Bandwidth Estimation in Prerecorded VBR-Video Distribution Systems Exploiting Stream Correlation

Transcription

1 Bandwidth Estimation in Prerecorded VBR-Video Distribution Systems Exploiting Stream Correlation G. Boggia, P. Camarda, and D. Striccoli Dipartimento di Elettrotecnica ed Elettronica Politecnico di Bari Via Orabona, BARI, Italy {g.boggia, camarda}@poliba.it, striccoli@de .poliba.it Abstract The main aspect of several multimedia applications is the transmission of Variable Bit Rate (VBR) video streams. Such applications require high bandwidth with stringent Quality of Service (QoS) guarantee. In such systems, a statistical bandwidth estimation is especially relevant as it is necessary for network infrastructure planning and admission control. In this paper, an original algorithm to estimate the aggregate bandwidth, needed by a given number of correlated video streams, is proposed and analyzed, by modeling the starting point and the permanence time in a bandwidth level of the smoothed video stream with known statistical distributions. The obtained analytical results are validated by simulation and compared with other numerical results already present in the literature, discussing the effectiveness of the proposed solution. 1 Introduction In the telecommunication world, multimedia applications like Video on Demand (VoD), Distance Learning, Internet Video Broadcast, and so on, will likely assume a growing importance [1, 2]. The core aspect of such applications is the transmission of video streams requiring a high bandwidth with stringent Quality of Service (QoS) guarantee, in terms of loss, jitter, delay, and so on. The scenario presented in this paper refers to a video distribution system where a great number of films stored in a server are transmitted across a network. All films are supposed to be 1

2 codified with the MPEG standards which in most cases produce a Variable Bit Rate (VBR) traffic. As an example, in Fig. 1 the bit rate of the first frames of a high quality MPEG-4 coded trace of the film Jurassic Park is reported [3]. Bit rate (kbit/s) Time (frames) Figure 1: frames of the Jurassic Park film. Given the nature of VBR traffic, a static conservative bandwidth assignment for statistically multiplexed video streams results in a non negligible waste of network resources. In fact, if all video stream starting points are randomly shifted in time, the sum of all stream bit rates will usually be lower than the sum of peak rates. Thus, a statistical approach is necessary to evaluate the bandwidth resources needed by a given number of video streams, respecting the QoS requirements. Workahead smoothing techniques can be exploited to reduce the total amount of bandwidth assigned to all video streams [4]-[10]. These techniques are based on the reduction of the peak rate and the bit rate variability of every stream present in the network. They consist in smoothing the bursty behavior of video streams by transmitting, ahead of playback time, pieces of the same film at a constant bit rate variable from piece to piece according to a scheduling algorithm. At the transmission side the stream rate is controlled by using a buffer, while at the receiving side frames are temporarily stored in a client buffer and extracted during the decoding process. Obviously, bit rates have to be chosen appropriately in order to avoid buffer overflow and underflow, ensuring a continuous playback at the client side. The size of smoothing buffers determines number and duration of the Constant Bit Rate (CBR) pieces that characterize the smoothed video stream. An increase of the smoothing buffer size produces a smaller number of CBR segments and a reduction of both the peak rate and the rate variability. As highlighted in [4, 5], a consistent gain in network resource utilization can be obtained by exploit- 2

3 ing this technique, which is likely to be implemented in real systems. Different types of smoothing techniques can be implemented: the Critical Bandwidth Allocation (CBA) algorithm minimizes the number of bandwidth increments [7], the Minimum Changes Bandwidth Allocation (MCBA) algorithm minimizes also the number of bandwidth decrements [8], the Minimum Variability Bandwidth Allocation (MVBA) reduces the variability of the bandwidth changes [9], while the Piecewise Constant Rate Transmission and Transport (PCRTT) algorithm divides the video flow into a piecewise CBR stream, with each piece lasting for a fixed time interval [9, 10, 11]. The choice of one of these algorithms depends on what has to be optimized, e.g., the peak rate, the number, the variability, and the periodicity of bandwidth changes [9]. As an example, in Fig. 2 the bit rate of the "Jurassic Park" film is reported smoothed with a client buffer of 512 kbytes and utilizing the MVBA approach [5]. The study developed in this paper is valid whatever smoothing technique is exploited and can be utilized also in the case of unsmoothed video streams. It is worth noting that smoothed video streams, characterized by long CBR pieces, represent exactly the source traffic model exploited in [12] Bit rate (kbit/s) Time (frame) Figure 2: frames of the Jurassic Park film, smoothed with the MVBA approach and a client buffer of 512 Kbytes. In such systems, a statistical bandwidth estimation is especially relevant as it is necessary for network infrastructure planning and for an efficient admission control [4, 13, 14]. As considered in [4], [15]-[18], and in many other works, here we will model network switches as bufferless multiplexers. 3

