Semi-Parametric Effective-Bandwidth Estimator Based on Buffer

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PAPER

Semi-Parametric Effective-Bandwidth Estimator Based on Buffer Measurements Shigeo SHIODA†a) , Member and Daisuke ISHII†∗ , Nonmember

SUMMARY The notion of effective bandwidth provides an elegant and powerful mathematical basis for the provision of QoS-assured services over IP networks. In this paper, we propose a semi-parametric estimator of effective bandwidth, called Gaussian estimator using buffer masurement, for superposition of sources in IP networks. In contrast to most existing proposals concerning the effective bandwidth estimator, our proposal works based on a small set of measurements of the workload in the buffer of a router. We analytically show the property of the proposed estimator with respect to the dependence on the service rate. We provide numerical results to show that our proposed estimator is more accurate than estimators that rely only on the amount of traffic from sources. key words: Internet, quality of service, effective bandwidth, traffic, extreme value analysis

1.

Introduction

IP-based networks are growing at a startling pace. While current IP networks generally carry data traffic on a besteffort basis, there is an increasing demand for the support of assured quality-of-service (QoS) characteristics. For example, voice over IP is an important service that requires the QoS assurance. Interactive data services, audio and video telephony over IP, and multimedia conferences may all be categorized as QoS-assured services. The estimation of the resource requirement, typically bandwidth requirement, of a source or superposition of sources is important in provisioning QoS-assured services on IP networks. Once resource requirement is obtained, the QoS assurance is possible by allocating the required resource through the combination of traffic shaping, special queueing scheme, and connection setup mechanism [1]. In fact, intensive work on resource allocation mechanism is currently in progress, and the approaches include integrated services (IntServ) [2], measurement-based admission control [3]–[5], differentiated services (DiffServ) [6], and multiprotocol label switching (MPLS) [7]. To devise the method for bandwidth requirement estimation, the notion of effective bandwidth has been developed by several authors in recent years [8]–[14]. The effective bandwidth is a mathematical formula that gives the necessary and sufficient bandwidth to be allocated to a source Manuscript received February 18, 2004. Manuscript revised June 2, 2004. † The authors are with the Dept. of Urban Environment Systems, Faculty of Engineering, Chiba University, Chiba-shi, 2638522 Japan. ∗ Presently, with Access Technology Dept., Softbank BB Coroperation. a) E-mail: [email protected]

in order to satisfy a QoS criterion of the type lim

x→∞

1 log P[B ≥ x] ≤ −θqos , x

(1)

or approximately P[B ≥ x] ≤ e−xθqos

for large x,

(2)

where B is the stationary workload in an infinite-capacity queue fed by the source. Note that θqos in (1) or (2) is closely related to the objective of packet-loss probability when the source sends packets to a queue with finite capacity xbu f . To see this, first notice that packet loss probability ≈ P[B ≥ xbu f ]. (Note that, in the above equation, P[B ≥ xbu f ] is the probability for an infinite-capacity queue.) Thus, if we let θqos = − log(ploss )/xbu f , then a QoS criterion packet loss probability ≤ ploss is equivalent to (2) when xbu f is sufficiently large. More detailed description about how the notion of effective bandwidth is related to actual performance metrics such as loss or delay is found in [12]. Since the effective bandwidth formula contains the logarithmic moment-generating function of the amount of traffic within this definition (see (3)), we should know the logarithmic moment-generating function of the source to obtain its effective bandwidth. Complete information about the logarithmic moment-generating function is, however, hard to obtain, because we need to obtain full statistical information on the source. Thus, it is natural to seek a method that allows us to estimate the effective bandwidth to a reasonable degree of accuracy on the basis of a small portion of the complete set of information about the source. Such a method should rely only on the minimal and accurate measurement on the source that most concisely captures its statistical behavior. The effective bandwidth estimation based on the partial information on the source has been studied by Duffield et al. [15], Chang [16], De Veciana et al. [10], and Courcoubetis et al. [17]. All of their proposals rely on the measurement about the amount of traffic from the source, and some use additional assumptions such that the source is Gaussian. It is not, however, obvious that the measurement on the traffic amount from the source is the most appropriate for the effective bandwidth estimation.

