A New Approach for the Dimensioning of Policing ... - Semantic Scholar

University of W
urzburg Institute of Computer Science Research Report Series

A New Approach for the Dimensioning of Policing Functions for MPEG-Video Sources in ATM-Systems O. Rose and M. Ritter Report No. 97

Abstract

January 1995

Institute of Computer Science, University of Wurzburg Am Hubland, 97074 Wurzburg, Germany Tel.: +49-931-8885507, Fax: +49-931-8884601 e-mail: [email protected]

The ATM transport concept of B-ISDNs allows for an ecient and exible use of network resources for variable bit rate services with high transmission rates. One major trac source which will make use of this capability will be video and multimedia applications. The changes in the cell rate of such sources are caused by the compression of the digitized video frames. The MPEG coding scheme is expected to be the compression algorithm for the rst ATM video applications. Thus, there is a need to nd source trac descriptors of MPEG video sources which can be eciently used for connection admission control and usage parameter control for these sources. Besides the peak cell rate, the sustainable cell rate may be used for these purposes. Both the ITU-T and the ATM Forum propose the GCRA to monitor these two cell rates. In this paper we develop a novel model for the cell stream of an MPEG video coder output and present a cell loss analysis for the GCRA monitoring its sustainable cell rate. A variety of ATM link capacities and sustainable cell rate parameters are studied to show the accuracy of the analysis and to test the appropriateness of the sustainable cell rate as connection trac descriptor of MPEG video sources.

1 Introduction The Asynchronous Transfer Mode (ATM) has been proposed by the ITU-T as the transport mechanism for the future Broadband Integrated Services Digital Network (B-ISDN). In ATM networks, information is transferred in small packets of xed size, called cells. Each cell consists of a payload eld of 48 Byte and a 5 Byte header. The concept of small cells and the asynchronous transfer technology allows for an e cient and exible use of network resources for Constant Bit Rate (CBR) and Variable Bit Rate (VBR) services as well as high transmission rates. The transmission rate of an ATM link depends on the medium used. Typical values are 150 Mbps and 600 Mbps. Video and LAN-to-LAN tra c are said to be the major tra c source in the rst implementation phase of ATM networks. These tra c sources are generally VBR sources and have a bandwidth demand ranging from some hundred kbps to a few Mbps. Due to the asynchronous resource sharing technology, statistical multiplexing between dierent VBR sources can be performed and therefore peak bit rate allocation is not necessary. This results in a lower total bandwidth requirement than in case of the traditional Synchronous Transfer Mode (STM). On the other hand, dierent tra c streams can now interfere with each other contrary to common STM networks. This interference leads to Cell Delay Variation (CDV) and may also cause cell losses in overload situations. Besides of Call Admission Control (CAC), priority control and additional congestion control functions, the Usage Parameter Control (UPC) is one control function de ned by the ITU-T. UPC is de ned as the set of actions performed by the network to monitor and control tra c at the user access. Its aim is to prevent the malicious or unintentional excessive usage of network resources which would lead to Quality of Service (QoS) degradation. At connection setup, a tra c contract is negotiated between the user and the network provider. The user speci es his bandwidth requirements by a number of parameters and the network commits to meet the negotiated QoS parameters as long as the user complies with its tra c contract. Mandatory source parameters are the Peak Cell Rate (PCR) and the CDV tolerance. In addition, the Sustainable Cell Rate (SCR) and the Burst Tolerance (BT) can optionally be speci ed if used, they must be speci ed together. One possible QoS parameter which the network commits to meet is the cell loss ratio. For video sources, the MPEG (ISO Moving Picture Expert Group) standard 8] is until now one of the most promising coding schemes which allows for a remarkable data reduction. However, a high degree of correlation on several time scales can be observed for MPEG coded sequences. Because of this, the dimensioning of UPC algorithms is a crucial task if a cell rate lower than the PCR should be policed. In 16] several UPC algorithms have been compared for a number of video sequences. However, the results are mainly based on simulation and the video data which was used was produced by non-standard experimental coders. In this paper we focus on the policing of real MPEG video tra c and present a new analytical approach for the dimensioning of the Generic Cell Rate Algorithm (GCRA) which is suggested by the ITU-T and the ATM Forum for cell stream monitoring in ATM networks. In Section 2 we describe the GCRA. A brief introduction to video sources is 1

given in Section 3 and a new MPEG video model is developed. The dimensioning problem of source parameters for MPEG video sources is addressed in Section 4. In Section 5 we present an analysis to estimate the cell loss probability if MPEG video sources are monitored by the GCRA. The algorithm is based on a discrete-time analysis technique developed in 21]. Numerical results to show the accuracy of the derived algorithm are discussed in Section 6 where we also look at some scaling eects occurring when MPEG video sources are policed. The paper ends with some concluding remarks. This paper is the generalized version of 18]. With the analysis presented in this paper, it is also possible to consider a spaced video cell stream, not only a video stream, where the frames are transmitted as a back-to-back burst of cells.

