Bandwidth Estimation in PrerecordedVBR-Video Distribution Systems Exploiting

Stream Correlation

G. Boggia, P. Camarda, and D. StriccoliDipartimento di Elettrotecnica ed Elettronica

Politecnico di Bari

Via Orabona, 4 – 70125 BARI, Italy

e-mail: {g.boggia, camarda}@poliba.it, striccoli@deemail.poliba.it

Abstract

The main aspect of several multimedia applications is the transmissionof Variable Bit Rate (VBR) video streams. Such applications require highbandwidth with stringent Quality of Service (QoS) guarantee. In such sys-tems, a statistical bandwidth estimation is especially relevant as it is nec-essary for network infrastructure planning and admission control. In thispaper, an original algorithm to estimate the aggregate bandwidth, neededby a given number of correlated video streams, is proposed and analyzed,by modeling the starting point and the permanence time in a bandwidthlevel of the smoothed video stream with known statistical distributions. Theobtained analytical results are validated by simulation and compared withother numerical results already present in the literature, discussing the ef-fectiveness of the proposed solution.

1 Introduction

In the telecommunication world, multimedia applications like Video on Demand(VoD), Distance Learning, Internet Video Broadcast, and so on, will likely assumea growing importance [1, 2]. The core aspect of such applications is the transmis-sion of video streams requiring a high bandwidth with stringent Quality of Service(QoS) guarantee, in terms of loss, jitter, delay, and so on. The scenario presentedin this paper refers to a video distribution system where a great number of filmsstored in a server are transmitted across a network. All films are supposed to be

1

codified with the MPEG standards which in most cases produce a Variable BitRate (VBR) traffic. As an example, in Fig. 1 the bit rate of the first 32.000 framesof a high quality MPEG-4 coded trace of the film “Jurassic Park” is reported [3].

0

500

1000

1500

2000

2500

3000

3500

0 5000 10000 15000 20000 25000 30000

Time (frames)

Bit

rat

e (k

bit

/s)

Figure 1: 32.000 frames of the “Jurassic Park” film.

Given the nature of VBR traffic, a static conservative bandwidth assignmentfor statistically multiplexed video streams results in a non negligible waste ofnetwork resources. In fact, if all video stream starting points are randomly shiftedin time, the sum of all stream bit rates will usually be lower than the sum of peakrates. Thus, a statistical approach is necessary to evaluate the bandwidth resourcesneeded by a given number of video streams, respecting the QoS requirements.

Workahead smoothing techniques can be exploited to reduce the total amountof bandwidth assigned to all video streams [4]-[10]. These techniques are basedon the reduction of the peak rate and the bit rate variability of every stream presentin the network. They consist in smoothing the bursty behavior of video streamsby transmitting, ahead of playback time, pieces of the same film at a constantbit rate variable from piece to piece according to a scheduling algorithm. At thetransmission side the stream rate is controlled by using a buffer, while at the re-ceiving side frames are temporarily stored in a client buffer and extracted duringthe decoding process. Obviously, bit rates have to be chosen appropriately in or-der to avoid buffer overflow and underflow, ensuring a continuous playback at theclient side. The size of smoothing buffers determines number and duration of theConstant Bit Rate (CBR) pieces that characterize the smoothed video stream. Anincrease of the smoothing buffer size produces a smaller number of CBR segmentsand a reduction of both the peak rate and the rate variability. As highlighted in[4, 5], a consistent gain in network resource utilization can be obtained by exploit-

2

ing this technique, which is likely to be implemented in real systems. Differenttypes of smoothing techniques can be implemented: the Critical Bandwidth Allo-cation (CBA) algorithm minimizes the number of bandwidth increments [7], theMinimum Changes Bandwidth Allocation (MCBA) algorithm minimizes also thenumber of bandwidth decrements [8], the Minimum Variability Bandwidth Allo-cation (MVBA) reduces the variability of the bandwidth changes [9], while thePiecewise Constant Rate Transmission and Transport (PCRTT) algorithm dividesthe video flow into a piecewise CBR stream, with each piece lasting for a fixedtime interval [9, 10, 11]. The choice of one of these algorithms depends on whathas to be optimized, e.g., the peak rate, the number, the variability, and the peri-odicity of bandwidth changes [9].

