On Long Range Dependence and Token Buckets - CiteSeerX

On Long Range Dependence and Token Buckets∗ Gregorio Procissi, Mario Gerla, Jinkyu Kim, Scott Seongwook Lee and M. Y. Sanadidi UCLA Computer Science Department 3803A Boelter Hall Los Angeles, CA 90095-1596 {gregorio,gerla,sslee,jinkyu,medy}@cs.ucla.edu

Keywords: LBAP processes, Token Buckets, Short and Long Memory Processes, Fractional Brownian Motion

Abstract The Long Range Dependence (LRD) property of actual traffic in today’s network applications has been shown to have significant impact on network performance. In this paper we consider the problem of optimally dimensioning token bucket parameters for LRD traffic. We first empirically illustrate the different behavior of token buckets when acting on LRD vs. SRD traffic with identical average and peak rates. The comparison shows that LRD traffic requires higher token rates and larger bucket sizes. In this paper we investigate the use of a statistical model to analytically determine optimal bucket parameters under various optimization criteria. The model is based on Fractional Brownian Motion and takes into account the degree of LRD. We apply the model to several aggregation scenarios of MPEG video sources. The analytic results are validated against empirical results. Minimum token bucket parameter curves obtained by analysis and via experiments match well. This is particularly true in the region relevant to the adopted optimization criteria. Thus, the analytic approach presented here is effective in optimally sizing token buckets for LRD traffic, and has potential in wider contexts under different traffic conditions, as well as for various optimization criteria.

I

INTRODUCTION

Since the beginning of the 90s, when the Long Range Dependent (LRD) nature of actual traffic was discovered [1], a large number of statistical models were developed for single and aggregated multimedia traffic characterization. Also, since then, the impact of the long memory characteristic of actual traffic on network performance has been widely investigated. ∗ This research was supported by NSF under Grant ANI-9983138 on High-Speed Networks Performance Measurements and Analysis

At the same time, a different philosophy to address traffic characterization was first developed in [2], introducing the notion of Linear Bounded Arrival Process (LBAP). A LBAP process is an arrival process constrained with a linear bound over the total amount of work produced by a traffic source over any time interval. Token buckets have been introduced for shaping traffic to conform to LBAP description. This technique, permits by just two parameters, (namely the slope of the straight line constraint - the token rate - and the additive constant - the bucket size) a complete ”worst case” characterization of the source. Intuitively, it is clear that, for a given source, the set of pairs (token rate, bucket size) that satisfy the LBAP constraint is infinite and that the determination of the minimal set of LBAP pairs, that is the closure of the aforementioned set, requires off-line processing of the whole trace. Furthermore, the choice of a suitable pair on this curve depends upon arbitrary strategies, though some heuristic criteria can be suitably followed as suggested in [3]. In this paper we first highlight the potential pitfalls of na¨ıve approaches to token bucket design when dealing with long range dependent traffic. An example of a na¨ıve approach, with respect to LRD traffic, is the use of empirical rules of thumb based on the knowledge of source average and peak rates. We then explore the effectiveness of an analytic model that takes into account the long memory property when choosing the LBAP descriptors for the case of aggregated MPEG sources. It is important to point out that the proposed analysis, based on the use of the Fractional Brownian Motion process, is not a statistical characterization of MPEG sources themselves. Instead, the main goal is to determine whether a model that adds the degree of long range dependence (in terms of the value of the Hurst parameter) to the average rate and the variance of actual traffic can be effectively used in the token bucket characterization. To this aim, we provide a summary of LBAP processes and long memory traffic characteristics in sections II and III. Then, in section IV we determine the impact of the long memory property on the LBAP curve by comparing an actual MPEG video trace with LRD, to a modified trace obtained by re-shuffling the data in order to destroy

the correlation structure. It is worth mentioning that the shuffling process maintains the distributions of interarrival times and of frame lengths, and hence the macroscopic descriptors like the source’s mean and peak rate. In section V, several criteria for the choice of token bucket descriptors from LBAP curves are suggested in the form of cost functions to be minimized, or predefined delay constraints to be obeyed. Section VI is dedicated to the definition of the analytic LRD traffic model, and to the solution of the optimization problems mentioned above. Finally, in section VII analytic results are compared to empirical results. The latter were obtained by processing several aggregation scenarios of MPEG video traces.

