Cell Scheduling and Bandwidth Allocation for a Class of VBR Video Connections

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1 Cell Scheduling and Bandwidth Allocation for a Class of VBR Video Connections Tao Yang *, Danny H.K. Tsang #, and Susan Y. Li * * Department of Industrial Engineering Technical University of ova Scotia Halifax, ova Scotia B3J 2X4 Canada # Dept. of Electrical & Electronic Engineering Hong Kong University of Sci. & Tech. Clear Water Bay, Kowloon, Hong Kong Abstract. In this paper, we study the bandwidth allocation problem for an ATM multiplexer loaded with heterogeneous low-motion VBR video sources. We propose a cell scheduling scheme which balances cell loss among different sources so that they experience nearly the same cell loss rate. To estimate the bandwidth required by a VBR video source, we characterize its traffic by four parameters: peak rate, mean rate, coefficient of variation, and the coefficient of the autocorrelation function. Based on the effective bandwidth approach and the stationary Gaussian approximation, we derive simple formulas for estimating the required bandwidth using the four traffic parameters. From numerical results, we conclude that the stationary Gaussian approximation seems to be a very promising bandwidth allocation scheme for VBR video sources. The effective bandwidth approach performs not as well as when it is applied to on-off sources due to its incapability in making full use of statistical multiplexing gain. Extra care must be taken to ensure that the QoS requirement is satisfied when using Gaussian approximation in bandwidth allocation because it does not always overestimate the required bandwidth. 1. Introduction Digital video communication is expected to be one of the major services supported by future broadband integrated services digital networks (B-ISDs). Depending on the specific video codec, video packets may be transmitted in constant bit rate (CBR) or in variable bit rate (VBR). Major tradeoffs between CBR video and VBR video are the network resource utilization and the complexity of the associated traffic control functions (e.g. connection admission control, traffic policing, and network resource allocation). VBR connections can take the advantage of statistical multiplexing gain offered by the asynchronous transfer mode (ATM), a transport technique recommended by ITU-IT for the future B-ISD, and therefore are potentially more efficient than their CBR counterparts in terms of network resource utilization. On the other hand, the traffic pattern or the bit rate of a VBR video connection varies over time in an unpredictable way and thus presents a greater challenge in the development of the associated traffic control and network resource allocation mechanisms [19], [22], [27]. We consider an ATM multiplexer load with a certain number of VBR video sources. Each source generates cells into the common buffer of the multiplexer and these cells are then transmitted through an output channel of a fixed capacity. Sources are not necessarily homogeneous and they may have different traffic characteristics. However, we restrict ourselves to lowmotion video sources or those with no abrupt scene changes. This type of sources have applications, for example, in video teleconferencing and video telephoning. In this paper, we shall discuss two issues that are related to bandwidth allocation for VBR video connections. The first issue is traffic interference. It has been pointed out by a number of researchers (see, for example, [17][18]) that when cells from sources of different traffic characteristics are multiplexed into a common buffer and then transmitted over a common channel, each source may experience a different cell loss rate, called individual cell loss rate from those of others and from the overall cell loss rate. For VBR video, even if all sources have the same statistical characteristics each of them may still experience a different cell loss rate [8]. The discrepancy in individual cell loss rates is largely due to traffic interference among different sources. An adverse effect of traffic interference is that some sources may experience a much higher cell loss rate than the overall cell loss rate [29]. If bandwidth allocation is based on the measure of overall cell loss rate, then the QoS (or the cell loss) requirement for some sources may not be satisfied even when the overall cell loss rate is within the QoS specification. One solution to this problem is to use individual performance measure when allocating bandwidth (see, for example, [17][18]). Another solution, proposed by Heyman et. al. [8], is to maintain the use of overall cell loss rate in bandwidth allocation and eliminate traffic interference by proper cell scheduling. In this paper, we propose another cell scheduling scheme based on the idea in [8]

