A DYNAMIC TIME SCALE APPROACH FOR ON-LINE MEASUREMENT-BASED CAPACITY ALLOCATION

Cunent high-speed networks handle a variety of services, requiring different QoS constraints. The choice of appropriately accurate but also practically implementab/e measurement algmithms in this context becomes crucial. In this paper, we perform a comparative study of alternative on-line bandwidth allocation algorithms, we analyze their complexity, and petf01m comparisons via simulation experiments. Moreover, we argue the relevance of a dynamic measurement time scale approach and demonstrate the effectiveness of this new approach over the static one. Our motivation is to use these algoxithms in the data plane of "'self-sizing" frameworks, and make use of their output in taking control plane decisions either locally or globally, in an on-line fashion. Previously, no such comprehensive comparison of relevant methods has been carried out, especially from a combined accuracy versus implementation complexity point of view and from the perspective of changing the measurement time scale "dynamically".

tl1e traffic demand.In this regard, the bandwidth allocation in high-speed QoS-oriented networks is critical and needs to be made dynamic, adaptive and measurement -based, rather than static, to attain a more efficient use of resources.Especially for network links shared through statistical multiplexing.adaptive bandwidth allocation algmithms based on traffic measurements can achieve important gains.
For these reasons, we perf01n1ed a comparison of online, dynamic measurement-based bandwidth allocation algorithms, which have low time complexity, and are based only on measurements instead of unreasonable assumptions about the incoming traffic.To the best of our knowledge, no such comprehensive comparison of relevant methods has been previously carried out, especially from a combined accuracy versus implementation complexity point of view.
We are also motivated by the fact that such algorithms can be used in the data plane of .. self-sizing" network frameworks such as [I, 2] and in which every node in the network mns a measurement-based bandwidth allocation algoritlun for every traffic class or "band".Periodically, the output of the algorithms, which give the required capacity demands of the traffic types, are collected, and either locally or globally, a control plane action is taken (i.e., the virtual links or scheduling allocations are re-calculated) so as to minimize a predefined objective such as bandwidth cost or maximize revenue.
In this context, it is very important to choose measurement methods that satisfy stringent constraints in terms of both accuracy and complexity.Most of the algorithms we used in this paper naturally originated from the effective bandwidth concept, since effective bandwidth is the amount of the required bandwidth to be allocated for the satisfaction of a QoS constraint.Furthermore, in the literature, we identified two research areas which are related to our aim.These are network traffic prediction [3,4,5,6] and measurement-based admission control (MBAC) [7,8,9].MBAC algorithms are composed of separate measurement procedure and admission criterion.We investigated the applicability of these separable measurement procedures for our purposes.However, other than the trivial Gaussian Approximation bandwidth estimator, such methods are not suitable for on-line re-sizing due to the fact that either they rely on unreasonable information (i.e., they presume that a priori traffic descriptors, such as present number of connections are known) or they have unacceptable computational complexity.
These incompatibilities stem from the different design considerations between measurement -based estimators in MBAC and se/f-sbng frameworks.First, MBACs are designed to operate only in the ingress nodes, where admission decisions are taken.Second, the period of execution of MBAC algorithms is at the connection level time scales.Our aim is to Fatih HaCJomeroglu and Michael Devetsikiotis A Dynamic Time Scale Approach for On-line Measurement-based Capacity Allocation obtain on-line algmithms, working on evety node in the network.Therefore their timescale and computational complexity are smaller than connection level timescales.Similar to MBAC algmitlm1s, traffic predictors do not suit our consideration either.The reason this time is not because they are centralized and computationally complex as in MBAC algotithms, but that they do not target a QoS constraint.
The algorithms in this paper take a window of traffic measurements as input, estimate the parameters they need in a bandwidth allocation calculation fommla and output the required amount of capacity to be reallocated.The measurements conespond to the amount of incoming traffic during a slot duration, istot• Consequently.the algmithm is called periodically every N * I slot seconds (the reallocation period), where JV is the window size.The choices of islot and JV are critical [ 1 0] and affect significantly the pelformance of algo-Iithms.The on-line measurement-based resource allocation algorithms are implemented and tested in a simulation scenario as shown in Figure 1.In our simulations, the performance metric is related to a QoS constraint, in our case packet loss probability.To quantify the amount of expanded resources, we use two cost meuics, namely, • Allocation Ratio (Average Capacity Allocation divided by Average Traffic Rate) • Average Queue Occupancy Our purpose is to obtain a feasible algorithm which is able to use bandwidth optimally (i.e., dynamically as in Figure 2) while still obtaining a performance close to the QoS target.The ideal algmithm should not require any know ledge or umeasonable assumptions on the traffic, and be based completely on measurements.
The remainder of the paper is structured as follows.After desctibing the algmithms in Section 2, we compare their response to different scenarios (i.e., vmiable buffer size, level of aggregation of sources, long-range dependence of traffic, computational and memory complexity requirements) and choose promising ones for further simulation study in Section 3. Then we explain our simulation methodology in Section 4. Later, simulation results highlighting the importance of measurement time scales in algorithm performance are presented in section 5. Section 6 introduces the use of a measurement-based time scale and shows its effectiveness over the static time scale approach.Finally, we conclude with 210 a discussion of the results and outline prospects for further enhancements.eb(s, t) = --'--'---:.:.. st