II

this curve as the LBAP curve, and we will denote it with the equation Λ(ρ, b) = 0. ρ

TOCKEN BUCKET

b

INPUT

OUTPUT

DATA BUFFER

LBAP PROCESSES AND THE LEAKY BUCKET REGULATOR Figure 1: General scheme of a Leaky Bucket regulator

A parsimonious characterization of source activity has always been a key challenge for traffic engineering. One of the most successful and widely adopted characterization makes use of the so-called Linear Bounded Arrival Processes (LBAPs), first introduced by Cruz [2]. A LBAP regulated source constrains the traffic it produces over any time interval τ by a linear function of the time interval. More specifically (for a complete overview see [3]), if we denote with A(τ ) the traffic the source transmits over the time interval τ , the traffic is said to be LBAP if there exist a pair (ρ, b) such that for any τ > 0;

6e+006 Simpsons MPEG video trace

(1)

where ρ represents the long-term average rate of the source and b is the maximum burst the source is allowed to send in any time interval of length τ . Note that b represents the maximum deviation a source may exhibit with respect to its long term average behavior. Operatively, a LBAP (ρ, b) source can be obtained by using a Leaky-Bucket regulator. A leaky bucket builds up tokens of fixed size (typically 1 byte each) at a constant rate ρ in a token bucket of fixed size b. The token bucket size b is often referred to as the token depth. The source is allowed to send an application layer AL-PDU (or packet if the leaky bucket operates at the network layer) if the bucket contains a number of tokens equal to or greater than the AL-PDU size. If the above condition is not met, the source data can be discarded or delayed in a data buffer until the bucket has enough token to permit its transmission. In the first case the leaky bucket acts as a policer, while in the latter case it acts as a shaper. From a theoretical point of view, the pair (ρ, b) that satisfies the LBAP constraint is not unique. On the contrary, the solutions of (1) lay on a 2D set whose border represents the curve of the infinite minimum (ρ, b) pairs that obey relation (1). Throughout the paper we will refer to

5e+006

4e+006 Bucket size (bits)

A(τ ) ≤ ρτ + b,

Intuitively, as ρ approaches the source average-rate from the right, simple queuing reasoning tells us that the corresponding value of b rapidly increases to infinity, while as ρ increases to reach the source peak-rate, b monotonically tends to zero. Figure 2 shows the LBAP curve obtained from the MPEG video trace “Simpson”, one of a set of videos used by Oliver Rose and available from the Web (see [4] for details).

3e+006

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Figure 2: LBAP curve for an MPEG video trace

While all the pairs (ρ, b) laying on the LBAP curve satisfy the LBAB description, one might think to exploit the spare degree of freedom to optimize some predefined criteria. We will get back to this idea in section V. Keshav pointed out [3], that for many common real traces, the LBAP curve exhibits a well defined ”knee area” outside of which, either ρ or b respectively grows quickly for slight variations of the other parameter. This fact suggests as a first suitable choice a pair (ρ∗ , b∗ ) that lies in the neighborhood of the knee region. Finally, it is worth mentioning that when the peak-rate

1.6e+006

1.6e+006 Re-shuffled trace

1.4e+006

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1.2e+006

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1e+006 bits/sec

bits/sec

Original trace

800000

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(a) LRD

0

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(b) SRD

Figure 3: Typical LRD and SRD traffic pattern

P of the source is known, a slightly refined characterization can be obtained, realizing that the total amount of traffic produced by the source in this case is bounded as follows:

- long range dependent (LRD), or with long memory, if: ∞ X

r(k) = ∞

(5)

k

A(τ ) ≤ min{ρτ + b, P τ }.

(2)

A source that satisfies such constraint is said to be Dual Leaky Bucket regulated, with descriptors (ρ, b, P ).

III

BASICS OF SHORT AND LONG MEMORY PROCESSES

In this section the terminology and fundamental concepts of short and long range dependent processes are summarized. For a detailed survey the reader is referred to [1] and [5]. Let A(t) be the total amount of traffic (in units of packets, bytes, or bits) produced by one or several sources over a time interval t. Let us divide the time interval t in nonoverlapping time units of length Tu . Let Xn represent the amount of traffic (also called ”work”) registered over the nth time unit; that is, Xn is the “increments process” of A(t). Suppose Xn is a wide sense stationary stochastic sequence with mean value m and autocovariance function: r(k) = E[(Xt+k − m)(Xt − m)].