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by synthetic model-driven sources and on real traffic data. Finally, in Sect. 5, we give our conclusions and make some remarks on points for future work. 2.

Preliminaries

2.1 Effective Bandwidth Consider a single-server queue being fed packets by a source or multiple sources. The queue is assumed to have an infinite buffer for which the service rate is a constant c. In this paper, we use the following definition of the effective bandwidth for traffic from a source (superposition of sources): def

1 log E[eθX(t) ] for θ > 0, t→∞ θt

α(θ) = lim Fig. 1

Packet swithcing in a network node.

In this paper, we propose an estimator of effective bandwidth of the superposition of sources multiplexed on a given link in IP networks. The network we consider is of the form depicted in Fig. 1. Here, we assume that information about the workload in the buffer (say, buffer C E in Fig. 1) dedicated to a link (say, link C E in Fig. 1) is collected through some form of measurement. In contrast to the existing proposals, our proposed estimator uses this buffer measurement results in estimating the effective bandwidth. In this paper, we will numerically show that using the buffer measurement significantly improves the accuracy of effective bandwidth estimation. The proposed estimator, as most of the existing estimators, assumes that the source is Gaussian, but the buffer measurement results are used to correct this assumption. In this sense, our proposal, which is called Gaussian estimator using buffer measurement in this paper, is a semi-parametric estimator. Note that De Veciana et al. [18] proposed estimating the inverse function of effective bandwidth by using a “virtual buffer method,” where the cell-arrival process is inputted to several virtual buffers with different service rates, and the asymptotic-decay rates (see Sect. 3.1) of these buffers are measured. Although this is similar to our proposed approach in the sense that both apply measurements of buffers in estimation, our proposal requires only a single virtual buffer and the asymptotic-decay-rate estimation of the virtual buffer can be conducted in a much simpler mechanism. Here is how the rest of this paper is organized. In Sect. 2, we show the definition of the effective bandwidth that we use and present examples of formulae derived from this definition for several traffic models. For later use, we also describe the heavy-traffic-asymptotic expansion of effective bandwidth that provides the theoretical basis for our proposal. In Sect. 3, we propose a semi-parametric estimator of effective bandwidth and discuss several properties of this estimator. Section 4 is devoted to a numerical examination of the performance of our proposal on traffic generated

(3)

where X(t) is the amount of traffic that arrives in the interval [0, t]. See Chang [8], de Veciana [18], Kesidis et al. [13], or Whitt [14] (we assume that the limit at the right-hand side of (3) exists for all positive θ)† . The parameter θ in the formula (3), which might be measured in units of bits−1 or bytes−1 , is related to the tail probability of the stationary workload in the queue. For example, if there exists a finite θ∗ such that def ∗ α(θ∗ ) = c, and α (θ∗ ) = dα(θ) dθ |θ=θ is positive and finite, then lim

x→∞

1 log P[B ≥ x] = −θ∗ . x

For details, see [8], [11]–[14]. Thus, if the QoS criterion (1) or (2) is imposed and α(θ) is nondecreasing in θ, then the effective bandwidth α(θqos ) gives the necessary and sufficient bandwidth that satisfies this QoS criterion when assigned to the source (superposition of sources). 2.2 Examples of Effective Bandwidth For later use, we present formulae for the effective bandwidth when sources generate traffic according to several stochastic models. 2.2.1 Gaussian Source First consider the case where the amount of traffic arriving in the interval [0, t] has the following simple form: X(t) = at + Z(t), †

Kelly [12] has proposed another definition of the effective bandwidth, which is also widely accepted: def

α(θ, t) =

1 log E[eθX(t) ] θt

for 0 < θ, t < ∞.