As an example, in Fig. 2 the bit rate of the "Jurassic Park" film is reportedsmoothed with a client buffer of 512 kBytes and utilizing the MVBA approach[5].

The study developed in this paper is valid whatever smoothing technique isexploited and can be utilized also in the case of unsmoothed video streams. Itis worth noting that smoothed video streams, characterized by long CBR pieces,represent exactly the source traffic model exploited in [12].

0

200

400

600

800

1000

1200

0 5000 10000 15000 20000 25000 30000

Time (frame)

Bit

rat

e (k

bit

/s)

Figure 2: 32.000 frames of the “Jurassic Park” film, smoothed with the MVBAapproach and a client buffer of 512 Kbytes.

In such systems, a statistical bandwidth estimation is especially relevant asit is necessary for network infrastructure planning and for an efficient admissioncontrol [4, 13, 14].

As considered in [4], [15]-[18], and in many other works, here we will modelnetwork switches as bufferless multiplexers.

3

This hypothesis can be supported by the following considerations. First of all,it has been shown [19, 20] that large network buffers are of substantial benefitin reducing losses due to rapid video bit rate changes and that the only effectiveway to reduce losses due to slow video bit rate changes is to allocate sufficientbandwidth for each network link. Since the smoothed traffic has mainly a slowbit rate variability (see Fig. 2), the adoption of large network buffers is not usefulin this scenario. A second consideration is that the employment of large buffersintroduces delays and jitters along the network nodes that can be intolerable forthe correct delivery of video streams guaranteeing QoS specifications.

In literature, there are various approaches for statistical estimation of band-width occupied by a number of video streams entering a bufferless multiplexer[4, 15, 16]. All these approaches are based on the assumption of independenceamong video streams. In particular, in [4] a Chernoff Bound based admissioncontrol method, for the general case of different video stream types, has been ex-ploited. Some variants of the Chernoff bound have been proposed in [16]. In [15]two different methods for admission control schemes exploiting the Normal Ap-proximation and Large Deviation (LD) approximations based on Chernoff boundhave been evaluated. They have been tested on the simple case of a single videostream type.

In this paper an original algorithm to estimate the aggregate bandwidth neededby a given number of correlated video streams is proposed and analyzed. Thesestreams are derived from a single source, respecting a specified loss probability,i.e., a QoS parameter defined as the probability that the aggregate bandwidth of allthe streams is above a given threshold [4]. Analytical results are compared withsimulation results and with the ones based on Chernoff bound [4], extending andgeneralizing previous studies on this topic, based on independence assumptions[4, 15, 16, 21, 22].The paper is structured as follows. In Section 2 the bandwidth estimation algo-rithm is described in detail. In Section 3 some numerical results are provided,comparing the proposed method with the Chernoff bound method and with simu-lation results. Finally, in Section 4 some conclusions are given.

2 Aggregate bandwidth estimation

In this section, an original algorithm to estimate the aggregate bandwidth of agiven number of correlated video streams is presented and discussed. Let us sup-pose to have a generic numberN of smoothed video streams deriving from asingle source (e.g., the “Jurassic Park” film of Fig. 1). All the considered videostreams present the same bit rate evolution. To characterize the source, the modelproposed in [12] can be utilized. That is, the film can be represented by a finite

4

number ofM bandwidth levelsλ1, λ2, . . . , λM , each of them identifying a CBRpiece of the smoothed film. The permanence time in a bandwidth level is con-sidered as a random variable with a statistical distribution characterized by thecumulative density function (cdf)F1(x) and the probability density function (pdf)f1(x). The marginal distribution of the bandwidth levels is defined asp(λi) = πi,i = 1, . . . , M . All these probabilitiesπi can be easily derived by dividing the per-manence time in theith bandwidth level by the total video stream duration. Thestate probability can be expressed by:

p(xN = λαN, xN−1 = λαN−1

, . . . , x1 = λα1) = p(λαN, λαN−1

, . . . , λα1) (1)

wherexi, i = 1, . . . , N , represents the bandwidth level of theith video stream andαi is an integer that can assume all values from 1 toM .