BANDWIDTH ESTIMATION ALGO-RITHMS
X (0, t) is the amount of incoming work dming a duration oft.The (s, t) pm•ameters are usually called the space and time parameters.Parameter s is calculated by using Large Deviations Theory <LDT) and by making a large buffer assumption.The overflow probability is calculated from an asymptotically exponential decrease assumption.
The time parameter t is related to time scales which are responsible for buffer overflow.It should be chosen small enough so that traffic is observed for buffer overflow analysis.Finally, the expectation in (I) is approximated by a time average, as suggested in [12].The empirical evaluation of (I) is simulated and compared with analytical effective bandwidth of known Poisson and ON-OFF source types in [13].

COURCOUBETIS EB ALLOCATION (CEB)
This method [ 14] is based on LDT and a large buffer assumption, similar to the DEB algorithm: The parameters m, B, s and I D are the mean rate, buffer size, space pmameter and index of dispersion of X[O, t].The space parameters is calculated using (2).
ID estimation process has a computational complexity proportional to space parameters in ( 1 ).An assumption of many sources is made, instead of a Large Buffer assumption, while using LDT to solve the problem of estimation of the space and time parameters (s, t).As described in [15], if M sources are multiplexed in the buffer B. rj is the percentage of streams of type j, and maximum allowed buffer overflow probability to be guaranteed is e-a, then minimum required bandwidth can be calculated by solving ' ' where In (6), the eb(s, t) term is found from (1).For a given t, the R( s, t) is a unimodal function of s, having a unique minimizer.Then, R( s, t) = R 1 ( s) is solved by using a golden section search method as described in [15].Tills process is repeated for a range of t values smaller than the measurement window time, and the maximum among them is taken as the required capacity to be allocated.

ON-OFF EB ALLOCATION (OOEB)
The idea is to obtain estimation values of an equivalent ON-OFF traffic model from measurements, and substitute them in the specific analytical effective bandwidth formula (7) for ON-OFF sources [16].
-sr +a+ b-1/2 ( -sr +ab)"-2 ba eb(s, t) = ?(7) ~s Parameters a, b and r denote a ON-OFF traffic model where ON and OFF periods m•e exponentially distributed with parameters a and b respectively, and r is the constant traffic generation rate in the ON state.These pm•ameters can be estimated by matching the first three moments of N data measurements falling into the window.These estimations have difficulties and require search algorithms, since direct solution of high order equations is not trivial.Moreover, this method is weak because of the limited fitting spectrum of ON-OFF model [17].

NORROS EB ALLOCATION (NEB)
In [18], besides introducing modeling of real traffic by fractional Brownian motion (FBM), an effective bandwidth formula (8) for FBM is also given. (8 )   where K(H) = HH (1 -H) 1 -H and m, H, Ploss, x and a are the mean, Hurst parameter, buffer overflow probability, buffer size and coefficient of variation respectively.Parameter a, is approximated by the index of dispersion.In fact, this is a valid assumption only when the traffic is short range dependent.
The Hurst pm•ameter can be set from a priori measurements.However. to react to unexpected traffic changes.a measurement-based on-line algorithm is favored.Difficulties of H estimation methods are analyzed in [I 9], where the comparison of several H estimation algorithms indicated that the Abry-Veitch estimator (AV estimator) based on Wavelet theory is a very effective approach [20].

