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IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 3, NO. 2, JUNE 2001

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Dynamic Resource Management Considering the Real Behavior of Aggregate Traffic José R. Gallardo, Dimitrios Makrakis, and Marlenne Angulo

Abstract—This paper explains why the theory of effective bandwidth is in general not applicable to characterize realistic traffic sources. Moreover, we show that a static allocation of network resources (bandwidth and/or buffer space) based on large deviation theory can be highly inefficient when the real statistical behavior of traffic is taken into account. As an alternative, we propose a dynamic resource management scheme based on prediction techniques. As a specific example, we apply this scheme to a Differentiated Service (DiffServ) Internet environment, in which the goal, besides policing the incoming traffic, is to optimize the use of network resources, thus minimizing the probability of occurrence of violations of contract guarantees. The performance of the proposed scheme is evaluated via simulations and our results show the superior performance of the new algorithm in terms of buffer overflows, output link utilization, and jitter, as compared to currently used policing and shaping mechanisms. Index Terms—Alpha-stable self-similar processes, dynamic resource management, effective bandwidth theory, prediction techniques, traffic modeling, traffic policing and shaping.

I. INTRODUCTION

S

IMULATION is used extensively these days in the planning process of telecommunications networks. Simulation allows the network designer to draw important conclusions and make the right decisions before major investments are made in production. The validity of the conclusions obtained related to the system requirements in order to provide quality-of-service (QoS) guarantees, greatly depends on how accurately the model captures the actual operation of the system under study. Self-similarity and long-range dependence have proven to be important features of aggregated traffic. One of the most relevant models presented in the past proposes the use of Fractional Brownian Motion to capture the behavior of the traffic [21]. As a further generalization, aimed at increasing the accuracy of the model, Fractional Stable Motion processes have been proposed [14]. These processes can capture not only the self-similarity of the traffic, but they can also match its level of burstiness. Manuscript received March 6, 2001. The associate editor coordinating the review of this paper and approving it for publication was Prof. Jenq-Neng Hwang. J. R. Gallardo was with the Broadband Wireless and Internetworking Research Laboratory, School of Information Technology and Engineering, University of Ottawa, Ottawa, ON, K1N 6N5 Canada. He is now with the Electronics and Telecommunications Department, CICESE Research Center, Ensenada, B.C, Mexico 22860 (e-mail: [email protected]). D. Makrakis is with the Broadband Wireless and Internetworking Research Laboratory, School of Information Technology and Engineering, University of Ottawa, Ottawa, ON, K1N 6N5 Canada (e-mail: [email protected]). M. Angulo is with the Faculty of Engineering, Autonomous University of Baja California, Mexicali, B.C., Mexico (e-mail: [email protected]. mx). Publisher Item Identifier S 1520-9210(01)04316-4.

The marginal behavior of Fractional Stable Motion processes is given by alpha-stable (long-tailed) distributions, of which the Gaussian distribution is a particular case. The theory of effective bandwidth, which is based on powerful and elegant mathematical results, basically states that the probability of buffer overflow in a stationary queueing system decreases exponentially with the buffer size as long as the “effective bandwidth” of the input traffic stream is smaller than the server capacity (see, for example, [7], [8], [16]). This statement is described in a more formal way in Appendix A. This theory has been used extensively to design telecommunications systems that have some QoS requirements (based mainly on the probability of buffer overflow). There are, however, certain conditions that must be satisfied in order for this theory to be applicable to a given system. The goal of this paper is to show that realistic traffic models and the concept of effective bandwidth, unfortunately, are not compatible and, therefore, cannot be combined. Our goal is also to propose an alternative dynamic approach to resource allocation and management, instead of the static solutions usually found using the effective bandwidth theory or similar large-deviation results. As a specific example, we apply our algorithm to an Internet boundary node carrying integrated traffic. We believe that these are important and timely results since the global character of the Internet and the increasing diversity of applications and services, demands that the Internet become a network capable of handling applications and services with diverse QoS requirements. The remainder of the paper is organized as follows. Section II explains the motivation for this work. Section III describes the proposed dynamic traffic control method, which is based on prediction techniques to increase the intelligence of the traffic controller. Section IV includes the simulation parameters and results. Section V concludes the paper. Appendix A summarizes the theory of effective bandwidth and includes a brief discussion on the applicability of this theory to realistic traffic sources. The prediction technique used in this work is described in Appendix B. Finally, a recursive algorithm to compute the prediction coefficients is described in Appendix C. II. MOTIVATION As mentioned above and explained in Appendix A, the notion that the probability of buffer overflow can always be upperbounded by an expression that decreases exponentially with the system buffer size is not accurate when the realistic behavior of the traffic is taken into account. In contrast, as shown in [14], the