Obviously, the bandwidth occupied by all the streams in a generic state isΛ = λα1 + · · · + λαN

. Thus, the aggregate bandwidthΛS required to satisfy agiven loss probabilitypl (considered as QoS parameter) can be statistically derivedby the following expressions:

p(Λ > ΛS) = pl =M∑

α1=1

M∑α2=1

· · ·M∑

αN=1

p(λαN, λαN−1

, . . . , λα1) (2)

withλα1 + · · ·+ λαN

> ΛS. (3)

The constraint (3) identifies a subsetS of all system states(λαN, λαN−1

, . . . λα1)such that the equation (2) is also verified.

The main issue is to evaluate the state probabilities represented by (1). Tosolve this problem, letni,j be the number of bandwidth level changes, startingfrom the levelλαi

and ending to the levelλαj. Let us suppose that the bandwidth

levels of the smoothed video steam are numbered progressively as they appear intheir temporal sequence, i.e.,λ1 is the first CBR piece of the smoothed stream,λ2

is the second, and so on. For example, ifαi = 3 andαj = 7, ni,j = αj − αi = 4.To derive the integersni,j whenαj < αi, it is supposed that the video streambegins again after its end, starting fromλ1 after the bandwidth levelλM . Giventhis assumption, it is clear that ifαj < αi thenni,j = M − (αi − αj). Taking intoaccount integersni,j, equation (1) can be rewritten as follows:

p(λαN, . . . , λα1) =

M−1∑n1,2=0

M−1∑n2,3=0

. . .

M−1∑nN−1,N=0

p(λα1 , . . . , λαN, n1,2, . . . nN−1,N)

and, exploiting conditional probabilities:

p(λαN, . . . , λα1) =

M−1∑n1,2=0

M−1∑n2,3=0

. . .

M−1∑nN−1,N=0

p(λαN|λαN−1

, . . . , λα1 , n1,2, . . . nN−1,N)·

5

·p(λαN−1|λαN−2

, . . . , λα1 , n1,2, . . . nN−1,N) . . . p(λα1|n1,2, . . . nN−1,N)·p(n1,2 . . . nN−1,N)(4)

We suppose that there is a strong correlation only between two consecutivelevelsλαi

andλαi+1. Thus, they depend only on the number of level changes

ni,i+1 . Furthermore, all theni,j are supposed to be independent from each other.For these reasons, (4) becomes:

p(λαN, . . . , λα1) =

M−1∑n1,2=0

M−1∑n2,3=0

. . .

M−1∑nN−1,N=0

p(λαN|λαN−1

, nN−1,N)·

·p(λαN−1|λαN−2

, nN−2,N−1) . . . p(λα2|λα1 , n1,2) · p(n1,2) . . . p(nN−1,N) =

=M−1∑

nN−1,N=0

p(λαN|λαN−1

, nN−1,N)p(nN−1,N) · · ·M−1∑

n1,2=0

p(λα2|λα1 , n1,2)p(n1,2)p(λα1)

and, in a more compact form:

p(λαN, . . . , λα1) = πα1

N−1∏

k=1

M−1∑nk,k+1=0

p(λαk+1|λαk

, nk,k+1)p(nk,k+1)

(5)