DRDMW (IMPROVED EMPIRICAL EB AL-LOCATION)
This method in [21] is an improved version of empirical effective bandwidth methods.A unified phenomenological framework to estimate overflow probability of both long range dependence (LRD) and short-range dependence (SRD) is put fmward by including the Hurst parameter in traditional analytical effective bandwidth methods.
Second, the difficulty of measming the effective bandwidth of real-time traffic online by using direct estimator [21] is alleviated by using an approach based on dual recursive algorithm with double moving windows (DRDMW).which is introduced in an empirical calculation of analytical effective bandwidth formula instead of using the direct estimator.

GAUSSIAN APPROXIMATION ALLOCA-TION (GA)
The simplest resource allocation method existing in literature is the GA method [22], where liak buffer is ignored and server capacity is set according to Gaussian arrival rate distribution: where m and (J' are the mean and standard deviation of the mrival rate distdbution.

COMPARISON OF THE ALGORITHMS
The first algorithm, namely Direct Effective Bandwidth Allocation algorithm, relies on the effective bandwidth formula, and possesses the problem of finding appropriate values for s and t, which depend on QoS requirements and the system parameters.The space parameter is estimated using the Large Buffer Assumption.The time parameter estimation is left somewhat arbitrary, for the time being.
The second algorithm uses (3).which is an altemative generic effective bandwidth definition in terms of the mean rate, index of dispersion, QoS parameter and buffer size.It is simpler, but it still doesn't address long range dependent traffic.
The Many Sources Asymptotic Effective Bandwidth algorithm relies on the effective bandwidth formula (I) and encounters the problem of estimation of (s, t).This method accomplishes it by solving a functional optimization problem.Although it is a very innovative approach, this may be A Dynamic Time Scale Approach for On-line Measurement-based Capacity Allocation too slow for our motivational self-sizing scenruio where every node takes on-line measurements of every traffic type.
The ON-OFF Effective Bandwidth formula (7) for the fom1h method is obtained by substituting an ON-OFF arrival process instead of X(O, t) in the analytical effective formula.
With regard to a practical usage of such expressions, we encountered other problems than estimation of (s, t) parameters, such as model parameter estimation, and goodness of fit of the model.
The Nanos Effective Bandwidth Allocation and Gaussian Approximation methods are alte1natives not including ( s, t) estimation.They are approximate expressions, which are derived independently of the effective bandwidth formula.The Gaussian Approximation algorithm assumes a bufferless link.This will overestimate required capacity.Moreover.the gaussian assumption is not valid for traffic formed by small number of sources.This places a constraint on the source type, however our aim is to have an algorithm capable of functioning without unreasonable assumptions.
The self-similarity is addressed only in the NEB and in the DRDMW method.Others do not discriminate between sh011 range dependence and long range dependence.Although the index of dispersion in the Courcoubetis fommla of the second algorithm stands for burstiness of the source, the fotmula is not for long range dependent traffic.Thus the effective bandwidth approximation on which the Courcoubetis fommla is based, (i.e., exponential decay of buffer overflow probability with increasing buffer size) is not valid for self-similar traffic (the decay is hyperbolic and slower than exponential).We provide a summary of our performance comparisons with respect to various network scenarios in Table 1.
The Gaussian Approximation algorithm is the easiest to implement, and suitable to be used as an algorithm setting an upper bound, since it does not consider buffer size.The Courcoubetis Effective Bandwidth Allocation is also another easy, and promising one, since this one takes into account buffer also.But neither of the previous two algorithms is designed with long range dependent traffic in mind.Norros effective bandwidth and DRDWM algorithms are the only ones incorporating the Hurst parameter, therefore addressing to long range dependent traffic.Although DRDMW is designed to alleviate the numerical overflows in the direct effective bandwidth allocation, that problem can not be completely alleviated due to the suucture of (1).Therefore, we picked the following three algmithms for further simulation analysis: • Gaussian Approximation (GA) • Courcoubetis Effective Bandwidth Allocation (CEB) • Norros Effective Bandwidth Allocation (NEB)

SIMULATION METHODOLOGY
The simulation scenario given in Figure 1 is a single setver queue simulation where the service rate is changed, in an on-line fashion, periodically based on recent traffic measurements.We used the Sup-FRP traffic model [23].The simulation flow slides packet by packet, emulating a real case scenario as in an Ethernet card passing packets to upper network layers.