In the above definition, t is the time scale of traffic fluctuation that gives the largest impact on the queueing phenomena. Although this definition has several desirable features that definition (3) lacks, it does not directly give the necessary and sufficient bandwidth for traffic from each source (or from superposition of sources). In addition to this, the time scale parameter t cannot be obtained in advance, and it should be determined through traffic measurement. Hence, in this paper, we focus on definition (3).

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where Z(t) is normally distributed with zero mean and a variance of vt. Kelly showed [12] that the effective bandwidth of such a Gaussian source is given by v α(θ) = a + θ. 2

For a packet network in general, the amount of traffic arriving in the interval [0,t] has the following form: N(t)


c = a(1 + g1 θ∗ + g2 θ∗2 · · ·) 1−ρ = g1 θ∗ + g2 θ∗2 · · · , ρ

(4)

2.2.2 Compound Poisson Process

X(t) =

and finite. First, observe that the expansion (6) yields or (8)

where ρ = a/c is utilization of the queue. Then, we use the reversion of the series (see Choudhury and Whitt [19]) to obtain    2 1 1−ρ g2 1 − ρ ∗ θ = +··· − 3 g1 ρ ρ g1 from the second relationship of (8). Thus, we have

Ln ,

1 log P[B ≥ x] x   2  1 1−ρ g2 1 − ρ =− + 3 + ···, g1 ρ ρ g1

n=1

lim

where N(t) denotes the number of packets arriving in the period [0, t] and Ln denotes the length of the nth packet. If packet arrival is according to a Poisson process with rate λ and L1 , L2 , · · · are independent random variables with the same distribution F, then X(t) is a compound Poisson process. According to Kelly [12], the effective bandwidth of such a compound Poisson process is given by  1 α(θ) = (eθx − 1)λdF(x). θ In particular, if L1 , L2 , · · · are exponentially distributed with mean l, then a , (5) α(θ) = 1 − θl

x→∞

which is the heavy-traffic approximation for the asymptotic decay rate [14], [19]. Remark 1. The relationships (7) hold if for complex z in the neighborhood of z = θ lim

t→∞

1 log E[ezX(t) ] = α(z) t

holds uniformly in z. For details on this, consult Choudhury and Whitt [19]. 3.

def

where a = λl is equal to the mean of X(1).

(9)

Semi-Parametric Estimator of Effective Bandwidth

3.1 Effective Bandwidth Estimator 2.3 Heavy-Traffic Asymptotic Expansions The effective bandwidths for most processes that have the finite limit of the right hand side of (3) (i.e. the effective bandwidths exist) are differentiable and have the following Taylor-series expansion about θ = 0: α(θ) = a(1 + g1 θ + g2 θ2 · · ·),

(6)

where the parameter a and the coefficients of expansion, g1 and g2 , are given by E[X(t)] , t E[(X(t) − E[X(t)])2 ] , and g1 = lim t→∞ 2at 3 E[(X(t) − E[X(t)]) ] g2 = lim . t→∞ 6at

a = lim

t→∞

(7)

Typical examples of processes for which the effective bandwidths have such an expansion are found in Courcoubetis and Weber [17], de Veciana and Walrand [10], and Choudhury and Whitt [19]. Note that, from expression (4), g1 = ν/2a and g2 = g3 = · · · = 0 for the Gaussian source. The expansion (6) is usually called the heavy-traffic asymptotic expansion. To illustrate the reason for this, assume that there exists a finite θ∗ such that α(θ∗ ) = c, and α (θ∗ ) is positive

The expansion (6) naturally leads to the following simple approximation for the effective bandwidth: αˆ gs (θ) = a(1 + g1 θ),

(10)

where E[X(t)] , and t E[(X(t) − E[X(t)])2 ] g1 = lim . t→∞ 2at

a = lim

t→∞

This is referred to as the Gaussian approximation in this paper because (10) is exact for the Gaussian source. The approximation (10) is appealing because they enable us to estimate the effective bandwidth solely from knowledge of the first and second moments of X(t). In fact, de Veciana [10] and Chang [16] suggested that this approximation is useful in the bandwidth design and performance evaluation of packet networks. This formula, however, is less accurate in general. In the following, we propose an effective bandwidth estimator which is based on (10), where g1 is determined through buffer measurements. Here, the asymptotic decay rate def

