Bandwidth Usage Distribution of Multimedia Servers using Patching

Transcription

1 Bandwidth Usage Distribution of Multimedia Servers using Patching Carlo K. da S. Rodrigues and Rosa M. M. Leão Federal University of Rio de Janeiro, COPPE/PESC CxP 68511, Phone , fax: Rio de Janeiro RJ, , Brazil Abstract Several multicast bandwidth sharing techniques have been proposed in the literature to provide more scalability to multimedia servers. These techniques are often analyzed in terms of the average bandwidth requirements they demand to satisfy client requests. However, average values do not always provide an accurate estimate of the required bandwidth. Therefore, they cannot be the only parameter used to guarantee a certain level of quality to the clients. In this work we propose a simple analytical model to accurately calculate the distribution of the number of concurrent streams or, equivalently, the bandwidth usage distribution considering the popular Patching technique. We show that the distribution may be modeled as a binomial random variable in the single object case, and as a sum of independent binomial random variables in the multiple object case. Through simulation we validate our results. Moreover, we also illustrate how these results may be practically used for instance (i) to allocate bandwidth to provide a given level of QoS, (ii) to estimate the impact on QoS when some system parameters dynamically change, and (iii) to configure the overall system. Key words: Patching, Multicast, Multimedia, Bandwidth, QoS This work is supported in part by grants from CPq and Faperj. address: kleber,rosam@land.ufrj.br ( Carlo K. da S. Rodrigues and Rosa M. M. Leão ). Preprint submitted to Elsevier Science 1 December 006

2 1 Introduction Multimedia servers for applications such as distance learning and video on demand have been the focus of recent studies in the literature. In particular, considerable attention has been given to the issue of providing scalability to these servers. The easiest way to service client requests in such applications is to schedule unicast data streams, one for each client request. The multimedia server bandwidth increases linearly with the client arrival rate since clients arrive and presumably stay in the system for some reasonable time (i.e., it increases linearly with the number of concurrent clients in the system). This increase rate notably precludes the use of this service for a very large number of clients and the study of bandwidth sharing techniques becomes essential to provide scalability. Bandwidth sharing techniques may be classified in multicast or periodic broadcast delivery techniques. Multicast techniques (e.g.,[1 3,5 10,1]) are reactive in the sense that data is delivered in response to client requests. Most of them provide immediate service and save server bandwidth by avoiding unnecessary transmission of data. Periodic broadcast delivery techniques (e.g., [13 1]) cyclically transmit object segments in a proactive way. They can guarantee a maximum start up delay. The idea is to divide the media object into a series of segments and broadcast each segment periodically on dedicated server channels. While the client is playing the current object segment, it is guaranteed that the next segment is downloaded on time and the whole object can be played out continuously. Besides multicast and periodic broadcast, an orthogonal scheme for reducing the load on network and video server is video caching (see [44] and references within). In this scheme a set of proxies are strategically placed close to the clients. Another approach to address the scalability issue in video applications is peer-to-peer streaming (see [46] and references within). In this approach clients can serve other clients by forwarding the video stream. Patching [] is a multicast technique and has been used in recent works [,45,3 8]. For example, in [] Patching is used to provide a peer-to-peer video streaming service, and in [45] it is integrated with proxy caching for media distribution. This technique is especially attractive because of its extreme simplicity and competitive efficiency in terms of bandwidth savings. The work of [9] for instance evaluated the complexity of several multicast delivery techniques. The analysis is based on the information the server has to keep and the work the server has to perform per client request. One of the main results they obtained is that Patching is one of the simplest schemes. And it was shown in the work of [7,9] that, for reasonably popular media objects, the average bandwidth requirements of Patching are very similar to those of much more complex techniques.

3 It is quite common to evaluate the efficiency of multicast techniques in terms of the average bandwidth requirements they demand to satisfy the client requests arriving in the system. A discussion about average bandwidth requirements may be found in [5 7,30 33]. However, average values do not always provide an accurate estimate of bandwidth requirements and therefore cannot be used solely to guarantee a given level of QoS to the clients. Tam et al. [9] were the first to measure the performance of multicast techniques using other metrics such as the maximum bandwidth and the bandwidth distribution. The authors have focused on three merge techniques: the dynamic Fibonacci tree [9,34], the Dyadic [10], and the ERMT (Earliest Reachable Merge Target) [7]. Their analysis was all based on simulations. The quality of service of multicast techniques was also evaluated in [8,33]. De Souza e Silva et al. [8] proposed an algorithm to compute the distribution of the bandwidth required by the Patching technique. Their analytical model estimates the distribution of the total requested bandwidth in one window of Patching. They showed that the requested data streams in a window are a good estimator of the data streams actually transmitted in a window as t. The computational cost of their algorithm is O(P ), where P is a function of the estimated error and is usually not very large. Their results indicated that if the server configuration is based on the average bandwidth required by each object, the probability of requiring more than the average value is large for a wide range of system parameters. In [33] approximate analytical models were developed to analyze the quality of service of the following schemes: Patching, Hierarchical Stream Merging (HSM) [7] and Bandwidth Skimming [8]. The authors used a closed queueing network model to estimate two performance measures: the mean client waiting time and the fraction of clients who leave without receiving service when the server is temporarily overloaded (i.e., the balking rate). The performance measures were computed from the iterative solution of a set of equations. Their results showed for instance that if server configuration is based on the average bandwidth there may be unacceptably high client balking rates. We may see that the analytical evaluation of the Patching mechanism done so far is based on the solution of one equation [8] or a set of equations [33], i.e., there is not a closed-form solution for the bandwidth usage distribution that may be practically employed. Moreover, in [8] they only consider the single object case, and in [33] they only analyze the multiple object case. In this paper we propose a simple analytical model to accurately calculate the distribution of the number of concurrent streams or, equivalently, the bandwidth usage distribution for the Patching technique. We consider both the single and the multiple object cases. We show that the distribution may be modeled in the form of a closed-form solution as a binomial random variable 3