4 This hypothesis can be supported by the following considerations. First of all, it has been shown [19, 20] that large network buffers are of substantial benefit in reducing losses due to rapid video bit rate changes and that the only effective way to reduce losses due to slow video bit rate changes is to allocate sufficient bandwidth for each network link. Since the smoothed traffic has mainly a slow bit rate variability (see Fig. 2), the adoption of large network buffers is not useful in this scenario. A second consideration is that the employment of large buffers introduces delays and jitters along the network nodes that can be intolerable for the correct delivery of video streams guaranteeing QoS specifications. In literature, there are various approaches for statistical estimation of bandwidth occupied by a number of video streams entering a bufferless multiplexer [4, 15, 16]. All these approaches are based on the assumption of independence among video streams. In particular, in [4] a Chernoff Bound based admission control method, for the general case of different video stream types, has been exploited. Some variants of the Chernoff bound have been proposed in [16]. In [15] two different methods for admission control schemes exploiting the Normal Approximation and Large Deviation (LD) approximations based on Chernoff bound have been evaluated. They have been tested on the simple case of a single video stream type. In this paper an original algorithm to estimate the aggregate bandwidth needed by a given number of correlated video streams is proposed and analyzed. These streams are derived from a single source, respecting a specified loss probability, i.e., a QoS parameter defined as the probability that the aggregate bandwidth of all the streams is above a given threshold [4]. Analytical results are compared with simulation results and with the ones based on Chernoff bound [4], extending and generalizing previous studies on this topic, based on independence assumptions [4, 15, 16, 21, 22]. The paper is structured as follows. In Section 2 the bandwidth estimation algorithm is described in detail. In Section 3 some numerical results are provided, comparing the proposed method with the Chernoff bound method and with simulation results. Finally, in Section 4 some conclusions are given. 2 Aggregate bandwidth estimation In this section, an original algorithm to estimate the aggregate bandwidth of a given number of correlated video streams is presented and discussed. Let us suppose to have a generic number N of smoothed video streams deriving from a single source (e.g., the Jurassic Park film of Fig. 1). All the considered video streams present the same bit rate evolution. To characterize the source, the model proposed in [12] can be utilized. That is, the film can be represented by a finite 4

