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1 Empirical Eective Bandwidths M. Falkner, M. Devetsikiotis, I. Lambadaris Department of Systems and Computer Engineering Carleton University 25 Colonel By Drive Ottawa, Ontario KS 5B6, Canada S. Tartarelli, S. Giordano Department of Information Engineering University of Pisa Via Diotisalvi 2, 5626 Pisa, Italy Corresponding author Michael Devetsikiotis tel: +{63{52{26 ext 573 fax: +{63{52{5727 March 8, 2 Abstract We analyze the accuracy of four methods for measuring eective bandwidths (EBs). We also seek to point out advantages and drawbacks of the four estimators. We nd that for nite time realizations of a process the measured eective bandwidth diers considerably from its analytical counterpart. We also show that increasing the trace length has little impact on the accuracy of the measurements. We explain this behavior as a consequence of the intrinsic characteristics of the observed trace. We subsequently introduce the notion of \empirical eective bandwidth" (EEB) as a measure of performance tailored to the actual values. We derive properties of the EEB which capture its behavior in the parameter space and we contrast these properties with the ones obtained for analytical eective bandwidths. Finally, we comment on the use of EEBs in the context of connection admission control. Symposium Title: High-Speed Networks General Conference Topic: Communications Systems Integration & Modeling

2 Empirical Eective Bandwidths Sandra Tartarelli, Matthias Falkner, Michael Devetsikiotis, Ioannis Lambadaris, Stefano Giordano Abstract We analyze the accuracy of four methods for measuring eective bandwidths (EBs). We point out advantages and drawbacks of the four estimators. We nd that for nite time realizations of a process the measured eective bandwidth diers considerably from its analytical counterpart. We also show that increasing the trace length has little impact on the accuracy of the measurements. We explain this behavior as a consequence of the intrinsic characteristics of the observed trace. We subsequently introduce the notion of \empirical eective bandwidth" (EEB) as a measure of performance tailored to the actual values. We derive properties of the EEB which capture its behavior in the parameter space and we contrast these properties with the ones obtained for analytical eective bandwidths. Finally, we comment on the use of EEBs in the context of connection admission control. I. Introduction The notion of eective bandwidths (EBs) has found wide applicability to the management and performance analysis of broadband networks. It has signicantly simplied connection admission control (CAC) algorithms for data trac. Consider for example a stationary and ergodic stochastic process X requiring a probabilistic guarantee on the buer occupancy in a queue with constant rate. Such a guarantee can be met if the eective bandwidth (:) of all streams feeding the buer satises the following condition [], [2], [3] since in this case we have for all, () := lim t! t log EeX[;t] < () () <() lim log P (Q >B),: (2) B! B Here X(;t) denotes the number of arrivals in a given interval (;t). A simple CAC-algorithm can then be based on the eective bandwidth. A newly arriving source X can be admitted to a virtual path (VP) if KX X (), i () (3) i= given that K sources, each with EB i (), are already admitted to the VP with QoS parameter [4]. Here X () denotes the eective bandwidth of the newly arriving stream. Equation () may also be used for buer sizing or to determine the parameters of a trac shaper [5]. In general the eective bandwidth function is dened by Kelly [6] as (; t) = t log E[eX[;t] ] <;t<: (4) Analytical forms of eective bandwidths for many classes of trac models have been calculated [7]. Interpretations for the parameters are given by [8]. However, applying denition (4) in a practical environment is not trivial. It requires a full characterization of the underlying process. The eective bandwidth may be modeled parametrically, but this requires to choose an analytical form of the source's eective bandwidth (AEB). The trac stream may then be used to estimate the parameters in order to fully characterize the stream. Both procedural steps introduce approximations leading to inecient resource allocation. Motivated by this remark we focus on the feasibility of direct EB measurements. Our main concern is that in a realistic context a given trace is only a nite time realization of a stochastic process. In such a trace we do not necessarily observe all the possible values of the underlying process, particularly the theoretical peak value. Using AEBs leads to a resource allocation where such values are taken into account, despite the fact that they may not be realized in the given trace. The amount of resources allocated thus may exceed the actual resource requirements for the given trace, compounding the ineciencies arising from parametric modeling itself. We therefore compare the resource allocation deriving from schemes based respectively on analytical and measured EBs. In particular we investigate whether the relevant parameters values lie in a range where analytical and measured EB are signicantly dierent. The above discussion motivates the importance of measuring the EB of a source. We review the estimators considered here in section II. In section III we analyze the accuracy of four dierent EB estimators for given, nite time trac traces: a direct estimator, a block estimator, an estimator based on the Kullback-Leibler distance and an estimator based on linear regression. Since one of our main targets is to compare measured EBs with their analytical counterparts, we