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equation that relates the probability of overflow to the traffic parameters and to the service rate and buffer size of the queueing system is give by the following expression: Fig. 1. Boundary node: functionality scheme.

when

A. Differentiated Service Overview

when

(1)

is the current buffer occupancy, is the total buffer size, and is the system service (or leak) rate. As for the traffic characteristics, is the mean arrival rate, and define the burstiis the Hurst paramness level (marginal distribution), and eter, which specifies the long-range dependence of the traffic behavior. In addition Cos and

As evidenced in (1), the rate of decrease of the overflow probability with respect to either the buffer size or the service rate is far slower than exponential. A direct implication of this fact is that very large service rates and/or buffer sizes are usually necessary in order to achieve an acceptable service. When the solutions found using (1) are not practical, dynamic traffic shaping and policing mechanisms have to be implemented, instead of statically allocating an excessive amount of resources to a specific traffic stream. III. DYNAMIC TRAFFIC CONTROL USING PREDICTIVE TECHNIQUES Now that we have established the motivation for this work (namely, that static resource allocation with QoS guarantees implies the reservation of prohibitively large amounts of bandwidth and/or buffer memory for each traffic stream), we will propose an alternative dynamic approach based on prediction techniques to increase the intelligence of the traffic controller. The prediction algorithm used in this work is described in Appendix B. As a specific example, we will apply our algorithm to an Internet boundary node carrying integrated traffic. In an effort to address the need for the provision of QoS guarantees, such as delay, packet loss probability, etc., the Internet Engineering Task Force (IETF) proposed the Differentiated Service (DiffServ) architecture, which is capable of implementing a scalable service differentiation by applying an aggregated traffic classification [4]. Our work addresses the boundary node, located at the edge of a DiffServ-capable network, since the complex functions of DiffServ are implemented within this particular node.

The DiffServ architecture was proposed to be implemented at the network layer. The packet privileges are marked in the IP-packet header [4], and those tags will indicate to the routers what treatment a packet should receive. Currently, the IETF is defining the following per-hop-behaviors (PHBs): expedited forwarding (EF), assured forwarding (AF), and default (DE). In this work, we will deal with the assured forwarding PHB since it is the one that demands more dynamic procedures and, at the same time, allows for a more efficient use of the resources. Assured forwarding defines four traffic classes; the aggregate traffic from each class is guaranteed to be provided with some minimum amount of bandwidth and buffering. There are three drop preferences in each class, which means that, when congestion occurs, packets with higher drop precedence will be discarded ahead of those with lower drop precedence. There are token bucket policers for each class. At the edge of the network, the boundary node performs the complex functions, such as packet classification and traffic conditioning. The rest of the routers are called cores and their function is solely to forward packets according to their class and drop preference. A crucial aspect of the DiffServ architecture is that a router’s forwarding behavior will be based only on packet markings, i.e., on the class of traffic (the aggregate behavior) to which a packet belongs. The boundary node functionality scheme proposed by IETF is depicted in Fig. 1. The packet classification is based on the IP header and will identify each packet as belonging to one of the four pre-specified classes. The meter is used to calculate and compare the packet-flow with the traffic agreement, and to conclude if the packet is in or out of contract. Bernet et al. propose in [3] for the meter function to use exponentially weighted moving average (EWMA) or token bucket-based mechanisms. Our work is based on the latter, which is described in detail in [17] and [18]. For the marker function, the Internet community has developed some proposals, such as the single rate three-color marker [18] (the approach we are using), and the two-color marker [17]. The classification rules are to be implemented by the network manager. In our case, the implementation of the marker is based on the use of two leaky buckets in cascade, and its function is to mark according to the contract parameters, such as sustainable rate and burst excess [18]. Buffers perform the shaper function, which delays one or all the packets in a traffic stream to bind them to agree with the traffic contract. Bernet et al. [3] recommend using different queues for each class of traffic and drop preference, in order to isolate the traffic streams. The dropper function discards some or all the packets in the traffic stream that do not conform to the traffic contract.

GALLARDO et al.: DYNAMIC RESOURCE MANAGEMENT

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Fig. 2.