4 in the single object case. In the multiple object case, it can be represented as a sum of independent binomial random variables, which may be easily implemented using Fast Fourier Transform [38]. Through simulation we validate our results. Moreover, we illustrate how our results may be practically used to (i) allocate bandwidth to provide a given level of QoS, (ii) estimate the impact on QoS when some system parameters dynamically change, and (iii) configure the overall system. The remainder of this paper is organized as follows. Section introduces some basic concepts and terminology. Section 3 is dedicated to the presentation of our analytical model. Analytical and simulation results are presented in Section 4. Lastly, conclusions and ongoing work are included in Section 5. Basic Concepts and Terminology Consider a multimedia server and a group of clients receiving object streams (e.g., a film, a video clip, etc.) across the Internet from this server. The paths from the server to the clients are multicast enabled. Multicast is not currently deployed over the global Internet. However, this can be alleviated through the use of proxies which can bridge unicast and multicast networks. One of the most common proposed architectures consists of a unicast channel from the server to the proxy and multicast channels from the proxy to the clients. Since, in general, the proxy and the clients are in the same local area network, multicast service can be easily implemented. Solutions like application level multicast using tunelling have also been used over the public Internet [9,4,43]. Still assume that Patching is used to provide immediate service, clients always request playback from the beginning of the object and watch it continuously till the end without any interruptions (i.e., sequential access), and client bandwidth is twice the object play rate. All these assumptions are considered in the original model of Patching presented in []. Also, consider that the client buffer is large enough to store a portion of the object size (to allow the stream synchronization). This supposition is based on the results presented in the works of [30,4,5]. These works show that, since the client bandwidth is twice the object play rate, in the worst case, the client will listen to two concurrent streams during an interval equal to half of the object size. However, since it is possible to compute an optimal threshold window [30,31], we may relax this initial supposition for the client buffer being half of the object size and set it to the optimal threshold window. Lastly, Table 1 summarizes some notation used in the rest of this work. The Patching technique operates as follows []. The server begins to deliver a full-object multicast stream upon the arrival of the first client. The following 4

5 Table 1 Key parameters λ j D j W j j B j d j T j w j u client arrival rate for object O j total length of object O j in units of time threshold window for object O j in units of time average number of requests for object O j that arrive during the period of length D j. It is referred as object popularity and is given by j = λ j D j average server bandwidth used to deliver object O j using the Patching technique in units of the object play rate unit length used for object O j total length of object O j in units d j threshold window for object O j in units d j total number of objects in the server clients who request the same object and arrive within a certain time interval, denoted as threshold window, retrieve the stream from the multicast channel (buffering the data) and obtain the missing initial portion, denoted as patch, directly from the server over a unicast channel; for clients arriving after the threshold window, a new session is initiated and the process restarts again. Both the multicast and patch streams deliver data at the object play rate so that each client can play the object in real time. Similar models have been proposed in [30,31] to estimate the average server bandwidth for the Patching technique. They assumed a Poisson client arrival process with rate λ j for the media object O j, where j = 1,..., u and u is the total number of objects in the server. The average server bandwidth for delivering object O j is [30,31]: j B j = D j+ λ j W W j + 1 λ j, (1) where D j and W j are the length and the threshold window, respectively, for object O j. The denominator of Equation (1) is the average time that elapses between successive full-object multicasts, i.e., the duration of the threshold window plus the average time until the next client request arrives. The numerator is the expected value of the sum of the transmission times of the full-object and the patch streams that are initiated during the interval. ote that the average number of patches that are started before the threshold window expires is 5

6 λ j W j, and the average duration of a patch is W j / [35]. ow differentiating Equation (1) with respect to W j and setting the result to zero gives: W j,optimal = j +1 1 λ j () As already mentioned, the client buffer can be equal to the optimal threshold window W j,optimal, instead of half of the object size. Substituting () into Equation (1) yields the following result for the average server bandwidth for the optimized Patching technique: B j,optimized = j + 1 1, (3) where j = λ j D j is the average number of client requests that arrive during D j. The parameter j is often denoted as the object popularity. For simplicity, we will refer to the optimized Patching technique as the Patching technique, and we will denote W j,optimal and B j,optimized as W j and B j, respectively. When the object popularity j is above (below) a certain threshold value, periodic broadcast delivery techniques achieve more (less) bandwidth savings than multicast ones [14]. Thus, it is reasonably to classify media objects in accordance with popularity in order to choose the most adequate type of delivery technique [30,14]. This is usually done as follows [8]: for hot objects (i.e., j 100), use periodic broadcast techniques; for cold objects (i.e., j < 10) and lukewarm objects (i.e., 10 j < 100), use multicast techniques. Since Patching is a multicast technique, we mainly focus on j values in the range of 10 to Bandwidth Usage Distribution In this section, we present our analytical model and derive the distribution of the number of concurrent streams needed by the Patching technique, or equivalently, the server bandwidth usage distribution. We initially address the case of the single object and then the case of multiple objects. 3.1 Single Object Consider a server implementing the Patching technique and that client request arrivals follow a Poisson process. Let C(t) be the stochastic process which de- 6