2 Usage parameter control in ATM environments Until now, the only source parameter which is speci ed in the standards is the PCR of a connection. The PCR of an ATM connection is de ned as the inverse of the minimum time between the generation instants of two cells of this connection. At the moment, there are discussions about the introduction of a second source parameter for VBR tra c, the SCR. It is thought as an upper bound of the average cell rate of a connection and when used, it must be speci ed together with the BT. Using a second source parameter like the SCR, the network operator may allocate less resources to a connection than if only the PCR is speci ed, while meeting the required QoS. PCR and SCR are de ned at the Physical Layer Service Access Point (PHY SAP) and the conformance of cell streams according to them is monitored at the UNI. For PCR/SCR monitoring, the GCRA was proposed by the ATM Forum 1]. There are two versions of the GCRA, namely the Virtual Scheduling Algorithm and the ContinuousState Leaky Bucket Algorithm, which are equivalent in that sense that both versions declare the same cells of a cell stream as conforming or non-conforming. We refer in this paper to the Virtual Scheduling Algorithm which is depicted in Figure 1. This algorithm was proposed rst by the ITU-T Draft Recommendation I.371 9] to monitor the PCR. In general, the GCRA uses a Theoretical Arrival Time (TAT) for the earliest time instant the next cell is expected to arrive. The TAT is initialized with the arrival time of the

rst cell of the connection ta(1). For PCR enforcement, cells should be spaced by I (the increment of the GCRA), but due to CDV a tolerance with limit L is employed. If cell number k arrives later than expected, the TAT for the next cell is given by the actual arrival time plus the increment. If cell number k arrives before its TAT but not before TAT ; L, then the TAT for the next cell is derived by incrementing the TAT for cell number k by I . Contrary, the TAT is not changed and the cell is declared as non-conforming if it arrives earlier than TAT ; L. For the enforcement of the SCR, the increment parameter I is set to Ts for a SCR of 1=Ts , and the limit parameter L = s corresponds to the maximum size of a burst that can be transmitted at PCR (cf. 1]). Cells which are identi ed as non-conforming can either be discarded or optionally be tagged to be discarded in case of network congestion. Here, we assume cell discarding. If a PCR of 1=T shall be monitored at the UNI, the CDV which is introduced between 2

arrival of cell k at time ta (k) next cell non-conforming cell

yes

ta(k) < TAT ; L

no

TAT = max(ta(k) TAT ) + I conforming cell

Figure 1: GCRA(I L) as Virtual Scheduling Algorithm. the PHY SAP and the UNI must be tolerated using the tolerance limit  . Thus, the PCR of an ATM connection can be monitored at the UNI using GCRA(T  ). The SCR 1=Ts can be monitored at the UNI by employing the BT s +  , i.e. with GCRA(Ts s +  ). The choice of the BT as s +  is motivated by the observation that a cell stream which complies with GCRA(Ts s) at the PHY SAP complies with GCRA(Ts s +  ) at the UNI (cf. 1]) if  is su cient to tolerate the CDV introduced. For video sources, the dimensioning of the parameter T to achieve conformance with GCRA(T 0) is a minor problem. A more crucial and interesting task is the dimensioning of a SCR which will generally lie between the PCR and the Average Cell Rate (ACR) of a connection. Before we address this problem, we rst focus on MPEG video source models and give some general comments about policing of video sources.

3 MPEG video source models Due to the high bandwidth needs of uncompressed video streams, several coding algorithms for the compression of video streams are in discussion. Since one of them, the MPEG (ISO - Moving Picture Expert Group) coding scheme 8, 10], will be used in a large variety of applications for the compression of video data, we will dedicate our interest to the modeling of the cell stream transmitted by an ATM adaptor fed by a video coder using this coding scheme. A video sequence consists of a series of frames, each containing a two-dimensional array 3

of pixels. The number of frames per second as well as the number of lines per frame and pixels per line depend on national standards. For each pixel, both luminance and chrominance information is stored. The compression algorithm is used to reduce the data rate before transmitting the video stream over communication networks. In MPEG coded streams, there are three types of frames, each using a slightly dierent coding scheme:

I-frames use only intra-frame coding, based on the discrete cosine transform and entropy

coding P-frames use a similar coding algorithm to I-frames, but with the addition of motion compensation with respect to the previous I- or P-frame B-frames are similar to P-frames, except that the motion compensation can be with respect to the previous I- or P-frame, the next I- or P-frame, or an interpolation between them.