Boundary node implementation.

The AF architecture does not restrict the use of any mechanism to perform the shaping and dropping functions, and it is at this point that our proposed dynamic resource management scheme is utilized. B. Our Model As shown in Fig. 2, we assume that this particular router services four different traffic classes, represented by aggregate video, voice, Ethernet (high volume data), and WWW (low volume data) traffic. Since the main feature of the DiffServ architecture, in order to have scalability, is that control is exerted on aggregate traffic streams, we need to use realistic models for aggregate traffic. According to [14], the video, Ethernet LAN, and WWW traffic sources are implemented based on alpha-stable self-similar stochastic processes. The aggregated voice traffic is produced using five individual sources, each of which is modeled as a two-state Markovian source [9]. The router is assumed to have a separate input buffer for traffic corresponding to each drop preference within each class (which gives a total of 12 input buffers) and a shared output buffer. In Fig. 2, however, only one input buffer is shown per traffic class to simplify the diagram. All the buffers are serviced using an FIFO discipline. For the input buffers, the service rate changes dynamically and is given by their corresponding token generator, according to the specific policing mechanism being used, as will be described in Section III-E. The service rate of the output buffer is assumed to be fixed and is, in our case, slightly larger than the sum of the sustainable rates for all the incoming streams (the assumed

load is 92% of the link capacity to test the system under heavy congestion). The basic idea is to try to preserve as many packets as possible at the input buffers, even if they violate their own service contract (if they do, they will be tagged accordingly), as long as their transmission does not affect the traffic corresponding to the other classes of service that are within the contract limits. If congestion at some point in time is too heavy, packets will be withdrawn from the input buffers and discarded because of the inability to store them in the output buffer to be eventually serviced. The existence of separate input buffers allows us to separate the control actions corresponding to the different classes of service. C. Time-Bounded Traffic Voice Traffic: Humans do not detect interactive audio application delays of up to 150 ms [5], [20]. For IP networks, delays between 150 ms and 400 ms are acceptable, but not ideal [5], [20]. Voice delays of more than 400 ms produce an unintelligible voice [5], [9], [20]. The maximum loss rate allowed for voice packets is 10 . In the special case of voice packets, the throughput is decreased considerably, because the payload that they carry is small, as compared to the header; the reason for that is that voice samples cannot be accumulated to be sent together because it would introduce excessive delay. Video Traffic: Acceptable delay for video traffic is in the range of 150–400 ms [6]. Generally, the loss requirements are between 10 and 10 [6]. The video traffic traces that we use in our analysis correspond to MPEG video, and are obtained using the model proposed in [14].

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D. Time-Insensitive Traffic Data Traffic: The traffic generated from Ethernet LANs and WWW users does not have, in general, restrictions for the endto-end delay. However, even if the network is supposed to give preference to the time-bounded traffic, it should not affect the minimum throughput agreed for the traffic that is not restricted in time.

TABLE I ACTUAL TRAFFIC PARAMETERS

TABLE II TRAFFIC CONTRACT (SERVICE LEVEL AGREEMENT)

E. Traffic Policing and Resource Allocation Algorithms This section describes two different policing mechanisms that are based on the leaky bucket idea: conventional leaky bucket and adaptive leaky bucket assisted by traffic prediction. The policies are applied locally (e.g., within a router) and their purpose is to optimize the resource utilization, which in turn intends to minimize the number of violations to the QoS requirements of the traffic. 1) Conventional Leaky Bucket (CLB): In the original leaky bucket, packets that arrive to the router are stored in a buffer. They will be forwarded only if they can draw a token from the token pool, otherwise they will remain in the buffer. If a new packet arrives when the buffer is full, it will be discarded. Tokens, in turn, are generated at the network’s average transmission rate and stored in the token pool, which has a finite size. After filling the token pool, additional tokens are discarded. The size of the pool can be seen as the maximum allowable burst length, since it is the maximum number of packets that can be transmitted back-to-back at a given time. 2) Adaptive Leaky Bucket Assisted by Traffic Prediction (PALB): In the adaptive version of the leaky bucket, the token generation rate is not constant anymore, but a function of the network load or network congestion. The packet buffer occupancy is monitored regularly and an appropriate action is taken after each observation. Two thresholds (a lower and an upper) are being used. When the buffer occupancy is between the two thresholds, the token generation rate remains unchanged. If the buffer occupancy falls below the lower threshold, the token generation rate is decreased. If the occupancy goes above the upper threshold, the token generation rate is increased. When using this approach, a decision is made to change the token generation rate after either too many or too few cells have been received, so that one of the thresholds has been crossed. This leads to a poor dynamic performance for the control mechanism, since the actions are often taken too late to be able to solve the situation. The thresholds can always be moved closer to each other to have the system react more rapidly, but this can go to the other extreme causing overreaction of the system. To alleviate this weakness, we propose to provide the router with the additional capability to predict traffic streams and to use that information to allocate the available bandwidth among the users. The adoption of traffic prediction gives us the advantage of being able to make decisions based on the future status of the router, which is inferred based on the current router status and the estimated amount of traffic that will arrive. IV. SIMULATION PARAMETERS AND RESULTS Tables I–IV summarize the simulation parameters. In Table III, the initial token generation rate is computed based on the sustainable packet rate. The upper and lower rates in

