N. G. Duffield, Economies of scale in queues with sources having power-law large deviation scalings, J. Appl. Prob. vol. 33, pp. 840--857, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....with the statement [9] This is clearly an issue of practical importance, and there is considerable scope for further work. However, until recently, there has been little study of the effect of multiplexing; most of the work has been theoretical studies of queueing behavior [9] 18] 19] [20], 21] 22] Also, through the experiences of network operators, it has been appreciated that the statistical variability of packet counts and byte counts, relative to the mean, decreases with the NAC; this is sometimes referred to as increased smoothness of the traffic. Recently, the statistical ....

N. G. Duffield, "Economies of Scale in Queues with Sources Having Power-Law Large Deviation Scaling," Queueing Systems, vol. 33, pp. 840--857, 1996.

....backbone links, the correlations [of longrange dependent traffic] while present, have little actual effect because the variance of the packet arrival process is quite small. In addition, there were theoretical discussions of the implications of increased multiplexing on queueing [13] 14] [15], 16] 17] But the problem with such theoretical study is that results depend on the assumptions about the individual traffic sources being superposed, and different plausible assumptions lead to different results. Without empirical study, it was not possible to resolve the uncertainty about ....

N. G. Duffield, "Economies of Scale in Queues with Sources Having Power-Law Large Deviation Scaling," Queueing Systems, vol. 33, pp. 840--857, 1996.

....because it can influence the amount of network resources required to provide a certain quality of service (QoS) to network users. This is due to the fact that the tail probability of the buffer distribution does not decay exponentially fast when the arrival process is long range dependent [18, 11, 9]. In some cases, such as real time video traffic, the effects of long range dependence can be captured through short range dependent Markovian models with a sufficient number of states [14, 21] In general, however, models that capture the longdependence property are essential to accurately ....

N. G. Duffield. Economies of scale in queues with sources having power-law large-deviations scalings. Journal of Applied Probability, 33, 1996.

....of sources are multiplexed together. Some authors contend that even high levels of aggregation would not mitigate the burstiness of long range dependent traffic [1] 2] On the other hand, others have shown that in the presence of multiplexing some smoothing does indeed take place [6] 7] [8], 9] 10] 11] 12] For instance in the context of traffic engineering for ATM multiplexers of VBR video sources it is shown in [9] that long term correlations do not have a significant impact on the cell loss rate when ATM buffers are of realistic dimensions. In [6] 7] 8] 10] 11] ....

....place [6] 7] 8] 9] 10] 11] 12] For instance in the context of traffic engineering for ATM multiplexers of VBR video sources it is shown in [9] that long term correlations do not have a significant impact on the cell loss rate when ATM buffers are of realistic dimensions. In [6] 7] [8], 10] 11] 12] a system with n independent identical sources feeding into a link with processing rate O(n) is considered and (under various assumptions on the input processes) a large deviations analysis is used to show that the steady state probability of the buffer J. Cao is with the ....

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N. G. Duffield, "Economies of scale in queues with sources having powerlaw large deviations scalings," Journal of Applied Probability, vol. 33, pp. 840--857, 1996.

....the fact that independent users may not be simultaneously active. In contrast, heavy tailed queueing behavior suggests that there may not be much scope to obtain multiplexing gains within a source, though there is considerable potential to achieve substantial multiplexing gains across sources ([3]) Heavy tailed service time distributions can also have significant performance and engineering impacts, even in situations in which classical teletraffic results are known to be highly robust. An example is the blocking performance of a group of servers whose holding times can span many time ....

N. G. Duffield. Economies of scale in queues with sources having power-law large deviation scalings. Preprint, 1994.

....with the statement [9] This is clearly an issue of practical importance, and there is considerable scope for further work. However, until recently, there has been little study of the effect of multiplexing; most of the work has been theoretical studies of queueing behavior [9] 18] 19] [20], 21] 22] Also, through the experiences of network operators, it has been appreciated that the statistical variability of packet counts and byte counts, relative to the mean, decreases with the NAC; this is sometimes referred to as increased smoothness of the traffic. Recently, the statistical ....

