Measured Effective Bandwidths

Transcription

1 Measured Effective Bandwidths S. Giordano, G. Procissi and S. Tartarelli Department of Information Engineering University of Pisa, Italy Abstract This paper studies different methods for measuring effective bandwidths: a direct estimator, a block estimator, an estimator based on the Kullback-Leibler distance, an estimator that uses a linear regression and a an estimator based on interarrival times. We apply these estimators to three finite time traces generated from a Poisson process, an On-Off fluid and from a Fractional Gaussian Noise process. We analyze their accuracy as compared to the analytical effective bandwidth, which is known for these models. We find that the asymptotic behaviour of measured effective bandwidths differs considerably from its analytical counterparts. The 95% confidence intervals fail to include the analytical values for a significant range of values of the parameter space. We also study the impact of the trace length on the measured effective bandwidth. We derive properties of the measured effective bandwidth which capture its behavior in the parameter space. We also contrast these properties with the ones obtained for analytical effective bandwidths. Finally, we present simulation results we run in order to test the efficiency of resource allocation schemes based on measured effective bandwidths. Introduction The notion of effective bandwidths (EBs) has found wide applicability for the management and performance analysis of broadband networks. The growing interest towards EBs is mainly due to the way they relate to queueing performance, through large deviation theory concepts []. In particular they allow to easily evaluate the amount of resources to allot to a stream in order to satisfy its Quality of Service (QoS) requirements. Effective bandwidths can also represent a measure of the level of congestion in a link. Consider for example a stationary and ergodic stochastic process X requiring a quality of service guarantee of the form P(Q > B) e,b () from an infinite buffer with constant service rate ρ. Here, Q represents the distribution of the queue length, is a quality of service parameter and B is the required buffer threshold. This guarantee can be met if: K j= N j α j () ρ (2)

2 Measured Effective Bandwidths 2 where N j is the number of sources of type j, α j is the effective bandwidth of traffic j and K is the number of traffic types [2, 3]. In this derivation, the effective bandwidth α() is defined as α() := lim t! t logeex[0;t] ; (3) with X (0;t) denoting the number of arrivals in a given interval (0;t). A simple connection admission control algorithm can then be based on the effective bandwidth. A newly arriving source X can be admitted to a buffer if α X () ρ, K i= α i () (4) given that K sources are already sharing the buffer with QoS parameter [4]. Here α X () denotes the effective bandwidth of the newly arriving stream. Equation () may also be used for buffer sizing. Alternatively, effective bandwidths may be used in a similar manner to size the parameters of a traffic shaper [5]. In general the effective bandwidth is defined by Kelly [6] as α(;t) = t loge[ex[0;t] ] 0 < ;t < : (5) Analytical forms for effective bandwidths for many classes of traffic models have been calculated [3]. However, applying definition (5) in a practical environment is not trivial. It requires a full characterization of the underlying process, since α(; t) is a function of the moment generating function. The effective bandwidth may be modeled parametrically, but this approach introduces the following problems. Parametric modeling requires to determine beforehand the statistical model for the analyzed stream. The traffic pattern may then be monitored to estimate the selected model traffic parameters. Finally the effective bandwidth is evaluated analytically. Both procedural steps introduce approximations leading to possible inefficient resource allocation. Since equation () is an upper bound for the probability of the workload of an infinite queue crossing a threshold B, allocation schemes based on analytical EBs are already conservative. Estimating the parameters of the analytical form for the EB introduces approximations. Using parametric models of the source s EB therefore only compounds the over-allocation of available resources. Furthermore, a given trace is only a finite time realization of a stochastic process. In such a trace we may not necessarily observe all the possible values of the underlying process, particularly the theoretical peak value. Using analytical EBs 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 might exceed the actual resource requirements for the given trace, compounding the inefficiencies arising from parametric modeling itself. The above discussion motivates the importance of measuring the effective bandwidth of a source. This report summarizes the results of our analysis on the estimation of the effective bandwidth function [7, 8]. We consider five different schemes: a direct estimator, a block estimator, an estimator based on the Kullback-Leibler distance, an estimator based on linear regression and a time estimator. We are particularly interested in studying their accuracy as