TABLE III TOKEN GENERATION INTERARRIVAL TIME

TABLE IV BUFFER ALLOCATION

every token generator are adjusted according to the traffic requirements. This gives the restriction for the maximum delay required by the sources. The performance analysis results that are presented below have been obtained through computer simulations, developed using the OPtimized Network Engineering Tools (OPNET) software package. All the results presented in this section correspond to 500 s of simulation time. Output Buffer Overflows: This statistic shows how packets are discarded because of their inability to enter the output buffer (see Fig. 2) as it gets full. The number of output buffer overflows gives us an indication of the system ability to cope with heavy congestion, as explained in Section III-B. Fig. 3 shows the cumulative number of losses in bytes. For the CLB algorithm, there were around 120 000 bytes lost at the end of the simulation. The PALB achieved a remarkable improvement, giving as the final result only 1400 bytes lost, which is a value almost 100 times smaller than the one obtained with the CLB algorithm. Notice that, as explained in Section III-B, the service rate of the output buffer is assumed to be slightly larger than the sum of the sustainable rates of the incoming streams (assumed load is 92% of link capacity) in order to test the system under heavy congestion conditions. That is why so many losses were observed with some of the algorithms; the same applies for the overall number of overflows, described below. Overall Router Overflows: The number of overflows as a function of time is displayed in Fig. 4. The number of lost bytes after the 500 s of simulation time, considering all the buffers for the CLB algorithm, while within the router, was of


Fig. 3. Cumulative number of overflows at the output buffer.

Fig. 4. Cumulative number of overflows in the whole router.

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Fig. 6.

Mean delay in voice packets.

Fig. 7. Mean delay in video packets. TABLE V PACKET LOSS RATIO

Fig. 5. Output link utilization.

it was of around bytes for the PALB technique. The achieved improvement is of about 20%. Output Link Utilization: The maximum output link utilization that can be reached in the ideal case, where there are no losses, is 0.92 (since the load is equal to 92% of the available bandwidth). As shown in Fig. 5, CLB offers a utilization of 88.5%, wasting approximately 4% of the link capacity. On the other hand, PALB increase the link utilization to 89.5%, reducing to 3% the proportion of unused bandwidth. This of course does not represent a great advantage, but is still a consequence of the intelligence introduced into the router. Delay in Time-Bounded Traffic: We present here the results corresponding to the mean delay experienced by time-bounded traffic, such as voice and video. Fig. 6 shows the voice delay for both of the policing schemes. The values obtained are 0.81 ms

for the CLB and 0.65 ms for the PALB algorithm, representing a difference of around 25%. Even though there is a considerable difference among them, both values are far below the 150 ms of maximum delay allowed for voice traffic. For video traffic, we obtained an average delay of 0.12 ms for the CLB algorithm and 0.23 ms for the PALB algorithm, as depicted in Fig. 7. These values, even though noticeably different, are again far below the 20 ms of allowed maximum delay for video packets. Packet Loss Ratio and Jitter: Table V summarizes the results obtained for the packet loss ratio corresponding to individual streams. The PALB algorithm satisfies the requirements for both video (10 ) and voice traffic (10 ) regarding this metric, even under the assumed condition of heavy congestion. CLB, on the other hand, achieves a very good performance for the video traffic, but not for the voice traffic. The performance of the two algorithms relative to the data traffic streams is similar. Notice, however, that in the case of CLB, it is the Ethernet traffic which contributes to most of the losses because of the high volume of data that it introduces to the system and because of its high variability; considering this high volume, even a small difference in the loss ratio implies a large difference in the actual number of losses.