3 restrict our analysis to three traces generated from stochastic models where the analytical eective bandwidth is known (Poisson, On-O, Fractional Gaussian Noise). The analysis for more realistic trac sources is beyond the scope of this paper. We particularly emphasize the niteness of these traces. We show that the estimators, being only asymptotically consistent, achieve a limited accuracy compared to the AEB. This limitation then leads us to introduce the concept of \empirical eective bandwidths" (EEBs) in section IV. We claim that a full characterization of AEBs is unnecessary for practical purposes and might even lead to an over-conservative resource allocation. EEBs on the contrary depend on observed values, therefore they consist of only the minimum required information. We also derive properties for the EEB which allow us to explain the behavior of measured eective bandwidths in the parameter space. In section V we verify the consistency of EEB-based resource allocation schemes. We conclude this paper in section VI with a brief summary and motivations for future research. II. Effective Bandwidth Estimators This section provides a brief overview of the methods and a discussion on their applicability. We also describe the traces used in our analysis. A. The direct estimator The EB can be estimated directly using equation (4) as shown by Gibbens [9]. Instead of considering the probabilistic expectation of the underlying process in equation (2), the temporal average is applied. The estimator takes the form ^(; t) = t log N, t Z N,t e P N i= xii(ti+t) dt (5) for N, t where N is the trace length. Since the direct estimator is based on the underlying denition of the analytical eective bandwidth, it is not restricted by any additional assumptions. Note that the integral in equation (5) may be hard to obtain numerically. Overows arising from a large number of arrivals in (; + t) may limit the applicability in both parameter spaces. B. The block estimator A block estimator has been proposed by [], []. It takes the form X ^(; t) = t log b N t c b N t c i= e P it k=(i,)t+ X k for a given time t. Note that this estimator is also based on equation (4). However, in contrast to the direct estimator, the estimator considers non-overlapping blocks of arrivals over an interval of length t. The estimator is based on the assumption that the block arrivals are realizations of independent and identically distributed (i.i.d.) random variables. This assumption limits the applicability of the estimator to short-range dependent streams. Dueld et al [] show that the estimator is asymptotically unbiased in this case as the number of samples in the trace N tends to innity. C. The Kullback-Leibler Distance (KLD) estimator The Kullback-Leibler distance (KLD) estimator [] is based on a virtual buer method, derived from equation (). If the arrival stream X feeds a single server queue with service rate, then the probability of the queue length exceeding a threshold B decays exponentially with rate. We can therefore estimate by observing the decay rate of the queue for a given service rate. In this case, the estimate for is obtained by minimizing the Kullback-Leibler distance between the observed queue size distribution for a given buer threshold B and p(b) =e,^, ()B. The KLD estimator evaluates to ^ = log( P b=b (b) P P b=b b(b) ): (7), B b=b (b) Note that the KLD estimator for () involves no time parameter t, since it is derived from equations () and (2). The KLD estimator is therefore only applicable for stationary, short-range dependent sources. Details can be found in [7], []. R The Kullback-Leibler distance is dened in general as I(f; ^f) := f(x) log( f(x) ^f(x) )dx. (6)

4 EEB (cells/slot) e-5 Analytical EB Block Estimator Direct Estimator θ (/cells). Analytical EB. KLD Estimator. LR Estimator. e θ (/cells) EEB (cells/slot) Fig.. Analytical and measured EBs for the Poisson trace with 95% CIs (-axis in log scale). 