(3)

The process Xn is said to be: - short range dependent (SRD), or with short memory, if: ∞ X k

r(k) < ∞

(4)

Thus, in a LRD process, the series associated to the autocovariance functions diverges. Also, it turns out that the autocovariance function itself decays to zero as k increases following the power form k −α , for some 0 < α < 1. That property determines several visible consequences allowing, through visual inspection or via quantitative analysis, the determination of whether a traffic process has LRD, and the ”degree” of LRD. For the sake of conciseness, we mention here some aspects out of many that could be outlined. Generally, a long memory process exhibits relatively long periods where the observations Xn tend to stay at high level, and long periods where the Xn stay at low levels (Joseph effect). Moreover, a short period analysis reveals cycles or local trends, while the complete time series looks stationary. Also, cycles seem to occur at almost all frequencies. Quantitatively, one can further observe that the variance of the sample mean decays to zero as a power 0 < α < 1 of the inverse of the sample size. This evidence is of particular interest as it is the basis of one of the most widely used tools for the detection and estimation of long memory processes, the so-called Variance-Time plot (see [5]). For historical reasons, the Hurst parameter H = 1 − α/2, for 1/2 < H < 1, is used instead of α. Figure 3-(a) and 3-(b) show the typical patterns (representing the number of bits per time units of 1 sec.) of a LRD trace (again the ”Simpsons” MPEG coded movie) and of a SRD one, obtained by destroying the correlation structure of the “Simpsons” trace via random re-shuffling. Figure 4-(a) and 4-(b) respectively show the autocovariance function and the Variance-Time plots for the two traces.

1

1e+011 Original trace autocov. function Re-shuffled trace autocov. function

0.8 1e+010

Variance

r(k)

0.6

0.4

1e+009

0.2 1e+008 0 VT Original trace Straight line H=0.8 VT Re-shuffled trace Straight line H=0.51 -0.2

1e+007 0

20

40

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1

10 Sample size

Lag (k)

(a) LRD

100

(b) SRD

Figure 4: Autocovariance functions and Variance Time plots for SRD and LRD traffics

LEAKY BUCKET PERFORMANCE WITH SRD AND LRD TRAFFIC

The aforementioned characteristics of long memory processes raise a number of issues about the LBAP description of LRD traffic. In particular, we are interested in the behavior of the LBAP curve in presence of traffic that has the same macroscopic parameters, average bit-rate and peak bit-rate, but that exhibits short and long memory respectively. Intuitively, the ”bursty” nature of long range dependent traffic would suggest that, for the same choice of ρ, the corresponding value of b for LRD traffic should be higher than the one for SRD traffic. As the value of b is generally related (as will be discussed in details in section V) to the buffer space pre-allocated at each intermediate node along the path at call setup time, a large b would waste buffer space, and also results in larger end to end delay. On the other hand, we expect that, for a fixed choice of the token depth descriptor b, LRD traffic would require a higher value of the token rate ρ in order to satisfy the LBAP constraint. To prove these hypotheses, refer to figure 5 where the LBAP curves of both the LRD trace ”Simpsons” and its SRD version obtained by re-shuffling are compared. The reshuffling procedure does not change average and peak rate of the sources and not even the marginal distribution of data size. The different behavior of the two curves is merely due to the correlation structure of the two processes. For the sake of completeness, the average-rate and the peak-rate are respectively about 0.445 Mbps and 5.768 Mbps, while the observed Hurst parameter of the original sequence is about 0.8. Figure 5 also shows the presence of a knee area for both curves though it appears more distinct for the SRD trace than for the LRD trace. In the LRD case, the knee is smoother and not as straight-

6e+006 Original trace Re-shuffled trace 5e+006

4e+006 Bucket size (bits)

IV

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1.5e+006 Token rate (bps)

2e+006

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Figure 5: LBAP curves for SRD and LRD traffic sources

forward to detect.

V

SETTING THE LBAPs PARAMETERS

So far, following [3] we have addressed the choice of the pair of token bucket descriptors pair (ρ∗ , b∗ ) in a heuristic manner through qualitative arguments about the knee point of the LBAP curve. While this approach is valid, a more formal statement of the problem is developed below. A first idea is to find a proper cost function φ(ρ, b) to be minimized (or in the dual formulation, a satisfaction function Σ(ρ, b) to be maximized) while satisfying the LBAP constraint Λ(ρ, b) = 0. The resulting point (ρ∗ , b∗ ) is the optimal solution of this constrained min-max problem with respect to the criterion specified by the cost function. If the mathematical expression of Λ(ρ, b) = 0 were provided, the problem could be solved by the well known method of Lagrange Multipliers. In the next section we formulate and

solve this problem when the statistical model of the traffic is given. Another important issue is the proper choice of the cost function φ(ρ, b). The heuristic idea of looking for the ”optimal” point in the vicinity of the knee point, and the decreasing nature of the LBAP constraint leads us to consider as a suitable cost function the distance (or its square) d(ρ, b) of the pair (ρ, b) from the point (m, 0), where m is the average bit-rate of the source. Hence, the cost function to consider is: φ(ρ, b) = ||(ρ, b) − (m, 0)|| =

p

(ρ − m)2 + b2 .