3 Measured Effective Bandwidths 3 compared to analytical results. For this reason, we restrict our analysis to three traces generated from traffic models where the analytical effective bandwidth is known (Poisson, On-Off, Fractional Gaussian Noise). We seek to point out possible assumptions which may restrict their use. Furthermore, we analyze the behavior in the parameter space. Some of the estimators require both a temporal and a spacial parameter. Others only require the spatial parameter. We intend to investigate the range for these parameters where the measurements are sufficiently accurate. In this report we place a particular emphasis on the finiteness of the traces. In any practical context, traffic can only be observed over a finite temporal window. The accuracy of asymptotically efficient estimators thus has to be investigated in this context. Our analysis suggests that the estimators fail to track the analytical effective bandwidth to within an acceptable confidence interval, at least in some region of the parameter space. Consequently we investigate the properties that determine the observed behavior. The remainder of the report is organized as follows: in section 2 we briefly review the five estimation methods considered in our analysis and we describe the traces used in our studies. Section 3 then presents the results of our comparisons between the measured effective bandwidth and their analytical counterparts. We only consider the first four estimators in this section. In fact the time estimator differs significantly from the other schemes, therefore we devoted a separate section to its analysis. In section 4 we derive properties for the empirical effective bandwidth which allow us to explain the behavior of measured effective bandwidths in the parameter space. We test the applicability of measured EBs to resource allocation schemes in section 5. To this end we simulate a single server queue designed in order to meet given QoS requirements. We compare results obtained providing bandwidth according to both estimated and analytical EB. Section 6 presents our preliminary analysis for the time estimator. We then summarize our findings in section 7. We conclude this report in section 8 with a brief summary of our results and open issues. 2 Effective Bandwidth Estimators In this section we present five such estimators, namely a direct estimator, a block estimator, an estimator based on the Kullback-Leibler distance, an estimator based on linear regression and a time estimator. Our aim is to investigate their applicability in a realistic setting. In particular, we study their accuracy using finite traces where the underlying analytical effective bandwidth is known. This allows us to determine the estimators accuracy. We also investigate their accuracy with respect to the trace length and parameter values. We particularly emphasize the finiteness of the traces, which is inherent in real traffic streams. The estimators can be classified by the underlying method. The direct- and block estimators are derived from equation (5). The estimators based on the Kullback-Leibler distance and linear regression on the other hand are based on equation (). They thus measure the decay rate of the queuing process arising from feeding the arrival stream X into a single-server queue with deterministic service rate ρ. Similarly to the direct and block schemes, the time estimator is related to definition (3). However, instead of measuring the arrival process, it considers the interarrival process. A change of variable is then applied in order to estimate the effective bandwidth of the arrival process. The next section provides a brief overview of these estimation methods. We also point out possible assumptions which may limit their applicability. We close section 3 with a brief description of the traces used in our analysis.

4 Measured Effective Bandwidths 4 To differentiate effective bandwidth measurements from their analytical counterparts, we use the notation ˆα(:; :) for the former and α(:; :) for the latter. We also use the terms measure and estimate synonymously. 2. The direct estimator Obviously, the effective bandwidth can be estimated directly using equation (5) as shown by Gibbens [9]. However, instead of considering the probabilistic expectation of the underlying process in equation (5), the temporal average is applied. The estimator takes the form where ˆα(;t) = t log t N,t Z tn,t 0 e X(τ;τ+t) dτ (6) X (τ;τ +t) = N i= x i I(τ t i τ +t) (7) for 0 τ t N, t and t N indicating the trace length expressed in time units. We thus replace the expectation in equation (5) by the sample mean of the arrivals measured over a sliding window of length t. Since the direct estimator is based on the underlying definition of the analytical effective bandwidth, it is not restricted by any additional assumptions. This estimator is applicable wherever the analytical effective bandwidth is defined. Note that the integral in equation (6) may be hard to obtain numerically due to finite precision arithmetic. A large number of arrivals over the interval (τ;τ +t) may lead to numerical overflows. Therefore the estimator is limited in both parameter spaces by such potential overflows. However in our simulations we managed to cover the whole range of interest. 2.2 The block estimator A block estimator for the effective bandwidth has been proposed by [0, ]. It takes the form where N is the trace length and ˆα(;t) = t log t N N=t i= e X i (8) X i := it k=(i,)t+ X k (9) for a given time t and for i N t. Note that this estimator is also based on equation (5). However, in contrast to the direct estimator, this estimator considers blocks of arrivals over an interval of length t. Thus, instead of using a sliding window of length t, the windows are non-overlapping. The estimator is based on the assumption that the block arrivals are realizations of independent and identically distributed (i.i.d.) random variables. Duffield et al [0] show that the estimator is asymptotically unbiased in this case as the number of samples in the trace N tends to infinity. However, this assumption limits the applicability of the estimator. In particular, it prevents the estimator to be applied to a self-similar traffic stream.