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A random variable

TABLE VI JITTER FOR TIME-BOUNDED TRAFFIC

is said to be bounded exponentially if (A2)

Finally, Table VI includes the results obtained for the jitter introduced by the two policing mechanisms. It can be observed that our new proposal achieves a much better performance in relation to this parameter, as well. V. CONCLUSIONS We have explained that the theory of effective bandwidth is not applicable to characterize realistic environments due to the long-tailedness of the marginal distribution of the processes that best model the statistical behavior of real traffic sources. We have also shown that a static allocation of network resources (bandwidth and/or buffer space) based on large deviation theory can be highly inefficient since the usual assumption that the probability of buffer overflow decreases exponentially with the buffer size is not applicable. The probability of buffer overflow decreases hyperbolically (much more slowly than exponentially) with either buffer size or allocated bandwidth. As an alternative, we propose a dynamic resource management scheme based on prediction techniques. The prediction technique is explained in detail and a fast algorithm to compute the prediction coefficients is also included. We have applied this scheme to an Internet environment in which DiffServ policies are implemented. The performance of the proposed scheme has been evaluated via simulations and our results show that the system is still capable of providing an acceptable service to real-time applications when the overall load is above 90%. The service provided to nonreal time applications (data) is also within reasonable limits. The system performance is evaluated in terms of buffer overflows, output link utilization, delay, nd jitter. APPENDIX A THE THEORY OF EFFECTIVE BANDWIDTH This theory was conceived of as a tool to provide a measure of resource usage that encompasses the traffic source characteristics and statistical description, and also depends on a certain QoS parameter, specified in terms of the probability of packet loss due to buffer overflow. The definition of the effective bandwidth of a traffic source, according to [19] is for

(A1)

is the cumulative number of packet arrivals during where . This definition is compatible with the the time interval concept of minimum envelope rate (MER) described in [7], ex(steady-state cept that in the latter case, the limit for value) is considered. Let us briefly review the conclusions that are made in [7], [8], and [16] regarding stability, queue length, and delay of queueing systems.

for some positive constants and . The basic conclusion obtained in the above-mentioned references is that, for a queueing system with a constant service rate, a necessary and sufficient condition for the queue length to be bounded exponentially is that the steady-state effective bandwidth of the input process be smaller than the service rate. Based on this result, it is shown that the system is stable in the sense that the delay is also bounded exponentially when a FIFO policy is implemented. In mathematical terms, the relevant result can be stated as follows. is a random process with stationary increAssume that ments and that the constant service rate of the queueing system . Then is for some if and only if the buffer occupancy at time , denoted by satisfies the following inequality:

(A3) , (A4)

is constant with respect to and . The where the function can be thought of as a QoS parameter, since the parameter probability of buffer overflow, which is given in (A4), depends on its value. In what follows, we will briefly discuss the applicability of the theory of effective bandwidth to characterize real aggregate traffic sources. As mentioned in Section I, it was shown in [14] that Fractional Stable Motion, which is an alpha-stable selfsimilar process with stationary increments, is capable of very accurately capturing the actual behavior of the traffic flowing be the cumulative number through today’s networks. Let , defined by of packet arrivals during the time interval (A5) is the mean input rate and is Fractional Stable where Motion. The very definition of effective bandwidth [(A1)] prevents the whole theory from being applicable to the most general case of alpha-stable processes, since the moment generating , defined as function of the process (A6) only exists when the stability parameter is 2 (Gaussian case). diverges for is very simple: if The proof that we assume that, for a fixed value of , there exists a value for which , then using the Chernoff bound [22] we can conclude that (A7) which is impossible due to the long-tail behavior of non-Gaussian alpha-stable random variables . The effective bandwidth can be calculated, though, for the , where Gaussian case. Suppose that is a Fractional Brownian Motion process such that Var , then [19] its effective bandwidth is given by (A8)