N. G. Duffield, "Economies of Scale in Queues with Sources Having Power-Law Large Deviation Scaling," Queueing Systems, vol. 33, pp. 840--857, 1996.

....work to reach a conclusion. Specifically, there has been insufficient empirical study of the detailed statistical properties of packet inter arrivals and packet sizes as a function of the load. There have been a number of theoretical discussions about possible implications of increases in load [1, 3, 10, 11]. The load depends on the amount of superposition (statistical multiplexing) of traffic sources, so one can study the implications of more and more superposition. The problem is that results depend heavily on the assumptions about the individual traffic sources being superposed, and theory can ....

N. G. Duffield. Economies of Scale in Queues with Sources Having Power-Law Large Deviation Scaling. Queueing Systems, 33:840--857, 1996.

....because it can influence the amount of network resources required to provide a certain quality of service (QoS) to network users. This is due to the fact that the tail probability of the buffer distribution does not decay exponentially fast when the arrival process is long range dependent [16, 9, 7]. In some cases, such as real time video traffic, the effects of long range dependence can be captured through short range dependent Markovian models with a sufficient number of states [12, 19] In general, however, models that capture the long dependence property are essential to accurately ....

N. G. Duffield. Economies of scale in queues with sources having power-law large-deviations scalings. Journal of Applied Probability, 33, 1996.

....size s required to ensure that the traffic stream can be transmitted with zero loss. We relax this constraint, by permitting a small non zero loss probability of e 0. Large deviations results have been produced for systems where the service rate and buffer size are scaled together [6] [7]. These results have indicated that scaling the amount of buffer space available proportionately with the number of multiplexed traffic streams does allow for less capacity to be provided. In this paper we extend these results downwards from the highly multiplexed cases for which large ....

....processes in Section VI. As [1] shows, where x 1 is not a valid assumption, effective bandwidths calculated in this fashion can be extremely inaccurate. If the traffic stream is long range dependent (LRD) the rate of decay of the queue length distribution may be much slower than exponential [7], 8] so Equation (2) is not applicable. III. EFFECTIVE BANDWIDTHS A NEW DEFINITION The conventional definition of effective bandwidths assumes that the buffer size k is a constant which is not scaled up as the capacity increases. We consider the possibility of the buffer size being ....

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N. G. Duffield, "Economies of scale in queues with sources having power-law large deviation scalings," Journal of Applied Probability, vol. 33, pp. 840--857, 1996.

....size L (1 Gammafl=2) B. 3 3.2 Related work The work in this chapter and the one preceding was motivated by the results of Courcoubetis and Weber [11] and Botvich and Duffield [4] who find large deviations rate functions for the amount of work in a queue and the event of overflow. Duffield [17] has treated separately the case of nonlinear scaling functions. These authors proved their results directly, but we will start with the sample path LDP and apply the contraction principle. Ours is a more general method, and it lets us fill in some gaps: in particular, we give the large deviations ....

....shows that this function is continuous on X for any C. By the Contraction Principle, this immediately gives Corollary 3. 1: an LDP for workload in queues with infinite buffers, which when simplified duplicates the results of Botvich and Duffield [4] for linear scaling functions v(t) of Duffield [17] for general scaling functions, and of Simonian and Guibert [52] for the special case of Markov modulated fluid sources. Corollary 3.1 Under the conditions of Theorem 2.7, if X L has mean rate less than C then Q(X L ) satisfies an LDP with good rate function I(b) inf x2XC :Q(x) b I(x) ....

N. G. Duffield. Economies of scale in queues with sources having power-law large deviation scalings. Journal of Applied Probability, 33:840--857, 1996.