5 Measured Effective Bandwidths The Kullback-Leibler Distance estimator The Kullback-Leibler distance (KLD) estimator [2] is the first of the two estimators based on a virtual buffer method. It is thus fundamentally different from the direct or the block estimators, in that it uses equation () instead of equation (5). If the arrival stream X is fed into a single server queue with service rate ρ, 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 ρ. Thus, instead of estimating ˆα() for a given value of,the value of is estimated for a given value of ρ = α(). In this case, the estimate for is obtained by minimizing the Kullback-Leibler distance between the observed queue size distribution for a given buffer threshold B and between p(b) =e, ˆα, (ρ)b. The Kullback-Leibler estimator for the value of corresponding to a given service rate ρ of the single server queue evaluates to b=b π(b) ˆ = log( b=b bπ(b), B ): (0) b=b π(b) The effective bandwidth function is thus measured for various values of ρ between the mean rate and the peak rate of the arrival stream 2. This implies that the mean and the peak values of the arrival stream are determined a priori. However, these values are typically easily obtained. As a result, the range of the values for the measured effective bandwidth ˆα() are known and need not be provided as parameters to the estimation function. Furthermore, since the values of are the result of the estimation, the domain of the effective bandwidth function is automatically determined. Note also that the KLD estimator for ˆα() involves no time parameter t. This results from using equation (), which is in turn based on the definition of the effective bandwidth given by (3). Details can be found in [3, ]. The KLD estimator is only applicable for stationary, short-range dependent sources, for which holds. 2.4 The linear regression estimator The linear regression (LR) estimator is very similar to the KLD estimator. It is also based on equation (). As above, the decay rate of the queue size resulting from feeding the arrival stream into a single server queue with service rate ρ 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. We can thus use linear regression to obtain an estimate for for a given value of ρ = α(). Thus, for a given value of ρ, the decay rate is estimated using ˆ =, M i= b iln(p(b i )), M M M i= x i M M i= ln(p(b i)) M i= b2 i, M( M M i= b i) 2 () Since the LR estimator is also based on equation (), the same restrictions as for the KLD estimator apply here. For long-range dependent traffic, a regression technique may be applied using the modified exponential bound proposed by Duffield et al. for an arrival sequence with Hurst parameter H > 0:5 [2] The Kullback-Leibler distance is defined in general as I( f ; ˆf) := R f(x)log( f (x) ˆf (x) )dx. 2 If ρ is less than the mean rate, the queue becomes unstable. If ρ exceeds the peak rate, the queue size is always empty.

6 Measured Effective Bandwidths 6 P(Q > B) e,b2,2h (2) In this case, the decay rate is no longer exponential. To apply a linear regression technique, we transform the problem into a logarithmic domain. Alternative estimation techniques for the effective bandwidth have been proposed by Gibbens [9] and by Vesilo and Solo [3]. The former scheme is basically a direct estimator to which a subsampling technique is applied in order to make it faster. The latter estimator has the desirable property of being adaptive. The effective bandwidth is estimated on-line, thus eliminating the requirement of collecting a large number of arrival observations. On the other hand it can be easily implemented only under quite restrictive aasumptions. These estimators are not considered in this report. 2.5 The time estimator The time estimator [4] differs significantly from the other schemes. Similarly to the direct and block estimators, it is related to definition (3). However, instead of measuring the arrival process, it considers the interarrival process. A change of variable is then applied in order to estimate the effective bandwidth of the arrival process. The motivation behind this approach is mainly to bypass the difficulty of detecting a proper time scale for the direct and block estimators (see section 3). In fact with the time estimator a block size A is set and the variable to be monitored is the time required to collect A cells. This implies that the time duration of the observation blocks depends on the burstiness of the stream: for high activity periods the block size will have a long duration while during low activity periods we will observe short blocks. This is in contrast with the direct and block estimators, for which the time interval is fixed and the observed variable is the number of cells in the given interval. As a consequence the time estimator will provide a function only of the variable. In the following we formalize the above concepts. The arrival time process T a is defined as the arrival time of the a th cell and the interarrival time process D a as the time between two successive arrivals: T a = in f ft > 0jA t ag and D a = T a, T a, (3) Let assume that the arrival process X (0;t) satisfies a Large Deviation Principle (LDP) with rate function I X (s) and scgf (scaled Cumulant Generating Function) λ(), defined as follows: I X (s) =, lim log P(X (0;t) > ts) (4) t t! λ X () =lim t! t logeex(0;t) = sup[s, I(s)] (5) Then T a is an adjoint process to the arrival process X (0;t) and satisfies a LDP with rate function I T (y) and scgf µ(φ) such that: I T (y) =yi X ( y ) (6)

7 Measured Effective Bandwidths 7 µ(φ) = lim log Ee φta = sup[φy, I T (y)] (7) a! The effective bandwidth can be evaluated either directly by using the scgf of the arrival process λ() or derived from the one of the interarrival process: y α() = λ X () where λ X () =,µ, (,) (8) By duality with the direct estimator to estimate µ(φ) we measure the time required to count a fixed number A of cells to arrive; the estimate of the scgf of the interarrival process is made by using N observations of the aggregated interarrival time: ( ˆµ N A (φ) = A log N N i= e φt i C ) where TC i = ic D k (9) k=(i,)c+ T i C is the time required to observe the ith set of A cells, a in equation (4) is the number of cells arrived in the observation interval [0;t) (a = At) and N = ba=ac is the number of observations. In this case the free parameters to be set are the sample size N, as before, and the value A. The time estimator works on measurement blocks with variable size, because instead of measuring the aggregated arrival process over blocks of fixed length t, it measures the aggregated interarrival process. In this way the size of the measurement blocks is inversely variable with the activity level of the monitored streams: when the traffic arrival rate is low, the cell interarrival times are relatively long and the prefixed number of interarrival times corresponds to a large block size; on the contrary when the traffic activity is high, the same number of interarrival times gives smaller block size. 2.6 Traffic streams In order to evaluate the different effective bandwidth estimators, we generate finite time traces from known traffic models. This allows us to compare the measured effective bandwidth against the analytical results. The three traffic models chosen for the finite time traces are Poisson On-Off Fluid Fractional Gaussian Noise (FGN) Note that both the Poisson and the FGN models generate discrete time arrivals, whereas the On-Off fluid model is in continuous time. The Poisson trace was generated with parameter λ = 4cells=slot. With a time-slot definition of 0:27 ms, this amounts to an average arrival rate of 6:25 Mbps. The default length of the trace is N = 00; 000 slots, corresponding to 27 seconds of traffic. Note that the trace length is varied in one of our experiments below. In this way, we are able to study the estimators as a function of the trace length.