Typically, I-frames require more bits than P-frames. B-frames have the lowest bandwidth requirement. The dierent ways in coding frames result in dierent statistical properties of each frame type. After coding, the frames are arranged in a deterministic periodic sequence, e.g. \IBBPBB" or \IBBPBBPBBPBB", which is called Group of Pictures (GOP) throughout the rest of the paper. These frames are packetized into ATM cells. We assume a payload of 48 Bytes per cell. All frame sizes mentioned in this paper are measured in cells. The experimental video data we use is the Star Wars movie sequence 5]. The sequence consists of 174126 frames, which corresponds to about 2 hours of movie. The GOP of this video is "IBBPBBPBBPBB". The size of the decoded video frames is 504 x 480 Pixels. Frame type Number Average Min Max CoV all 174126 41.12 2 483 1.15 I 14511 157.74 31 483 0.33 P 43531 60.58 6 454 0.63 B 116084 19.25 2 169 0.65

Table 1: Statistical data of Star Wars sequence. Table 1 shows some statistical data of the video stream and Figure 2 shows the histograms of the number of cells per frame for the dierent frame types. In addition to the distributions of the frame sizes, there are the following correlation properties of MPEG coded video streams, which can be used for modeling: 4

0.10 Relative frequency 0.02 0.04 0.06 0.08 0.0

I Frames P Frames B Frames

0

100

200 300 No. of cells

400

500

Figure 2: Distributions of the I-, B-, and P-frame sizes of the Star Wars sequence.

 Dependences introduced by the coding algorithm due to the use of a certain GOP (short-term correlations),  Long-term correlations within the frame process of a single stream due to the content of the lm.

The GOP plays the most important role concerning autocorrelation eects of an MPEG video stream coded with dierent frame types, because it xes the periodic nature of the stream. This unique property of MPEG coded videos prevents us from using video models which are based on statistical data from video sequences which have only one frame type or ignore the GOP structure, like in 2, 4, 6, 11, 12, 13, 14, 19] and 20]. Thus, there is a need to develop a new model, which describes the number of cells per frame of the coder output. The basic idea behind our model is to describe the coder output process by a cyclic array of frame size distributions of the speci c GOP owned by a video sequence. From the Star Wars sequence we will obtain therefore a sequence of 12 dierent distributions. The only frame-by-frame correlation information which is used in our model is the order of the frame size distributions xed by the GOP pattern. The long-term dependences among frames of consecutive GOPs, e.g. the correlations introduced by similar pictures of one movie scene, seem also to be less important in our case. To sum up, the GOP is the only correlation information which is used for our model. To describe the cell stream produced by the ATM adaptor of the MPEG coder, the following is assumed:

 a single-layer coder is used, 5

 the ATM adaptor and the transmission link have the same capacity,  the ATM adaptor transmits the cells with a given inter-cell distance,  the rst cell of a frame is transmitted at the beginning of the frame. This means that one frame at a time will arrive at the ATM layer, the packetization takes place, and the ATM cells are transmitted with the maximum rate of the adaptor taking into account the given spacing distance. The modeled cell stream can be described by the following parameters:

 frame duration D, which is measured in cells and can be calculated by D = FC ,

where C denotes the maximum output rate of the ATM adaptor in cells=sec and F denotes the frame rate of the video sequence in frames=sec. Of course, the maximum frame size of the encoded video sequence has to be always smaller than D cells.  inter-cell distance dcell , i.e. each used slot is followed by dcell ; 1 empty slots. If dcell = 1, the cells are transmitted back-to-back. The maximum value of dcell is D c, where A b Amax max is the number of cells of the largest frame.  frame size distributions a1(i) : : : aG(i) for a sequence with a GOP of G frames, which are sampled from real MPEG-coded video data. aPic(i) denotes the probability that frame number Pic of a GOP has a length of i cells. In Figure 3 the cell streams of a simple \IP"-GOP sequence are shown as an example, i.e. G = 2, with frame size distributions a1(i) and a2(i). Every D = 13 cells a new frame is starting, where its size in cells is computed from its distribution. Figure 3 a) shows a cell stream with dcell = 1, and b) a cell stream with dcell = 2.

4 Policing of MPEG video sources As with other services, a video connection which was accepted by the CAC mechanism has to be monitored by the UPC to check whether it ful lls the tra c contract negotiated with the network or not. Unlike with other services, for video tra c it is hard to determine the key tra c parameters. The only parameter which is available without di culties is the PCR of the coder adaptor to the ATM network, i.e. the transmission capacity of the ATM adaptor. If a more detailed description of the cell stream of the video connection is needed, one has to know the video sequence in advance, but this will only be the case for movie broadcast services or video data base retrievals. Video connections which consist of live video transmission, like broadcasts of sports events, will suer from a lack of information about the cell stream. 6

a)

A1

A2

I

P

I

D b)