7 notes the number of Patching windows opened in the interval (0, t). It thus directly follows that C(t) is a renewal process [35] whose renewal points correspond to the instants at which the server schedules a new full-object multicast stream, i.e., the start of a new window. Moreover, we have that a cycle of C(t) has an average duration of W j + 1/λ j = W j + D j / j, and then this is the base interval we consider in the analysis to follow. ow assume that the object O j is divided into time units of length d j. For example, a two-hour object can be divided into 0 units of 6 minutes each, or divided into 100 units of 7 seconds each. ow let T j = D j /d j and w j = W j /d j. Then, we have that T j and w j are, respectively, the object size and the window size measured in number of discrete units of length d j. Also consider that if there is at least one client arrival in a time unit, the server initiates a stream (patch or full-object multicast) at the beginning of the next time unit to serve this client. We stress that the value of d j can be as small as desired to obtain a very small probability of more than one arrival in the same time unit and a guarantee of immediate service. ote that since the media object and time in our model are divided into units of length d j, the base interval of our analysis W j + D j / j may alternatively be rewritten as w j + T j / j and Equation () as: w j = T j( j + 1 1) j (4) The interval w j +T j / j is composed by a total of w j +T j / j units of length d j. Hence, this interval can still be expressed in the following form: [1, w j +T j / j ]. Let us now focus on each one of its w j + T j / j units individually in order to identify a distribution for the number of concurrent streams in each one of them. Once we obtain these individual distributions, we try to identify a unique distribution that may be deployed for the entire base interval of analysis [1, w j + T j / j ]. ote that O j may be any object in the multimedia server, then we drop the subscript j from now on. Consider the k-th interval [1, w + T/] of the process C(t) (see Figure 1). In this interval we have transmissions due to fullobject multicasts and also due to patches. Then, we have that the number of concurrent streams in each unit of length d of the k-th interval is given by a multicast component plus a patch component. Let us denote the multicast component as M C. We consider M C as a constant value and estimate it using Little s result [36]: M C = / + 1 (i.e., the average number of multicast streams is given by the arrival rate times the duration of the media object). Although the real number of multicast streams is not constant due to the possible variation of the arrival rate λ (and conse- 7

8 quently of ), M C is a good approximation since changes on the arrival rate λ, within certain limits, do not severely impact on the value of M C. Consider >> 1, which refers to the scenarios we are interested in. In this case M C can be approximated by M C = / = Thus we have that the number of multicast streams increases approximately with the square root of the arrival rate. The patch component is associated to client arrivals and we estimate it as a random variable. Our goal now is to determine the distribution of this random variable. Let us examine two successive intervals: the (k-1)-th and k-th intervals. We want to determine how many units of the k-th interval can transmit a patch initiated in the (k-1)-th interval. ote that we only need to consider the (k-1)-th interval because the patches generated in other previous intervals will finish before the k-th interval starts. Consider that the k-th interval is divided into two sub-intervals: [1, w T/] and [w T/ + 1, w + T/]. In the former we may have transmissions initiated in the (k-1)-th interval, while in the latter we do not have any transmission initiated in the previous interval. Figure 1 shows a scenario where an arrival occurs in the last unit of the (k-1)-th interval. This arrival generates the longest possible patch which is equal to w. We note that the number of units in the k-th interval, transmitting a patch stream initiated in the (k-1)-th interval, decreases as the value of T/ increases. In the limit, when T/ = w (see Equation (4)), all patches are initiated and transmitted in the same interval. ote that we do not need to consider scenarios where T/ > w because this condition occurs only for = 1 (see Equation (4)). Bandwidth sharing techniques are mostly useful when > 1. We mainly focus on the values of in the range of 10 to 100, as already mentioned. arrivals 4-th unit patch=4 w-th unit patch=w w-d d w-t/ Time w T/ w T/ (k-1) interval k interval Fig. 1. Patches in two consecutive intervals: (k-1) and k. We proceed in three steps to obtain the distribution of the number of concurrent patch streams in each unit of the k-th interval. We first compute the number of concurrent streams in the i-th unit of the (k-1)-th and k-th intervals assuming that there is a client arrival in each unit of the window w of the (k-1)-th interval. In the second step we compute the number of concurrent streams in the i-th unit of the k-th interval considering that there is a client arrival in each unit of both windows w of the (k-1)-th and k-th intervals. Finally, in the third step, we analyze the impact of the Poisson client request 8

