D. D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queuing Syst., vol. 20, pp. 293--320, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....concludes 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 ....

D. D. Botvich and N. G. Duffield, "Large Deviations, the Shape of the Loss Curve, and Economies of Scale in Larger Multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....as on Internet 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 ....

D. D. Botvich and N. G. Duffield, "Large Deviations, the Shape of the Loss Curve, and Economies of Scale in Larger Multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....multiplexing, scheduling, queuing analysis I. INTRODUCTION B OUNDS on the probability of buffer overflow in a network node (element) fed with independent arrival processes (inputs, flows) that are each constrained by arrival curves are obtained in [1] 2] 3] 4] 5] 6] 7] 8] [9], 10] 11] 12] under various assumptions. We say that a flow is regulated, or constrained, by an arrival curve if the number of bits observed on the flow during any time interval of duration is at most . Leaky bucket regulation corresponds to an affine function . ....

D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....in [7] Proposition 1: many sources asymptotic upper bound] Suppose that lim inf J t (0) log t 0. Then, we have, for any t, log P (t) Nb # I(b) b# 0, 4) where I(b) is given by I(b) inf #. 5) The assumption (3) is shown to be more general than the one used in [8] in that (3) holds even for on off sources with heavy tailed on time distributions (see [7] For the classical treatment of the many sources asymptotic results, we refer to the papers [8] 9] B. Problem description and model assumptions We consider a two stage queueing system shown in Figure ....

....is given by I(b) inf #. 5) The assumption (3) is shown to be more general than the one used in [8] in that (3) holds even for on off sources with heavy tailed on time distributions (see [7] For the classical treatment of the many sources asymptotic results, we refer to the papers [8], 9] B. Problem description and model assumptions We consider a two stage queueing system shown in Figure 2. In this figure, the upstream queue (with queue length (t) represents a node that is capable of serving a large number of traffic flows in a network, while the downstream queue (with ....

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D.D.BotvichandN.Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....to obtain the expected c and the long run total rate of revenue for the case of highly variable VBR layers, i.e. when the additive effective bandwidth approach becomes overly conservative. Our method relies on the extensive literature on refined loss calculations at multiplexers, e.g. [53], 54] 55] 56] 57] 58] With these refined loss calculations we perform admission control to enforce a limit on the probability of loss (i.e. buffer overflow) at the bottleneck link, where typically, A refined loss calculations give very accurate estimates of the loss ....

....in the proxy ( V Me ) we check whether the loss probability on the bottleneck link would exceed the prespecified limit when the layers are added to the current link load k. Given the frame sizes B of the prerecorded videos, this is straightforward by applying the techniques in [53], 54] 55] 56] 57] 58] If the loss probability limit continues to be met with the additional layer(s) there is no blocking. We increment the earned revenue by and update N and k. Otherwise, i.e. if the loss probability limit would be exceeded with the additional layer(s) we count ....

D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....multiplexing, scheduling, queuing analysis I. INTRODUCTION B OUNDS on the probability of buffer overflow in a network node (element) fed with independent arrival processes (inputs, flows) that are each constrained by arrival curves are obtained in [1] 2] 3] 4] 5] 6] 7] 8] [9], 10] 11] 12] under various assumptions. We say that a flow is regulated, or constrained, by an arrival curve if the number of bits observed on the flow during any time interval of duration is at most . Leaky bucket regulation corresponds to an affine function . Existing ....

D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....B curve for a specific transmission rate p provides a way to obtain the resulting loss rate when using a given buffer size to send a video source at a certain transmission rate. We define by loss curve the e versus B curve described above. The term has been used in the literature in a similar way [6]. Using the loss curve, buffer space can be traded off at the expense of higher loss rate in realistic networked video applications. Besides, there are cases in which it is necessary to obtain the burstiness curve under the constraint that at most B bits of buffer space is available in the queue ....

....the effect of statistical mul tiplexing, and hence allocating network resources based on these models would be conservative. Both Choudhury et al. 14] and Shroff et al. 66] verified the limitations of the effective bandwidth techniques. Based on these results, new schemes have been proposed [6, 11, 12, 43] to calculate the bandwidth used for admission control that improve network utilization significantly without the need for large amount of buffering. 37 Statistical Service with Rate Renegotiation: Two recent schemes have been proposed in the literature based on this approach. Renegotiated CBR ....