8 Measured Effective Bandwidths 8 The On-off fluid trace was generated based on the Bellcore specification for VBR I traffic. According to this specification, both on and off phases are geometrically distributed with a mean of 240 slots and 7 slots respectively, where a slot is defined to have a length of 2:83 µs. A cell arrives with period of 6 slots during the on phase. The average number of cell arrivals per on-phase is thus 40 cells. The average rate of the stream evaluates to 6:24 Mbps. Since we use an On-Off fluid model for comparison, we converted these parameters into continuous time measured in ms. Both on and off phases are now exponentially distributed with E[T ON ]= 0:679ms and E[T OFF ]=2:037ms respectively. Cells arrive at a rate of 58:9 cells/ms, giving again a total of 40 cells per on-phase or 6:24 Mbps for the stream. The default trace length was set to N = 25;000 cycles 3, which amounts to about minute and 8 seconds of traffic. Again, the trace length is varied in some experiments to study its impact on the accuracy of the estimators. The FGN trace has also been parameterized to match the Bellcore rates. We again model the number of cell arrivals in a discrete time slot of length 0:27 ms. The mean number of arrivals and the corresponding variance were set to m = 4 cells/slot and σ 2 = 0:36 cells/slot respectively. We used a Hurst parameter of H = 0:7. With these values, the average arrival rate of the stream is 6:25 Mbps. Where not differently specified, the number of slots in the trace was set to N = ;048;576, corresponding to 6 minutes and 44 seconds. As before, we vary the trace length in one of the experiments to assess the sensitivity of the estimation results on N. The FGN trace was generated using a fast Fourier transform method based on Paxon s generator [5] (available from c.html). We also test the sensitivity of the measurement results to the traffic generator, by using a random-midpoint displacement (RMD) algorithm for the generation of the trace. 3 Accuracy of Measured Effective Bandwidths In this section we assess the accuracy of the first four estimation methods presented in the previous section. For each of the three traffic traces we apply the estimators to a trace of length N, as specified in the last section. We also construct 95% confidence intervals by repeating the estimation times, each time with a trace length N. We present our findings for the Poisson trace, the On-Off trace and the FGN trace in succession. 3. Poisson traffic stream We start our analysis in this case by applying the four estimators to a single trace of length N = 00;000 slots. The estimated effective bandwidth surfaces for the direct and the block estimator are shown in Figure (). The -values in the plot are varied between 0:0000 and 5 with a multiplicative step of :5. The time values range between and slots, with a step of. The estimated effective bandwidth is thus shown in a log scale in Figure () in the -dimension. Note that both surfaces are almost identical. This is to be expected in this case, because of the independence of arrivals in each slot. Applying a sliding window technique, as is done in the direct estimator, thus yields the same empirical distribution of the number of arrivals X (τ;τ + t) in an interval of length t as the block method. However, as time increases, the estimated effective bandwidth also decreases. This seems to be unexpected in this case. In fact, Kelly [6] shows that if the arrival process has independent 3 We define a cycle as the duration of an on phase plus an off phase.