I

P

I

D

Figure 3: Examples of the model cell stream. One possibility to overcome this problem is the de nition of video categories with dierent safety bandwidth requirements, for example categories with respect to the frequency of scene changes or the set of tolerated camera actions. But even if we would be able to compute a variety of parameters of the video cell stream in advance, we have to decide which of these parameters will be used for CAC and UPC. At the moment, only the PCR is already standardized as source tra c descriptor 9]. The ACR, which would be a nice tra c descriptor for CAC, is however useless, since it can not be policed e ciently by any UPC function due to the burstiness of video tra c. Studies concerning ACR policing can be found in e.g. 17] and 3]. Therefore, the introduction of a SCR as source tra c descriptor is discussed in the standardization bodies. In this paper, we investigate whether it is advantageous to use the SCR as control parameter of video cell streams and how to dimension the required parameters. Originally, standard coding schemes like MPEG-I, H.261, and MPEG-II/H.262 are not designed for the compression of videos which are transmitted on a medium where a loss of data, e.g. cells in ATM systems, is possible. Therefore they have no built-in mechanisms to classify parts of the coder output stream as important and less important, i.e. there is no possibility to have priority classes with respect to cell loss of the packetized coder cell stream. This leads to the disadvantage that any UPC function can only carry out blind cell discarding actions, and cells which are important for decoding the video are discarded even if there were cells in the stream, which are less important. This behavior can only be changed, if new coding schemes are developed, which allow the packetizing algorithm of the ATM layer to produce cells with priorities or the establishment of two VCs with dierent cell loss requirements. Until now, such coders are in development and will not be available in the near future. Therefore, it is very likely that the MPEG coding scheme will be used for the rst video applications in ATM networks. In the next section we present a new analytical approach for the dimensioning of the 7

UPC function in ATM networks if MPEG video sources are considered.

5 Cell loss analysis The algorithm we present in the following is of iterative nature and based on the discretetime analysis of the GI X ]=D=1 ; S queueing model presented in 21]. This analysis technique has been applied in 7] to analyze the GCRA if the input tra c is assumed to follow a renewal process. An extension which deals with ON/OFF sources was presented in 17]. Based on this, we describe an algorithm to cope with the cyclic occurrence of frames of dierent types in MPEG coded video sequences due to the coding in GOPs. The actual state of the GCRA(Ts s ) is described by a discrete-time random variable Z (t), which represents the remaining time until the next cell is expected to arrive 7]. A cell arriving at time t0 seeing the GCRA in state Z (t0) = i is considered to be conforming for i  s, otherwise non-conforming. Since we have to deal with dierent frame types in the GOP, we use the following notation:

APic ; ZPic
k + ZPic
k

discrete random variable for the size of frame number Pic in the GOP measured in cells, Z (t) just before the beginning of the k-th slot in frame number Pic in the GOP, Z (t) just after the beginning of the k-th slot in frame number Pic in the GOP.

; and Z + we use the terms z ; (i) and z + (i), respecFor the distributions of ZPic
k Pic
k Pic
k Pic
k tively. The frame sizes APic are assumed to follow renewal processes with distributions aPic(i). For the sake of simplicity the indices denoting the iteration steps are omitted. Let us consider a particular frame in the GOP, say frame number Pic and assume a frame size of dD=dcell e cells. This corresponds to a frame which has the maximum possible size to be transmitted with an inter-cell spacing of dcell cells. If k mod dcell = 0, i.e. a cell is + is determined out of Z ; by arriving at slot k, then ZPic
k Pic
k

8< ; ; > ZPic
k : ZPic
k s + = ZPic
k ; + Ts : Z ;  s : : ZPic
k Pic
k

(1)

+ is computed by Otherwise, ZPic
k + = Z; : ZPic
k Pic
k

(2)

According to equation (1), the corresponding distributions can be obtained by

8 >< 0 : 0  i  s + ; zPic
k (i) = > zPic
k (i) : s < i < Ts : zPic
k ; (i) + z ; (i ; T ) : T  i  T +  s s s s Pic
k 8

(3)

for s < Ts and for s  Ts the following holds:

8 >< 0 : 0  i < Ts + ; (i ; Ts ) : Ts  i  s zPic
k (i) = > : zPic
k : zPic
k ; (i) + z ; (i ; T ) :  < i  T +  s s s s Pic
k

(4)

In case of equation (2), i.e. no cell arrival in slot k (k mod dcell 6= 0), we get the distribu+ (i) by tions zPic
k + (i) = z ; (i): zPic
k Pic
k

(5)

; The computation of ZPic
k +1 is driven by the decrease of Z (t) by one each slot until it reaches zero, i.e. ; + ZPic
k +1 = maxf0 ZPic
k ; 1g:

(6)

Therefore, the distributions are determined by

8> + + (1) : i = 0 zPic
k (0) + zPic
k < ; + (i + 1) : 0 < i < T +  : zPic
k zPic
k s s +1 (i) = > : 0 : i = Ts + s

(7)