(6)

The tight relationship between the token bucket descriptors and the bandwidth and buffer size allocated in the network to the source suggests another cost function. Depending upon the condition and the topology of the network, either bandwidth or buffer size may have different marginal utility. Network operators might prefer to allocate additional amount of buffers over bandwidth or the other way around. This suggests a cost function defined as a linear combination with arbitrary weighting of the parameters ρ and b; that is: φ(ρ, b) = ωρ (ρ − m) + ωb b.

(7)

The geometric interpretation of this problem is straightforward and is represented by the point at which the LBAP curve is tangent to the constraint line ωρ (ρ − m) + ωb b = k, where k is the cost to be minimized. Finally, as suggested in [6], a different strategy in finding the token bucket descriptor over the LBAP curve can be derived from analytical known results obtained for a network of Latency-Rate schedulers [7]. These schedulers constitute a wide class that include all the work conserving schedulers such as WFQ, WF2Q+, Virtual Clock, SCFQ, WRR and so fourth. In particular, suppose an amount of bandwidth c and a amount of buffer space B is allocated to a dual leaky bucket regulated source with parameters (ρ, b, P ) throughout all the schedulers in the network. In [7], Stiliadis and Varma proved that for a network of K Latency-Rate schedulers, the maximum end to end delay of packets belonging to the source is bounded by: D≤ Sj

µ

P −c P −ρ

¶µ

B ρ

¶

+

K X

Sj

Θ

(8)

to statistical modeling of sources. Thus, as a first approximation, the contribution of the latency term can be neglected. Moreover, if the bandwidth and buffer allocated to the source in each node are ρ and b respectively, the condition (8) is met, provided: Dmax ≤

(9)

That gives us a new criterion for choosing the (ρ, b) pair as the intersection between the straight line b = Dmax ρ and the LBAP curve. It is worth noticing that this criterion applies even in the case in which the peak-rate is not known and the source is simply Leaky Bucket regulated [7].

VI

A LRD MODEL FOR AGGREGATED VIDEO SOURCES

All the techniques summarized in the previous section require the knowledge of the LBAP curve which, in fact, has to be determined off-line. In this section we develop an analytic model for LRD traffic to be used in the framework of an a-priori dimensioning of the token bucket for the aggregated video traffic streams. It is worth pointing out that the goal of this investigation is not the statistical modeling of aggregated video streams itself. The goal of this investigation is rather to develop a model that takes into account the long memory property of the aggregated traffic to determine optimal bucket parameters for such LRD traffic. To this end we consider a fluid model for the total amount of work produced by the sources based on the Fractional Brownian Motion (FBM) and we derive the corresponding LBAP curve as well as the optimal token bucket pair according to the strategies summarized in section V. In section VII these results will be compared with results derived empirically aggregating actual video streams under three different scenarios. Let us assume the following model for the aggregated amount of work produced by the sources in the time interval [0, t]: A(0, t) = mt + B H (t), t ≥ 0 (10) where mt is the mean value over the interval (0, t) and B H (t) is FBM process defined as [8]: B H (t) =

j=1

where Θ is the latency of the jth node that is introduced to take into account the difference between real schedulers and the ideal reference fluid model represented by the Generalized Processor Sharing (GPS) scheduler. The result (8) is obtained using worst-case analysis and therefore is generally quite conservative. Moreover the LBAP characterization itself is conservative with respect

b . ρ

σ C1 (H)

Z

t

f (t, y)dB(y),

t≥0

(11)

−∞

where: a) 0 < H < 1 is the Hurst parameter; b) f (t, y) is the kernel of the representation, given by: f (t, y) = (t − y)H−1/2 − (−y)H−1/2 I(−∞,0) (y), (12) and IA is the Indicator function of the set A;

c) C1 (H) =

½Z

0

∞

¡

H−1/2

(1 + y)

H−1/2 2

¢

− (y)

1 dy + 2H

¾1/2 (13)

d) and dB(y)is the underlying White Gaussian Noise. The FBM process is a non stationary, long memory process, with Gaussian marginal distribution N (0, σ 2 t2H ), stationary increments and B H (0) = 0. The amount of work produced over any time interval τ is then: A(t, t + τ )

= =d

mτ + B H (t + τ ) − B H (t) mτ + B H (τ ).