θdecay (c) = − lim

x→∞

1 log P[B ≥ x] x

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is assumed to be measured instead of the second moment of X(t). (In Sect. 3.3, we show that the extreme-value analysis enables us to measure the asymptotic decay rate easily.) Note that θdecay (c) is a non-decreasing function of the service rate c [16]. By setting g2 = g3 = · · · = 0 in (9), we obtain the following estimate of g1 : gˆ 1 =

1−ρ 1 − a/c = . ρθdecay (c) aθdecay (c)/c

Substituting g1 = gˆ 1 and gn = 0 (n ≥ 2) into (6) yields the following estimator of effective bandwidth: def

αˆ gsbm (θ,  c; θdecay (c)) = a(1 + gˆ 1 θ) θ 1 − a/c =a 1+ . a/c θdecay (c)

(11)

We call this the Gaussian estimator using buffer measurement. 3.2 Properties of the Proposed Estimator As expression (11) indicates, the proposed estimator has dependence on the service rate. In this section, we analytically show the property of the proposed estimator with respect to the dependence on the service rate. In the following analysis, we do not consider error in the measurement of the asymptotic decay rate. Theorem 1. If a < c, then  a < αˆ gsbm (θ, c; θdecay (c)) < c       αˆ gsbm (θ, c; θdecay (c)) = c       αˆ gsbm (θ, c; θdecay (c)) > c Proof. See Appendix.

1. Initially (for example at time 0), set the service rate of the queue in the simulator at the link speed. 2. At time T 1 , estimate the effective bandwidth based on the buffer measurement results during (0, T 1 ]. 3. At the same time, let the service rate of the queue be equal to the estimated effective bandwidth. 4. Repeat the procedures 2 and 3 at every succeeding adjustment period. Theorem 1 gives theoretical justification to such use of the proposed estimator. Actually, we confirmed that, with this type of recursive substitution, αn quickly converges to a value, which is close to the exact effective bandwidth. In detail, please see [20]. The next result shows that if X(t) behaves like the process of output from a leaky bucket, the estimator αˆ gsbm (θ, c; θdecay (c)) is bounded from above. This property assures that the estimator αˆ gsbm (θ, c; θdecay (c)) does not make too large estimates, which property is preferable when the estimator is used to search the bandwidth requirement as explained above. Theorem 2. Suppose that X(t) has the following form: X(t) = at + H(t), where H(t) is a bounded stochastic process with zero mean such that H(t) ≤ h for all t.

θdecay (c) > θ,

Furthermore, suppose that the effective bandwidth α(θ) satisfies α(θdecay ) = c; then

θdecay (c) = θ, θdecay (c) < θ.

αˆ gsbm (θ, c; θdecay (c)) < a + ahθ.


To illustrate the implication of Theorems 1, consider a QoS criterion of the type 1 log P[B ≥ x] ≤ −θqos . x If this QoS criterion is not satisfied (i.e., θqos > θdecay (c)), then αˆ gsbm (θqos , c; θdecay (c)) provides estimated values which are above the current service rate c. On the other hand, if the QoS criterion is satisfied (i.e., θqos < θdecay (c)), then αˆ gsbm (θqos , c; θdecay (c)) provides estimated values which are below the current service rate c. The values derived with the approximate formulae, αˆ gs (θ), however, does not have this property. It is quite promising to use our proposal in the traffic measurement tools that have the function of simulating the input of measured patterns of packets into a single server queue, which emulates an output buffer of a router, to search the bandwidth requirement that satisfies a given QoS criterion. We have implemented such a tool, where the service rate of the queue in the simulator is periodically adjusted to the effective bandwidth estimated by our proposal at intervals of fixed length T 1 [20]. More precisely,




Proof. See [21].