5 number of M bandwidth levels λ 1, λ 2,..., λ M, each of them identifying a CBR piece of the smoothed film. The permanence time in a bandwidth level is considered as a random variable with a statistical distribution characterized by the cumulative density function (cdf) F 1 (x) and the probability density function (pdf) f 1 (x). The marginal distribution of the bandwidth levels is defined as p(λ i ) = π i, i = 1,..., M. All these probabilities π i can be easily derived by dividing the permanence time in the i th bandwidth level by the total video stream duration. The state probability can be expressed by: p(x N = λ αn, x N 1 = λ αn 1,..., x 1 = λ α1 ) = p(λ αn, λ αn 1,..., λ α1 ) (1) where x i, i = 1,..., N, represents the bandwidth level of the i th video stream and α i is an integer that can assume all values from 1 to M. Obviously, the bandwidth occupied by all the streams in a generic state is Λ = λ α1 + + λ αn. Thus, the aggregate bandwidth Λ S required to satisfy a given loss probability p l (considered as QoS parameter) can be statistically derived by the following expressions: M M M p(λ > Λ S ) = p l = p(λ αn, λ αn 1,..., λ α1 ) (2) with α 1 =1 α 2 =1 α N =1 λ α1 + + λ αn > Λ S. (3) The constraint (3) identifies a subset S of all system states (λ αn, λ αn 1,... λ α1 ) such that the equation (2) is also verified. The main issue is to evaluate the state probabilities represented by (1). To solve this problem, let n i,j be the number of bandwidth level changes, starting from the level λ αi and ending to the level λ αj. Let us suppose that the bandwidth levels of the smoothed video steam are numbered progressively as they appear in their temporal sequence, i.e., λ 1 is the first CBR piece of the smoothed stream, λ 2 is the second, and so on. For example, if α i = 3 and α j = 7, n i,j = α j α i = 4. To derive the integers n i,j when α j < α i, it is supposed that the video stream begins again after its end, starting from λ 1 after the bandwidth level λ M. Given this assumption, it is clear that if α j < α i then n i,j = M (α i α j ). Taking into account integers n i,j, equation (1) can be rewritten as follows: p(λ αn,..., λ α1 ) = n 1,2 =0 n 2,3 =0... n N 1,N =0 and, exploiting conditional probabilities: p(λ αn,..., λ α1 ) = n 1,2 =0 n 2,3 =0... n N 1,N =0 p(λ α1,..., λ αn, n 1,2,... n N 1,N ) p(λ αn λ αn 1,..., λ α1, n 1,2,... n N 1,N ) 5

6 p(λ αn 1 λ αn 2,..., λ α1, n 1,2,... n N 1,N )... p(λ α1 n 1,2,... n N 1,N ) p(n 1,2... n N 1,N ) (4) We suppose that there is a strong correlation only between two consecutive levels λ αi and λ αi+1. Thus, they depend only on the number of level changes n i,i+1. Furthermore, all the n i,j are supposed to be independent from each other. For these reasons, (4) becomes: = p(λ αn,..., λ α1 ) = n 1,2 =0 n 2,3 =0... n N 1,N =0 p(λ αn λ αn 1, n N 1,N ) p(λ αn 1 λ αn 2, n N 2,N 1 )... p(λ α2 λ α1, n 1,2 ) p(n 1,2 )... p(n N 1,N ) = n N 1,N =0 p(λ αn λ αn 1, n N 1,N )p(n N 1,N ) and, in a more compact form: N 1 p(λ αn,..., λ α1 ) = π α1 k=1 n k,k+1 =0 p(λ α2 λ α1, n 1,2 )p(n 1,2 )p(λ α1 ) n 1,2 =0 p(λ αk+1 λ αk, n k,k+1 )p(n k,k+1 ) (5) Each factor of (5) in square brackets can be easily derived from real video traces. In particular, from the bit rate evolution of the smoothed film, it can be observed that the bandwidth level λ k+1 can be reached from λ k only after n k,k+1 bandwidth changes. No other value of is permitted. Thus, the (5) can be modified as follows: N 1 p(λ αn,..., λ α1 ) = π α1 k=1 p(λ αk+1 λ αk, n k,k+1 )p( n k,k+1 ) (6) The probability p(λ αk+1 λ αk, n k,k+1 ) assumes the value 1, since, given the bandwidth level λ αk and n k,k+1, only the bandwidth level λ αk+1 can be reached. Then, the formula (6) becomes: N 1 p(λ αn,..., λ α1 ) = π α1 k=1 p( n k,k+1 ) (7) To evaluate the state probability by (7), it is necessary to derive the probabilities p( n k,k+1 ) from a statistical model. To find these probabilities, for each 1 k N 1, let us suppose that the first video stream starts at t = 0, the second one starts after a random temporal shift t 1, the third one after a temporal shift of t 2 (if compared to the first one), and so on. Now, fixing a generic time instant t and observing the bandwidth levels of the N video streams of the same type in t, it is 6