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Figure 2 gives a visual representation of how algorithms adjust setvice rates, tracking fluctuations in the incoming traffic rates, so as not to waste resources.We performed simulations with 5 different tslot values (0.01, 0.05, 0.1, 0.5, 1 s) and 5 window sizes (N) values (3, 6, 30, 60, 300 slots) in every method.Therefore, we had 25 simulations per method.As a total, we present here results of 75 simulations.Also note that the measured statistics in this paper resulted after 30 simulation replications and confidence intervals are insignificant.
In all of the simulations in this section, we generated traffic with the same mean value of 20 Kbytes/s, the same Hurst parameter of 0.7 and the same buffer size of 5 Kbytes.We set the QoS target packet loss probability to 10-3 , so as to have a common ground for the performance comparisons of algorithms in the simulation scenario (Figure 1 ).
Note that the average traffic rate of 20 Kbytes/s is the product of an average packet size of 200 bytes and average packet arrival rate of 100 packets/s.Therefore, the average time between two consecutive packets is 0.01 seconds and the tslot values chosen in the simulations, which are (0.01, 0.05, 0.1, 0.5, I s), correspond to cases where 1, 5, 10, 50 and 100 packet rurivals take place on average in a slot time duration, respectively.Also note that the choices for tslot and 1V are made deliberately to have simulations where reallocation takes place in every 3 seconds, but with different measurement resolution in the recent history of the measurement data.For example, a simulation with (0.01s, 300 slots) includes 300 measurements, whereas the one with (Is, 3 slots) includes three measurements in the same recent 3 seconds history.
We did not implement an on-line H estimation [24].We provided the value of H (i.e., 0.7) to the algorithms beforehand, so that we can examine the perfonnance of the bandwidth allocator, independent from the performance of the H estimator.The combined, on-line Hurst parameter and EB estimation is beyond the scope of tllis paper and is left as future work.

Perfect
Perfect Requirement Table 1.Performance Comparisons

SIMULATION RESULTS
In this section, we first present performance and cost plots.We demonstrate and observe the impmtance of time scale choice in measurement-based algorithms.At the end of this section, we provide the processing time plots with respect to the window sizes.

PERFORMANCE AND COST RESULTS
Tables 2, 3 and 4 show respectively the performance and cost results of the GA, CEB and NEB algorithms.We observe how the algodthms response against different tslot and JV values in the simulation scenario given in Figure 1.
In Table 2, we observe that the QoS target, which is 10-3 , is satisfied in every ( ts/ot, N) combination with the exception of (tslot = 1 s, N = 3 slots).Note that GA is used as an upper band of resource allocation for comparison purposes.It does not consider buffer size.In fact, it assumes there is no buffer.This is why, it is expected to be more generous than other algmithms.The fact that a violation of QoS took place in this method implies trouble for other methods.The average allocation ratio and the average queue occupancy values agree with Pzoss values and show that as islot is increased for a constant 1V, the allocation ratio decreases towards 1.We also observe that for constant tslot• increasing f.l results in a larger capacity allocation.But tills rate of increase in capacity allocation depends on the tslot value.So, once tslot is pmperly chosen, choosing N looses its impm1ance, since the change in the ratio values are much smaller.
Table 3 tells us that the CEB's performance changes similar to GA against tslot and JV variations, but Ptoss values are relatively about an order of magnitude higher.The QoS target is violated for the following (tslot, N) pairs: (0.5 s, 3 slots), (I s, 3 slots), (0.5 s, 6 slots) and (I s, 6 slots).This algorithm, considering the presence of buffer, theoretically pennits lesser resource usage than GA.Compared to GA. we observe that biggertslot values lead to better allocation ratios (i.e., ratios closer to I) and the choice of N has a greater effect on CEB.With proper choice oft slot and JV, the same per-formance can be achieved with lesser resource usage.Here, similar toGA, we see the impm1ance of choosing tslot properly.But unlike GA, here choosing N is also important.T11is is because the rate of increase of ratio values when JV is increased is much more significant.Choosing a large JV leads to serious over-allocation.
From Table 4, we observe that the loss probability results of NEB are between the ones of GA and CEB for N values of 3 and 6.However, when N is either 30, 60 or 300, Pzoss is smaller than other algorithms, which implies an overallocation of bandwidth, which is justified by the values of the allocation ratio metric.When it comes to the choice of istot.as in the previous methods, a bigger tstot resulted in smaller resource allocation.This method is the only one which allocates more capacity to the traffic possessing higher longrange dependence.Overall, it can be said that the NEB includes similar performance and cost changes as the ones of CEB, but performance values are around one order of magnitude better, and consequently, cost values are higher.This shows that the NEB has a tendency of allocating more resources than the CEB, and results in better QoS constraint satisfaction.