9 Measured Effective Bandwidths 9 increments, then the analytical effective bandwidth α(; t) does not depend on time. We would thus expect the estimated effective bandwidth to be constant over time, which is clearly not the case in the measured effective bandwidth. This can be attributed to the finiteness of the trace length. In fact the observed scaled maximum (the maximum of the aggregated arrivals divided by the time interval) decreases. Since the measured effective bandwidth tends to this value, it decreases as the aggregation interval increases. We will discuss this phenomenon in detail in section 4. Estimated EB (Block) Estimated EB (Direct) EB EB log time - log time Figure : Measured effective bandwidths for the Poisson trace using the direct and the block estimators. Note that the -axis is in log scale. To compare the estimated effective bandwidth for the block and the direct schemes with the results obtained from the other two estimators and with the analytical results, we have to collapse the time dimension. The above comparison between the measured effective bandwidth and the analytical effective bandwidth leads us to select t =. At this value, the measured effective bandwidth is closest to its analytical counterpart. In fact the smoothing effect on the scaled maximum increases with the aggregation time interval, as already mentioned above. Recall that this time value implies that the estimates from both are identical (with a window size, a sliding window is equivalent to a jumping window). Figure (2) shows the measured effective bandwidth with 95% CIs for the direct / block method and the KLD /LR method on the left and right respectively. Note that the axes for the virtual buffer based methods are reversed. In this case, we estimate for given values of ρ = α(). The CIs are thus for the estimated -value. The axis representing the estimated effective bandwidth is again in log scale. All CIs were computed over sample traces, each with a length of N = 00;000 slots. The tightness of the 95% CIs confirms the consistency of the above analysis. All estimators seem to track the analytical effective bandwidth adequately for values of <. However, none of the estimation methods is able to track the analytical effective bandwidth for large values of. The point at which the estimators start to deviate is almost identical for the different methods. The direct / block estimators seem to perform better, degrading gradually. This irregular behavior of the virtual buffer based estimators as increases, can be attributed to the difficulty in obtaining a satisfactory empirical distribution for the queue length. If the virtual queue is served at a high service rate (compared to the number of arrivals), the maximum queue size remains small. Consequently, only few observations are available to fit a line through the log of the empirical queue size distribution. The accuracy of the fitted line is thus very poor. Similarly for the KLD estimator. The estimators may be improved by implementing a heuristic, requiring for example, a minimum number of observations in order to compute an estimate. Furthermore the execution time for the direct and block methods is shorter than for the remaining two methods. The direct method took on average 640 ms to compute a single esti-

10 Measured Effective Bandwidths Analytical EB Block Estimator Direct Estimator e-05 Analytical EB KLD Estimator LR Estimator e Figure 2: Analytical and measured effective bandwidths for the Poisson trace with 95% CIs mation point on a Sun Ultra. The estimation time for the KLD method on the same machine varied between 500 ms for large and :02 seconds for small Comparison with truncated Poisson process We now show that the discrepancies between the analytical effective bandwidth and the measured effective bandwidth are resulting from the finiteness of the trace. To this end we compare the estimated effective bandwidth with the effective bandwidth of a truncated Poisson process. The mean rate and the truncation level of the truncated process are set to the observed mean and the observed maximum value (over a time slot) respectively. In this way, we approximate the empirical histogram of the number of arrivals per slot with a truncated Poisson process. Denoting the truncated process by X T, the corresponding distribution function by P T (X T = x) and the finite time realization of the Poisson process by ˆX,wehave where P T (X T = x) = cp(xt = x) if x max( ˆX ) 0 otherwise c = e λ max( ˆX ) i=0 : λ i i! The analytical effective bandwidth for this process then evaluates to α T () = log max( ˆX) i=0 (λe ) i =i! max( ˆX ) i=0 λ i =i! The resulting comparison is shown in Figure (3). We illustrate the measured effective bandwidth for the block estimator, as well as the analytical effective bandwidths of both the Poisson process and the truncated Poisson process. In this case, the maximum observed number of arrivals per slot was 5. The three effective bandwidth functions are very close for small values of. However, as increases, the effective bandwidth of the Poisson process tends 4 Recall that time is measured in slots here. A slot is defined to be 0:27 ms. ()

11 Measured Effective Bandwidths EB 25 EB Truncated Poisson Estimated EB EB Poisson Theta Figure 3: Comparison of the measured EB against the analytical EB for both the Poisson and the truncated Poisson processes. to infinity, whereas its truncated counterpart tends to the maximum value. The graph shows that the truncated Poisson process is a more accurate model for finite time trace than a pure Poisson process. This experiment leads us to conclude that the finiteness of arrival streams is an important characteristic which has to be considered for practical applications of effective bandwidths Dependency on the trace length Figure (4) shows the estimated effective bandwidths for the block estimator and the KLD estimator for the trace lengths N = 024 slots and N = 65; 536 slots respectively. In both cases, the two measured curves are remarkably similar. In particular, the measured effective bandwidth seems to be consistent for a significant portion of the range. This holds for both the block and the KLD estimators. Increasing the trace length only seems to affect the measurements for larger values of. The implications of this results are twofold. First, we conclude that a reasonable estimate of a traces effective bandwidth can be obtained by a relatively small trace. A trace length of N = 024 slots corresponds in our case to 277 ms. Obtaining reasonably accurate estimates for the effective bandwidth is important when applied to CAC. Secondly, this result supports our argument that the behavior of the effective bandwidth 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 effective bandwidth. However, those observations which are exceeding the current maximum are responsible for increasing the tail of the estimated effective bandwidth curve for large values of. The measurements for intermediary trace lengths of N = 4096 slots and N = 6;384 slots confirm these conclusions. They are not shown here to simplify the presentation of our findings. 3.2 On-Off fluid traffic stream We now present the results for measured effective bandwidths for the On-Off fluid. Again, we start by analyzing the direct and the block estimator. As before, we collapse the time dimension of these two methods to incorporate the results into a comparison of all four estimators against the analytical values. Finally, we present results of confidence intervals.