Each factor of (5) in square brackets can be easily derived from real video traces.In particular, from the bit rate evolution of the smoothed film, it can be observedthat the bandwidth levelλk+1 can be reached fromλk only afternk,k+1 bandwidthchanges. No other value of is permitted. Thus, the (5) can be modified as follows:

p(λαN, . . . , λα1) = πα1

N−1∏

k=1

p(λαk+1|λαk

, nk,k+1)p(nk,k+1) (6)

The probabilityp(λαk+1|λαk

, nk,k+1) assumes the value 1, since, given the band-width levelλαk

andnk,k+1, only the bandwidth levelλαk+1can be reached. Then,

the formula (6) becomes:

p(λαN, . . . , λα1) = πα1

N−1∏

k=1

p(nk,k+1) (7)

To evaluate the state probability by (7), it is necessary to derive the probabilitiesp(nk,k+1) from a statistical model. To find these probabilities, for each1 ≤ k ≤N − 1 , let us suppose that the first video stream starts att = 0, the second onestarts after a random temporal shiftt1 , the third one after a temporal shift oft2(if compared to the first one), and so on. Now, fixing a generic time instantt∗ andobserving the bandwidth levels of theN video streams of the same type int∗, it is

6

easy to note that the second video stream will be in a bandwidth level that can bederived simply counting the number of bandwidth changes int1, starting from thebandwidth level of the first video stream. Similarly, the third video stream will bein a bandwidth level obtained counting the bandwidth changes int2, starting fromthe bandwidth level of the first video stream. The same can be done for the restof theN video streams. Then, the probabilityp(n1,2) represents the probability tohaven1,2 bandwidth changes betweenλα1 andλα2 in a time interval equal tot1.Let Tα1 be the permanence time of the first video stream in the bandwidth levelλα1, Tα1+1 the permanence time inλα1+1, and so on. There will ben1,2 bandwidthchanges int1 if and only if:

tr + Tα1+1 + · · ·+ Tα1+n1,2 ≤ t1 ≤ tr + Tα1+1 + · · ·+ Tα1+n1,2+1

wheretr represents the residual time in the bandwidth levelλα1. This parameterhas to be introduced, since the temporal windowt1 can start in a random pointwithin the permanence timeTα1. The probability ofnk,k+1 bandwidth changesp(nk,k+1) can be obtained by:

p(nk,k+1) = p(tr+Tαk+1+· · ·+Tαk+nk,k+1≤ t1 ≤ tr+Tαk+1+· · ·+Tαk+nk,k+1+1)

that is:p(nk,k+1) = p(t1 ≥ tr + Tαk+1 + · · ·+ Tαk+nk,k+1

)−−p(t1 ≥ tr + Tαk+1 + · · ·+ Tαk+nk,k+1+1). (8)

Given thatt1 is the time interval between the starting points of two video streams,it can be modelled with a random variable with known statistical distribution.The (8) can be derived exploiting the Laplace transform method illustrated in[23]. In particular, let us suppose to have the independent random variablesX0,X1, X2, . . . Xk, X, whose pdfs have Laplace transform given byf ∗X0

(s), f ∗Xi(s)

and f ∗X(s) respectively. If the Laplace transforms are all analytic over the setDσ = {s|<(s) ≥ σ}, whereσ is a real number, it can be proved that

p(X0 + · · ·+ Xk ≤ X) =1

2πj

∫ σ+j∞

σ−j∞

f ∗X0(s)[f ∗Xi

(s)]k

sf ∗X(−s)ds

and, applying the Residue Theorem:

p(X0 + · · ·+ Xk ≤ X) = −∑p∈σp

Ress=p

{f ∗X0

(s)[f ∗Xi(s)]k

sf ∗X(−s)

}(9)

whereσp is the set of poles off ∗X(−s), while the notationRes{·}s=p

denotes the

residue at pole s=p [23]. Applying (9) to (8) and considering that the permanence