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D. D. Botvich and N. G. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, Theory and Applications, vol. 20, no. 3-4, 293-320, 1995.

....a super additive service curve . Then, under (A1) A4) 4R476 8 L8 s v ; #eH T U k . T # U (5) Proof: Appendix I. Note that (5) and other bounds in [9] satisfy the economy of scale, a notion originally introduced by Botvich and Duffield [16]. It means that if we scale , then the probability to overflow decays exponentially with . We also note that with fixed aggregate arrival curve, the bound in (5) is tightest for all the inputs having identical arrival curves X Y 476 8 l . We call this the economy of equality; it ....

D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....service curve . Then, under (A1) A4) for any , it holds DR, d R, q (q R ( R SC e i = 3) 0 ; N 0 . Proof: Appendix A. The bound (3) and our other bounds in [9] satisfy the economy of scale, a notion originally introduced by Botvich and Duffield [15]. It means that if we scale Y , then the probability to overflow decays exponentially . We also note that with fixed aggregate arrival curve, the bound in (3) is tightest for all the inputs having identical arrival curves 9 : 9 ( We call this the economy of equality; it ....

D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....a large number 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] ....

....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] 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 ....

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D. D. Botvich and N. G. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, pp. 293--320, 1995.

...., 4 4 6 8 8 s v ; H (5) for any , and any M M M . Proof: Appendix I. Note that (5) and other bounds in [9] satisfy the economy of scale, a notion originally introduced by Botvich and Duffield [16]. It means that if we scale and 3 as 4 r 8 , then the probability to overflow decays exponentially with r . We also note that with fixed aggregate arrival curve, the bound in (5) is tightest for all the inputs having identical arrival curves X Y 4 6 8 l .We call this the economy of ....

D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....in the context of the manysources asymptotics [20] 28] We next compare this approach with an approximation based on the many sources asymptotics. The many sources asymptotics have been widely studied and can be found in many papers on queueing analysis using large deviation technique [5] [30], 31] 32] Most of the papers deal with the tail distribution rather than the loss probability. In [9] the authors developed the first result on the loss probability based on the many sources asymptotics. We call this the Likhanov Mazumdar (L M) approximation for loss. Since the L M result was ....

D. D. Botvich and N. G. Du#eld, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....concludes 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 ....

D. D. Botvich and N. G. Duffield, "Large Deviations, the Shape of the Loss Curve, and Economies of Scale in Larger Multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....level x within a relatively small interval around the DTS. For further results and explanations on this, see [12] 13] and references therein. For fairly general processes, there has recently been a large body of work that employs the notion of time scale based on large deviation theory [8] [14], 15] This effort has focused on the asymptotic behavior of the buffer overflow probability when the number of sources, the queue length, and the service rate are all proportionally sent to infinity. Let L(C; B; n) be the buffer overflow probability of a buffer size B with service rate C and n = ....

.... b) st J X j=1 n j j (s; t) 4) where j (s; t) log Efexp(sX j [0; t] g=st is the effective bandwidth of a source of type j (see [16] for details) This equation is referred to as the many sources asymptotic and has been proven for discrete time in [15] and for continuous time in [14]. From this equation, the overflow probability can be written as PfQ Nbg = e NI o(N) e NI : 5) In order to use the many sources asymptotic, we need to calculate the extremizing parameters in (4) Let s and t be the extremizing parameters over the sup and inf, respectively. The ....

D. D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....In brief, 2.1, 2. 2 review and extend recent work suggesting that the cumulative log moment generating function L t (q) logE exp[qA(0; t] q 2 R; t 0 of the arrivals over time intervals is a useful representation of the traffic which translates directly to buffer overflow characteristics [1, 15]. We will see that two parameters q t ; t define the space and time scales of interest to determine overflow probabilities in a buffered link. Next we consider the case where only the second order characteristics are important in determining performance, and, in particular, the key traffic ....

....and practical significance of such results. Finally we consider the case of Gaussian arrivals processes where results take a particularly simple form. 2.1 Large deviations We begin our study by briefly discussing an asymptotic study of a multiplexer supporting a large number N of i.i.d. streams [1, 3]. Let A N t = N i=1 A i ( Gammat ; 0] denote the aggregate arrivals over an interval of length t. As shown in Fig. 1 resources are also scaled in N, thus we identify a buffer and capacity per stream, b;c. Using large deviations, for large N we can estimate the probability that over a time ....