The next step is the derivation of the state of the GCRA(Ts s) at the beginning of the ; assuming a frame size equal to dD=d e next frame boundary. Since we computed ZPic
k cell cells, we now have to take into account the dierent possible frame sizes. The state of the GCRA(Ts s ) just before the frame boundary in dependence of the size j of the current frame is given by  ;
; (D ; j  )g ZPic
D
j = maxf0 ZPic
j

(8)

if we de ne j  = (j ; 1)  dcell + 1. The reason for this is that there are no cell arrivals in the last (D ; j ) slots of the frame period. We obtain the corresponding distributions by

8 DP ;j
; >> zPic
j
(l) : i = 0 < l=0  zPic
D
j (i) = > zPic
j : ;
(i + (D ; j )) : 0 < i  T +  ; (D ; j ) s s >: 0 : Ts + s ; (D ; j ) < i  Ts + s

(9)

 Using ZPic
D
j , the system state just before the next frame boundary is given by ;  ZPic +1
0 = ZPic
D
APic 

(10) 9

; where (Pic + 1) is computed modulo G. To obtain zPic +1
0 (i), we have to multiply the  system state distributions just before the frame boundary zPic
D
j (i) by the probabilities of observing a frame of size j for the frame type Pic. This leads to the following equation: ; zPic +1
0 (i) =

dD=d Xcelle j =1

 aPic (j )  zPic
D
j (i):

(11)

Now, the distributions in equilibrium can be derived by applying iteratively the equations presented above with respect to the GOP used and the current slot in each frame. With the equilibrium system state distributions just before the cell arrivals, the probability pPic (k) to observe a non-conforming cell at slot k (k = 0 : : : D ; 1) in a frame of type Pic is

pPic (k) =

TX s+s i=s +1

; (i) zPic
k

for

Pic = 1 : : :  G:

(12)

To derive the probability PPic to observe a non-conforming cell in a given frame in the GOP, the pPic (k)'s have to be multiplied with the complementary cumulative probability distribution acPic(i) of the frame size distribution aPic(i). This has only to be done for values of k, where cell arrivals are possible, i.e. k mod dcell = 0. Furthermore, a normalization is required: dD;1P=dcelle

PPic =

k=0

pPic (k  dcell )  acPic(k) dD;1P =dcell e ac (k) k=0

for

Pic = 1 : : :  G:

(13)

Pic

The overall cell loss probability can now easily be obtained by

PG P  EA ] Pic Pic Pic =1 Ploss = PG EA ] : Pic=1

(14)

Pic

6 Numerical results 6.1 Parameters and conguration In this section we present numerical results based on simulation and analysis to show the eectiveness of SCR monitoring of video cell streams and point out some interesting properties for dimensioning the UPC function. We focus on four ATM adaptor capacities: 150 Mbps, 75 Mbps, 37.5 Mbps, and 34 Mbps. For the considered Star Wars sequence, 10

Capacity D Tpeak Tmean 150 Mbps 14740 30 360 75 Mbps 7370 15 180 37.5 Mbps 3685 7 90 34 Mbps 3340 6 81

Table 2: ATM adaptor and SCR parameters. this leads to the frame durations and minimum (Tpeak) and maximum values (Tmean) for the SCR parameter Ts shown in Table 2. It is important to point out that all simulation results in this paper were produced using directly the sequence of the frame sizes of the Star Wars movie and not by means of the coder output model.

6.2 Simulation study We rst give simulation results for a capacity of C = 150 Mbps to show some eects for dimensioning of the parameters Ts and s on the cell loss probability. Figure 4 shows the loss curves for Ts ranging from 30 to 110, where s is measured in multiples of the frame duration D. For small values of Ts the losses decrease very fast, whereas for values of Ts larger than 60 a knee in the curve can be observed. The knee is always located at a value of s where the UPC function tolerates bursts which are as long as the maximum frame length of the video sequence. If a certain value of s is reached, e.g. s  3:8D for Ts = 80, the loss probability drops to zero, i.e. there are no losses as soon as the parameter s is large enough to force the UPC function to accept consecutive bursts of several frames. We proved this assumption by using only the frame data of GOPs with a high mean frame length, i.e. worst case GOPs, for the simulation and received the same drop-down locations of the curves. With all values of Ts in Figure 4, small loss probabilities can be achieved, but it should be noted that for Ts = 90 already a buer capacity in the network elements of about 1000 cells is needed to store the burst of this single connection that is tolerated to achieve a cell loss probability smaller than 10;6 . Generally, buer sizes in ATM networks are in the order of 103 cells. Therefore, the value Ts should be chosen to be close to Tpeak to obtain realistic values for the required buer size, e.g. for Ts = 60 a buer size of about 500 cells is needed. Figure 5 shows that the SCR owns a certain scalability property. The cell loss curves remain almost identical, if the parameters Ts and s are scaled by the same factor as the adaptor capacity. In Figure 5 two groups of curves for Ts = 60 and Ts = 100 are shown. We start with a capacity of 150 Mbps and use the scaling factors 1.0, 0.5, and 0.25, i.e. Ts = 60 for 150 Mbps, Ts = 30 for 75 Mbps, and Ts = 15 for 37.5 Mbps. To allow for a comparison of these curves, the horizontal axis has to be scaled accordingly. Note, the 11

