution of. From upper left to lower right, the curves in Fig. 10(a) correspond to Mbytes, whereas the curves in Fig. 10(b) correspond to Mbytes. Consistent with the Lemma 3.2, each peak-rate curve is piecewise-linear convex. Large values of limit smoothing during regions with large frames, since the combined ingress egress buffers are nearly full. At the other extreme, small values of limit the number of frames available for workahead transmission, resulting in larger peakrate requirements; when, the peak rate equals the maximum frame size, since no smoothing can occur. Following the left-h side of the curves in Fig. 10 shows how the minimum window size changes as a function of the rate. Consistent with the discussion in Section IV-A, the minimum start up latency does not depend on the total buffer size similar to the trends in Fig. 9. Interestingly, the E.T. plots in Fig. 10(a) are nearly symmetric in while the Star Wars curves in Fig. 10(b) exhibit significant variations as the start up latency grows. These irregularities are a by-product of the video encoding

12 REXFORD AND TOWSLEY: SMOOTHING VARIABLE-BIT-RATE VIDEO IN AN INTERNETWORK 213 (a) a window of frames. The buffer requirement is consistently just over half of the available memory. Conceptually, the optimal start up delay is typically the smallest window size that removes the buffer size constraints, resulting in. Smaller values of limit smoothing since the ingress node cannot transmit sufficient data to capitalize on the egress buffer space; similarly, larger values of result in a large buffer requirement that strains the storage space in the ingress buffer. Hence, the service could effectively utilize an -byte buffer budget by assigning by selecting the minimum start up delay that satisfies. This serves as a useful guideline for minimizing the peak variability of the bwidth requirements for the smoothed video stream. (b) Fig. 11. Optimal start up delay w opt the corresponding minimum buffer requirement W opt for different buffer sizes M for a motion-jpeg encoding of E.T. In (b), the solid line corresponds to M =2W. scheme. MPEG compression results in a mixture of large frames (typically) smaller frames within a group of pictures (GOP); the Star Wars video in Fig. 10(b) has a GOP size of 12 frames. For MPEG video, a small change in can substantially increase the minimum buffer requirement ( ), limiting the ability to smooth regions with large frames. In contrast, motion-jpeg compression does not introduce much short-term variability in bwidth requirements, since each frame is encoded independently. Investigating the relationship between start up delay buffer size more closely, Fig. 11 plots the window size that minimizes the peak rate across a range of buffer sizes for the E.T. trace; experiments with other video traces show similar trends. As expected, size grows as a function of the total buffer size where the optimal configuration has. Plots of consistently have a staircase shape since the start up delay is an integer number of frames. Except for this discretization effect, these plots are nearly linear, corresponding to the even spacing of the peak-rate curves in Fig. 10(a). To minimize the peak rate, the start up delay must be large enough to permit aggressive smoothing without allowing a window of large frames to consume too much of the aggregate buffer space. Fig. 11(b) investigates the relationship between the total memory size the minimum buffer requirement for VII. CONCLUSIONS AND FUTURE WORK In this paper, we have investigated bwidth smoothing across a sequence of nodes, where a service provider has control over a subset of the path between the video server the client site. Starting with a single-link system, with buffer space at ingress egress nodes, we show that a simple binary search can determine the buffer allocation start up latency that minimize the peak rate in the transmission schedule. Drawing on these results, we develop efficient techniques for minimizing the buffer sizes start up latency, subject to a constraint on the transmission rate. Then, we generalize these results to the tem model, consisting of a sequence of multiple nodes links. We first show how to compute the transmission schedule on each link for a given buffer allocation. Then, we show that the optimal resource allocation places buffers at the ingress egress nodes. This allows us to apply the results of the single-link system to the tem model. Finally, experiments with full-length MPEG motion-jpeg traces verify the analytic results provide guidelines for allocating buffer space start up latency in a real system. These initial simulation results focus on the important special case where the ingress node receives the original unsmoothed version of the video stream (i.e., ). In a realistic internetworking environment, the arrival vector may depend on the network conditions at the upstream nodes, or on the policies peering arrangements of the neighboring service provider. Experiments with other arrival vectors can allow us to investigate how the traffic characteristics of the incoming stream affect resource requirements across the tem of nodes. Along similar lines, we have applied our smoothing model to the online smoothing problem, where the ingress node does not have a priori knowledge of the frame sizes [22]. This study shows that an ingress node does not need to have prior knowledge of frame sizes to achieve the performance gains of bwidth smoothing. With good estimates of a small window of future frame sizes, an online smoothing algorithm can approximate the results in Section VI. Based on these promising results, we are in the process of building a prototype of a proxy smoothing service for transmitting variable-bit-rate video in an internetworking environment.