7 easy to note that the second video stream will be in a bandwidth level that can be derived simply counting the number of bandwidth changes in t 1, starting from the bandwidth level of the first video stream. Similarly, the third video stream will be in a bandwidth level obtained counting the bandwidth changes in t 2, starting from the bandwidth level of the first video stream. The same can be done for the rest of the N video streams. Then, the probability p( n 1,2 ) represents the probability to have n 1,2 bandwidth changes between λ α1 and λ α2 in a time interval equal to t 1. Let T α1 be the permanence time of the first video stream in the bandwidth level λ α1, T α1 +1 the permanence time in λ α1 +1, and so on. There will be n 1,2 bandwidth changes in t 1 if and only if: t r + T α T α1 + n 1,2 t 1 t r + T α T α1 + n 1,2 +1 where t r represents the residual time in the bandwidth level λ α1. This parameter has to be introduced, since the temporal window t 1 can start in a random point within the permanence time T α1. The probability of n k,k+1 bandwidth changes p( n k,k+1 ) can be obtained by: p( n k,k+1 ) = p(t r +T αk +1+ +T αk + n k,k+1 t 1 t r +T αk +1+ +T αk + n k,k+1 +1) that is: p( n k,k+1 ) = p(t 1 t r + T αk T αk + n k,k+1 ) p(t 1 t r + T αk T αk + n k,k+1 +1). (8) Given that t 1 is the time interval between the starting points of two video streams, it can be modelled with a random variable with known statistical distribution. The (8) can be derived exploiting the Laplace transform method illustrated in [23]. In particular, let us suppose to have the independent random variables X 0, X 1, X 2,... X k, X, whose pdfs have Laplace transform given by fx 0 (s), fx i (s) and fx (s) respectively. If the Laplace transforms are all analytic over the set D σ = {s R(s) σ}, where σ is a real number, it can be proved that p(x X k X) = 1 σ+j 2πj σ j fx 0 (s)[fx i (s)] k f s X( s)ds and, applying the Residue Theorem: p(x X k X) = p σ p Res { f X0 (s)[f X i (s)] k s } fx( s) (9) where σ p is the set of poles of fx ( s), while the notation Res{ } denotes the residue at pole [23]. Applying (9) to (8) and considering that the permanence 7

8 times T αi are independent and identically distributed, we have: p( n k,k+1 ) = p σ p Res fr (s)[f (s)] n k,k+1 1 ft 1 ( s) + s + fr (s)[f (s)] n k,k+1 f Res t1 ( s) s p σ p = = fr (s)[f (s)] n k,k+1 1 (1 f (s))ft Res 1 ( s) s p σ p (10) where f r (s) is the pdf Laplace transform of the residual permanence time in a bandwidth level, f (s) is the pdf Laplace transform of the permanence time in a bandwidth level, and f t 1 (s) represents the pdf Laplace transform of the time interval t 1 in which there are n k,k+1 bandwidth changes. It is known that [24] f r (s) = η [1 f (s)] s (11) where 1/η is the mean permanence time in a bandwidth level. Replacing (11) in (10) we obtain: p( n k,k+1 ) = η [f (s)] n k,k+1 1 [1 f (s)] 2 ft Res 1 ( s). (12) s 2 p σ p The probability (12) depends on the pdf Laplace transforms of the permanence time in a bandwidth level, f (s), the pdf of the starting time of video streams, ft 1 (s), and the mean permanence time in a bandwidth level, 1/η. To consider a high degree of correlation between the starting points of smoothed video streams, that is the purpose of this study, we suppose to model the random variable t 1 with the sum of two independent exponential random variables, with pdf f 1 (t) = µ 1 e µ 1t and f 2 (t) = µ 2 e µ 2t respectively. The pdf of t 1 is obviously f t1 (t) = f 1 (t) f 2 (t), and the mean value of t 1 is t 1 = (1/µ 1 ) + (1/µ 2 ). The correlation degree between the starting points of video streams can be set simply by modifying the parameters µ 1 and µ 2. Furthermore, the pdf Laplace transform ft 1 (s) is a rational function with two simple poles s 1 = µ 1 and s 2 = µ 2. The probability p( n k,k+1 ) can be evaluated using (12). With a bit of algebra, it can be derived that: [ η µ1 p( n k,k+1 ) = f (µ ) n k,k (1 f (µ 2 )) 2 µ 1 µ 2 µ 2 µ ] 2 f (µ ) n k,k (1 f (µ 1 )) 2 (13) µ 1 8