Notice that the steady-state effective bandwidth (the limit of ) diverges when , that is, when the (A8) for has long-range dependent increments. Since all the process conclusions related to the stability of queueing systems (e.g., those reported in [7], [8], and [16]) are based on the assumption that the steady-state effective bandwidth of the input process is finite, they are only applicable to a very reduced subset of alpha-stable self-similar processes. Namely, to short-range de) Gaussian processes, which are not realistic, pendent ( i.e., they do not reflect the actual behavior of today’s network traffic. Therefore, we can conclude from here that neither the queue length nor the delay will be exponentially bounded in a real environment. There is a generalized form of large deviations theory, proposed by Duffield and O’Connell in [11] and [24], in which it is proved that, for a queueing system with a long-range dependent input traffic, it is still possible to obtain an expression for the effective bandwidth of a properly scaled version of the input random process, which in turn allows us to obtain the asymptotic rate of decay of the overflow probability. That generalized theory is easily applicable to Fractional Brownian Motion processes, giving the same result obtained by Norros in [21], i.e., that the probability of buffer overflow has a Weibullian decay relative to the buffer size, and therefore decreases more slowly than exponentially. However, the theory is again not applicable to the real traffic in the most general case since a scaled version of an alpha-stable random variable is still an alpha-stable random variable, and its moment generating function will always diverge. There may be an even more general transformation (different from multiplying by a scaling factor) that can yield a meaningful definition of effective bandwidth for the general alpha-stable self-similar processes but, to the authors’ knowledge, such a transformation has not been proposed yet. A first attempt to make the effective bandwidth theory useful for the alpha-stable traffic case is reported in [1]. In that paper, the authors propose to truncate the infinite integral that defines the moment generating function of the traffic process in order to obtain a finite value. The proposal is not mathematically rigorous and, moreover, it does not constitute a useful practical tool either, according to its own authors.

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to minimize the prediction error because of the lack of second moments for alpha-stable distributions. In the proposed method, a distance is specified between stable random variables, which is based on the sample autocorrelation function for this kind of processes. The predicted value is then selected as a linear combination of previous samples in such a way that the distance between itself and the actual value of the process is minimized. To the authors’ knowledge, nonlinear prediction is not feasible at this point due to the lack of a metric function (to define distance) for arbitrary nonlinear functions of FSN samples. Fractional lower-order moments (FLOM) could potentially be used for this purpose, but the lack of closed-form expressions for the probability density function of stable random variables prevents their use in an analytical form. Even though FSN processes are originally defined by means of a stochastic integral, they can also be specified using the following discrete Moving-Average expression [15]: (B1) is a set of independent identically where , distributed (iid) alpha-stable random variables according to the notation used in [23], and the coefficients are all nonnegative. For balanced LFSN, anti-balanced LFSN, and Log-FSN, the coefficients in (B1) satisfy (B2) Let us define a Hilbert space [2] for the samples of a FSN process by defining an inner product as follows:

(B3) Since the inner product depends only on the difference let us define for any value of

,

and (B4)

APPENDIX B PREDICTION OF REAL TRAFFIC Since the model for the cumulative traffic arrivals within is Fractional Stable Motion, which is an the interval alpha-stable self-similar process with stationary increments, Fractional Stable Noise (FSN) will be the model for the stationary incremental process, which describes the number of arrivals during a time period of unit length. In other words, a sample of a FSN process will describe the number of arrivals and , for any integer value of . between In this Appendix, we will describe a method to predict the value of a sample of a FSN process based on the information extracted from previous samples of the same process and from its dependence structure. Least square error methods are not used

Even though the autocorrelation function of FSN processes does not converge, it is shown in [10] that their normalized sample autocorrelation function does converge in probability to the summation in (B3)—provided that it itself converges—as the sample size increases to infinity. The norm will then be given by the square root of the inner product of a random variable with itself, as follows: (B5) From here, we can define the distance between two random variables as the norm of the difference between them. Notice that (B5) can be interpreted as a measure of the spread of the random

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variable , somehow equivalent to the standard deviation of a Gaussian random variable. th Now, assume we want to estimate the value of the sample of an FSN stochastic process, where the lag is a nonnegative integer, based on samples from the past, as follows:

APPENDIX C RECURSIVE COMPUTATION OF PREDICTION COEFFICIENTS In this Appendix, we will describe a recursive and very efficient algorithm to compute the prediction coefficients mentioned in Appendix B. Using matrix notation and taking into is an even function of ,(B10) becomes account that

(B6) where the s are constants to be determined. The prediction will be given by error

(B7) From (B5) and (B7), we can see that the distance between the predicted and the actual value or, equivalently, the norm of the satisfies the following equation: prediction error

(C1) where ; ; , for . In what follows, we will use a more explicit notation for the s, specifying the order of the predictor to which they correspond and the lag between the value to predict and the availth constant corresponding to a able samples. That is: and lag , . Likewise, predictor of order to denote the inner-product matrix of order . we will use From (C1) we can conclude that