3.3 Estimating the Asymptotic Decay Rate through the Extreme-Value Analysis

lim

x→∞

Using the extreme-value analysis, Zeevi and Glynn [22] showed that if a stationary process (X1 , X2 , · · ·) satisfies some appropriate weak-dependence conditions, then Mn 1 → ∗ log n θ

as n → ∞

almost surely where def

Mn = max{X1 , X2 , · · ·}, 1 def θ∗ = − lim log P[X > x]. x→∞ x We took advantage of the power of the above results, in our experiments through simulation (see Sect. 4), by using the following approach to estimate the asymptotic decay rate: let Bn denote the workload in the queue at the instant of the nth packet’s arrival, N(t) denote the number of packets arriving during (0, t], and

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modulated Poisson process) and input them into a singleserver queue. Then, over successive fixed-length measurement windows, we periodically estimated the effective bandwidths in the following way: at the end of every measurement window, we used the method explained in Sect. 3.3 to estimate the asymptotic decay rate and, on the basis of the result, used the proposed estimator (11) to obtain an estimate of the effective bandwidth. Specifically, at the end of the nth measurement window, the estimate of the effective bandwidth was given by αˆ gsbm (θqos , c; θˆdecay ((n − 1)T 1 , nT 1 )) Fig. 2 Relationship between estimated asymptotic decay rate and the number of arrivals. def

θqos = −

The estimate of the asymptotic decay rate during (t1 , t2 ), θˆdecay (t1 , t2 ), was (12)

The above estimator is very useful in the sense that it does not have any parameters that require tuning for accuracy of estimation. The accuracy of the asymptotic-decay-rate estimator (12) would depend on the number of packets arriving during (t1 , t2 ]. We numerically investigated this dependence when a Poisson or two-state MMPP source generated packets. The dependence of estimator (12) on the number of arrivals for one run is depicted in Fig. 2. The mean input rate of the Poisson process is 5 Mbps. For the two-state MMPP source, the mean input rate in state 1 (state 2) was 20 Mbps (1 Mbps) and the mean duration time of state 1 (state 2) was 0.1 s (30 s). (These sources were also used for numerical experiments shown in Sect. 4.1.) From the figure, we see that, for the Poisson process, the convergence of (12) to the exact value was fast and (12) yielded reasonably accurate estimates when more than 1000 packets arrived. For the MMPP process, however, the convergence of (12) to the exact value was relatively slow and (12) required at least 25000 arrivals of packets not so as to yield inaccurate estimates. When the average load of the link is 1 Mbps and the average length of packets is 500 byte, the average number of packets arriving during one minute is 15000. So, for bursty sources, the estimation interval should be a few minutes or longer. 4.

log 10−3 [bit−1 ]. 1.0 × 106 Note that the effective bandwidth, α(θqos ), gives an approximation of the bandwidth which is necessary and sufficient to satisfy this QoS criterion: def

MB (t1 , t2 ) = max{BN(t1 )+1 , · · · , BN(t2 ) }.

log(N(t2 ) − N(t1 )) . θˆdecay (t1 , t2 ) = MB (t1 , t2 )

where θˆdecay (t1 , t2 ) is the estimate of the asymptotic decay rate based on the buffer measurements during (t1 , t2 ], T 1 is the length of the measurement window, and

Numerical Experiments by Simulation

4.1 Simulation Using Synthetic Model-Driven Sources To evaluate the accuracy of our proposal, we first conducted three simulation experiments using synthetic model-driven sources, whose effective bandwidths are exactly known. In each of simulation experiments, we sequentially generated packets using a synthetic source (Poisson or Markov

P[B ≥ xth ] ≤ e−θqos xth = 10−3 , where xth = 1 Mbit. For comparison, we also evaluated the effective bandwidth of the sources using two existing estimators. The first one is the Gaussian approximation αˆ gs (θ) defined by (10), which was proposed by de Veciana [10] and Chang [16]. The second one, which was proposed by Duffield et al. [15], estimates the effective bandwidth by directly measuring the cumulant generating function (cgf) of the amount of traffic over the period of length T 2 . In this paper, we call it cgfmeasuring estimator and denote it by αˆ cg f (θ). To explain the cgf-measuring estimator in more detail, define def