9 The equation (13) holds only if n k,k+1 0. The probability p(0) has to be calculated in a different way. In fact, p(0) = p(t 1 < t r ) = 1 p(t r t 1 ), thus: p(0) = 1 + p σ p Res [ f r (s)ft ] 1 ( s) s = 1 + p σ p Res [ η(1 f (s))ft ] 1 ( s) and using the same procedure adopted to derive (13): [ η µ1 p(0) = 1 (1 f (µ 2 )) µ ] 2 (1 f (µ 1 )) µ 1 µ 2 µ 2 µ 1 s 2 (14) (15) Given all the probabilities calculated by (13) and (15) and the marginal probabilities π αi, directly derived from video traces, all of the state probabilities given by (7) can be easily calculated. The aggregate bandwidth Λ S is then derived, given a loss probability p l, by exploiting (2) and taking into account the condition (3). For very low values of p l, as usual in practical cases of interest, the number of combinations of state probabilities, respecting (3), to be added in (2) significantly decreases, resulting in a faster computation of Λ S. 3 Numerical results In this section some numerical results are presented, in order to test the effectiveness of the proposed algorithm if compared with other algorithms already present in literature. In this study, bandwidth results obtained with the algorithm developed in this paper are compared with the Chernoff bound algorithm and with some simulation results, in the particular scenario of a given number of video streams, all of the same type, whose starting points are strongly correlated in time. Permanence times in a bandwidth level can be modeled with the Inverse Gaussian distribution, called also Wald distribution [25]. It can be verified that this distribution provides a very good matching, in terms of pdf, with the permanence times distribution directly derived from real smoothed video traces. The Wald distribution pdf, with parameters λ and µ, is the following: λ f(x) = 2πx e λ(x µ) 2 3 2µ 2 x. The parameters λ and µ are evaluated through the maximum likelihood estimators (see [25] for further details), taking into account samples of permanence times derived from real smoothed video traces. The Laplace transform of the Wald pdf can not be evaluated in a closed form, but it has to be calculated by numerical evaluation. Using the maximum likelihood estimators with n samples of permanence 9

10 times x 1, x 2,...x n, as derived from real video traces, the parameters λ and µ can be calculated as follows [25]: λ = n i=1 n 1 x i x ; 1 µ = x where x is the arithmetic mean of x 1, x 2,...,x n. The mean of the Wald distribution is 1/η = µ. N video streams of a single type have then been aggregated; their starting points have been modeled with a random variable whose statistical distribution is the sum of two independent exponential random variables with parameters µ 1 and µ 2, as described in Section 2. The mean time T m = 1/µ 1 + 1/µ 2 is chosen to quantify the degree of correlation among video streams, and the variance is chosen as var = T m+1 2. Thus, the parameters µ 2 1 and µ 2 can be derived simply choosing the value of T m, under minimum variance conditions. Lower values of T m mean a stronger streams correlation, in terms of starting points. Applying (13) and (15) the probabilities p( n k,k+1 ) can be easily calculated, exploiting the Laplace transform of the Wald pdf. The proposed algorithm has been compared with the Chernoff bound method and with some simulation results through different scenarios, varying number and type of video streams, their smoothing buffer and the mean starting time T m of all streams. In Figure 3 this comparison is illustrated. 20 video streams of type Star Wars, of total lenght video frames, have been aggregated. Their starting points have been modeled with the sum of two negative exponentials distribution whose mean time has been chosen as T m = 40s, to simulate a strong stream correlation. Aggregate bandwidth with the Chernoff bound method has been obtained following the method illustrated in [4]. Its input parameters are the bit rates λ 1,...,λ M and their corresponding probabilities p 1,...,p M. Each probability p i has been calculated by dividing the number of video frames with bit rate λ i by the total number of video stream frames. Simulation results are obtained by aggregating N video streams, whose starting points have been randomly chosen following the same statistical distribution used for algorithm calculation, and with the same mean time T m. In each discrete time instant, the simulated aggregate bandwidth is obtained simply summing the bit rates of all video streams. From Figure 3 it can be noted that algorithm results follow much better simulation results than the Chernoff bound method, resulting at the same time slightly conservative than simulation. The Chernoff bound method instead fails its estimation, since its curve lies under the simulation curve. This is due to the correlation hypothesis, taken into account by the proposed algorithm, but not by the Chernoff bound. In fact, under stream correlation hypothesis, it will be more likely that a great number of flows will be in a high bit rate, if compared with independence 10