cell rejection probability

100 10;1 10;2 10;3 10;4 10;5 10;6 10;7

Ts = 30

60

80

2

4

90

100

110

10;8 10;9

0

6

8

10 12 14 s in frame durations Figure 4: Dependence of cell rejection probability on Ts (C = 150 Mbps). video sequence, i.e. the number of cells per frame, which was used to create the input for the UPC function was the same for each scaling factor. Figure 5 shows that the curves of the two groups are matching well. In general, the curves are matching better for large values of Ts. By means of this scalability property the SCR parameters for a large variety of adaptor capacities can be calculated easily by multiplying a constant, if the parameters for a single capacity is known.

6.3 Analytical results Now, we investigate the accuracy of the analysis presented in Section 5. Generally, the results are of exact nature if the frame sizes in the GOP follow renewal processes. In reality however, correlations can be observed. In Figure 6 the overall cell loss probability for dierent choices of dcell is plotted over s to verify the accuracy of our analysis. We use a adaptor capacity of 34 Mbps and a SCR with Ts = 15. The relative dierence between the analytical and simulation values is always smaller than 1%. Since the accuracy does not depend on the choice of the inter-cell distance (cf. Figure 6), we use dcell = 1 for the following numerical examples. Furthermore, loss curves for I, P, B frames only, as well as for all frames are shown. 12

cell rejection probability

100

C = 150 Mbps C = 75 Mbps C = 37:5 Mbps

10;1 10;2 10;3 10;4 10;5

Ts = 100

10;6 10;7

Ts = 60

10;8 10;9

0

2

4

6

8 10 s in frame durations Figure 5: Scalability properties of policing parameters.

In Figure 7 results for the same adaptor capacity of 34 Mbps and a higher SCR with Ts = 10 are presented. A relative dierence of 1% can also be observed for the loss curves of the single frame types I, P, and B. This leads to the conclusion that our simple video coder output model is appropriate for the estimation of the cell losses for this type of UPC function. For the parameter set of Figure 7, the B frames always experience less losses than the P-frames, and the P-frames less losses than the I-frames. This seems to be obvious, because the mean frame size of the B-frames is smaller than the one of the P-frames, and the mean frame size of the P-frames smaller than the one of the I-frames. However, as presented in Figure 8, crossing of the loss curves of dierent frame types is possible. This behavior is depending on the long-term correlations of the video sequence which is used. Moreover, there is no crossing of the curves, if the value of Ts is chosen to be close to Tpeak and s can be chosen small, too. As soon as the value of s is larger than about two frame durations the analytical results underestimate the cell losses, because the MPEG model does not take into account long-range dependences within the video sequence. This eect is not problematic, however, since large values of s would lead to large buers within the network. For useful values of Ts and s the analysis is very accurate (cf. Figure 7 and 9). Figure 9 shows the curves for a 150 Mbps ATM adaptor and a value of Ts = 30. The behavior of the curves is similar to that of Figure 7. 13

cell rejection probability

100

•• •• •

10;1

•• • •

10;2

• • •

10;3

•

10;4

• •

•

•

• • •

10;5

simulation dcell = 1 dcell = 2 dcell = 4 dcell = 6

•

10;6 10;7 10;8 10;9

• • •

cell rejection probability

0:5

•

•

•

2:0 2:5 s in frame durations Figure 6: Approximation accuracy (Ts = 15, C = 34 Mbps).

100

0:0

•

1:0

1:5

•

10;1

• • •

10;2 10;3

• • •

•

• •

10;4

•

• • •

•

10;5

all frames I-frames only P-frames only B-frames only simulation

• •

10;6 10;7 10;8 10;9

0:0

0:5

1:0

1:5 2:0 s in frame durations Figure 7: Approximation accuracy (Ts = 10, C = 34 Mbps, dcell = 1). 14

• •

10;1

all frames I-frames only P-frames only B-frames only simulation

• • •

10;2

• •

•

10;3

•

•

10;4

•

•

•

•

•

•

10;5

•

•

•

•

•

10;6

•

•

• •

10;7 10;8 10;9

4 5 6 s in frame durations Figure 8: Approximation accuracy divergence (Ts = 20, C = 34 Mbps, dcell = 1).

cell rejection probability

cell rejection probability

100

0

1

100

2

3

• • • •

10;1 10;2

• •

10;3

•

10;4

•

•

• • •

all frames I-frames only P-frames only B-frames only simulation

10;5 10;6 10;7 10;8 10;9

0:0

0:5

1:0

1:5 2:0 s in frame durations Figure 9: Cell rejection analysis (Ts = 30, C = 150 Mbps, dcell = 1). 15