2 The second issue to be addressed in this paper is the estimation of bandwidth required by a connection (or a group of connections multiplexed into a common channel) to satisfy its QoS requirement. This issue has been the focus of intense research in recent years and, as a result, numerous approaches have been proposed from both industry and academia. An extensive literature review is beyond the scope of this paper but interested readers may refer to some recent survey papers [1, 2, 5, 7, 9-11, 13, 14, 20, 24, 26] and books [21, 23, 25] as well as the references cited there. However, many of these methods are proposed for on-off sources and bandwidth allocation for VBR video sources remains to be a nearly unexplored area. In this paper, we extend the effective bandwidth approach [3, 4, 6, 12] and the stationary Gaussian approximation [6], originally proposed for on-off sources, to the case of VBR video sources. Through numerical studies, we find that the effective bandwidth approach performs much poorer than the Gaussian approximation. Also, Gaussian approximation seems to work much better when it is applied to VBR video sources than when it is applied to on-off sources. Finally, Gaussian approximation is not always conservative even when there is a large number of sources. The paper is organized as follows. In the next section, we describe the traffic model and discuss traffic characteristics of VBR video sources. In section 3, we discuss the proposed cell scheduling scheme and compare it with the one proposed in [8] through numerical studies. In section 4, we derive simple formulas for the calculation of the amount of bandwidth required by a VBR video source using the effective bandwidth approach and Gaussian approximation. In section 5, we present some numerical results and discuss applicability of both approaches in the case of VBR video sources. In section 6, we draw conclusions from this study. 2. The Traffic Model We consider an ATM multiplexer loaded with low-motion VBR video sources. The multiplexer has a buffer space for K cells and cells are transmitted on firstcome-first-served basis through an output link with a capacity of C cells per second (Figure 1). We use the sequence {X (i) n, n 0} to represent the ith source, where X (i) n is the number of cells in the nth frame from the i th source. Each source sends 30 frames per second and within each frame interval the cells are equally spaced. Observations on a sufficiently long sequence of data from a real video teleconference suggest that the sequence can be considered as stationary [8]. We shall henceforth assume that the class of VBR video traffic considered in this paper possesses the stationary property. VB R Video Source 1 VB R Video Source 2 VB R Video Source M MUX Buffer K cells Output Channel C cells/s Figure 1. The Traffic Model We characterize a video source {X (i) n, n 0} by its peak rate i (cells/frame), mean rate µ i (cells/frame), the coefficient of variation c i, and the autocorrelation function A i (m). By standard definitions, these statistics are given by: i = sup n 0 {X(i) n }, µ i = E(X (i) n ), c i = σ i /µ i, and A i (m) = E((X (i) n+m µ i)(x (i) n µ i))/σ 2 i, (1) where σ 2 i = Var(X(i) n ) is the variance of {X(i) n, n 0}. It is well known (see, for example, [8]) that the autocorrelation function A i (m) can be well approximated by an exponential function which takes the form A i (m) = (1 a i ) m, (2) where 0 < a i < 1 is called the coefficient of the autocorrelation function. Therefore, the four parameters ( i,µ i,c i,a i ) characterize a VBR video source and will be used hereafter to represent a source. On frame basis, the aggregate traffic can be approximated by the sequence {X n, n 0}, where (i) X n = X n, n = 0,1,2,. (3) Let (,µ,c,a) be the characteristics of {X n, n 0}. Then, we can easily obtain = i, µ = µ i, and c = µ 2 i c 2 i / µ 2.(4) To compute the coefficient a, let A(m) be the autocorrelation function of {X n, n 0}. Then, we have µ 2 2 A(m) = i c i (1 a i ) m, (5) where σ 2 = µ i 2 c i 2 σ 2 is the variance of {X n, n 0} (see [16] for details). Clearly, A(m) is not exactly exponential but it can be approximated by A(m) (1 a) m, where a = µ 2 i c 2 i a i / σ 2 (6) is the weighted average of all a i. It should be noted that if all a i are equal then A(m) becomes an exponential