12 Measured Effective Bandwidths N=024 N= N=024 N=65536 e Figure 4: Comparison of the measured effective bandwidth for different trace lengths for the Poisson trace using the block and the KLD estimator. Figure (5) illustrates the estimated surfaces of the direct and block estimators for the On- Off fluid trace. It also shows the analytical effective bandwidth surface for an On-Off fluid process. The -values in the plot are varied between 0:0000 and 5 with a multiplicative step of :5. The time values range between and, again with a step of. Note that the -axis is thus shown in a log scale. As in the case for Poisson traffic, the estimated surfaces from the direct and the block estimators are remarkably similar, showing 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 analytical effective bandwidth 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 again for large values of. For the direct estimator, the measurements start to degenerate with t 5. For the block estimator, the measured effective bandwidth degenerates with t 4. The analytical effective bandwidth on the other hand does not exhibit such a behavior. These inaccuracies are further illustrated in Figure (6) showing the squared error between the analytical effective bandwidth and the measured effective bandwidth. Our explanation offered in the last section also seems to apply for On-Off traces. As the time parameter increases, the observed maximum value of A(0; t), scaled by t, decreases. Since the measured effective bandwidth tends to this value as increases, the estimates degenerate in this parameter region. Analytical EB Estimated EB (Direct) Estimated EB (Block) 50 AEB40 t EB 40 EB log 0 t 0-3 log 0 t 0-3 log 0 Figure 5: Analytical and measured effective bandwidths for the On-Off fluid trace using the direct and the block estimators To compare the measured effective bandwidth with the KLD- and LR estimators, we again collapse the time dimension using equation (25). In this case, the choice of a representative time scale is less obvious. Without any prior assumption about the application of the effective bandwidth, we resort to the following heuristic: we restrict ourselves to the range of values for

13 Measured Effective Bandwidths 3 (AEB-EEB)^2 (Direct) (AEB-EEB)^2 (Block) 50 EBDiff^ t log EBDiff^ t log 0 Figure 6: Squared error of the effective bandwidth measurements for the On-Off fluid trace using the direct and the block estimators Analytical EB Direct Estim Block Estim LR Estim KLD Estim e Figure 7: Analytical and measured effective bandwidths for the On-Off fluid trace t where the measurements do not degenerate, giving us t 5. We then examine the squared error plots shown in Figure (6) and select the largest value of t for which the squared error is small. We thus arrive at a value of t = 5 for our comparison. The results are shown in Figure (7). The -dimension is again shown in a log scale. In this case, the measured effective bandwidth seems to be sufficiently close to the analytical effective bandwidth for all four methods. Note how our argument explaining the discrepancy between the analytical and measured effective bandwidths for the Poisson trace is less apparent here. For a fixed time interval t, the arrivals are limited by the peak rate of the source. For On-Off fluid sources, this peak rate is clearly finite. Consequently, the analytical effective bandwidth varies between the mean and the finite peak rate. The consistency of the measured effective bandwidth is confirmed by Figure (8), showing 95% CIs for all four methods. Again, we separate the plots for the direct / block estimators and the KLD / LR estimators, with the axes on the latter plot reversed. As before, we computed the 95% CIs over estimations, each with a length of N = 25;000 cycles 5 per estimation. The tight error bars in figure (8) indicate high consistency for the measured effective bandwidth. For the direct / block estimators, the CIs include the analytical values only for small -values. For > 0:05, the analytical effective bandwidth once again falls outside the 95% CI. The KLD /LR estimators on the other hand seem to achieve a better accuracy for this traffic type. But 5 Recall that we define a cycle as the duration of an on-phase plus and off-phase.

14 Measured Effective Bandwidths 4 Estimated Effective Bandwdith Analytical EB Block Estimator Direct Estimator e-05 Analytical EB KLD Estimator LR Estimator e Figure 8: Analytical and measured effective bandwidths for the On-Off fluid trace with 95% CIs even in this case, a closer examination of the CIs reveals that for parts of the -range, the analytical effective bandwidth falls outside the 95% CI (in this case, for < 0:05). Consistent with our previous findings, the measured effective bandwidths are below the analytical EB as increases. Both functions tend to the peak value of the source. In comparison with each other, we find again that the direct / block and the KLD / LR estimators perform similar. 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. The direct estimator runs between 0:6 seconds and 0:38 seconds for small and large values of respectively. The corresponding values for the block estimator are 0:032 seconds and 0:079 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. The virtual buffer estimators for fluid sources are more complicated to implement. Much of the time is spent on arranging the data structures to compute workload areas and on the subsequent integration. We thus do not place too much significance on the actual value. However, we do note that the virtual buffer estimators, although being more accurate, take significantly longer to obtain a single estimation point. This is a considerable disadvantage of these methods. For On-Off fluid traces,the trade-off between run-times and accuracy leads us again to prefer the block estimator Dependency on the trace length The results of comparing trace lengths N = 500 cycles and N = 32;768 cycles are shown in Figure (9), with the block estimation shown on the left and the KLD estimations shown on the right. For the block estimation, an increase in the trace length has again the effect of pushing the tail of the estimated effectve bandwidth curve up. However, even with a small trace length the estimator performs surprisingly well. In the case of the KLD estimator increasing the trace length has a different effect in this case. Rather than affecting the tail of the effective bandwidth function, its shape improves for smaller values of. Note that even for small traces, the bursty nature of On-Off traffic allows us to observe the peak value of the traffic stream. We attribute the accuracy of the KLD estimators for high -values to this fact. Considering the short length of the trace N = 500 cycles, we conclude that the KLD estimator is acceptable. Again we confirmed these results with traces of length N = 48 and N = 892 cycles