9 arrival process. Let us consider the first step. As just mentioned, we only observe the patches started in the (k-1)-th interval and assume that there is a client arrival in each unit of the window w of this interval. Figure 1 shows that an arrival in the j-th unit of the interval generates a patch of length j units. Then, there is a patch in the i-th unit of the (k-1)-th interval if, and only if, j i, j = 1,,..., i 1, and a client arrival occurs in the j-th unit. We can also see from Figure 1 that the longest possible patch initiated in the (k-1)-th interval starts at w + 1 and ends at w. Therefore the interval we have to evaluate is [1, w]. We denote by m i,w the number of concurrent streams in the i-th unit of the interval [1, w], where i = 1,,..., w. The following lemma determines an expression for m i,w. Lemma 1 Considering one client arrival in each unit of the interval [1, w] and i = 1,,..., w, the number of concurrent patch streams m i,w in the i-th unit of the interval [1, w] is given by: Case 1: i [1, w]. m i,w = i 1 j=1 1{j i} = i 1 (5) Case : i [w + 1, w]. m i,w = w j=1 1{j i} = w (i 1) (6) where 1 denotes the indicator function, i.e., 1{x} = 1 if x is true, and 1{x} = 0 if x is false. Proof See Appendix A. So far we obtained expressions to calculate m i,w, i.e., the number of concurrent streams in the (k-1)-th and k-th intervals considering the patches exclusively initiated in the (k-1)-th interval. We now present the second step of our analysis. Let l i,w be the number of concurrent streams in the k-th interval [1, w + T/], considering the patches initiated in this interval as well as in the previous one. We show that l i,w can be easily computed by properly superposing the expressions obtained in Lemma 1. Figure illustrates the superposition of two successive intervals: the (k-1)-th and k-th intervals. From this figure we can see that the number of concurrent patches in the first unit of the k-th interval is equal to m 1,w + m w+t/+1,w, in the second unit it is equal to m,w + m w+t/+,w, and so on. The final formulation for l i,w is presented in Lemma. 9

10 w T/ w (k-1)-th interval 1 m w+t/+1,w m w,w w T/ w k-th interval k-th interval with patches initiated in the (k-1)-th and k-th intervals m 1,w m w+t/+1,w + m 1,w mw-t/,w w-t/ m w,w + m w-t/,w w+t/ m w-t/+1,w Fig.. Superposition of the (k-1)-th and k-th intervals. Lemma Considering one client arrival in each unit of both windows w of the (k-1)-th and k-th intervals and i = 1,,..., w + T/, the number of concurrent streams l i,w in the i-th unit of the k-th interval is given by: Case 1: i [1, w T ] l i,w = w T + 1, if (w T ) and i are even (7) w T, otherwise Case : i [w T + 1, w] l i,w = i 1 (8) Case 3: i [w + 1, w + T ] Proof l i,w = w (i 1) (9) See Appendix B. We now proceed to the third step. Assume that client arrivals are represented by a Poisson process. Let X i be a random variable that denotes the number of concurrent streams due to patches in the i-th unit of the interval [1, w +T/], where i = 1,..., w +T/. Consider the scenario of Figure 3 and let us examine the possible values taken, for example, by X 4. To simplify the figure we do not represent patches eventually generated in the previous interval. X 4 may be 10

11 equal to 0, 1 or, depending on the client arrivals in the previous units. X 4 = 1 when there is one arrival in the second or in the third unit (see Figures 3(a) and 3(b), respectively). In Figure 3(c) we have the case where X 4 =, i.e., when there is an arrival in both the second and third units. Lastly, X 4 = 0 when there are arrivals neither in the second nor in the third units. ote that we do not need to make assumptions about the first unit because only the nd and 3 rd units of this interval generate patches long enough to extend over the 4 th unit. patch arrival patch arrival patch arrivals patch w w w w + T/ (a) one arrival in the second unit w + T/ (b) one arrival in the third unit w + T/ (c) one arrival in the second and third units Fig. 3. Concurrent streams in the 4 th unit of an interval [1, w + T/]. The following theorem determines the probability distribution function of X i. Theorem 1 X i has binomial distribution with parameters p = 1 e /T and n i,w, i = 1,..., w + T/. The parameter n i,w is given by: (1) Case 1: w = T, then: n i,w = i 1, if i [1, w] w (i 1), if i [w + 1, w + T ] (10) () Case : w > T, then: i [1, w T ], then: n i,w = w T + 1, if (w T ) and i are even (11) w T, otherwise i [w T + 1, w], then: n i,w = i 1 (1) 11