COMPLEXITY
Figure 3 shows the processing times of the algorithms as a function of window size values.The processing time is the time required for the algorithm to re-calculate capacity.The processing time is seen to be related to JI.,T for CEB and NEB, with a complexity of O(JV 2 ).This result is in parallel with our expectations.CEB and NEB calculate autocorrelations of measurements falling into the measurement window of size JV, and this requires a processing time propmtional to 1V 2 .Whereas, GA uses only the mean and variance of the measurements, whose calculations are fully on-line.As a result, GA has a complexity of 0(1) and regardless of N, its execution time remains much smaller compared to other algorithms.

Fatih Hamomeroijlu and Michael Devetsikiotis
A Dy;_  can be explained intuitively from the stmcture of the formulas used in the capacity allocation algorithms.To illustrate, consider the Gaussian Approximation Algorithm's fotmula (10), where m and u are the mean and standard deviation of the most recent JV measurements, [.X 1 , X 2 , ... ,~YN]-Each . .Xi represents incoming traffic load in consecutive tslot durations.By the law of large numbers, as tslot increases, say tstot --+ A, where A is a time parameter, which is large (dependent on the traffic charactetistic), then xi approaches m * A for all i.This causes u in (I 0) to go to zero, and the capacity value, eb to approach to the mean rate, nt.A similar reasoning can be given for other methods, in which not only standard deviation, but also autoconelations of measurements, [.X 1 , .X 2 , ••• , XN] are used.
On the other hand, we also observe that taking tslot arbitrarily small leads to over-allocation of resources.As tslot decreases, the measurement history (tslot * 1V) decreases too.This decreases the confidence and increases the randomness in the formula parameter estimations, yielding overallocations.
We could obtain empirical istot values from the performance and cost metrics plots, so that the QoS is satisfied with minimum resource allocation.However this particular istot value would be useful only for the traffic that we used in our simulations.Consider using a traffic whose mean is m * J( (that is J{ times bigger).This time, on average Atimes more traffic load will fall on average into the slots.Relatively, tills is the same experiment as using ]( times bigger tslot measurement slots, with mean traffic rate m.In other words, the measurement time scale is relative to the traffic characte1istics.A static tslot may conespond to cases where we described previously as small or large, depending on the incoming traffic.
As a result, we believe that the measurement time scale istot should also change dynantically based on measurements [XJ,X2, ... ,XN], in order to keep the algorithms always working close to their best.
In [25], the Maximum Time-Scale (MaxTS = t') is used as the time scale of interest for queueing systems feed by a fractal Brownian motion (fBm) process: where k = J -2 *In (Floss), m is the mean traffic rate, u is the standard deviation of the traffic rate and Cis the capacity of the server.The value of C is de1ived from (12), where A.H(t) is the probabilistic envelope process of the fBm cumulative ar-~val process AH(t) (AH(O) = 0), such that P(AH(I) >
-d-,-converges to t 1e mean an1va rate. .H mcreases with a decreasing rate after t*.Tills means that the probability that the average arrival rate exceeds the link capacity decreases for t > t'.
In the remaining of the paper, we test using a dynamic time scale by estimating t* using the recent 1V measurements and taking tstot = t* as the measurement slot duration for the next 1V measurements.In other words, besides effective capacity, tslot is also recalculated after every 1V measurements.