....discussion concludes with the statement [12] This is clearly an issue of practical importance, and there is considerable scope for further work. However, until recently, there has been little study of the effect of multiplexing; most of the work has been theoretical studies of queuing behavior [1, 4, 11, 12]. Also, through the experiences of network operators, it has been appreciated that the statistical variability of packet counts and byte counts, relative to the mean, decreases with the ACL; this is sometimes referred to as increased smoothness of the traffic. Recently, the statistical properties ....

N. G. Duffield. Economies of Scale in Queues with Sources Having Power-Law Large Deviation Scaling. Queueing Systems, 33:840--857, 1996.

....[9] noticed a change in the Hurst parameter as the network utilization changes. There have been some theoretical treatments of the effect of a changing amount of superposition on queueing. Queue length changes with an increasing number of superpositions while holding the utilization fixed [4, 2, 12, 11]. However, in these treatments, the probability of exceeding a level that grows with k, the number of superpositions, is studied for a fixed interval of time; but it seems to us more appropriate to let the time interval get smaller with the rate since we would not expect buffers to get arbitrarily ....

N. G. Duffield. Economies of Scale in Queues with Sources Having Power-Law Large Deviation Scaling. Queueing Systems, 33:840--857, 1996.

.... , 2) for an increasing positive function w( Compared to the variational problem of (1) the optimum is attained at the same t # and a value of # # that is t # w(t # ) times as large. 9 The function w( in Corollary 3.4 is usually called a scaling function. It was introduced in [14, 16] to enable large deviations analysis in situations where there is no exponential decay in the bu#er size. For the model with M G # input, it was proposed in [33] to use w(t) log IP(D # t) where D # is the residual session duration. In the sequel we use the scaling w(t) v(t) v A ....

....(i) For all x # (#, r) lim t## J t (x) x # r # . ii) The convergence in (i) is uniform on compact subsets of (#, r) Proof (i) Notice that J t (x) is the Legendre Fenchel transform of the cumulant function (3) The following result is established in the proof of Theorem 2 of [14]. Let f t be a sequence of convex functions that converge pointwise to f on the interior of the e#ective domain of f . Then also the Legendre Fenchel transforms f # t (x) sup # (#x f t (#) converge to f # (x) sup # (#x f(#) on the interior of the e#ective domain of f # . Therefore, ....

N. Duffield. Economies of scale in queues with sources having power-law large deviation scalings. Journal of Applied Probability, 33: 840--857, 1996.

....work to reach a conclusion. Specifically, there has been insufficient empirical study of the detailed statistical properties of packet inter arrivals and packet sizes as a function of the load. There have been a number of theoretical discussions about possible implications of increases in load [1, 3, 10, 11]. The load depends on the amount of superposition (statistical multiplexing) of traffic sources, so one can study the implications of more and more superposition. The problem is that results depend heavily on the assumptions about the individual traffic sources being superposed, and theory can ....

N. G. Duffield. Economies of Scale in Queues with Sources Having Power-Law Large Deviation Scaling. Queueing Systems, 33:840--857, 1996.

....include: Traffic aggregation effects. The aggregate network traffic, even if from multiple independent sources, will still be self similar. Bursts will exist across many time scales and positive correlations in traffic will adversely affect the quality of service (QOS) provided to network users [7, 8]. The cell loss ratio (CLR) in a network with self similar traffic may be several orders of magnitude higher than that predicted by the traditional Markovian traffic models [3] Buffer ineffectiveness. Increasing the buffer sizes used in the network will have marginal impact on the cell loss ....

N. Duffield, "Economies of Scale in Queues with Sources Having Power-Law Large Deviation Scalings", Journal of Applied Probability, Volume 33, pp. 840-857, 1996.