3 function, i.e., A(m) = (1 a i ) m and (6) gives the exact value for a. 3. Cell Scheduling When cells from different sources are multiplexed into a common buffer and then transmitted through a common output channel, the cell loss rate experienced by each source is in general different from those of other sources and from the average cell loss rate of the aggregate traffic. This may happen even if all sources have the same traffic characteristics [8]. The discrepancy in cell loss rates among different sources has some adverse effect on bandwidth allocation if it is based on overall cell loss rate of the aggregate traffic. For example, a particular source may experience a much higher cell loss rate than the overall cell loss rate and the QoS for this particular source may not be satisfied even when the overall cell loss rate is within the QoS specification. One solution discussed in [8] is to balance cell loss among all sources by a proper cell scheduling scheme. This scheduling scheme, however, does not change the overall cell loss rate of the aggregate traffic but it brings individual cell loss rates closer to each other and to the overall cell loss rate as well. The basic idea of the cell scheduling scheme in [8] is as follows. The multiplexer has a counter for each source which records the number of lost cells belonging to that source by the time. Each time when the buffer is full and there is an arriving cell, the multiplexer discards the cell which is from the source that has the smallest number of lost cells. otice that the discarded cell may not be the arriving cell. If this is the case, then one of the cells already in the buffer is ejected and the arriving cell joins the end of the queue. In any event, the counter for the source which loses one cell is increased by one. It is evident that this cell scheduling scheme does not change the total number of lost cells and therefore it does not affect the overall cell loss rate of the aggregate traffic. Although the above scheme balances, to a certain degree, the individual cell loss rates its effectiveness still depends on traffic characteristics of the sources. For example, consider the case of two sources: source 1 and source 2. Assume that source 1 has an average cell rate much smaller than that of source 2. Then, under the above cell scheduling scheme, source 1 may experience a larger cell loss rate than source 2 does because the scheduling tends to balance the number of lost cells from each individual source instead of the individual cell loss rates. We therefore propose a modified cell scheduling scheme which uses two counters for each source: one records the number of lost cells and the other counts the total number of cells generated. The ratio of the former to the latter gives the cell loss rate for that source. When the buffer is full and there is an arriving cell, the multiplexer discards the cell which is from the source that has the smallest cell loss ratio computed from the two counters. In this way, the scheduling scheme attempts to balance the individual cell loss rates. Table 1. Characteristics of Sources Source # i µ i c i a i Mbits/s Mbits/s 1,6, ,7, ,8, ,9, ,10, In order to compare these two cell scheduling schemes, we have done some simulation studies. Here, we present some of the results from a simulation run that involves 15 VBR video sources. The simulation ends after each source has generated 40,000 frames. The traffic characteristics of the 15 sources are given in Table 1. In the simulation, the traffic from each source is generated according to the D-MMDP source model to be discussed in the next section. In Table 2, we show the individual cell loss rates together with the