Yk = X(kT 2 ) − X((k − 1)T 2 ) The cgf-measuring estimator relies on the assumption that the amounts of traffic over the period of length T 2 , {Y1 , Y2 , Y3 · · ·}, are identically distributed and independent from each other. Under this assumption, n 1 log E[e k=1 θYk ] n→∞ nθT 2 1 = lim log(E[eθY1 ])n n→∞ nθT 2 1 = log E[eθY1 ]. θT 2

α(θ) = lim

In the simulation, the amounts of traffic during the successive periods of length T 2 were periodically measured. Thus, for example, at the end of the first measurement window, we have T 1 /T 2 measurement results; that is, Y1 , Y2 , · · · , YT1 /T 2 . (We assume that T 1 is a multiple number of T 2 .) Then, the cgf-measuring estimator yields the estimate by the following:  T /T  1 2    def 1  T 2
θYk   αˆ cg f (θ) = log  e  .     θT 2 T 1 k=1

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Fig. 3

Estimation results: Poisson source.

In the first simulation, we used a source generating packets according to a Poisson process by setting the mean input rate at 5 Mbps. The length of packet was exponentially distributed with a mean of 500 byte. The effective bandwidth formula for such a source is given by (5) and, for the above mentioned parameter values, the effective bandwidth is equal to 5.19 Mbps. Figure 3 depicts estimates of the effective bandwidth for one run when T 1 = 300 s, T 2 = 3 s, and the service rate is 5 Mbps. Clearly, the proposed estimator and cgf-measuring estimator are very accurate while Gaussian approximation rather overestimates the effective bandwidth. In the second simulation, we used a source generating packets according to a two-state Markov-modulated Poisson process (MMPP). The mean input rate in state 1 (state 2) was 20 Mbps (1 Mbps) and the mean duration time of state 1 (state 2) was 0.1 s (30 s). The length of packet was also exponentially distributed with a mean of 500 byte. Note that the effective bandwidth formula of a two-state MMPP is given by

1 (λ1 + λ2 )θl − (ν1 + ν2 ) α(θ) = 1 − θl   2θ   (λ1 − λ2 )θl  + { − (ν1 − ν2 )}2 + 4ν1 ν2  ,   1 − θl where λ1 (λ2 ) is the packet arrival rate in state 1 (state 2), and ν1 (ν2 ) is the inverses of the mean duration of state 1 (state 2) [8], and, for the above mentioned parameter values, the effective bandwidth is equal to 19.7 Mbps. Figure 4 depicts estimates of the effective bandwidth and P[B ≥ xth ] for one run when T 1 = 300 s, T 2 = 3 s, and the service rate is 15 Mbps. Figure 4 indicates that Gaussian approximation overestimates the effective bandwidth when the system is heavily loaded (P[B ≥ xth ] is large) and underestimates when the system is lightly loaded (P[B ≥ xth ] is small). The cgf-measuring estimator always underestimates the effective bandwidth. Compared with these existing proposals, the proposed estimator is more accurate and stable. In the third simulation, we used a two-state MMPP source, where the mean duration time of each state was longer than the estimation interval T 1 . More precisely, the mean duration time of state 1 (state 2) was 10 s (100 s), T 1

Fig. 4 Estimation results: MMPP source (the mean duration time of state 1 (state 2) was 0.1 s (30 s)).