....Normal distribution. Using the approximation 1 Gamma Phi(y) exp( Gammay 2 =2) he suggests as an approximation: P(Z x) exp( Gamma (C Gamma m) 2H 2H 2H (1 Gamma H) 2 Gamma2H am x 2 Gamma2H ) 2. 3) Using the theory of large deviations, Duffield and O Connell [21] see also [20]) have shown that the approximation (2.3) is logarithmically accurate for large x. For H = 1=2, FBM reduces to Brownian motion and the Weibull distribution in (2.3) reduces to an exponential distribution. A third way to introduce LRD in an input process is to take a fluid queue fed by a single ....

....Brownian Motion: see (2.2) and (2.3) Brichet et al. show via some scalings and a limiting operation, how the FBM model of Norros is related to their fluid queue, thus providing a further physical motivation for the FBM model. 6. Conclusions and suggestions for further research 33 Duffield [20] studies the workload of a buffer fed by N sources which may be of a very general nature (not necessarily on off fluid) and which may display long range dependence. More specifically, he assumes that the input process of each source has a power law large deviation scaling (as is the case in FBM) ....

N.G. Duffield. Economies of scale in queues with sources having power-law large deviation scalings. Journal of Applied Probability, 33:840--857, 1996.

.... drifts due to Choe and Shroff [2] ONOFF inputs with long tailed ON periods Jelenkovic and Lazar [5] and Heath, Resnick and Samorodnitsky [4] M=G=1 type of inputs with long tailed G distributions by Parulekar and Makowski [10]and Liu, Nain, Towsley and Zhang [8] The early work of Duffield [3] must also be mentioned in this context where bounds on the logarithmic decay of the complementary workload distribution were obtained. An excellent account of the state of the art can be found in the recent survey of Boxma and Dumas [1] There have been several type of large buffer asymptotics ....

....Proof: This follows from the fact that in the homogeneous case : J 0 = b C Gamma ae r c 1 and is unique. The result (III.31) coincides exactly with the result in Liu et al. [Corollary 4.1] 8] where an additional assumption that r (C Gamma ae) has been made. An earlier paper of Duffield [3] showed that the upperboundon the logarithmic equivalent was 1 Gammafi(C Gamma ae) assuming r = 1. Clearly for the case r = 1 it is trivial to check that our equivalent which is given by Gamma(bC Gamma aec 1)fi 1 Gamma fi(C Gamma ae) which implies that our result is tighter and moreover ....

N. G. Duffield; Economies of scale in queues with sources having powerlaw large deviation scalings, Journal of Applied Probability, 33, 1996, pp. 840-857.

....q in (5) see, e.g. 6, 13, 20] where can be viewed as an indication of the statistical multiplexing gains. In the same vein, more recent work has been conducted in investigating effective bandwidth approximations to loss probability when there are a large number of sources (see, e.g. [9, 3, 11]) The class based call admission control schemes proposed in this paper can be modified to incorporate this new approximation so that the statistical multiplexing gain due to large number of sources can be exploited. 2.2 Generalized Processor Sharing (GPS) Scheduling Generalized Process Sharing ....

N. G. Duffield, Economies of Scale in Queues with Sources Having Power-Law Large Deviation Scalings, Preprint, March 1995.

....aggregation effects. The aggregate network traffic, even if from multiple independent sources, will still be self similar. Bursts will exist across many time scales and positive correlations in traffic will adversely affect the quality of service provided to network users [Duffield et al., 1995; Duffield, 1996]. The cell loss ratio (CLR) in a network with self similar traffic may be several orders of magnitude higher than that predicted by the traditional Markovian traffic models used in most network planning studies [Chen et al., 1995] Buffer ineffectiveness. Increasing the buffer sizes used in the ....

N. Duffield, "Economies of Scale in Queues with Sources Having Power-Law Large Deviation Scalings", Journal of Applied Probability, Volume 33, pp. 840-857, 1996.

....because it can influence the amount of network resources required to provide a certain quality of service (QoS) to network users. This is due to the fact that the tail probability of the buffer distribution does not decay exponentially fast when the arrival process is long range dependent [18, 11, 9]. In some cases, such as real time video traffic, the effects of long range dependence can be captured through short range dependent Markovian models with a sufficient number of states [14, 21] In general, however, models that capture the longdependence property are essential to accurately ....

