A GREEDY POliCY FOR THE ADMISSION CONTROL OF SELF-SIMILAR TRAFFIC

Admission control is a fundamental mechanism for the provisionof Quality of Services in multiservice networks. The control of long-range dependent traffic demands specially designed mechanism. In tlus paper, a greedy policy for the admission control of self-similar traffic is introduced. This policy accepts flows into the network according to a decreasing order of revenue. It is shown that results produced by this new policy are close to the one produced by and optimum policy.

The concept of equivalent bandwidth is intimately connected with admission control and associated service requirement [6].The equivalent bandwidth of a connection (flow) is a characterization of the minimum required bandwidth of that connection such that QoS requirements are met.There has been a great interest in the concept of equivalent bandwidth, since it promises to bridge the design of statistical multiplexing networks to the familiar design of circuit-switched networks.Although there is a remarkable collection of results of equivalent bandwidth [6]- [9], very few results are available for traffic with long-range dependencies [7]- [8].
A new traffic model called the fractional Brownian motion (fBm) envelope process for the characterization of an LRD flow has been proposed [13].A new framework for computing probabilistic delay bounds for a deterministic queueing system has also been developed.It was shown that the delay bounds were in agreement with known results obtained via large deviation theory.This traffic characterization has made a more intuitive understanding of the dynamics of queueing systems possible.Time-scales for the complete characterization of the behavior of a queueing system fed by self-sinlilar flows has also been derived [14].The accuracy of the fractional Brownian motion (fBm) envelope process for modeling real network traffic has been extensively validated [12].
Recent studies [10] [II] have shown that IP traffic, as well as video traffic, presents multi-scaling characteristics which are properly modeled by multifractal processes.For time scales larger than a cut-off value, IP and video traffic should be considered monofractal (self-sinlilar).For IP traffic, the cut-off time scale is of the order of a round trip time (512ms to I sec); while for video traffic, the cut~off time scale is of the order of the transmission time of a frame.
Besides the provision of QoS requirements, network providers are mostly interested in increasing revenues obtained from the acceptance of new flows (connections).In this paper, a new adnussion control policy for flows presenting LRD is proposed.This policy accepts new flows in decreasing order of their associated revenue.Results given by this new policy are compared to ones provided by an optimal policy which maximizes the long-term revenue by using a knapsack type of solution.It is shown that the use of the proposed policy generates revenues close to the one generated by the optimal policy.Tills paper is organized as follows.In Section II.an envelope process for self-sinlilar traffic is presented.In Section Ill, the time scale of interest for a queueing system fed by a self-sinlilar process is derived.In Section IV, the statistical Revista da Sociedade Brasileira de Telecomunicayoes Volume 18, Nlimero 2, Outubro de 2003 multiplexing of heterogeneous self-similar flows is shown.In Section V, a new admission control policy is introduced.Finally, Section VI presents the conclusions.
It It should be noted that the flow does not necessarily need to be self-sintilar in order to fit this characterization: what is important is that it matches the behavior of the envelope process for the time scale of interest.The accuracy of the fBm envelope process representation is investigated by inspecting how well it can model the worst-case behavior of real network traffic.Assuming that the input traffic is characterized by a trace with N sample points, defined by A.(t), where A(t) represents the cumulative number of cells aniving up to time t,t E [1, 2, ... NJ, a very simple method for computing the fBm envelope process parameters for this trace is proposed, which computes the optimal envelope process of the trace.The advantage of this approach lies in the fact that it is not necessary to estimate the Hurst parameter of the trace accurately.The optimal envelope process for this trace is defined by Y(t-s) = max,<t(A(t)-A.(s)).Assume that the process is stationary, yields Y(T), T = ts, which defines the maximum number of cells aniving in a given interval of size T. Therefore, the fBm envelope process parameters A(.) can be chosen to match the behavior of Y(.).  1.The optimal envelope process, i.e Y(.), is then computed and H chosen so that .4(.) matches the behavior of Y(.).This example shows that the fBm envelope process matches closely the behavior of the LRD traces.The effectiveness of the fractal Brownian motion envelope process was extensively validated by utilizing synthetic traces as well as traces derived in real networks.A report on the validation can be found in [12].

TIME SCALE OF INTEREST
Tills section characterizes the time until a queue reaches its maximum occupancy, in a probabilistic sense.The queue size at this time provides a simple delay bound [ 13].A rigorous mathematical de1ivation of the delay bound can be found in [14], and a heuristic derivation is introduced in keeping with the precepts explored here.Consider a continuoustime queueing system, with deterministic service given by c.