7

timesTαiare independent and identically distributed, we have:

p(nk,k+1) = −∑p∈σp

Ress=p

f ∗r (s)[f ∗(s)]nk,k+1−1f ∗t1(−s)

s+

+∑p∈σp

Ress=p

f ∗r (s)[f ∗(s)]nk,k+1f ∗t1(−s)

s=

= −∑p∈σp

Ress=p

f ∗r (s)[f ∗(s)]nk,k+1−1(1− f ∗(s))f ∗t1(−s)

s(10)

wheref ∗r (s) is the pdf Laplace transform of the residual permanence time in abandwidth level,f ∗(s) is the pdf Laplace transform of the permanence time ina bandwidth level, andf ∗t1(s) represents the pdf Laplace transform of the timeintervalt1 in which there arenk,k+1 bandwidth changes. It is known that [24]

f ∗r (s) =η [1− f ∗(s)]

s(11)

where1/η is the mean permanence time in a bandwidth level. Replacing (11) in(10) we obtain:

p(nk,k+1) = −∑p∈σp

Ress=p

η [f ∗(s)]nk,k+1−1 [1− f ∗(s)]2 f ∗t1(−s)

s2. (12)

The probability (12) depends on the pdf Laplace transforms of the permanencetime in a bandwidth level,f ∗(s), the pdf of the starting time of video streams,f ∗t1(s), and the mean permanence time in a bandwidth level,1/η. To consider ahigh degree of correlation between the starting points of smoothed video streams,that is the purpose of this study, we suppose to model the random variablet1with the sum of two independent exponential random variables, with pdff1(t) =µ1e

−µ1t andf2(t) = µ2e−µ2t respectively. The pdf oft1 is obviouslyft1(t) =

f1(t) ∗ f2(t), and the mean value oft1 is t1 = (1/µ1) + (1/µ2). The correlationdegree between the starting points of video streams can be set simply by modify-ing the parametersµ1 andµ2. Furthermore, the pdf Laplace transformf ∗t1(s) is arational function with two simple poless1 = µ1 ands2 = µ2.

The probabilityp(nk,k+1) can be evaluated using (12). With a bit of algebra,it can be derived that:

p(nk,k+1) =η

µ1 − µ2

[µ1

µ2

f ∗(µ2)nk,k+1−1(1− f ∗(µ2))

2−

−µ2

µ1

f ∗(µ1)nk,k+1−1(1− f ∗(µ1))

2

](13)

8

The equation (13) holds only ifnk,k+1 6= 0. The probabilityp(0) has to be calcu-lated in a different way. In fact,p(0) = p(t1 < tr) = 1− p(tr ≤ t1), thus:

p(0) = 1+∑p∈σp

Ress=p

[f ∗r (s)f ∗t1(−s)

s

]= 1+

∑p∈σp

Ress=p

[η(1− f ∗(s))f ∗t1(−s)

s2

](14)

and using the same procedure adopted to derive (13):

p(0) = 1− η

µ1 − µ2

[µ1

µ2

(1− f ∗(µ2))− µ2

µ1

(1− f ∗(µ1))

](15)

Given all the probabilities calculated by (13) and (15) and the marginal probabil-itiesπαi

, directly derived from video traces, all of the state probabilities given by(7) can be easily calculated. The aggregate bandwidthΛS is then derived, givena loss probabilitypl, by exploiting (2) and taking into account the condition (3).For very low values ofpl, as usual in practical cases of interest, the number ofcombinations of state probabilities, respecting (3), to be added in (2) significantlydecreases, resulting in a faster computation ofΛS.

3 Numerical results

In this section some numerical results are presented, in order to test the effective-ness of the proposed algorithm if compared with other algorithms already presentin literature. In this study, bandwidth results obtained with the algorithm devel-oped in this paper are compared with the Chernoff bound algorithm and with somesimulation results, in the particular scenario of a given number of video streams,all of the same type, whose starting points are strongly correlated in time.