12 Proof i [w + 1, w + T ], then: n i,w = w (i 1) (13) As we know, the interval [1, w + T/] is divided into units of length d. Let A be a random variable which represents the number of arrivals in one unit of the interval [1, w + T/]. We have that P [A > 0] = 1 e /T since the arrivals follow a Poisson process and λ is normalized with regard to the value of d. Then we have a sequence of units, each one with probability p = 1 e /T of success (where success means at least one arrival in the unit). The total number of units which can generate a patch to be transmitted in the i-th unit is given by: Case 1: w = T All patches generated in an interval are transmitted in the same interval, then n i,w is given by Equations (5) and (6) obtained in Lemma 1. Case : w > T Some patches generated in an interval can extend over the next interval. In this case n i,w = l i,w obtained in Lemma. Thus, it follows that X i has a binomial distribution with parameters n i,w and p = 1 e /T, where i = 1,..., w + T/. X i is a random variable that denotes the number of concurrent streams due to patches in the i-th unit of the interval [1, w + T/]. ote that the random variables X 1, X,..., X w+t/ have the same parameter p = 1 e /T while the parameter n i,w depends on the value of i and w. Then, we have a binomial distribution with a different parameter in each of the units of the interval. We want to determine a unique distribution to model the number of concurrent streams due to patches in the base interval of analysis [1, w + T/]. Let us examine the values of n i,w for the Case of Theorem 1. We analyze this case because w is larger than T/ for the situations we are interested in, i.e., (see Equation (4) for reference). From the equations of Case we observe that the computation of n i,w varies according to the sub-interval. For the first sub-interval, i.e., [1, w T/], n i,w does not depend on i and can assume at most two values. For the two other sub-intervals, n i,w depends on i and may possibly assume more than two values. To illustrate this aspect consider T = 10 4 (recall that w = W/d and T = D/d). We calculate the proportion of the interval [1, w + T/] corresponding to the first subinterval [1, w T/] and the two other sub-intervals 1

13 together [w T/ + 1, w + T/]. Table shows the results obtained for = 10, 5, 50, 100, 00, 400, 700. It can be seen that the percentage of the first sub-interval (where n i,w assumes at most two values) increases as the value of increases. Table Size of the sub-intervals for different values of [1, w T/] [w T/ + 1, w + T/] 10 56% 44% 5 7% 8% 50 80% 0% % 14% 00 90% 10% % 7% % 5% For example, Figure 4(a) shows P [X i = k] for = 50, T = 10 3 and k = 1,. The value of P [X i = k] is constant for the sub-interval [1, w T/] since n i,w = w T/ for these units. In Figure 4(b) the value of P [X i = k], for = 100, T = 10 3 and k = 1,, oscillates between two values for the subinterval [1, w T/] (n i,w is computed from Equation (11)). In both figures, at the end of the interval, we see that P [X i = k] shows more variability. As already explained, this behavior is due to the larger variability of the parameter n i,w in the sub-interval [w T/ + 1, w + T/] P[Xi=k] 0.16 k= P[Xi=k] 0.05 k= k=1 0.0 k= (a) =50, T=1000 i (b) =100, T=1000 i Fig. 4. P [X i = k] for the interval [1, w + T/]. Based on the analysis presented above, we approximate the distribution of the number of concurrent patches in the interval [1, w + T/] by a binomial distribution with parameters p = 1 e /T and the expected value of n i,w. ote that the larger is, the more accurate our approximation tends to be since the variability of the parameter n i,w tends to reduce. More formally, let 13

14 X be a random variable which represents the number of concurrent patches in the interval [1, w + T/]. We thus approximate the distribution of X by a binomial distribution with parameters p = 1 e /T and n w = w+t / n i=1 i,w. w+t/ Finally, we have to consider the full-object multicasts to obtain the distribution of the total number of concurrent streams required by Patching. It is the sum of the concurrent patches and the concurrent full-object multicast streams. Let Z denote the random variable which represents the total number of concurrent streams. Z has the same distribution of X shifted to the right by M C. As previously mentioned, the number of concurrent fullobject multicasts is approximated by M C = / + 1. Thus it follows that P (Z M C 1) = 0. The probability distribution function of Z is given by Equation (14). We note that both X and Z only depend on the parameters and T of the object O. 0, 0 k v 1 P (Z k) = ( ) kj=v nw l (1 e /T ) l (e /T ) nw l, v k n w + v (14) w+t / n i=1 where n w = i,w, v = M w+t/ C = / + 1, and l = j v. Our model is closer to the original continuous model of Patching as the value of d decreases. In the limit, when d 0, our model becomes identical to the original Patching model since users have immediate service. ow, we define two criteria to estimate a value for d. The first is based on the probability of more than one arrival in a unit of length d, and the second is based on the relative error between the expected value of Z and the average bandwidth B computed from Equation (3). Figure 5 shows the probability of more than one arrival in a unit of length d for several values of and T. Recall that T = D/d. From this figure we note that, for T 10 4, the probability of more than one arrival is smaller than 10 4 for all values of. Let us now evaluate the second criterion. The expectation of Z is given by: E[Z] = n w (1 e /T ) + / + 1 (15) We define the relative error between E[Z] and B as E MB = B E[Z]. In Figure 6, we plot the relative error as a function of T for = 10, 5, 40, B 75 14

15 P[A>1] e-05 T=10 3 T=10 4 1e-06 1e-07 T=10 5 1e-08 1e-09 T=10 6 1e-10 1e Fig. 5. Probability of more than one arrival in a unit of length d. and 100. We also show the complementary cumulative distribution function (CCDF) of Z for = 100 and several values of T. As already expected, the value of the relative error decreases as the value of T increases. However, we note that the relative error has no significant reduction for T > Moreover, the CCDFs of Z are practically the same for T EMB =40 =10 =5 P[ Z > k ] T=10 5, x = = T= T= (a) Relative Error T k (b) CCDF of Z Fig. 6. Analysis of the value of T. 3. Multiple Objects A multimedia server usually services client requests for more than one object simultaneously and so the bandwidth usage distribution in this case plays an important role in the overall system configuration. In the last subsection, we show that the random variable Z, which represents the distribution of the number of concurrent streams for a single object, depends on two parameters: and T. ote that we can consider T equal to a constant value computed in accordance with the two criteria previously defined (see last subsection). If T is set to a constant, the random variable Z depends exclusively on the parameter. 15