4 D. The linear regression (LR) estimator The linear regression estimator is also based on equation (). As above, the decay rate of the buer occupancy resulting from feeding the arrival stream into a single server queue with service rate is obtained. Since the decay rate is exponential with rate, the log of the probability of the queue size exceeding a threshold B is linear and we can estimate using linear regression. The same restrictions as for the KLD estimator apply here. For long-range dependent trac, a regression technique may be applied using Dueld's bound for an arrival sequence with Hurst parameter H > :5 [2]. In this case, a logarithmic transformation allows us to obtain a linearly decaying function of the log of the complementary queue size distribution. E. Trac streams In order to evaluate the dierent EB estimators, we generate nite time traces from Poisson, On-O Fluid and FGN trac models. This allows us to compare the measured EB against analytical results. The performance of measured EBs for more realistic trac sources is considered by ongoing work. We derive the parameters for all traces from the Bellcore specication for VBR I sources. We set the average arrival rate to 6:25 Mbps and dene a slot length of :27 ms for discrete-time trac. We thus obtain = 4 cells/slot for the Poisson trace. For the On-o source, both phases are exponentially distributed with average sojourn time E[T On ]=:68 and E[T Off ]=2:4 respectively. The arrival rate during the on-phase is set to 58:9 cells/ms. For the FGN trace, we set the mean number of arrivals and the corresponding variance to m = 4 cells/slot and 2 =:36 cells/slot respectively. We used a Hurst parameter of H =:7. The FGN trace was generated using a fast Fourier transform method based on Paxon's generator [4] (available from c.html). Unless specied otherwise, we use trace lengths of N = ; slots and N =;48; 576 slots respectively for the Poisson and the FGN trac. The Poisson trace thus corresponds to 27 seconds of real trac, whereas the FGN trace amounts to 6:44 minutes. For the uid source we generated a trace of N =25; cycles 2 or :3 minutes of trac. III. Accuracy of measured effective bandwidths In this section we assess the accuracy of the four dierent estimation methods for EBs. For each of the three trac traces we apply the estimators to a trace of length N with 95% condence intervals by repeating the estimation 3 times. A. Poisson trac stream Figure () shows the measured EB against the AEB for the four methods for : <<5. Note that the axes for the virtual buer based methods are reversed. In this case, we estimate for given values of = (), thus showing CIs for the estimated -value. The results for the direct / block estimators were obtained for t =. This choice is motivated by the observation that the EB is time-independent if the underlying process has independent increments [6]. All estimators seem to track the AEB adequately for small values of. However, none of the estimation methods performs well for large values of. The point at which the estimators start to deviate is almost identical across methods (at around ). The erratic performance of the virtual buer based estimators can be attributed to the diculty in obtaining a satisfactory empirical distribution for the queue length at large service rates. The consistency of our analysis is conrmed by the values of the traditional 95% CIs. In general, the degradation of the estimators in the -space can be explained by the property that the EB approaches the maximum value of X(; + t) as!(see [6]). We prove this property in section VI below. Whereas the AEB 2 We dene a cycle as the duration of an on phase plus an o phase.

5 .. Analytical Effective Bandwidth Estimated Effective Bandwidth(Direct) AEB(cells/ms) t (ms) θ (/cells).97 EEB (cells/ms) t (ms) θ (/cells).. Fig. 2. Measured EBs for the on-o uid trace (-axis in log scale). tends to innity asgrows, the measured EB tends to the maximum of the observed values. For the underlying Poisson process, P (X(;t)=k)>for all k =;2;:::, in particular for large values of k. By the argument of Laplace the analytical EB tends to innity. In contrast, using a particular realization of this process we observe a maximum number of arrivals for X(;t)<. In our case, the measured maximum number of arrivals was observed to be 5. We therefore investigated the implications of this deviation on resource allocation. For a QoS requirement of>2:5 in this case, a CAC method based on AEB would require a bandwidth capacity of 7.9 cells/slot. This is clearly inappropriate when the maximum observed value of the trace is only 5 cells/slot. If AEBs are used in the CAC procedure, the current trac stream may well be rejected even though sucient bandwidth may beavailable to carry it. However, at least for the Poisson trace considered here, such big values for are not relevant in practice. Such values for correspond in fact to very small loss probabilities and small buer sizes. We can derive a realistic range for from equation () by assuming that P (Q >B), foramultiplexing buer 3. For small buer sizes (B cells), evaluates to =, log, = = :23. Even for high loss probabilities of,2, the corresponding -value is :46. A conservative upper bound for is thus <, which is precisely the region where all estimators perform well. Figure () also illustrates the similarities between the dierent estimation methods. With t =, the direct estimators and the block estimators are identical. The LR estimator performs insignicantly better than the KLD estimator for small values of. Both virtual buer based estimators degenerate sigicantly for >. This result favors the direct /block method. B. On-o uid trac stream Figure (2) illustrates the estimated surfaces of the direct estimator for the on-o uid trace, as well as the AEB surface. The results obtained from the block