(14)

Dealing with a stochastic process, the LBAP constraint can be expressed as a bound on the probability of nonconformance. The bound is denoted here by e−γ for convenience, as we will show below the solution to be also exponential. The constraint is expressed as: P (ρ, b, τ ) = P r {A(τ ) ≥ ρτ + b} ≤ e−γ

for all τ

with the corresponding LBAP curve: © ª (ρ, b) : P (ρ, b, τ ) = e−γ for all τ .

(15)

(16)

The condition (16) can be reformulated as:

P (ρ, b) = sup P (ρ, b, τ ) ≤ e−γ

(17)

τ

where for P (ρ, b) there exists an asymptotic expression firstly derived by Norros [9] and later on by means of Large Deviations Techniques by Duffield et al. (see [10], [11]) given by: log P (ρ, b) ≍ −α(σ, H)(ρ − m)2H b2−2H with α(σ, H) =

1 1 . 2σ 2 H 2H (1 − H)2−2H

(18)

Λ(ρ, b) = α(σ, H)(ρ − m)

2H−2

− γb

(A) Minimum distance cost function. the cost function is given by p φ(ρ, b) = (ρ − m)2 + b2 .

Determining The Optimal Bucket Parameters In the following, the analytic expression of the optimal pair (ρ, b) according to the different optimal criteria outlined in section V are derived. The solutions are obtained using the Lagrange Multipliers method. Namely, we define the function:

In this case

(22)

The optimal parameters in this case are given by: s µ ¶H+1 H ∗ ρ = m + σ(1 − H) 2γ 1−H and b∗ = σ(1 − H)

s

2γ

µ

H 1−H

¶H

.

Note that the optimal pair lies on the straight line: sµ ¶ 1−H (ρ − m). b= H The corresponding minimum cost is given by: s µ ¶H H ∗ ∗ . φ(ρ , b ) = σ 2γ(1 − H) 1−H

(23)

(24)

(25)

(26)

(B) Linear cost function. The linear cost function with arbitrary weighting of the two parameters is: φ(ρ, b) = ωρ (ρ − m) + ωb b.

(27)

The optimal pair is given by the following expressions: ρ =m+H ∗

µ

ωρ ωb

and (20)

(21)

and solve the equations obtained by setting the partial derivatives of F with respect to ρ, b and λ equal to zero. We now consider each of the cost functions defined Section V, and solve for the optimal token bucket parameter for each function.

(19)

Finally, the LBAP curve can be written as: 2H

F (ρ, b, λ) = φ(ρ, b) − λΛ(ρ, b)

b = (1 − H) ∗

µ

¶H−1 p

ωρ ωb

¶H p

2γσ 2

2γσ 2

(28)

(29)

In this case, the optimal pair lies on the straight line: µ ¶ ωρ 1 − H b= (ρ − m) (30) ωb H and the minimum cost is given by: φ(ρ , b ) = ωb ∗

∗

µ

ωρ ωb

¶H p

2γσ 2 .

(31)