16 P[ Z > k ] Assume that there are n different objects stored in the multimedia server. These objects have lengths D 1, D,..., D n, respectively. We may compute the corresponding values T 1, T,..., T n using the two criteria previously defined (see last subsection). otice that T = max{t 1, T,..., T n } may be used for all these n objects. This is because the larger T is, the more accurate our model becomes, as already observed. We emphasize that to consider T as a constant and the same for all objects stored in the server does not mean that the objects must have a same length D, as we explain in the following. We know that T = D/d (see Table 1 for reference). Since we propose an analytical model and have a steady-state analysis, the value of D does not affect the real distribution. For example, in Figure 7 we plot the CCDF curves (obtained via simulation) for an object of popularity = 5 and considering D =100 min, 00 min, 500 min. We may see that the curves are the same. Similar behavior is observed for other different values of D and a same in the interval [10, 100] =5 and D=100 min, 00 min, 500 min k Fig. 7. Distribution of Z for = 5 and different values of D. Still we consider that the n objects stored in the multimedia server have the same play rate. This is a quite common assumption of previous work (e.g., [9,33,18,15]) and is mainly supported by recent researches (e.g., [41,11]) which analyze real media server workloads. Let Y be the random variable that denotes the number of concurrent patches due to the delivery of all objects of popularity, {1,..., max }, where max is the highest object popularity value in the server. Once we consider T as a constant and the same for all objects stored in the server, we have that Y is a sum of identical binomial random variables, with parameters p = 1 e /T and n,w = m w+t/ i=1 objects of popularity in the server [37]. n i,w w+t/, where m denotes the total number of ow we want to estimate the total required bandwidth due to patches in the multimedia server. Let Y be the random variable that denotes the total number of concurrent patches due to the delivery of all objects in the server. Then, 16

17 Y is the sum of potentially max binomial random variables with different parameters. There is no closed-form solution for the distribution of this sum. However the distribution of Y may be calculated based on the convolution of the individual probability mass functions (PMFs) of Y, for = 1,..., max. This convolution may be efficiently implemented with the Fast Fourier Transform [38]. The total number of concurrent streams is the sum of patches and full-object multicasts. The number of full-object multicasts is equal to f = max m =1 +1. Let S be the random variable that represents the total number of concurrent streams due to the delivery of all objects in the server. The distribution of S is the distribution of Y shifted to the right by f. Similarly to the analysis in the single object case, the shift is because we assume that the total number of full-object multicast streams is a constant given by f, which corresponds to the sum of all individual multicast components related to the objects in the server. Thus it follows that P (S f 1) = 0. 4 Results In this section, we validate our analytical model using simulation and show for instance how we may practically use the bandwidth usage distribution we obtained (i) to allocate bandwidth to provide a given level of QoS, (ii) to estimate the impact on QoS when some system parameters dynamically change, and (iii) to configure the overall system. We mention that the simulation results are obtained using the modeling environment Tangram-II [39,40], and have 95% confidence intervals that are within 5% of the reported values. Client arrivals are generated according to a Poisson process and the media objects are 100 minutes long. Lastly, since Patching is a multicast technique, we mainly focus on the values of in the range of 10 to 100. Tangram-II is an environment for computer and communication system modeling and experimentation developed at Federal University of Rio de Janeiro (UFRJ), with participation of UCLA/USA, for research and educational purposes. It combines a sophisticated user interface based on an object oriented paradigm and new solution techniques for performance and availability analysis. The user specifies a model in terms of objects that interact via a message exchange mechanism. Once the model is compiled, it can be either solved analytically, if it is Markovian or belongs to a class of non-markovian models, or solved via simulation. There are several solvers available to the user, both for steady state and transient analysis. The organization of this section is detailed in the following. Subsection 4.1 shows the model accuracy and how to use the bandwidth distribution of a 17

18 single object to dynamically reserve a given number of channels for a group of clients. In Subsection 4., we study the behavior of the bandwidth distribution of a single object considering different values of the threshold window w for a given object popularity. We evaluate the required bandwidth when the server operates using a window that is not the optimal one and quantify the impact on QoS. This analysis is motivated by the fact that the computation of the optimal window to be used at time t depends on the accurate estimation of the arrival rate λ at time t. Finally, Subsection 4.3 evaluates the use of the bandwidth distribution of multiple objects to compute the total server capacity to achieve a given level of QoS to the clients. 4.1 Model Validation and Bandwidth Reservation To evaluate the accuracy of our model we set T = 10 4 and use two important criteria: the relative error and the mean squared error (MSE). As for the first criterion, we compare the average bandwidth B, computed from Equation (3) (theoretical), with that computed from our model E[Z] (analytical - Equation (15)). We calculate the relative error from Error = B E[Z]. Figure 8 shows B and E[Z], and Figure 9 presents the obtained relative errors B for We may see that they lie between 10 and 10 3 and are thus satisfactory. In Table 3 we illustrate the parameters p and n w considering some values of. 14 Band theoretical band analytical band Fig. 8. Average bandwidth. As for the second criterion, we compute the MSE values for the single object and multiple objects cases from MSE single = M i=1 (1 F S(s)) (1 F Ss (s)) M i=1 (1 F Z(z)) (1 F Zs (z)) MSE multiple =, respectively, where M is the number M of samples and the subscript s of the random variables Z and S is used to refer to simulation results. Recall that Z and S represent the number of concurrent streams for the single object and multiple object cases, respectively. Figure 11 shows the complementary cumulative distribution function (CCDF) of the number of concurrent streams considering = 10, 5, 40, 50, 75 and 100. In M and 18