The cumulative arrival process is represented by A.H(t).
Let AH(t), continuous and differentiable, be the probabilistic envelope process of A.H(t), such that Pr(A.H(t) > AH(t)) S <.
Dming a busy period, which starts at time 0, the number of cells in the system at timet is given by q(t).Thus, q(t) = AH(t)-ct ~ 0.
The maximum delay in a FIFO queueing system is given by the maximum number of cells in the queue dming the busy period, which can be defined as follows: Therefore, Pr(q(t) > qmax) S Pr (q(l) > <)(t)) $ <, Pr (q(l) > qmax) "'<.
The queue length at timet.q(t), will thus only exceed the maximum queue length qmax with probability t.In other words, only when the arrival process exceeds the envelope process, will the maximum number of cells in the system exceed the estimated value.Intuitively, by bounding the behavior of the arrival process, it is possible to transform the problem of obtaining a probabilistic bound for the stochastic system defined by q(t) = AH(t)-ct 2: 0, into an easier problem of finding the maximum of a detenninistic system, described by ij(t) = iiH(t)-ct 2 0.
For the fBm process, the envelope process defined above is inserted into Equation 2, which gives the following: (3) In order to compute Qma.c• it is necessary to find t* such that or equivalently, Hence, t"* is given by The time scale of interest is defined by the time until a queue size reaches its peak, i.e., t*.This is denoted the Maximum Time Scale (MaxTS), and it defines the point in time where the unfinished work in the queueing system achieves its maximum in a probabilistic sense.This means that the average anival rate has dropped below the link capacity so that the queue size will start decreasing.The average arrival rate converges to mean arrival rate of the flow on the basis of the law of large numbers.Consequently, there is no need to worry about anything other than the time scale for which the flow rate still exceeds the link capacity.In other words, after a period of time, the probability that the average arrival rate exceeds the link capacity is negligible; hence, the arrival model need no longer reproduce the flow behavior for these time scales.This is the most important time scale in terms of traffic modelling.As a rule of thumb, to choose the parameters of an input flow to match the :fBm envelope process, it is necessary to find MaxTS analytically, while the parameters of the fBm process must be chosen to match the flow optimal envelope process of the flow at MaxTS.
Substituting t* back into Equation 2, it can be concluded that: Since the fBm process does not exceed with probability 1 -<, the maximum number of cells will be bounded by Qmax with the same probability.C must be found such that Gmax is equal to K. In other words, a buffer of size I\ will overflow with probability < if the link capacity is c.Therefore, c is given by

220
This result was previously obtained by NoiTos [4] [19] and Duffield [5].However, the framework here makes it possible to achieve the same results predicted by the large deviation theory without making restlictive assumptions.It has also been possible to reduce the sensitivity of the estimation process by using a bound rather than attempting to estimate the parameters directly from the full trace.
A compmison between the MaxTS and the so-called Critical Time Scale (CTS) [20] is provided in [14].It is shown that this CTS is the most likely time scale when buffer overflow occurs; but, it does not furnish infmmation about the marginal distribution of the random vmiables which describes the overflow process.

STATISTICAL MULTIPLEXING OF SELF-SIMILAR FLOWS
In this section, Ma'\TS is used to de1ive expressions for predicting the equivalent bandwidth and buffer requirements of an aggregate of self-similar flows.Essentially, a method for computing the bandwidth necessmy to suppmt requirements of buffer overflows is proposed, as well as for detennining the maximum probabilistic delay for an aggregate of heterogeneous flows.The problem in tills section can be stated as follows: Given a set of flows with mean ai, standard deviation Ui and Hurst parameter Hi. what is the link capacity needed so that the maximum queue she will be bounded by Gmax with probability<?
Assume N independent flows A},(t) defined by the following parameters: mean ai , standard deviation Ui and Hurst parameter H; for i E [ 1, N].Let the aggregate traffic be denoted by AH(t) = 2::;;:, 1 A[,(t).The envelope process of each flow is given by A},(t), and the envelope process of the aggregate traffic is given by AH(t).The Gma.r of a queue with heterogeneous flows can be computed by finding t* for the envelope process of the aggregate stream.The mean of the aggregate traffic is given by the sum of the mean of the individual flows.Similarly, since the flows are independent, the variance of the aggregate traffic is also furnished by the sum of the variance of the individual flows.Hence, the envelope process of the aggregate traffic is defined as follows: Replacing in Equation 4, gives the following: N =c-Lai.combining Equations 5 and 6 results in the following: (7) Using Equations 6 or 7 makes it possible to answer the fundamental question posed at the beginning of tills section.
For the special case of multiplexing 1Y identical flows, the envelope process is given by: since the Hurst parameter is preserved when aggregating JV identical flows [18].In this case, Equation 6is reduced to the following: Using the same approach as above, it is possible to find t* and qmaa,•: iV ----rr=r-rJmax, ti = [ ""H ]6