overall cell loss rate for three cases: (1) First-Come-First-Out (FIFO) or no scheduling; (2) the scheduling scheme proposed in [8] (Original); and (3) the modified scheduling scheme proposed in this paper (Modified). From Table 2, we can see that the original scheduling scheme reduces the difference between individual cell loss rates comparing with the case in which no cell scheduling is practiced. However, individual cell loss rates differ significantly even under the original scheduling scheme. When the modified scheme is applied, we can see that individual cell loss rates have virtually no difference and they are all very close to the overall cell loss rate. Table 2. Overall cell loss rate = Source # Cell loss Rate 10 4 FIFO Original Modified

4 4. Bandwidth Allocation In this section, we consider the problem of bandwidth allocation for low-motion VBR video connections. The essence of the problem is to estimate the amount of bandwidth required by a connection based on its traffic characteristics and QoS requirement. Once the buffer size is fixed, the cell loss rate becomes the most important measure of QoS. When the multiplexer is equipped with the cell scheduling scheme discussed in the previous section, it only needs to ensure that the overall cell loss rate instead of the individual cell loss rate is below a pre-specified value. There are two well established approaches to bandwidth allocation based on the overall cell loss rate. One is the effective bandwidth approach and the other is stationary Gaussian approximation. To extend the former to the case of VBR video connections, we need to use a certain source model with which we can calculate the effective bandwidth required by a source based on its traffic characteristics and QoS requirement. For this purpose, we review the concept of the discrete-time Markov modulated deterministic process (D-MMDP) [15,16]. A D-MMDP is quite similar to its continuoustime counterpart which has been used to model the aggregate traffic of on-off sources [28, 29]. The only difference is that the modulating process is a discretetime Markov chain in the former while it is a continuous-time chain in the latter. More specifically, a D-MMDP is an arrival (or a point) process whose rate is controlled by a discrete-time Markov chain {X ~ n, n 0} with a state space {R 0, R 1,, R M }, where R 0, R 1,, and R M are nonnegative integers. If X ~ n = R i, then there are precisely R i cells which are equally spaced over the n th frame interval (with no arrival at the beginning of the interval and one arrival at the end of the interval). A D-MMDP is completely determined by the vector R = ( R 0, R 1,, R M ) and P, the transition probability matrix of {X ~ n, n 0}. We now consider the simplest two-state D-MMDP {Z n, n 0} with R = ( 0, R) and P = 1 α α β 1 β, where R is a positive integer. This basic source will be referred to as a mini-source. otice that {Z n, n 0} is completely determined by parameters (R,α,β). We approximate a VBR video source {X (i) n, n 0} with traffic characteristics ( i,µ i,c i,a i ) by the D-MMDP { X (i) n, n 0}, where M i X (i) (k ) n = Z, n k =1 (k and {Z ) n, n 0} are identical, independent minisources with parameters (R i ). { X (i) n, n 0} can be considered as the superposition of M i identical, independent mini-sources and it is completely determined by parameters (M i, R i ). We should point out that the arrivals of cells from {X (i) n, n 0} within one frame are equally spaced over that frame interval. In order to match {X (i) n, n 0} with { X n (i), n 0}, we choose the values of the parameters (M i, R i ) so that { X (i) n, n 0} has the same traffic characteristics as those of {X (i) n, n 0} (i.e., ( i,µ i,c i,a i )). After some algebraic manipulations (see [15] for details), we have M i = i / µ i 1 2 c, (7) i 2 R i = i c i i / µ i 1, (8) α i = a i µ i / i, (9) and β i = a i (1 µ i / i ), (10) where [x] is the integer closest to x. Once we have the source model for {X (i) n, n 0}, we can easily compute its effective bandwidth e i (cells per frame) using the formula provided in [12]: e i = M i δ ln 1 (1 α i + (1 β i )e δr i (1 α i ) 2 2(1 α i β i α i β i )e δr i + (11) +(1 β i ) 2 e 2δR i )], where δ = ln p / K, p is the cell loss requirement, and (M i, R i ) are computed using equations (7), (8), (9), and (10). One important property of effective bandwidth is that it is additive. That is, when cell streams of sources are superimposed to form an aggregate traffic, the effective bandwidth of the latter is simply the sum of the effective bandwidths of all individual sources. Therefore, the bandwidth required by the aggregate traffic of sources is estimated to be