(B8) does not depend on , we will denote it by . Since In order to optimize our predictor, we need to define the values in such a way as to minimize of the prediction coefficients . Notice that, since the norm is a positive function, minis equivalent to minimizing . Thus, the imizing solution of the set of equations

(C2) From (C1), using a predictor of order (C2) we have

, and subtracting

(C3) for (B9)

where (C4)

can be reduced to solving the set of linear equations

On the other hand, from (C1) the following expression holds as well: (B10) are as defined in (B4) where the coefficients above. Appendix C describes a recursive algorithm to compute the prediction coefficients, which is faster than the common approach of matrix inversion. This approach takes advantage of . the symmetry of the inner product It is simpler to find the prediction coefficients when the algorithm described in this Appendix is used than it is with the algorithm described in [15]. However, it was verified in [13] that they both give very similar performances. The difference between the two algorithms is the definition used for the distance between random variables.

(C5) where (C6) Now, from (C5)

(C7)


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From (C3) and (C7), and assuming that is a nonsingular matrix (we need this matrix to be nonsingular in order for the solution to (C1) to be unique) we obtain (C8a)

(C8b) Equation (C8) and the initial condition allow us to recursively compute the constants , . starting from It can be shown that the complexity of this algorithm is , whereas the common approach of matrix inversion [12]. An added advantage of this has a complexity of are approach is that the results for intermediate values of readily available, allowing the user to decide which final value of to use, based on the behavior of the norm of the prediction . error REFERENCES [1] S. Bates and S. McLaughlin, “The effective bandwidth of stable distributions,” in Proc. IEEE ICASSP, Seattle, WA, May 12–15, 1998. [2] S. K. Berberian, Introduction to Hilbert Space. London, U.K.: Oxford Univ. Press, 1961. [3] Internet Engineering Task Force, Draft in progress, Y. Bernet, A. Smith, and S. Blake, “A conceptual model for DiffServ routers,”, June 1999. [4] S. Blake, D. Black, M. Carlson, and Internet Engineering Task Force, RFC 2475, “An architecture for differentiated services,”, ftp://ftp.isi.edu/in-notes/rfc2475.txt, Dec. 1998. [5] J. C. Bolot and A. Vega-Garcia, “Control mechanisms for packet audio in the Internet,” in Proc. IEEE Infocomm, 1996, pp. 232–239. [6] J. C. Bolot and T. Turletty, “A rate control scheme for packet video in the Internet,” in Proc. IEEE Infocomm, 1994, pp. 1216–1223. [7] C. S. Chang, “Stability, queue length and delay of deterministic and stochastic queueing networks,” IEEE Trans. Automat. Contr., vol. 39, no. 5, pp. 913–931, May 1994. [8] C. S. Chang and J. A. Thomas, “Effective bandwidth in high-speed digital networks,” IEEE J. Select. Areas Commun., vol. 13, no. 6, pp. 913–931, Aug. 1995. [9] J. N. Daigle and J. D. Langford, “Models for analysis of packet voice communications systems,” IEEE J. Select. Areas Commun., vol. SAC-4, no. 6, pp. 847–854, Sept. 1986. [10] R. Davis and S. Resnick, “Limit theory for the sample covariance and correlation functions of moving averages,” The Annals of Statistics, vol. 14, no. 2, pp. 533–558, 1986. [11] N. G. Duffield and N. O’Connell, “Large deviations and overflow probabilities for the general single-server queue, with applications,” Math. Proc. Camb. Philos. Soc., vol. 118, pp. 363–374, 1995. [12] M. Friedman and A. Kandel, Fundamentals of Computer Numerical Analysis. Boca Raton, FL: CRC Press, 1994. [13] J. R. Gallardo, “Fractional stable noise processes and their application to traffic modeling and fast simulation of broadband telecommunications networks,” D.Sc. dissertation, The George Washington University, Washington, DC, 2000. [14] J. R. Gallardo, D. Makrakis, and L. Orozco-Barbosa, “Use of alphastable self-similar stochastic processes for modeling traffic in broadband networks,” Perform. Eval., vol. 40, no. 1–3, pp. 71–98, Mar. 2000. [15] , “Prediction of alpha-stable long-range dependent stochastic processes,” in IEEE/IEE ICT, vol. 2, Cheju, Korea, June 15–18, 1999, pp. 222–226. [16] P. W. Glynn and W. Whitt, “Logarithmic asymptotics for steady-state tail probabilities in a single-server queue,” J. Appl. Prob., vol. 31A (special issue), pp. 131–156, 1994. [17] K. Hyogon, “A Fair Marker,” Internet Engineering Task Force, Internet Draft, Work in progress, Apr. 1999. [18] H. Juha, “A single-rate three-color marker,” Internet Engineering Task Force, Internet Draft, Work in progress, Mar. 1999.