Fig. 5 Estimation results: MMPP source (the mean duration time of state 1 (state 2) was 10 s (100 s)).

was 3 s, and T 2 was 0.3 s. The mean input rate in state 1 (state 2) was 20 Mbps (1 Mbps). Figure 5 depicts the estimates of the effective bandwidth and P[B ≥ xth ] for one run. Figure 5 indicates that all estimators capture the variation of packet arriving rate due to the change of the MMPP state. Furthermore, in state 1, the cgf-measuring estimator and the proposed estimator precisely estimate the effective bandwidth while the Gaussian approximation largely overestimates the effective bandwidth. In state 2, the Gaussian approximation and the proposed estimator slightly overestimate the effective bandwidth while the cgf-measuring estimator slightly underestimates the effective bandwidth. Similar results were obtained for other MMPP sources that have different parameter values. Remark 2. The cgf-measuring estimator αˆ cg f (θ) is strongly dependent on T 2 . We found through simulation experiments that αˆ cg f (θ) is decreasing with T 2 . The cgf-measuring estimator, however, always underestimated the effective bandwidths of a MMPP source for any values of T 2 in our experiments.

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Fig. 6

Input rates per 0.1 s: data set 1.

Fig. 7

Input rates per 0.1 s: data set 2.

4.2 Simulation Using Real Traces

Table 1

Fig. 8

Estimation results: data set 1.

Fig. 9

Estimation results: data set 2.

Service rate dependence of the estimates: data set 1.

4.2.1 LAN Traffic Next, we conducted the experiments by simulation based on two sets of real traffic data (data sets 1 and 2), which were collected on a 100-Mbps Ethernet cable of a LAN at Chiba University. The data was collected by using a packet monitoring system, which was implemented based on pcap library [23], when 30 users used the subnetwork. Each data set consists of the lengths and arrival times of packets that appeared on the cable over 2000 s. Number of packets recorded in each data set is about 120000. Figures 6 and 7 respectively show the sequences of input rates per T second, {a(0, T ), a(T, 2T ), · · ·}, for data sets 1 and 2, where T = 0.1 and a(x, y) is defined by def

a(x, y) =

X(y) − X(x) . y−x

These figures confirm that the traffic patterns recorded in each of the data sets had been very bursty and thus displayed a typical characteristic of LAN traffic (see Leland et al. [24]). We have estimated the effective bandwidth for each data set when T 1 = 300 s and T 2 = 3 s. Note that, in the simulation, we set the service rate (capacity of the link) at 4 Mbps because, compared with the input rate, the capacity of the original link (100 Mbps) was too large to model a busy

and congested interface. Resultant estimates and P[B ≥ xth ] for data sets 1 and 2 are plotted in Figs. 8 and 9, respectively. The results are similar with the one when an MMPP source was used (in Sect. 4.1). That is, Gaussian approximation gave the largest estimate, and the cgf-measuring estimator gave the smallest estimate. Note that the estimates by Gaussian estimator using buffer measurement were dependent on the service rate. We depict this dependence in Tables 1 and 2, where, for a given pair of estimator and service rate, the greatest estimate among those obtained in the simulation is listed. The tables indicate that the bandwidths (service rates) which are necessary and sufficient to satisfy the QoS criterion P[B > x] ≤ 10−3 are between 6 Mbps and

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Service rate dependence of the estimates: data set 2.

Fig. 10

Table 3

Input rates per 0.1 s: WIDE US-Japan-link traffic.

Estimation interval dependence of the estimates: data set 1.

7 Mbps for data set 1 and between 7 Mbps and 8 Mbps for data set 2. The tables also indicates that αˆ gsbm is the most accurate estimator. Compared with this, both αˆ gs and αˆ cg f are less accurate: in particular, αˆ gs is too conservative and αˆ cg f is too optimistic. Note that αˆ gsbm does not always yield better estimates than other estimators when the link is too lightly loaded (please see the case when the service rate is 100 Mbps in Table 2). When the estimator αˆ gsbm is used within a traffic measurement tool having the queue simulation function as explained in Sect. 3.2, the service rate of the queue in the simulation should be adjusted so that the queue is not too lightly loaded. The estimation results have the dependence on the estimation interval T 1 . Table 3 summarizes the estimation interval dependence for data set 1 when T 2 = T 1 /100 and the service rate is 6 Mbps. The table indicates that the proposed estimator has least dependence on the estimation interval compared with other estimators. These experiments by simulation lead us to conclude that αˆ gsbm is quite promising as an estimator of the bandwidth requirement for a bursty source typical of IP networks. 4.2.2 WIDE US-Japan-Link Traffic Finally, we conducted a simulation experiment using a 15minute-long traffic trace collected at an US-Japan 100-Mbps link of WIDE backbone on October 27, 2003 [25]. Figure 10 show the sequence of input rates per 0.1 second. We have estimated the effective bandwidth when T 1 = 180 s and T 2 = 1.8 s. Resultant estimates and P[B ≥ xth ] are