5 C = e i. (12) The additive property of effective bandwidth simplifies the bandwidth allocation procedure but it fails to fully utilize the statistical multiplexing gain when there is a large number of sources multiplexed into the same channel. To resolve this problem, Guerin et al [6] introduced the stationary Gaussian approximation. In the following, we extend this method to VBR video sources. Stationary Gaussian approximation is aimed at taking the advantage of statistical multiplexing gain when there is a large number of sources. Here, the aggregate traffic {X n, n 0} is approximated by a stationary Gaussian process which is characterized by its mean and variance. As discussed in section 2, (i) X n = X n can be considered as the sum of independent random variables. According to the law of large numbers, X n can be approximated by a Gaussian random variable with mean µ and standard deviation σ = cµ, where µ and c are given by formula (4) in section 2. This approximation improves as the number of sources increases. To compute the amount of bandwidth required by {X n, n 0} to meet the cell loss requirement p, the overall cell loss rate is approximated by P{X n > C}. The amount of bandwidth Ĉ required by {X n, n 0} is given by Ĉ = µ ( 1 + c ln p ln(2π) ), (13) where Ĉ approximately satisfies P{X > Ĉ} = p n. 5. umerical Examples and Discussions In this section, we examine, through numerical examples, the performance of the two bandwidth allocation schemes discussed in the previous section. First, we consider the case of homogeneous sources. The traffic characteristics of each source are: i = 74.0 Mbits/s, µ i = Mbits/s, c i = , and a i = These values are measurements of the sequence of data (consisting of about 37,000 frames) from a real videoconference [15]. The multiplexer has a buffer space of 100 cells and the QoS (cell loss rate) requirement is p = In Figure 2, we plot the estimated bandwidth required by the aggregate traffic against the number of sources. Three curves are plotted: the effective bandwidth, the stationary Gaussian approximation, and the "exact" which is obtained from the exact queueing analysis of the D-MMDP/D/1/K queue where the aggregate traffic {X n, n 0} is approximated by its matching D-MMDP { X n, n 0} (see [15] for details). In Figure 2, we can see that the effective bandwidth approach performs poorly and overestimates the required bandwidth by a large margin. On the other hand, the stationary Gaussian approximation performs remarkably well. It slightly underestimates the bandwidth. Required Bandwidth (Mbits/s) Effective Bandwidth Gaussian Approximation Exact umber of Sources Figure 2. Homogeneous Sources We next examine the case of heterogeneous sources. We consider two classes of sources. For class 1 sources, we assume i = 74.0 Mbits/s, µ i = Mbits/s, c i = , and a i = 0.02, and for class 2 traffic, we have i = 36.0 Mbits/s, µ i = 12 Mbits/s, c i = 0.7, and a i = In Figure 3, we plot the acceptance region based on three estimations: the effective bandwidth, the Gaussian approximation, and the exact. Here, we assume that the buffer size K = 100 cells, the channel capacity C = 700 Mbits/s, and the QoS requirement is p = In Figure 3, we can see that the acceptance region of the effective bandwidth approach is much smaller, due to over estimation, than those of the Gaussian approximation and the exact. We also observe that the Gaussian approximation is slightly underestimate the bandwidth required by the aggregate traffic. umber of Class 2 Sources Effective Ba ndw idth ++++ G aussian Approximati on Exact umber of Class 1 Sources Figure 3. Heterogeneous Sources: Acceptance Region Comparing with the results reported in [6] where both the effective bandwidth approach and the Gaussian approximation are examined for the case of on-off sources, we observe that these two approaches perform differently when they are applied to VBR video sources.