estimator were almost identical. The estimated surfaces show the same behavior in both the time and the space. For small values of and over the entire time-range, the measurements seem to track the AEB reasonably well. Similarly, the accuracy seems to be satisfactory for large values of and small time values. However, as the time window increases, the estimators start to degenerate for large values of. As the time parameter increases, the observed maximum value of A(;t), scaled by t, decreases. Since the measured eective bandwidth tends to this value as increases, the estimates degenerate in this parameter region. The AEB on the other hand does not exhibit such a behavior. To compare the measured EB with the KLD- and LR estimators, we collapse the time dimension using equation (4). We restrict ourselves to the range of values for t where the measurements do not degenerate, giving us t 5, and then select the largest value of t for which the squared error is small. We thus arrive atavalue of t = 5 for our comparison. The results are shown in Figure (3). In this case, the measured EB seems to be suciently close to the AEB for all four methods. Note how our argument explaining the discrepancy between the analytical and measured eective bandwidths for the Poisson trace is less apparent here. For a xed time interval t, the arrivals are limited by the peak rate of the source. For On-O uid sources, this peak rate is clearly nite. Consequently, even the AEB varies between the mean and the nite peak rate. The estimators capture this peak rate well with an appropriate choice for t. The tight error bars in gure (3) indicate high consistency for the measured eective bandwidth. However, a closer examination of the CIs reveals that for parts of the -range, the AEB falls outside the 95% CI (in this case, for <:5). Consistent with our previous ndings, the measured EBs are below the AEBs as increases. In comparison with each other, we nd again that the direct / block and the KLD / LR estimators perform similarly. For the direct / block estimators, the CIs are overlapping over the entire range. The same holds true for the KLD / LR estimators for smaller -values. 3 We assume that the arrival stream seeks admission into such amultiplexing buer. We do not assume here that the network supports per stream queueing for a large number of arrival streams.

6 EEB (cells/ms) Analytical EB Block Estimator Direct Estimator (/cells) θ.... e-5 Analytical EB KLD Estimator LR Estimator e θ (/cells) EEB (cells/ms) Fig. 3. Analytical and measured EBs for the On-O uid trace with 95% CIs EEB(cells/sec) x 5 x log θ (/cells) -4 log θ (/cells) log t (sec) -5-4 EEB (cells/sec) log t (sec) Fig. 4. Analytical and measured EB for FGN. C. FGN trac stream Our analysis of measured EBs diers signicantly for the FGN trace. Since most estimation methods are not suitable for this kind of trac, we restrict ourselves to the direct estimator. Figures (4) show the AEB and the measured EB of the FGN trace described in section III. The gures clearly show that for large values of, the measured and the analytical EBs are once again signicantly dierent. For large values, the AEB increases in time, whereas the measured EB decreases. This latter observation is consistent with our ndings from the On-O and Poisson sources. For small values of, the plots in (4) seem to indicate reasonable accuracy of the estimator. However, our detailed analysis [5] shows that such discrepancies also exist in this range. Even for small values of and t, the discrepancies between the analytical and the measured EBs are signicant. D. Comments on the estimators The block estimator is by far the fastest of the considered schemes. We report here some indicative results, we obtained by running the estimators on a Sun Sparc Ultra for the.3 minute On-O trace, consisting of N =25; cycles. The direct estimator runs between :6 seconds and :38 seconds per -value for small and large values of respectively. The corresponding values for the block estimator are :32 seconds and :79 seconds. The run times for the KLD estimators are between 2 seconds and 45 seconds for a single estimation point. The LR estimator takes between 3 and 43 seconds. Note that these time values are dependent on the implementation and the trace length 4. The above values highlight the dierences between the estimators in obtaining a single estimation point. We note that the values are heavily dependant on the trance length. Considering a trace length of.3 minutes, we believe that the block estimator performs particularly well. The improved accuracy for high values of achieved by the direct scheme as compared to the block estimator does not compensate for the much longer run time. However, 'intermediate' schemes that use partially overlapping windows are also feasible. The KLD and LR estimators have the advantage to estimate the EB directly at the time scale of interest for queuing 4 In particular, the KLD and LR estimators require a histogram. In our implementation, we analyze the arrival stream rst to obtain good values for the histogram. This increases the computational overhead signicantly for large traces.