7 Conclusions In this paper a new model for MPEG video sources and an analytical algorithm for the cell losses of the GCRA monitoring the Sustainable Cell Rate of an MPEG video cell stream was presented. The results show that the analysis using this simple model is very accurate compared to the simulation results based on real MPEG video data. A minor drawback of the analysis technique used in this paper is that the computation time depends on the frame duration D, i.e. the ATM adaptor capacity. Large capacities lead to time consuming computations. For all parameter sets considered in this paper no numerical problems have occurred but the computation time increases rapidly with the link capacity. However, for large link capacities this can be avoided if we make use of the scalability property of the SCR parameters. As far as the dimensioning of the GCRA parameters Ts and s is concerned, the analytical and simulation results lead to several conclusions. To deal with reasonable buer sizes of the network elements it is necessary to keep the parameter Ts close to the PCR of the video sequence considered. In this case, both the required buer can be kept small and small values of s can be achieved. The parameter s should always be chosen at least as large as the maximum frame size of the video sequence times Ts to obtain small loss probabilities. Unfortunately, the loss curves show that the I-frames which contain the most important information of the MPEG frames experience higher losses than the other frame types. Discarding of cells on a frame-type basis could therefore lead to an improvement of the video quality 15]. For video sequences with rapidly changing scene contents like action movies or sports events the SCR will lie generally close to the PCR if s is chosen reasonably. This implies a poor multiplexing gain. For sequences like video conferencing or video telephony, however, the SCR can be dimensioned remarkably lower than the PCR due to minor changes in the scene content.

Acknowledgement The authors would like to thank Mark Garrett (Bellcore, Morristown, NJ) for providing the Star Wars data. The authors appreciate the support of the Deutsche Bundespost Telekom (FTZ).

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32] R. V. Book, J. H. Lutz und K. W. Wagner. On complexity classes and algorithmically random languages. August 1991. 33] F. Hubner. Queueing analysis of resource dispatching and scheduling in multi-media systems. September 1991. 34] H. Gold und G. Bleckert. Analysis of a batch service system with two heterogeneous servers. September 1991. 35] H. Vollmer und K. W. Wagner. Complexity of functions versus complexity of sets. Oktober 1991. 36] F. Hubner. Discrete-time analysis of the output process of an atm multiplexer with periodic input. November 1991. 37] P. Tran-Gia und O. Gropp. Structure and performance of neural nets in broadband system admission control. November 1991. 38] G. Buntrock und K. Lorys. On growing context-sensitive languages. Januar 1992. 39] K. W. Wagner. Alternating machines using partially dened \AND" and \OR". Januar 1992. 40] F. Hubner und P. Tran-Gia. An analysis of multi-service systems with trunk reservation mechanisms. April 1992. 41] U. Huckenbeck. On a generalization of the bellman-ford-algorithm for acyclic graphs. Mai 1992. 42] U. Huckenbeck. Cost-bounded paths in networks of pipes with valves. Mai 1992. 43] F. Hubner. Autocorrelation and power density spectrum of atm multiplexer output processes. September 1992. 44] F. Hubner und M. Ritter. Multi-service broadband systems with CBR and VBR input trac. Oktober 1992. 45] M. Mittler und P. Tran-Gia. Performance of a neural net scheduler used in packet switching interconnection networks. Oktober 1992. 46] M. Kowaluk und K. W. Wagner. Vector language: Simple description of hard instances. Oktober 1992. 47] B. Menzel und J. Wol von Gudenberg. Kommentierte Syntaxdiagramme fur C++. November 1992. 48] D. Emme. A kernel for funtions denable classes and its relations to lowness. November 1992. 49] S. O hring. On dynamic and modular embeddings into hyper de Bruijn networks. November 1992. 50] K. Poeck und M. Tins. An intelligent tutoring system for classication problem solving. November 1992. 51] K. Poeck und F. Puppe. COKE: Ecient solving of complex assignment problems with the propose-and-exchange method. November 1992. 52] Th. Fritsch, M. Mittler und P. Tran-Gia. Articial neural net applications in telecommunicationsystems. Dezember 1992. 53] H. Vollmer und K. W. Wagner. The complexity of nding middle elements. Januar 1993. 54] O. Gihr, H. Gold und S. Heilmann. Analysis of machine breakdown models. Januar 1993.  55] S. Ohring. Optimal dynamic embeddings of arbitrary trees in de Bruijn networks. Februar 1993. 56] M. Mittler. Analysis of two nite queues coupled by a triggering scheduler. Marz 1993. 57] J. Albert, F. Duckstein, M. Lautner und B. Menzel. Message-passing auf transputer-systemen. Marz 1993. 58] Th. Stock und P. Tran-Gia. Basic concepts and performance of high-speed protocols. Marz 1993. 59] F. Hubner. Dimensioning of a peak cell rate monitor algorithm using discrete-time analysis. Marz 1993. 60] G. Buntrock und K. Lorys. The variable membership problem: Succinctness versus complexity. April 1993. 61] H. Gold und B. Frotschl. Performance analysis of a batch service system working with a combined push/pull control. April 1993. 62] H. Vollmer. On dierent reducibility notions for function classes. April 1993.  63] S. Ohring und S. K. Das. Folded Petersen Cube Networks: New Competitors for the Hyepercubes. Mai 1993.  64] S. Ohring und S. K. Das. Incomplete Hypercubes: Embeddings of Tree-Related Networks. Mai 1993.  65] S. Ohring und S. K. Das. Mapping Dynamic Data and Algorithm Structures on Product Networks. Mai 1993. 66] F. Hubner und P. Tran-Gia. A Discrete-Time Analysis of Cell Spacing in ATM Systems. Juni 1993. 67] R. Dittmann und F. Hubner. Discrete-Time Analysis of a Cyclic Service System with Gated Limited Service. Juni 1993. 68] M. Frisch und K. Jucht. Pascalli-P. August 1993. 69] G. Buntrock. Growing Context-Sensitive Languages and Automata. September 1993.  70] S. Ohring und S. K. Das. Embeddings of Tree-Related Topologies in Hyper Petersen Networks. Oktober 1993.  71] S. Ohring und S. K. Das. Optimal Communication Primitives on the Folded Petersen Networks. Oktober 1993.