19 Error e Fig. 9. Relative errors for average bandwidth. this figure, we have both analytical and simulation results (solid and dotted lines, respectively). And in Figure 10 we have the computed MSE values for MSE e-04 1e Fig. 10. Mean squared error (MSE) values. ote that the values of MSE are all smaller than This indicates that the analytical results are satisfactorily close to those obtained using simulation. From Figures 11 and 10 we can observe that the accuracy of the model tends to increase with the value of. In other words, the proposed model tends to be more accurate for the most popular objects, i.e., the objects that will significantly impact on the server bandwidth. We stress that this is a tendency that has been confirmed by the overall results obtained in the experiments. Therefore we may not state that for any given value of, the larger it is, the more accurate the model is. We recall that we use the expected value of n i,w to parameterize the binomial distribution. The parameter n i,w exhibits more variability in the sub-interval [w T/ + 1, w + T/] as shown in Section 3.1. Then, the accuracy of the model is mainly affected by the size of the sub-interval [w T/ +1, w+t/] and the variability of n i,w in this subinterval. The size of the sub-interval [w T/ + 1, w + T/] decreases with the value of. However, the variability of 19

20 n i,w does not always decrease with the value of. For example, the coefficient of variation of n i,w for = 10 is smaller than that for = 5. The behavior of the MSE is a consequence of the variability of n i,w. Figure 11 also demonstrates that, in general, the analytical results underestimate the simulation ones. The explanation for this result is that we use average values to determine the multicast component (M C ) as well as to calculate the parameter n w of the binomial distribution. Consider two binomial random variables Y 1 and Y with the same parameter p (probability of success) and distinct parameters m 1 and m (number of trials), where m 1 < m. We have that Y 1 always underestimates Y [37]. In our model we approximate the number of concurrent patches by a binomial random variable X with parameters p and the expected value of n i,w. We recall that the number of patches in the i-th unit of the interval [1, w + T/] is represented by the random variable X i with parameters p and n i,w. We thus have the condition stated above, i.e., the random variable X underestimates the random variables X i with n i,w > n w. Similarly, the average value provided by M C underestimates some of the real instantaneous values of the multicast component. Table 3 Values of parameters p and n w p n w e e e e e e In Figure 1 we have the CCDF for the multiple object case. In this example we considered the same six objects analyzed in Figure 11. The analytical and simulation curves are quite close to each other, and the MSE is equal to.3e-04 which is quite satisfactory. We may also note that this value is smaller than the values obtained for 40. This occurs because although the bandwidth consumed for delivery of a given object may significantly vary over time, the total bandwidth (i.e., the sum of individual bandwidths) to deliver a large number of objects simultaneously will have lower coefficient of variation over time for independently requested objects and fixed client request rate [6,37]. In the following we focus on applications concerning the single object case and leave the discussion of the multiple object case for Section

21 P[ S > k ] P[Z > k] 1 simulation analytical P[Z > k] 1 simulation analytical = =5 = =50 = = k (a) =10,5,40 (b) = 50, 75, 100 k Fig. 11. CCDF of the number of concurrent streams for a single object. 1 simulation analytical k Fig. 1. CCDF of the number of concurrent streams for multiple objects. Using our analytical results we have the analysis that follows. With the distribution of the number of concurrent streams we can reserve bandwidth for a group of clients such that these clients may have a very low probability of not being immediately served. Assume, for example, that we reserve k channels in the server for a group of users retrieving object O such that P [Z > k] = 10 3, that is, the probability of the number of concurrent streams be larger than k is equal to The value of k may be easily obtained from Table 4 for = 5, 50, 100. This table shows the estimated bandwidth (EB), the probability of the number of concurrent streams to exceed the estimated bandwidth (P [Z > EB]), and the needed increase with respect to the average bandwidth (I) for = 5, 50, 100. The values of EB in the first line of the table are equal to the average bandwidth computed from Equation (3). We may conclude that if bandwidth reservation is based on the average bandwidth, the probability of the number of concurrent streams to exceed the reserved value is greater than 0.38 (Table 4) for all values of. In others words, the probability that a client arrives and has to wait until the server may allocate a channel for him is equal to 0.38 for = 50. This means that clients have a high probability of waiting until the server may allocate a channel to serve them. We note that for = 5, approximately 50% of the clients will not be immediately served. 1