7 performance. In particular, since the mean and peak rates are assumed to be estimated beforehand, the eective bandwidth range is already determined. This indirectly also determines the range for. If the entire eective bandwidth function is estimated using the direct or the block estimators, the range for has to be xed a priori. On the other hand, the block and direct estimators allow to perform the estimate directly at the value of of interest, while for the virtual buer estimators represents the output. This feature is relevant for CAC schemes, where the value of is determined beforehand, based on the buer size and the loss probability requirement. The direct and block estimators can easily be implemented on-line. We report here the on-line version for the block estimator. It processes data within a moving window, that moves with step t. ^ n =^ n,,a n +B n where : An = t log t N ex[(n,)t+;nt] B n = t log t N ex[n+(n,)t+;n+nt] n is the estimated EB at time n> and N is the window size measured in trac samples. E. Trace length evaluation of measured eective bandwidths Our experiment consists of re-running the above measurements over the same parameter space for dierent trace lengths. We henceforth restrict ourselves to the block estimators for the Poisson and the On-O uid traces. For the FGN trace, we use the direct estimator. Figure (5) shows the estimated EB for the block estimator for the Poisson and On-O traces. In both cases, the two measured curves are remarkably similar. In particular, the measured EB seems to be consistent for a signicant portion of the range. Increasing the trace length only seems to aect the measurements for larger values of. The implications of this result are twofold. First, we conclude that a reasonable estimate of a trace's EB can be obtained by a relatively small trace. A trace length of N =;24 slots corresponds in our case to 277 ms. This is important for instance to size the measurement window for on-line CAC. Secondly, this result supports our argument that the behavior of the EB for large values of is dominated by the maximum observed value. Additional observations which are below the maximum observed with a small trace seem to add little information to the estimated EB. However, those observations which are exceeding the current maximum are responsible for increasing the tail of the estimated eective bandwidth curve for large values of. The measurements for intermediate trace lengths conrm these conclusions. They are not shown here to simplify the presentation of our ndings. EEB (cells/slot) N=24 N=65536 EEB (cells/ms) cycles 52 cycles e e θ (/cells) θ (/cells) Fig. 5. Comparison of the measured EB for dierent trace lengths for the Poisson and on-o traces using the block estimator. The relatively scarce sensitivity of the measured EB to the trace length partially applies to the FGN trace. In this case, we evaluated traces of length N = 52 and N = ; 48; 576 slots. Figure (6) illustrates the estimated EB for time t = and t 2 = 6. For small time scales, the performance of the estimators is almost independent of the trace length. The estimated EBs are very close for signicant sections of the range. However, as we increase the time scale, the dierence in the tail of the estimated eective bandwidths also increases. This suggests that additional observations are more valuable for larger time scales. IV. Empirical Effective Bandwidths The above limitations lead us to formally dierentiate measured and analytical eective bandwidth by introducing the notion of \empirical eective bandwidths" (EEB). Let X again be a given, nite time realization of a weakly-dependent, stationary stochastic arrival process of length N. We dene the EEB of such a trace with parameters and t as

8 EEB (cells/sec) e-5 "N=,48,576" "N=52".... θ (/cells) EEB (cells/sec) e-5 "N=,48,576" "N=52".... θ (/cells) Fig. 6. Measured EB for N =;48; 576 and N = 52 at t = and t = 6 slots. a(; t; N) = t log b EN [e X(;t) ] <;<t<n (8) where X(;t) indicates the aggregate number of cell arrivals in an interval of length t slots and EN b [e X(;t) ] is the measured log moment generating function over a trace consisting of N samples. To simplify the notation, we restrict ourselves to the discrete time case, where X t denotes the number of cell arrivals in time slot t of the nite trace of length N slots. We note however that all results derived below carry over to the continuous time case. We relate equation (8) to the probability P (Q >B) of the queuing process Q exceeding a given threshold B. We proceed to derive basic properties of the EEB for a given trac trace X t, t = :::N, with sample mean m and sample variance s 2. We particularly contrast these properties with their analytical counterparts. Throughout this section, we assume that (6) has been used as estimator for Ee b X(;t) and hence that the block sums are approximately independent. Following the notation in [] we denote by Xi ~ P it := j=(i,)t+ X j;i=:::b N t cthe number of cell arrivals over an interval of length t. We also let M t = b N t c. A. Behavior in the space Property : For a xed value of t<, XM t lim a(; t; N) = ~X i = m (9)! tm t i= Proof: Since the limit as! for the numerator and the denominator equals, l`h^opital`s rule holds and so we have d P lim! a(; t; N) =lim! d log Mt M t i= e Xi ~ P Mt d = i= X ~ i = m:2 lim! d t tm t A similar argument for the AEB shows that it tends to E[X]. Since m E[X], the AEB and the EEB thus exhibit a similar behavior as!. Property 2: For a xed value of t<, lim a(; t; N) =! t max[ Xi ~ ] () Proof: Since the limit as!for the numerator and the denominator equals, l'h^opital's rule holds again and so we have lim! The numerator can be written as d t log M t d log M t XM t i= XM t i= e ~ Xi = lim! d d log M t P Mt i= e ~ Xi lim! d d t X e ~ Xi d d log M t M t e max[ Xi] ~ i= = d d log emax[ ~ Xi] = max[ ~ Xi ]:2

9 In this case the result for the analytical eective bandwidth is signicantly dierent. A similar derivation shows that lim (; t) = sup fx : P fx(;t)>xg>g ()! t The AEB thus tends to the essential supremum of the process, which may be innite. However, for a particular nite time realization of the process the probability of observing the theoretical supremum becomes less likely as t increases. Note how this property explains the discrepancy between the AEBs and their measured counterparts observed in the last section. This is our principal justication for the introduction of EEBs. B. Behavior in the t space The last observation also allows us to describe the dierence between the EEB and the AEB in the t-space. To further illustrate our point using an example, let us assume that X is a Markov-modulated arrival process with peak rate H and mean rate m < H. Then, P (X(;t) = Ht) is positive. However, for any nite time realization this probability tends to zero as t increases, since lim t!n X(;t)=mt<Ht. The analytical EB is limited by the supremum 5 of X(;t)=t. However, the EEB is limited by the maximum of X(;t)=t, which tends to mt as t increases. In the special case where the central limit theorem (CLT) holds for X i,we can characterize the behavior of the EEB by the following additional property. Property 3: Assume that the trace X n is a nite realization of a weakly-dependent stochastic process with nite population mean and population variance 2. Assume also that t is large enough such that the block sums ~ Xi ;i = :::M t are independent and that the CLT holds. For large N; t with t<n and for a given the EEB degenerates towards lim a(; t; N) t!n =m+s2 2 Proof: By the central limit theorem, the block sums ~ Xi N(t; t 2 );i = :::M t for t suciently large [7]. Furthermore, in this case the sample mean m and the sample variance s 2 are unbiased estimators for and 2 of a function of a random variable. Equation (8) can then be written as (2) lim a(; t; N) = lim t!n t!n t log E[e Xi ~ ] 22t = lim log et+ 2 = + 2 t!n t 2 : The result follows from the unbiasedness of m and s 2. 2 Note that this condition excluded the trivial case of a constant source. The behavior of AEBs in the t-space is typically non-decreasing. C. Behavior with respect to the trace length Property 4: As N! lim a(; t; N) =(; t): (3) N! Proof:The proof is given by Dueld in []. 2 Note that the last property holds in both the t and spaces. V. Application of empirical effective bandwidths to CAC To show the eectiveness of the EEB concept in a practical environment we now report simulation results. We compare the allocated resources using EEBs with the results obtained from using the analytical counterpart. Note that we also assume that the entire nite time realization is known, as is the case for pre-recorded video, for example 6. For each trac type, we generate a test trace of length N. We then select the block estimator. We assume that the given arrival stream wishes to gain access to a single server queue with various buer sizes B and a specied quality of service requirement, expressed as a cell loss ratio (CLR). We use equation () to evaluate the required -value, called. Since we only estimate the EB for a limited number of -values, we apply a linear interpolation to derive the numerical value at. We subsequently determine the corresponding service rate using a( ;t ;N)=. The selection 5 Which may indeed be increasing in t. 6 If the trace is not known, measured EBs can however be employed to CAC schemes. In this case the EEB can be used to evaluate the bandwdith required to satisfy the QoS for the existing connections, while the new call declares an a priori parameter, typically the peak rate.