Instead of the ( Cm) te1m in (II), we used L * m, where Lis taken as a constant L = (A.llocationRatio)-1.The reason is that we allocate capacity dynamically and do not have a constant C 1 .Table 5 shows the improvement of using a dynamic tslot = t* against static tstot choices (GA is used as the capacity allocation algorithm, and the ?toss target is set to 10-3 as in the previous simulations).As the mean rate increases, the performance of the static tslot cases changes (the ratio decreases and Ptoss increases), whereas the performance of the dynamic tstot = t* case remains the same.This shows that on-line measurement-based algorithms with constant measurement intervals are heavily dependent on the incoming traffic's mean rate, whereas the ones with dynamic measurement intervals are more robust 2 .To illustrate the benefits visually, we generated a traffic trace of 1000 s, where the mean rate of traffic between 200 1 We tried using the avernge capacity allocation instead of C. but this caused a multiplicative effect, such that.when capacity allocation increases, C -m term in (11) decreases.But decreasing tslot results in increased capacity allocation measurement in the next window.and this loop ends up having t* ::::::: 0.
2 The particular performnnce figures for dynamic tslot case in Table 5 are dependent on the value of L (we used L = 1.5).However, note that the choice of L does not affect the robusmess of the algorithm.
and hLi.-, ncan rate at the remaining intervals.Figure 4 i>i.H._;< ,,,,--namic capacity allocations.Note that the allocation ratios m the static istot cases change in the region of traffic with high mean rate.But the allocation ratio remains roughly the same in the dynamic time scale case.Tills is achieved by adjusting islot as shown in Figure 6.
The number of packets dropped increases when the mean is increased gradually in Figure 5 (fort slot = 0.01 s, no loss oc-cmTed, due to over-allocation).The t"' case peiforms again in between the static islot cases.But note that when islot = t*, the algmithm can self-adjust and perfom1 similarly against traffic mean changes.whereas the petfonnance of an algorithm with static tslot is dependent on the traffic.To illustrate, a method with static tslot = 0.01 s case will over-allocate significantly when the mean rate decreases much below of 4 Kbytes/s, and a method with static tstot = 2 s case will suffer significant degradation of the QoS tm•get when the mean rate increases much above I 00 Kbytes/s.

SUMMARY AND CONCLUSIONS
In this paper, we presented and compared measurementbased on-line capacity allocation algorithms and proposed a way to improve their robustness.
We distinguished such algmithms fi•om MBAC and traffic predictors due to their smaller time scales and QoS-oriented use.We observed that their performance is directly dependent on the involved measurement time scales.Mainly, we saw that when the time slot length is increased while the window size is kept constant, due to increasing aggregation of packets in the slot interval, the variations between the measurements in the measurement window decrease and the allocated capacity approaches the mean traffic rate.This causes the loss probability to increase.
Since the measurement time scale is directly related to the measured traffic, the result of a measurement-based algorithm using constant time scale is open to the performance degradations due to the changes in traffic trends.However, our goal was to obtain an algorithm which does not require any a priori traffic knowledge, and which is based fully on the measurements.Therefore we incorporated the Maximum Time-Scale (MaxTS) parameter and tested successfully adapting the measurement time scales based on measurements themselves.
To sum up, in this paper, we • identified on-line measurement-based capacity allocation algorithms, • compared their performances analytically, • simulated promising ones, • observed significant affects of the choice of measurement time scale, • proposed to vary measurement time scale adaptively, • through an example, showed the perfonnance robust- The outcomes of this study can be used for choosing algorithms to be implemented in real switches, taking into account trade-offs of complexity, accuracy and robustness.time.

Figure 2 .
Figure 2. A visual example view of dynamic capacity allocation.

Table 2 .
-ie Approach for On-line Measurement-based Capacity Allocation i Window Results for GA

Table 3 .
Results for CEB

Table 4 .
Results for NEB

Table 5 .
Performance Metrics vs.Time Scale