Permanence times in a bandwidth level can be modeled with the Inverse Gaus-sian distribution, called also Wald distribution [25]. It can be verified that thisdistribution provides a very good matching, in terms of pdf, with the permanencetimes distribution directly derived from real smoothed video traces. The Walddistribution pdf, with parametersλ andµ, is the following:

f(x) =

√λ

2πx3e−λ(x−µ)2

2µ2x .

The parametersλ andµ are evaluated through the maximum likelihood estimators(see [25] for further details), taking into account samples of permanence times de-rived from real smoothed video traces. The Laplace transform of the Wald pdf cannot be evaluated in a closed form, but it has to be calculated by numerical evalu-ation. Using the maximum likelihood estimators withn samples of permanence

9

timesx1, x2,...xn, as derived from real video traces, the parametersλ andµ canbe calculated as follows [25]:

λ =n∑n

i=11xi− x−1

; µ = x

wherex is the arithmetic mean ofx1, x2,...,xn. The mean of the Wald distribu-tion is 1/η = µ. N video streams of a single type have then been aggregated;their starting points have been modeled with a random variable whose statisticaldistribution is the sum of two independent exponential random variables with pa-rametersµ1 andµ2, as described in Section 2. The mean timeTm = 1/µ1 + 1/µ2

is chosen to quantify the degree of correlation among video streams, and the vari-ance is chosen asvar = T 2

m+12

. Thus, the parametersµ1 andµ2 can be derivedsimply choosing the value ofTm, under minimum variance conditions. Lowervalues ofTm mean a stronger streams correlation, in terms of starting points. Ap-plying (13) and (15) the probabilitiesp(nk,k+1) can be easily calculated, exploitingthe Laplace transform of the Wald pdf.

The proposed algorithm has been compared with the Chernoff bound methodand with some simulation results through different scenarios, varying number andtype of video streams, their smoothing buffer and the mean starting timeTm ofall streams. In Figure 3 this comparison is illustrated. 20 video streams of type“Star Wars ”, of total lenght 40.000 video frames, have been aggregated. Theirstarting points have been modeled with the sum of two negative exponentials dis-tribution whose mean time has been chosen asTm = 40s, to simulate a strongstream correlation. Aggregate bandwidth with the Chernoff bound method hasbeen obtained following the method illustrated in [4]. Its input parameters are thebit ratesλ1,...,λM and their corresponding probabilitiesp1,...,pM . Each probabil-ity pi has been calculated by dividing the number of video frames with bit rateλi

by the total number of video stream frames.Simulation results are obtained by aggregatingN video streams, whose start-

ing points have been randomly chosen following the same statistical distributionused for algorithm calculation, and with the same mean timeTm. In each discretetime instant, the simulated aggregate bandwidth is obtained simply summing thebit rates of all video streams.

From Figure 3 it can be noted that algorithm results follow much better simu-lation results than the Chernoff bound method, resulting at the same time slightlyconservative than simulation. The Chernoff bound method instead fails its estima-tion, since its curve lies under the simulation curve. This is due to the correlationhypothesis, taken into account by the proposed algorithm, but not by the Chernoffbound. In fact, under stream correlation hypothesis, it will be more likely that agreat number of flows will be in a high bit rate, if compared with independence

10

0

2

4

6

8

10

12

1,54

77E-0

3

1,49

86E-0

3

1,40

95E-0

3

1,37

39E-0

3

1,35

12E-0

3

1,28

82E-0

3

1,26

24E-0

3

1,22

86E-0

3

1,19

94E-0

3

1,16

56E-0

3

Loss probability

Ag

geg

ate

ban

dw

idth

(M

bit

/s)

AlgorithmSimulationChernoff

Figure 3: Results for 20 streams of the “Star Wars” film, smoothed with a clientbuffer of 1024 Kbytes.

hypothesis, so the aggregate bandwidth in correspondence of a given loss prob-ability will be generally higher (see [4]). The Chernoff bound is based on thestream independence hypothesis, thus it will probably underestimate simulationresults. This situation has been experimented in several other scenarios. In Figure4, in fact, the same situation is represented, but the mean time has been chosen asTm = 60s, to simulate a weaker stream correlation.