10 TABLE I Simulation results for the On-Off Fluid trace Buer a( ;t ;N) ^P(Q>b) CI e-5 (.2e-5,.5e-5 ) e-5 (.86e-5,.92e-5) e-5 (3.89e-5, 3.98e-5) e-5 (5.37e-5, 5.5e-5 ) (9.8e-5,.) e-5 (3.e-5, 4.e-5 ) e-5 (2.24e-5, 2.9e-5) e-5 (.4e-5,.65e-5) Buer ( ) ^P (Q >b) CI e-7 (8.77e-7, 9.72e-7) e-6 (.68e-6,.87e-6) e-6 (3.5e-6, 3.4e-6) e-6 (4.8e-6, 5.32e-6) e-5 (.83e-5, 2.5e-5) e-5 (2.75e-5, 3.57e-5) e-5 (.89e-5, 2.46e-5) e-5 (8.93e-6,.44e-5) TABLE II Simulation results for the Poisson trace: block estimator and analytical EB Buer ^( ;t ;N) ^P(Q>b) CI e-5 (2.7e-5, 2.8e-5) e-5 (6.35e-5, 6.58e-5) e-5 (8.3e-5, 8.6e-5) e-5 (5.89e-5, 6.43e-) e-5 (4.6e-5, 4.83e-5) e-5 (9.37e-6,.5e-5) Buer ( ) ^P (Q >b) CI e-5 (.6e-5,.69e-5) e-5 (7.7e-5, 7.3e-5) e-5 (8.54e-5, 9.4e-5) e-5 (6.39e-5, 6.96e-5) e-5 (4.89e-5, 5.64e-5) e-5 (.42e-5, 2.29e-5) of a proper time scale is still under investigation, therefore we adopt here a conservative approach. t indicates the time scale at where the EEB takes the maximum value, i.e. t = arg supft 2 R + : a( ;t;n)g: (4) Finally, we feed the given trace into the single server queue with the given buer size B and the determined service rate. We observe the simulated CLR and compare it against the specied requirement. In the case of Poisson and On-O uid traces we repeat this experiment using the AEB to establish a comparison. In order to obtain condence intervals for our procedure, we then repeat this experiment 3 times. A. Results for the On-O uid and Poisson traces Table I lists the simulation results for our experiment for various buer sizes for the On-O uid trace. The quality of service criterion for this experiment was set to = :. Again, this requirement is clearly met using both the EEB and the analytical eective bandwidths. Both methods exceed the requirement by at least one order of magnitude. This can be explained by the inequality in equation () and the resulting upper bound on which the allocation is based. Recall that our analysis in section III showed that the estimated and the analytical eective bandwidths are very close, therefore we expected the two schemes to have similar performance. However, simulations for the Poisson case yielded similar results (see Table II). In this case the QoS requirement was set to =:. Only as the buer size gets smaller the analytical EB starts being signicantly larger than the measured one. However, as already observed in section III-A, these small values for the buer size are not realistic for a multiplexing buer scenario.

11 VI. Conclusion and Future Research In this paper we have thoroughly analyzed eective bandwidth estimators and their applicability to bandwidth allocation schemes. The methods considered here are able to match the analytical eective bandwidth to a high degree of accuracy only for small values of the range. This can be explained by the property of the measured eective bandwidth to approach the scaled maximum observed value over a time interval. Demanding strict QoS requirements (very low loss probabilities or cell delays) might require bandwidths in the range where is large. However, for the trac sources considered in this paper, the relevant range of values for lies in the region where the EEB matches its analytcal counterpart. For the direct and the block estimators, which are dependent on time, we observe a degeneration of the measured eective bandwidth in time for suciently large values of. We attribute this phenomenon to the niteness of the arrival stream. Furthermore, we show that eective bandwidth estimates can be obtained with a relatively small number of observations. Increasing the trace length improves the estimate most at large values of and t. These limitations of measuring the eective bandwidth of a source induced us to dene the notion of EEB, which are obtained through such measurements. We show that EEB have dierent properties than their analytical counterparts. In the future, we intend to further investigate the signicance of the relevant time scale for applications such as CAC. Ongoing work is devoted also to compare EEBs with parametric approaches, with particular emphasis on realistic trac sources. Acknowledgments The authors would like to acknowledge the support of Nortel Networks, NSERC (Natural Sciences and Engineering Research Council of Canada), CITO (Communications and Information Technology Ontario), and MITACS (the Network of Centres of Excellence on Mathematics for Information Technology and Complex Systems). 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