15 Measured Effective Bandwidths cycles 52 cycles cycles 500 cycles e Figure 9: Comparison of the measured effective bandwidth for different trace lengths for the On-Off trace using the block and the KLD estimator. Effective Bandwidth x log log 0 t 0 x log log 0 t 0 Figure 0: Theoretical EB of an FGN process respectively, but we supress these here to simplify our presentation. 3.3 FGN traffic stream In this section we consider FGN streams, characterized by a Hurst parameter H > 0:5. Therefore the block estimator cannot be employed, since the blocks would not be independent of each other. Also the KL estimator, the way it is formaulated in section 2, is not directly applicable. For this reason, in this section we restrict ourselves to the direct estimator (in section 5 we will also consider the LR estimator for the FGN trace). In addition to the comparison between the analytical and the measured results, we examine the sensitivity of the measurement results to the traffic generator and the behavior for large values of. Figures (0) shows the analytical effective bandwidth and the measured effective bandwidth of the FGN trace described in section 2.6. The -values vary between :0000 and 0 /cells with a multiplicative factor of :5. The range for the t-values is instead between :27 ms and 0 seconds, with multiplicative factor of 2. The figures clearly show that for large values of, the measured effective bandwidth and the analytical effective bandwidth are once again significantly different. In this case, they differ by many orders of magnitude. For large values, the analytical effective bandwidth increases in time. On the other hand, the measured effective bandwidth decreases as t increases. This latter observation is consistent with

16 Measured Effective Bandwidths 6 Squared Error log log 0 t Figure : Squared Error e-05 RMD generator fft generator e-05 RMD generator fft generator Figure 2: Comparison RMDfft generators for t = andt = 32 slots, N = 3;072. our findings from the On-Off and Poisson sources where the measured effective bandwidth also degenerates in time. For small values of, the plots in (0) seem to indicate reasonable accuracy of the estimator. However, this is not at all the case, as illustrated in the squared error plot shown in Figure (). Note that the plot cover a very narrow range for. Even for small values of and t, the discrepancies between the analytical and the measured effective bandwidths are significant. Recall that the average traffic rate is 4;760:5 cells/sec. To demonstrate that this finding does not depend on the underlying traffic generator, we generated a second trace with the same parameter values using the RMD method [6]. Since the FGN generator is only asymptotically unbiased, we hope to demonstrate in this way that the observed behavior is independent of the selected generation scheme. We compared the estimated effective bandwidth for both traces (FGN and RMD) with length N = 3;072 slots. To illustrate the differences, we restrict ourselves to two time values only: t = slotandt 2 = 32 slots. The results are illustrated in Figure (2). For t, the estimated effective bandwidths are very close. For t 2, the measurements differ slightly as increases but are never the less still sufficiently close to one another. We therefore conclude that the differences between the measured and the analytical effective bandwidth are independent of the underlying traffic generator.

17 Measured Effective Bandwidths Asymptotic behavior in In this section we provide evidence for our hypothesis that the estimated effective bandwidth tends to the measured maximum aggregated value A max over the time interval (0;t) as grows. The proof is given in section 4. In this experiment, we fix = 9:707, its largest value considered here. Tables, 2, 3 show the estimated effective bandwidth at this value, as well as the observed maximum A max for different values of time and trace lengths of 52, 3,072 and,048,576 slots respectively. In table we also report the value of the analytical effective bandwidth at = 9:707. As shown, the measured A max depends both on the trace length and on t. The observed maximum aggregated value A max decreases as the time parameter increases. The asymptotic value of the estimated effective bandwidth gets smaller and smaller as time increases. This behavior is in sharp contrast with the theoretical effective bandwidth for FGN processes, which grows extremely fast with time. The longer the trace length, the bigger A max. However, from tables, 2, 3 we can also infer that increasing the trace length even by several orders of magnitude does not affect A max significantly. time(ms) â( max ;t) A max Theor. EB Table : Behavior of an FGN trace for large values of,n = 52. time(ms) â( max ;t) A max Table 2: Behavior of an FGN trace for large values of, N = 3; Dependency on the trace length The scarce sensitivity of the measured effective bandwidth to the trace length is further witnessed by the following experiments, where we tested traces of length N = 52 slots and N = ;048;576 slots. Figure (3) illustrates the estimated EB for fixed times t = slotand t 2 = 6 slots. For small time scales, the performance of the estimators is almost independant of the trace length. The estimated effective bandwidth are very close for significant sections