N. G. Duffield. Economies of scale in queues with sources having power-law large-deviations scalings. Journal of Applied Probability, 33, 1996.

.... analysis) Starting with the work by Norros [35] there has been mounting evidence that clearly shows that the performance of queueing models with self similar inputs can be radically different from the performance predicted by traditional traffic models, especially by Markovian models (e.g. see [9, 8, 12]) Here we complement this evidence by illustrating the practical relevance of our findings for (i) parsimonious traffic modeling for high speed networks, ii) efficient simulation of actual network traffic, and (iii) analyzing queueing models and protocols under realistic traffic scenarios. Two ....

....concept of equivalent bandwidth [10] used in a number of proposed call admission control schemes. At the same time, the presence of the Joseph Effect in measured traffic does not preclude economies of scale (i.e. statistical multiplexing gains) by multiplexing a large number of such sources (see [8]) In view of our physical explanation that the Joseph Effect in aggregate LAN traffic is caused by the Noah Effect in the individual ON OFF sources that generate the aggregate stream, understanding the impacts of the Noah Effect in simple ON OFF source models on queueing performance becomes ....

N. G. Duffield. Economies of Scale in Queues with Sources having Power-Law Large Deviation Scalings. Preprint, 1994.

....efficacy of our FPP models in evaluating queueing performance of various types of real traffic sources exhibiting fractal behavior. 3 Gaussian Approximation of FPPs Recently, queueing analysis with self similar processes has received a considerable interest both from researchers and practitioners [5, 11, 14, 17]. While several new queueing results appear using different types of self similar processes, those based on the Gaussian property of input processes are well known since those cases permit tractable analysis [14, 15, 17, 24] One such process is the fractional Gaussian process (fGn) Y j fY n ; n = ....

N. G. Duffield. Economies of scale in queues with sources having power-law large deviation scalings. J. Appl. Prob., 33:840--857, 1996.

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N. G. Duffield, Economies of scale in queues with sources having power-law large deviation scalings, J. Appl. Prob. vol. 33, pp. 840--857, 1996.

....at high utilization, even in the presence of high variability. This has been established by studies using trace driven simulation [9] by model based simulation [28] and supported by the analysis of the queueing properties of models of long range dependent traffic sources and their aggregates [14]. But in order to provide statistical guarantees, it is necessary to know those characteristics of the traffic which determine the bandwidth required by the aggregate in order to obtain the desired QoS. Whereas this may be predicted from models, in practice the parameters of the model are not ....

....we apply it in the Markovian case. iv) The framework is sufficiently general to allow extensions to other regimes. In Section 5 we extend the theory to cover admission to buffered resources in the many source asymptotic in the large deviation formalism that has been described by several authors [2, 7, 14, 42, 45]. The CAC algorithm is based upon estimation of the central objects used in these cited papers, namely the transient cumulant generating function of the arrival process. This approach has been implemented in [35] vi) Finally, in Section 6 we consider the problem of characterizing the effective ....

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N. G. Duffield, Economies of scale in queues with sources having power-law large deviation scalings, J. Appl. Prob., 33:840--857, 1996.

....with multiplexing a large number of sources. The everincreasing network bandwidth implies that more and more sources will be able to be multiplexed. This gain is generally possible, even in the presence of long tailed distributions and more general long range dependence; e.g. see Duffield [8, 9] for demonstration of the multiplexing gains available for long range dependent traffic in shared buffers. As the scale increases, describing the detailed behavior of all sources becomes prohibitively difficult, but fortunately it becomes easier to describe the aggregate, because the large numbers ....

N. G. Duffield, "Economies of scale in queues with sources having power-law large deviation scalings", J. Appl. Prob. vol. 33, pp. 840--857, 1996.

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