[19] F. P. Kelly, “Notes on effective bandwidths,” in Stochastic Networks: Theory and Applications. ser. Royal Statistical Society Lecture Notes, F. P. Kelly, S. Zachary, and I. B. Ziedins, Eds. London, U.K.: Oxford Univ. Press, 1996, vol. 4, pp. 141–168. [20] T. Kostas, M. Borella, I. Sidhu, G. Shuster, and J. Grabiec, “Real time voice over packet-switched networks,” IEEE Network, Jan./Feb. 1998. [21] I. Norros, “A storage model with self-similar input,” Queueing Syst., vol. 16, no. 3–4, pp. 387–396, 1994. [22] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991. [23] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes. London, U.K.: Chapman & Hall, 1994. [24] N. G. Duffield and N. O’Connell, “Large deviations and overflow probabilities for the general single-server queue, with applications,” Dublin Institute for Advanced Studies—Applied Probability Group, Tech. l Rep. DIAS-STP-93-30.

José R. Gallardo received the B.Sc. degree in physics and mathematics from the National Polytechnic Institute, Mexico City, Mexico, in 1988, the M.Sc. degree in electrical engineering from CICESE Research and Higher Education Center, Ensenada, Mexico, in 1991, and the D.Sc. degree in electrical engineering from the George Washington University, Washington, DC, in 2000. From 1991 to 1993, he worked as a member of the Academic Staff at CICESE. From February 1997 to April 2000, he worked as a Research Associate at the Advanced Communications Engineering Centre of the University of Western Ontario, London, ON, Canada. From May to December 2000, he worked as a Postdoctoral Fellow at the Broadband Wireless and Internetworking Research Laboratory of the University of Ottawa. He is currently with the Electronics and Telecommunications Department of CICESE Research Center. His main areas of interest are traffic modeling, traffic control, and performance evaluation of broadband communications networks.

Dimitrios Makrakis received the Diploma Ing. degree from the University of Patras, Greece, the MA.Sc. degree from the University of Toronto, ON, Canada, and the Ph.D. degree from the University of Ottawa, ON, Canada, all in electrical and computer engineering. From August 1995 to June 1996, he was an Assistant Professor in the Department of Electrical Engineering of Lakehead University, Thunder Bay, ON. From July 1996 to June 2000, he was with the Electrical and Computer Engineering Department of the University of Western Ontario, London, ON, initially as Assistant and later as Associate Professor and as Director of the Advanced Communications Engineering Centre, a research facility established in partnership with Bay Networks (presently NORTEL) and Bell Canada. Since January 2000, he has been with the School of Information Technology and Engineering (SITE) of the University of Ottawa, where he holds the rank of Associate Professor. His research activities are in the areas of wireless and optical networks, Internet-related technologies, network control and grade/QoS, teletraffic modeling, performance analysis, and multimedia over wireless. He led or participated in the past in several major projects and has led successfully large research groups. He maintains very close collaboration with the industry and has received considerable industrial funding. He has published more than 110 papers in refereed technical journals and conference proceedings. Dr. Makrakis was recently awarded a Canada Foundation for Innovation (CFI) grant, an Ontario Research and Development Challenge Fund (ORDCF) grant, and received the Premier’s of Ontario Research Excellence Award (PREACH).

Marlenne Angulo received the Diploma Ing. degree from the Sonora Institute of Technology, Obregon, Mexico, in 1996 and the M.Sc. degree in electrical engineering from CICESE Research and Higher Education Center, Ensenada, Mexico, in 2000. From March to December 1999, she was a Research Assistant at the Advanced Communications Engineering Centre of the University of Western Ontario, London, ON, Canada, working on projects related to network modeling, simulation, and performance analysis. She is currently with the Faculty of Engineering of the Autonomous University of Baja California, Mexicali, BC, Mexico. Her main areas of interest are next-generation Internet, differentiated services, ATM networks, and multimedia traffic modeling.