As it can be noted from Figure 4, the Chernoff curve is closer to simula-tion curve if compared with Figure 3, given the minor correlation degree amongstreams; nevertheless, the Chernoff bound always underestimates simulation re-sults, while the proposed algorithm correctly slightly overestimates them. In Fig-ure 5 the same comparison has been performed, but considering a different typeof film (the “Jurassic Park” film), with length of 90.000 frames, smoothed with aclient buffer of 512 Kbytes, andN = 10 video streams aggregated; the value ofTm has been chosen of60s.

In Figure 6 the same “Jurassic Park” film has been considered, but with asmoothing buffer of 64 Kbytes.N = 30 video streams have been aggregated andthe chosen mean starting time isTm = 20s.

4 Conclusions

In this paper, a new bandwidth estimation algorithm for an aggregate of smoothedvideo streams, all derived from a single source, and whose starting points arestrongly correlated in time, has been analyzed and discussed. The QoS parameter

11

0

2

4

6

8

10

12

7,51

69E-0

4

7,25

81E-0

4

6,77

40E-0

4

6,59

86E-0

4

6,47

40E-0

4

6,15

97E-0

4

6,02

35E-0

4

5,85

75E-0

4

5,70

92E-0

4

5,54

32E-0

4

Loss probability

Ag

gre

gat

e b

and

wid

th (

Mb

it/s

)

AlgorithmSimulationChernoff

Figure 4: Results for 20 streams of the “Star Wars” film, smoothed with a clientbuffer of 1024 Kbytes.

6

7

8

9

10

11

12

13

2,49

04E-0

3

2,46

29E-0

3

2,37

31E-0

3

2,36

55E-0

3

1,50

66E-0

3

1,43

74E-0

3

1,42

29E-0

3

1,41

20E-0

3

1,36

87E-0

3

1,36

25E-0

3

Loss probability

Ag

gre

gat

e b

and

wid

th (

Mb

it/s

)

AlgorithmSimulationChernoff

Figure 5: Results for 10 streams of the “Jurassic Park” film, smoothed with aclient buffer of 512 Kbytes.

12

0

10

20

30

40

50

60

70

1,34

41E-0

6

1,05

05E-0

6

1,01

29E-0

6

9,78

42E-0

7

9,05

65E-0

7

8,13

92E-0

7

7,71

95E-0

7

7,71

54E-0

7

7,71

14E-0

7

Loss probability

Ag

gre

gat

e b

and

wid

th (

Mb

it/s

)

AlgorithmSimulationChernoff

Figure 6: Results for 30 streams of the “Jurassic Park” film, smoothed with aclient buffer of 64 Kbytes.

taken into account for statistical bandwidth evaluation is the loss probability. Theproposed algorithm has been compared with the Chernoff bound method, alreadypresented and discussed in literature, and with some simulation results. It hasbeen verified that the Chernoff algorithm underestimates simulation results, dueto the stream independence hypothesis, while the proposed algorithm, taking intoaccount the correlation hypothesis, slightly overestimates them, at the same timefollowing very well simulation curves. In conclusion, we can assert that the pro-posed method can be fruitfully exploited for bandwidth estimation and admissioncontrol purposes in case of correlated streams.

References

[1] A. S. Tanenbaum,Computer Networks, 3rd ed. Prentice-Hall, 1996.

[2] J. Kurose and K. Ross,Computer Networking: A Top-Down Approach Fea-turing the Internet. Addison Wesley Longman, 2000.

[3] “MPEG-4 and H.263 video traces for network performance evaluation,”http://www-tkn.ee.tu-berlin.de/research/trace/trace.html.

[4] Z.-L. Zhang, J. Kurose, J. D. Salehi, and D. Towsley, “Smoothing, statisticalmultiplexing, and call admission control for stored video,”IEEE Journal on

13

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