Fig. 11 Table 4 traffic.

Estimation results: WIDE US-Japan-link traffic.

Service rate dependence of the estimates: WIDE US-Japan link

plotted in Fig. 11 when we set at the service rate (capacity of the link) at 19 Mbps. We also summarize the service-rate dependence of the estimate by the proposed estimator in Table 4. The table indicates that the bandwidths (service rates) which are necessary and sufficient to satisfy the QoS criterion P[B > x] ≤ 10−3 are between 19 Mbps and 20 Mbps. The result of this case is similar with the one when a Poisson source was used (in Sect. 4.1). That is, the proposed estimator and cgf-measuring estimator are accurate while Gaussian approximation overestimates the effective bandwidth. This implies that the traffic on the backbone link is smooth and its burstiness is comparable to that of Poisson process. Table 4 also implies that the estimate by the proposed estimator does not largely depend on the service rate for such a smooth traffic.

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5.

Conclusion

In this paper, we have proposed a semi-parametric estimator of effective bandwidth that exploits buffer measurements. The proposed estimator will be applicable to various engineering tasks to do with IP networks, including bandwidth design, the dynamic bandwidth adaptation of label-switched paths in MPLS-based networks in response to bandwidth requirements, and admission control for VoIP connections. Note that application of the proposed estimator will require implementation of the measurement of bufferoccupancy levels and this is not implemented in IP routers that are currently on the market. There are, however, several traffic measurement tools, which include the functions of simulating the input of measured patterns of packets into a single server queue to evaluate performance [20]. Thus, implementing our estimators in such measurement tools is possible, and thus is quite a promising approach in terms of traffic-engineering applications. Making our proposal applicable to cases where a scheduling algorithm (e.g., weighted fair queueing) is used in an IP router remains a challenging problem. Extensions to cover the case where the packet arrival process has a long-range dependence, so that the limit on the right-hand side of (3) does not exist, also remains a subject for further study. Acknowledgment We would like to thank the anonymous reviewers for their comments, which have significantly improved the quality of the paper.

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Appendix:

Proof of Theorem 1

Observe that αˆ gsbm (θ,c; θdecay (c))  θ 1 − a/c = a 1+ a/c θdecay (c) θ θ )a + c, = (1 − θdecay (c) θdecay (c) from which the desired conclusion readily follows.

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Shigeo Shioda received the B.S. degree in physics from Waseda University in 1986, the M.S. degree in physics from University of Tokyo in 1988, and the Ph.D. degree in teletraffic engineering from University of Tokyo, Tokyo, Japan, in 1998. In 1988 he joined NTT, where he was engaged in research on measurements, dimensioning and controls for ATM-based networks. Since March 2003, he has been an Associate Professor in the Department of Urban Environment and Systems, Chiba University. His research interest includes the Internet traffic modeling, measurement and performance analysis. He received Network System Research Award of IEICE in 2003 and Information Network Research Award of IEICE in 2004. Prof. Shioda is a member of the ACM, the IEEE, and the Operation Research Society of Japan.

Daisuke Ishii received the B.E. and M.E. degrees in communication systems from Chiba University, Chiba, Japan, in 2002 and 2004. He joined Softbank BB Cooperation in 2004 and has engaged in research and development on network operation and management for IP networks. He recieved Network System Research Award of IEICE in 2003.