(c-ii) '
iimax = A(t*)-ctiwhere ti and tfmax corresponds to a queueing system fed by a single flow.
This specific case of a single flow is shown first.Figure 2 shows the overflow probability as a function of the maximum buffer size for a link utilization of 60% and for H = 0.63 and CT 2 = 0.64.When the overflow probability given by the analytical models is compared to that with the overflow probability observed in the simulation experiments, it can be seen that the two models provide reasonably consistent results.It can be observed that the precision of results found here increases with the Hurst parameter as well as with the link utilization.
To evaluate the effectiveness of the equivalent bandwidth expressions (Equations 617), it is necessary to define multiplexing gain as the ratio between N times the equivalent bandwidth of a single flow and the equivalent bandwidth of 1V identical flows.It can be seen that a significant multiplexing gain can be achieved when multiplexing homogeneous flows.In Figure 3, the gain for a link capacity of !50 Mbps, and flows with a mean arrival rate of l.!Mbps for various Hurst parameters is plotted.Figure 3 •----~------... fact that Equations 6/7 take into consideration the existence of long periods with no anivals in streams with high Hurst parameters.As a consequence, Equations 6/7 demand less bandwidth when multiplexing several flows than would be necessmy in a non-multiplexing approach.It was also shown that the multiplexing gain increases not only with the vmiance and the Hurst parameter, but also with that of the number of multiplexed flows.A comparison of the equivalent bandwidth expressions shown here with those derived by Kelly [7], and by Stathis and Maglaris [21] is made, using Equation 3.35 in [7] and Equation 6 in [21].The real flow traffic parameters as described in [21] are utilized.In Figure 4, the number of accepted connections as a function of the overflow probability are presented, considering flows with a mean rate of 1.4 Mbps, " = 0.28 Mbps and an H = 0.85.The buffer size was set to 1000 ATM cells.It can be observed that the number of flows admitted as predicted by the present approach is the same as that predicted by Kelly (overlapping curves).In other words, the same result predicted by the large deviation theory is achieved with much less computational effort.[7] and [21] for Different Overflow Probability Requirements

A GREEDY POLICY FOR ADMISSION CONTROL
Whenever a request for admission anives, an admission contt•oller needs to verify both if it will be possible to provide the required QoS and to continue to fumish QoS to all other already accepted flows.In order to maximize revenue, the admission controller should collect requests during time intetvals, to decide which requests should be granted.
The policy proposed here accepts requests according to their decreasing order of associated revenues.However, this simple policy may not maximize the total revenue, since it considers arrivals during short periods and not in the long run.In the long run, a single flow with an associated high revenue can consume the same amount of resources and provide lower total revenue than a set of flows each with an associated low revenue.An optimal policy should find a combination of flows that, in the long run, gives the highest possible revenue subject to the available bandwidth.This is a classic knapsack problem in which the available bandwidth is the knapsack, and the bandwidth demand of each flow is the size of each object.This knapsack problem can be stated as: where Riis the revenue of flow i; Hiis the Hurst parameter of flow i: a -is the mean rurival rate of flow i; ai -is the standard deviation of flow i: t* -is the Maximum Time Scale (MaxTS): To verify the extent to which the proposed greedy policy leads to the same results produced by the knapsack approach, extensive simulation experiments have been carried out.I were used.Note that the mean arrival rate is nmmalized to the channel capacity.The mean duration of a flow varies between 20 and 500 seconds, and the buffer size corresponds 1000 ATM cells.In Figure 5, the total revenue is displayed as a function of the mean interanival time of the flows.The total revenue is computed by the accumulated revenue of all accepted flows during the simulation experiment.The exact value of the revenue is immaterial in the sense that it changes according to the choice of a and b.The lower the interarrival time, the higher the load.Note that for moderate to high loads, the revenue produced by the greedy policy is close to the one produced by the optimal approach.Under low load conditions, this trend is not observed.
In Figure 6, the total revenue as a function of the mean flow duration is shown for a utilization value of 0.7.It can be seen that the greedy policy provides revenues close to the one given by the optimal approach irrespective of the flow duration (for moderate to high loads).

CONCLUSIONS
Admission control is a fundamental mechanism for !he provision of Quality of Service in multiservice networks such as IP Diffserv and ATM networks.Such provision is sensitive to the long-range dependencies existing in several types of traffic.In this paper, a simple admission control policy was introduced.This new policy admits flows into the network according to the decreasing order of revenue provided by these flows.It was shown that for moderate to high loads tills simple policy produces revenues in the same range as rev-    enues produced by an optimal policy.The simplicity and ease of policy implementation introduced here make it a potential candidate for realtime admission control in QoS-miented networks.

FigureFigure 1 .
Figure I shows the accuracy of the fBm envelope when compared to Bellcore's LAN trace.The sample average arrival rate and the sample variance are computed and inserted

i=l ( 6 )
Equation 6 can be solved numerically to find t*, which is then insetted in Equation 5 to compute Gmax• Moreover, ' Revista da Sociedade Brasileira de Telecomunicayoes Volume 18, Numero 2, Outubro de 2003 considers flows with CT 2 = 0.3.Results indicate that the multiplexing gain also increases with the Hurst parameter, especially for streams with moderate to high variance.This can be explained by the

Figure 2 .Figure 3 .
Figure 2. Accuracy of the Predicted Overflow Probability by Equations 6/7 for a Single flow as a Function of the Buffer Size

Figure 4 .
Figure 4. Comparison of the Number of Admitted FlowsGiven Equations 6n and the Work in[7] and[21] for Different Overflow Probability Requirements

Figure 5 .
Figure 5. Revenue x Average Flow Interarrival Time

Figure 6 .
Figure 6.Revenue x Average Flow Duration

Table 1 .
Traffic parameters of video streams.
The revenue function used is aT+ bY, where Tis time duration,