Name Terminator Bond Simpsons Lambs Dino Asterix Starwars Sbowl

Average bit-rate (m)(Mbps) 0.262 0.572 0.446 0.175 0.314 0.536 0.223 0.564

Peak-rate (P) (Mbps) 1.90 2.66 5.76 3.22 2.87 3.54 2.995 3.38

Table 2: MPEG aggregate traces description

Name Aggregate 1 Aggregate 2 Aggregate 3

VII

Average bit-rate (m)(Mbps) 1.46 1.64 3.1

Peak-rate (P) (Mbps) 36.9 38.8 45.5

Hurst parameter (H) 0.79 0.77 0.79

NUMERICAL RESULTS

To test the effectiveness of this approach, we compare the results obtained from the model with those achieved using real traces. In particular we consider the aggregation of MPEG traces in three different scenarios, varying the number of traces and the traces themselves. Table 1 summarizes the real traces used as well as their average and peak-rate. Using the estimated parameters in the table, we compute the optimal token bucket descriptors for each aggregated traffic, and compare them to the analytic model; the comparison criterion being the effectiveness of analytic and empirical parameters in matching the actual LBAP constraints. We produce three different traffic aggregations by respectively multiplexing the first four traces (“Aggregate 1”), the last four (“Aggregate 2”), and all eight traces together (“Aggregate 3”). For each aggregated trace, we evaluate all the parameters needed to compare with the theoretical model. The following table shows the average and peak-rate of each aggregated traffic, as well as its estimated Hurst parameter. In the following we will assume unitary weights for the linear cost function (7). A critical aspect of the analysis is the choice of the maximum allowed non-conforming traffic probability: e−γ . Since the real LBAP curve is drawn for the deterministic constraint (1), the γ parameter is chosen in a range of values close to the absolute value of the natural logarithm of the minimum non-conforming packet probability measurable from actual traces, that is 1/(trace

length). In our traces, as the trace length were respectively given by 160,000 and 320,000 frames, we let γ assume values from 11 to 13. Aggregate 1. The first scenario we consider here is the one in which the traces Terminator, Bond, Simpsons and Lambs were multiplexed to get an aggregate trace. Figure 6 represents the theoretical and empirical LBAP curves and the straight line used to identify the optimal pair under the three different criteria for γ = 13. The results show that the optimal pairs computed over the theoretical curve are substantially close to those obtained in the empirical LBAP curve. Moreover, they meet the initial condition we defined; that is, they lay on the knee region of the actual LBAP curve. 3e+006 Real LBAP curve Model LBAP curve Distance Cost Criterion Linear Cost Criterion Delay Criterion

2.5e+006

2e+006 Bucket size (bits)

Table 1: MPEG real traces description

1.5e+006

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Figure 6: Model vs. Empirical curves for the Aggregate 1 trace

Aggregate 2. In the second scenario, we multiplexed the real traces of Dino, Asterix, Starwars and Sbowl to get the second aggregated traffic. The result shown in Figure 7 confirms the effectiveness of the analytic approach. The theoretical optimal points still lay on the knee point of the actual LBAB curve. Aggregate 3. As a final case we consider the aggregation of all the traces listed in Table 1. As the number of sources increases, we expect that the Gaussian model would get closer to the real LBAP curve. Indeed that happens and the curves (Figure 8) look much closer than in the first two cases over a larger area, not only in the vicinity of the knee point. Note that after the knee region, the actual LBAP curve stays constant for increasing values of the token rate up to the peak rate. This is simply due to the packetized nature of actual traffic. The constant flat value of b corresponds to the maximum packet size of the trace. In contrast, in the analytic model, the traffic is treated as a fluid and thus the analytic curve monotonically decays to zero.

video traffic tout court, but with the goal of checking how an analytic model that takes into account the long memory property could be effective in determining optimal LBAP descriptors within the set of LBAP pairs of interest. The analytic and empirical results show a good match and validate a promising analytic approach for network resource dimensioning.

3e+006

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Bucket size (bits)

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Real LBAP curve Model LBAP curve Distance Cost Criterion Linear Cost Criterion Delay Criterion

1e+006

ACKNOWLEDGMENTS

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5e+006 6e+006 7e+006 Token rate (bps)

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Figure 7: Model vs. Empirical curves for the Aggregate 2 trace

The authors wish to thank Dr. Rosario G. Garroppo and Dr. Stefano Giordano from the NetGroup of the Information Engineering Department of the University of Pisa for the frequent and stimulating discussions during the preparation of this work.

3e+006 Real LBAP curve Model LBAP curve Distance Cost Criterion Linear Cost Criterion Delay Criterion

2.5e+006

REFERENCES

Bucket size (bits)

2e+006

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0 4e+006

6e+006 Token rate (bps)

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Figure 8: Model vs. Empirical curves for the Aggregate 3 trace

In any case, a visual inspection of the results confirms the effectiveness of the analytic model in the definition of the optimal token bucket descriptor. Also, it is interesting to note that the higher level of multiplexing has a good return in the optimal pair computation: good match between the analytic and the empirical curves are achieved for γ = 11, while for larger values of γ the analytic model would have produced slightly more conservative results.

VIII

CONCLUSIONS

In this paper we investigated the impact of the long memory property of multimedia traffic on the choice of token bucket parameters. Starting from the definition of LBAP, we compared token bucket curves drawn for an LRD trace and for the corresponding SRD trace obtained by reshuffling the data. We then developed and used a statistical LRD model based on the Fractional Brownian Motion process to determine optimal token bucket parameters for aggregated MPEG sources. A number of optimization criteria or cost functions were suggested. This was not done for the purpose of a statistical characterization of aggregated

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