6 Firstly, the Gaussian approximation improves significantly when it is applied to VBR video sources. The major reason is that the stationary distribution of a VBR video source {X (i) n, n 0} is much closer to Gaussian distribution [8] than an on-off source. Therefore, the approximation is quite accurate even when the number of sources is small. Secondly, the effective bandwidth approach performs poorly for VBR video sources in comparison with its performance for on-off sources. Due to its additive property, the effective bandwidth approach does not fully make use of the statistical multiplexing gain which in general becomes quite significant when there is a large number of sources. From source modeling point of view, a VBR source is equivalent to several on-off (mini-) sources. Therefore, there is more statistical multiplexing gain in VBR video sources than on-off sources and the effective bandwidth approach fails to take the full advantage. Thirdly, it was pointed out in [6] that the Gaussian approximation overestimates the required bandwidth. However, our numerical results suggest that this is not always true, especially when buffer size is small. Therefore, caution must be taken when using the stationary Gaussian approximation. 6. Conclusions In this paper, we have studied the bandwidth allocation problem for an ATM multiplexer loaded with heterogeneous low-motion VBR video sources. We have proposed a cell scheduling scheme which balances cell loss among different sources so that they experience nearly the same cell loss rate. To estimate the bandwidth required by a VBR video source, we have characterized its traffic by four parameters: peak rate, mean rate, coefficient of variation, and the coefficient of the autocorrelation function. We have extended the effective bandwidth approach and the stationary Gaussian approximation to the case of VBR video sources and derived simple formulas for estimating the required bandwidth using the four traffic parameters. From numerical results, we can conclude that the stationary Gaussian approximation seems to be a very promising bandwidth allocation scheme for VBR video sources. The effective bandwidth approach performs not as well as when it is applied to on-off sources due to its incapability in making full use of statistical multiplexing gain. The Gaussian approximation does not necessarily always overestimate the required bandwidth. When the buffer size is relatively small, it can underestimate the required bandwidth. Therefore, extra care must be taken to ensure that the QoS requirement is satisfied when using Gaussian approximation in bandwidth allocation. 7. References [1] J.J. Bae and T. Suda, "Survey of traffic control schemes and protocols in ATM networks" Proc. of IEEE 79(2) (1991) [2] C.A. Cooper and K.I. park, "Toward a broadband congestion control strategy" IEEE etwork Magazine (May 1990) [3] A.I. Elwalid and D. Mitra, "Effective bandwidth of general Markovian traffic sources and admission control of high speed networks," IEEE/ACM Trans. on etworking, Vol. 1, pp , June [4] R.J. Gibbens and P.J. Hunt, "Effective bandwidth for multi-type UAS channel," Queueing Systems, Vol. 9, pp , [5] H. Gilbert, O. Aboul-Magd and V. Phung, "Developing a cohesive traffic management Strategy for ATM networks", IEEE Communications Magzine (October 1991) [6] R. Guerin, H. Ahmadi, and M. aghshineh, "Equivalent capacity and its application to bandwidth allocation in high-speed networks" IEEE JSAC 9(7) (1991) [7] I.W. Habib and T.. Saadawi, "Controlling flow and avoiding congestion in broadband networks" IEEE Communications Magazine (October 1991) [8] D.P. Heyman, A. Tabatabai, and T.V. Lakshman, "Statistical analysis and simulation study of video teleconference traffic in ATM networks", IEEE Trans. on Circuits and Systems for Video Technology 2 (1), (1992). [9] D. Hong and T. Suda, "Congestion control and prevention in ATM networks" IEEE etwork Magazine (July 1991) [10] J.Y. Hui, "Resource allocation for broadband networks" IEEE JSAC 6(9) (1988) [11] R. Jain, "Congestion control in computer networks: Issues and Trends" IEEE etwork Magazine (May 1990) [12] G. Kesidis, J. Walrand, and C.-S. Chang, "Effective bandwidth for multiclass Markov fluids and other ATM sources," IEEE/ACM Trans. on etworking, Vol. 1, pp , August [13] A.A. Lazar and G. Pacifici, "Control of resources in broadband networks with quality of service guarantees" IEEE Communications Magazine (October 1991) [14] J.-Y. Le Boudec, "The asynchronous transfer mode: a tutorial" Computer etworks and ISD Systems 24 (1992) [15] S.Y. Li, T. Yang and D.H.K. Tsang, "Source Modeling and Queueing Analysis ofa Class of VBR Video Traffic in ATM etworks," Working Paper 94-03, Department of Industrial Engineering, Technical University of ova Scotia, Halifax, ova Scotia, B3J 2X4, Canada, February 1994.

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