22 Table 4 Bandwidth evaluation for = 5, 50, 100 =5 =50 =100 EB P [Z > EB] I EB P [Z > EB] I EB P [Z > EB] I 6 4.9e e e e % 10.3e % 14.7e % 8 1.3e % 11 1.e-01.% e % 9 5.e % 1 5.5e % e % e % 13.3e % e % e % e % e % 1 1.6e % e % 1 1.5e % e % e % 5.6e % 4. Analysis for Different Values of the Threshold Window Consider a server that dynamically estimates the value of λ and then determines the threshold window to be used in accordance with the value of the optimal window (Equation ()). Suppose that the value initially obtained for is equal to 75. From Figure 13, the bandwidth needed in order to have P [Z > k] 10 is 17. ow assume that the value of λ changes and the server does not immediately detect it. The server continues to operate with the optimal window determined for = 75 and keeps the same previously reserved bandwidth. Considering this scenario, we would like to answer the two following questions: (i) What is the quality of service provided by the previously reserved bandwidth to the clients accessing the server now? (ii) What is the impact of operating with a threshold window that is not the optimal one? And what are the bandwidth requirements of this threshold window? Let us analyze the first question. In our example we assume that = 75, the reserved bandwidth is equal to k channels, and the threshold window is equal to the optimal value computed for = 75. If the bandwidth and the threshold window are computed in accordance with a popularity equal to 75 and the arrival rate λ increases (decreases), we can observe from Figure 13 that the probability of the number of concurrent streams to be greater than the reserved value increases (decreases). This is an expected result, but using our model we can precisely quantify the QoS offered to the users when their arrival rate changes and the server does not detect it. Suppose, for example, that the number of reserved channels is 17. We have P [Z > 17] = 6.8e-03 for = 75. If λ decreases so that = 50 and the server does not detect it,

23 we have that P [Z > 17] = 1.1e-04. Thus, the probability that we need more channels, besides the reserved ones, decreases by one order of magnitude. On the other hand, if λ increases so that = 100, we have that P [Z > 17] = 7.e-0, i.e., the probability increases by one order of magnitude. (ote that the bandwidth distributions for = 50 and = 100 in Figure 13 were computed using the optimal window defined for = 75.) 1 P[Z > k] 0.1 = = = e k Fig. 13. Bandwidth usage distribution for different values of and the optimal window w defined for = 75. Figure 14 presents another example that illustrates the quality of service perceived by the clients when the arrival rate changes. In this example we set an initial value for the arrival rate and evaluate the QoS as the arrival rate increases. In Figure 14(a) the server is initially configured for = 10 and in Figure 14(b) it is initially configured for = 50. We note that, for = 10 and = 50, an increase of approximately 0% in the arrival rate does not significantly affect the quality offered to the clients. The same behavior is observed for other values of in the interval [10, 100]. P[Z > k] P[Z > 5] P[Z > k] P[Z>1] P[Z > 7] P[Z>14] P[Z > 8] P[Z>17] (a) For =10 : P[Z>5] = 10 - ; P[Z>7] = 10-3 ; P[Z>8] = 10-4 (b) For =50 : P[Z>1] = 10 - ; P[Z>14] = 10-3 ; P[Z>17] = 10-4 Fig. 14. Quality of service as the value of increases. ow, we present the analysis of the second question. Figure 15 shows the distribution of the number of concurrent streams Z when the threshold window varies around 0% of its optimal value for = 50 and = 100. We may notice that these distributions are quite similar. Hence, if the server does not 3

24 detect changes in the value of λ, and consequently does not update the value of the threshold window, the required bandwidth for this non optimal window is quite similar to that computed for the optimal one. In other words, the threshold window may vary within certain limits around the optimal window value without significantly affecting the bandwidth requirements. Qualitatively similar results are obtained for other values of in the interval [10, 100]. 1 1 P[Z > k] window 0% smaller than the optimal one optimal window window 0% larger than the optimal one P[Z > k] window 0% smaller than the optimal one optimal window window 0% larger than the optimal one (a) =100 k (a) =50 k Fig. 15. Distribution of the number of concurrent streams for = 50 and = 100 when the threshold window w varies around 0% of the optimal window value. In summary, we are able to predict the bandwidth requirements for a given object, or for a group of clients, in order to guarantee a certain quality of service based on the distribution of the number of concurrent streams. This distribution also allows us to predict the quality of service when the client arrival rate changes and the server does not detect it quickly. Moreover, increases of 0% in the arrival rate do not considerably impact on the quality offered to the clients. Another important point is related to the value of the threshold window. We observe that changes around 0% of the optimal window value do not significantly impact on the distribution. 4.3 Server Configuration As already mentioned, the distribution of the number of concurrent streams in the case of the multiple objects may be used, for example, to configure the server. To exemplify this point, we consider four specific scenarios based on the real case studies of [41]. They showed that the distribution of access frequencies for all media objects of two educational servers (BIBS and e-teach) can be approximated by a concatenation of two Zipf distributions. The first Zipf is used for the most popular objects and the second for the least requested objects. We only consider the first Zipf distribution since the access frequency of the least requested objects is very low and therefore has little or almost no influence on server bandwidth. The scenarios we studied are summarized in Table 5. It presents the number of objects in each scenario for each value of 4