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72] O. Rose und M. R. Frater. A Comparison of Models for VBR Video Trac Sources in B-ISDN. Oktober 1993. 73] M. Mittler und N. Gerlich. Reducing the Variance of Sojourn Times in Queueing Networks with Overtaking. November 1993. 74] P. Tran-Gia. Discrete-Time Analysis Technique and Application to Usage Parameter Control Modelling in ATM Systems. November 1993. 75] F. Hubner. Output Process Analysis of the Peak Cell Rate Monitor Algorithm. January 1994. 76] K. Cronauer. A Criterion to Separate Complexity Classes by Means of Oracles. January 1994. 77] M. Ritter. Analysis of the Generic Cell Rate Algorithm Monitoring ON/OFF-Trac. January 1994. 78] K. Poeck, D. Fensel, D. Landes und J. Angele. Combining KARL and Congurable Role Limiting Methods for Conguring Elevator Systems. Januar 1994. 79] O. Rose. Approximate Analysis of an ATM Multiplexer with MPEG Video Input. Januar 1994. 80] A. Schomig. Using Kanban in a Semiconductor Fabrication Environment | a Simulation Study. Marz 1994. 81] M. Ritter, S. Kornprobst, F. Hubner. Performance Analysis of Source Policing Architectures in ATM Systems. April 1994. 82] U. Hertrampf, H. Vollmer und K. W. Wagner. On Balanced vs. Unbalanced Computation Trees. May 1994. 83] M. Mittler und A. Schomig. Entwicklung von "Due{Date\{Warteschlangendisziplinen zur Optimierung von Produktionssystemen. Mai 1994. 84] U. Hertrampf. Complexity Classes Dened via k-valued Functions. Juli 1994. 85] U. Hertrampf. Locally Denable Acceptance: Closure Properties, Associativity, Finiteness. Juli 1994. 86] O. Rose, M. R. Frater. Delivery of MPEG Video Services over ATM. August 1994. 87] B. Reinhardt. Kritik von Symptomerkennung in einem Hypertext-Dokument. August 1994. 88] U. Rothaug, E. Yanenko, K. Leibnitz. Articial Neural Networks Used for Way Optimization in Multi-Head Systems in Application to Electrical Flying Probe Testers. September 1994. 89] U. Hertrampf. Finite Acceptance Type Classes. Oktober 1994. 90] U. Hertrampf. On Simple Closure Properties of #P. Oktober 1994. 91] H. Vollmer und K.W. Wagner. Recursion Theoretic Characterizationsof Complexity Classes of Counting Functions. November 1994. 92] U. Hinsberger und R. Kolla. Optimal Technology Mapping for Single Output Cells. November 1994. 93] W. Noth und R. Kolla. Optimal Synthesis of Fanoutfree Functions. November 1994. 94] M. Mittler und R. Muller. Sojourn Time Distribution of the Asymmetric M=M=1==N { System with LCFS-PR Service. November 1994. 95] M. Ritter. Performance Analysis of the Dual Cell Spacer in ATM Systems. November 1994. 96] M. Beaudry. Recognition of Nonregular Languages by Finite Groupoids. Dezember 1994. 97] O. Rose und M.Ritter. A New Approach for the Dimensioning of Policing Functions for MPEG-Video Sources in ATM-Systems. Januar 1995.

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