18 Measured Effective Bandwidths 8 time(ms) â( max ;t) A max Table 3: Behavior of an FGN trace for large values of, N = ;048; e-05 N=,048,576 N= e-05 N=,048,576 N= Figure 3: Estimates taken over traces,048,576 and 52 samples long, at t = andatt = 6 slots respectively. of the range. As we increase the time scale, the difference in the tail of the estimated effective bandwidths also increases. This suggests that additional observations are more valuable for larger time scales and large values. This phenomenon can be explained by considering that the number of samples over which the estimate is taken decreases as t grows. Therefore also the probability of observing large aggregated values gets smaller. 4 Properties of Measured Effective Bandwidths In this section we establish some properties of measured effective bandwidths for a given traffic trace X t, t = :::N, with sample mean m = N N i= X i and sample variance s 2 = N, N i= (X i, m) 2. We contrast the results obtained for measured EBs with the results for the analytical effective bandwidth. In particular, we show that the behavior of the measured EB as! differs significantly from analytical effective bandwidths. Throughout this section, we assume that (8) has been used as estimator for E b N e X(0;t) and hence that the block sums are independent. Following the notation in [0] we denote by X i := it j=(i,)t+ X j;i = :::b N t c the number of cell arrivals over an interval of length t. Wealsolet M t = b N t c. The previous section has highlighted how the estimate depends on the number of samples, especially for large values of and large time scales. For this reason we explicitely indicate N

19 Measured Effective Bandwidths 9 in the expression of the estimate, i.e. ˆα(; t; N). Property : For a fixed value of t <, lim M t ˆα(;t;N)!0 = tm t X i = m (2) i= Proof: Since the limit as! 0 for the numerator and the denominator equals 0, l hôpital s rule holds and so we have lim!0 t log M t M t i= e X i = lim!0 d d log M t M t i= e X i lim!0 d d t = t lim!0 M t M t i= X i e X i M t M t i= e X i M t = tm t X i = m: i= A similar argument for the analytical effective bandwidth shows that it tends to the expected value of the arrival process. Note that the sample mean is an unbiased estimator for the population mean. The analytical effective bandwidth and the measured effective bandwidth thus exhibit a similar behavior as! 0. Property 2: For a fixed value of t <, lim ˆα(;t;N) =! t max[ X i ] (22) Proof: Since the limit as! for the numerator and the denominator equals 0, l hôpital s rule holds again and so we have lim! t log M t M t i= In this case the numerator can be written as e X i = lim! d d log M t M t i= e X i lim! d d t d d log M t M t i= e X i d d log M t = d d logemax[ X i ] = max[ X i ] M t e max[ X i ] i= The result follows. In this case, the result for the analytical effective bandwidth is significantly different. A similar derivation shows that

20 Measured Effective Bandwidths where X(0;t) is defined by Kelly [6] as lim! α() = t X(0;t) (23) X (0;t) =supfx : PfX (0;t) > xgg (24) The analytical effective bandwidth thus tends to the essential supremum of the process, which may be infinite. However, for a particular finite time realization of the process the probability of observing the theoretical supremum becomes less likely as t increases. For example, for the underlying process the probability of observing the peak rate over the entire interval (0;t) can be positive, even as t grows large. For a finite time realization of the process such observations are less likely, in particular as t grows large. As a result the asymptotic behavior of the measured EB is determined by the observed maximum value of X i ;i = :::M t as!. Note how this property explains the discrepancy between the analytical effective bandwidths and their measured counterparts observed in the last section. The differences between the analytical effective bandwidth and the measured effective bandwidth in the space may be summarized as follows: the analytical effective bandwidth varies between the mean µ and the supremum of X (0;t)=t for any fixed t. On the other hand, the measured effective bandwidth varies between the sample mean m and the measured maximum max[ X i ]. Property 2 explains the behavior pointed out from the estimation in section 3. In particular it is apparent the reason of the observed degeneration with time of the estimated EB as compared to its analytical counterpart. In fact as the aggregation time interval enlarges the probability of oserving the supremum of X (0; t)=t over a finite trace decreases. Therefore the difference between analytical and measured EBs increases with time. 5 Simulation Results To show the applicability of measured EBs in a practical environment we now report some simulation results. We fix a required CLP for a given buffer size and we determine the amount of bandwidth to be allocated using both measured and analytical EBs. As in the previous sections, we consider finite time realizations of On-Off fluids, Poisson and FGN arrival processes. Note that we also assume that the entire finite time realization is known, as is the case for pre-recorded video, for example. We present our results by first describing our experimental methodology, followed by a separate presentation of the results for the Poisson, the On-Off and the FGN traces. 5. Simulation Methodology To show the applicability of measured EBs we proceed as follows: we generate a test trace of length N. We considered the block estimator for On-Off and Poisson traffic and the LR estimator for the FGN trace. We subsequently apply the measured EB to our CAC problem. We assume that the given arrival stream wishes to gain access to a single server queue with various buffer sizes B and a specified quality of service requirement δ, expressed as a cell loss probability. We use equation () to evaluate the required -value, called. Note that refers