11 12 Aggegate bandwidth (Mbit/s) Algorithm Simulation Chernoff 1,5477E-03 1,4986E-03 1,4095E-03 1,3739E-03 1,3512E-03 1,2882E-03 1,2624E-03 1,2286E-03 1,1994E-03 1,1656E-03 Loss probability Figure 3: Results for 20 streams of the Star Wars film, smoothed with a client buffer of 1024 Kbytes. hypothesis, so the aggregate bandwidth in correspondence of a given loss probability will be generally higher (see [4]). The Chernoff bound is based on the stream independence hypothesis, thus it will probably underestimate simulation results. This situation has been experimented in several other scenarios. In Figure 4, in fact, the same situation is represented, but the mean time has been chosen as T m = 60s, to simulate a weaker stream correlation. As it can be noted from Figure 4, the Chernoff curve is closer to simulation curve if compared with Figure 3, given the minor correlation degree among streams; nevertheless, the Chernoff bound always underestimates simulation results, while the proposed algorithm correctly slightly overestimates them. In Figure 5 the same comparison has been performed, but considering a different type of film (the Jurassic Park film), with length of frames, smoothed with a client buffer of 512 Kbytes, and N = 10 video streams aggregated; the value of T m has been chosen of 60s. In Figure 6 the same Jurassic Park film has been considered, but with a smoothing buffer of 64 Kbytes. N = 30 video streams have been aggregated and the chosen mean starting time is T m = 20s. 4 Conclusions In this paper, a new bandwidth estimation algorithm for an aggregate of smoothed video streams, all derived from a single source, and whose starting points are strongly correlated in time, has been analyzed and discussed. The QoS parameter 11

12 12 Aggregate bandwidth (Mbit/s) Algorithm Simulation Chernoff 7,5169E-04 7,2581E-04 6,7740E-04 6,5986E-04 6,4740E-04 6,1597E-04 6,0235E-04 5,8575E-04 5,7092E-04 5,5432E-04 Loss probability Figure 4: Results for 20 streams of the Star Wars film, smoothed with a client buffer of 1024 Kbytes. 13 Aggregate bandwidth (Mbit/s) Algorithm Simulation Chernoff 2,4904E-03 2,4629E-03 2,3731E-03 2,3655E-03 1,5066E-03 1,4374E-03 1,4229E-03 1,4120E-03 1,3687E-03 1,3625E-03 Loss probability Figure 5: Results for 10 streams of the Jurassic Park film, smoothed with a client buffer of 512 Kbytes. 12

13 70 Aggregate bandwidth (Mbit/s) Algorithm Simulation Chernoff 0 1,3441E-06 1,0505E-06 1,0129E-06 9,7842E-07 9,0565E-07 8,1392E-07 7,7195E-07 7,7154E-07 7,7114E-07 Loss probability Figure 6: Results for 30 streams of the Jurassic Park film, smoothed with a client buffer of 64 Kbytes. taken into account for statistical bandwidth evaluation is the loss probability. The proposed algorithm has been compared with the Chernoff bound method, already presented and discussed in literature, and with some simulation results. It has been verified that the Chernoff algorithm underestimates simulation results, due to the stream independence hypothesis, while the proposed algorithm, taking into account the correlation hypothesis, slightly overestimates them, at the same time following very well simulation curves. In conclusion, we can assert that the proposed method can be fruitfully exploited for bandwidth estimation and admission control purposes in case of correlated streams. References [1] A. S. Tanenbaum, Computer Networks, 3rd ed. Prentice-Hall, [2] J. Kurose and K. Ross, Computer Networking: A Top-Down Approach Featuring the Internet. Addison Wesley Longman, [3] MPEG-4 and H.263 video traces for network performance evaluation, [4] Z.-L. Zhang, J. Kurose, J. D. Salehi, and D. Towsley, Smoothing, statistical multiplexing, and call admission control for stored video, IEEE Journal on 13

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