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D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers, " Tech. Rep. DIAS-APG-94-12, Dublin Institute for Advanced Studies, 1994.

.... distribution has focused on the asymptotic tail behavior of this distribution; i.e. the asymptotic behavior of P( #X# x ) or equivalently P( Q x ) The theory of Large Deviations has been widely used providing very general and elegant results on the asymptotic behavior of log P( #X# x )[3], 10] 11] see [9] for more details about general Large Deviation techniques) For example, in [11] using Large Deviation techniques, it has been shown for a large class of stochastic processes that log P( #X# x ) x## # #x, 2) where the asymptotic decay rate, #, is a positive constant ....

....paper) is about x asymptotics i.e. the asymptotic behavior of P( Q x ) as the queue length x increases. There has been recent work that focuses on the asymptotic behavior of P( Q x )when the number of sources, the queue length, and the service rate are all proportionally sent to infinity (e.g. [3], 16] We classify these studies as M asymptotics, where M represents the number of sources in the system. In particular, Montgomery and De Veciana [16] have significantly strengthened the corresponding 0 7803 5420 6 99 10.00 (c) 1999 IEEE log similarity relation in [3] using Bahadur Rao ....

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D. D. Botvich and N. G. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....approaching traffic overload, without the need for source traffic characterization, is provided by the Virtual Queue mechanism: a resource marks packets or not depending on the state of a fictitious queue, of lower capacity than the real queue. In Section III D we use the many sources asymptotic [27], 28] 29] to provide important insights into the relevant packet level timescales, and thus into the robustness of such mechanisms. Finally, in Section IV, we conclude. II. Connection level network models We now explore more fully how an end system or user might decide whether or not to enter ....

....traffic types, at the cost of a more elaborate notation. Suppose a connection generates a workload for the resource of X[0; t] in time [0; t] assume X has stationary increments, and that the workloads generated by different connections are independent. Then the many sources asymptotic regime [27], 28] 29] shows that for large systems log P (c; b; n) sup t inf s fstnff(s; t) Gamma s (b ct)g : 26) where ff (s; t) 1 st log E h e sX[0;t] i 0 s; t 1 (27) is the effective bandwidth of an individual source. Let (s ; t ) be an extremal pair for Equation (26) then t ....

D. D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

....of the buffer contents normalized by the number N of users is given by the Lindley equation, W t 1 = max[0; X t W t Gamma f 0 s ] 6) where f 0 s = f s = N 53 8) is the per user service rate in cells sec. Iterating on this equation, we have the following for the stationary queue content [9], W = max t [S t Gamma tf 0 s ] 7) where S t = P t i=1 X i . Now define S t (r) Efe r S t g; 8) 2,s 21 12 1,s High Low l a a l Fig. 5. 2 State MMPP Source Model. and assume the following are true: ffl Assume that the limit (r) lim t 1 1 t ln S t (r) 9) exists ....

.... principle [10] with good rate function, I(a) given by the Fenchel Legendre transform of the limitinglog moment generating function, r) and is given by, I(a) sup r [r a Gamma (r) 11) If the sequence fSn=ng satisfies the large deviation principle with rate function I then it follows that [9], 11] 12] P (W b) e Gammab fl ; 12) where b = B=N is the normalized buffer size on a per source basis and fl is the solution to the following equation, fl = supfr : r) rf 0 s g: 13) Specifically, for identical MMPP sources fed into a multiplexer with constant service rate, we ....

[Article contains additional citation context not shown here]

D. D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Tech. Rep. DIASSTP -94-12, Dublin Institute for Advanced Studies, The Applied Probability Group, Dublin, Ireland, 1994.

....[16, 4] may also be used to establish results for the continuous time GPS system. The paper deals only with the large buffer asymptotics under the GPS scheduling. Another future direction is to study the asymptotic behavior of the GPS scheduling with a large number of sources a la the methods of [22, 3]. Zhi Li Zhang Large Deviations and GPS for a Two Queue System 17 Acknowledgments I am indebted to Richard S. Ellis for teaching me Large Deviation Theory and to my advisor, Don Towsley, for encouragement and guidance. I am also grateful to the anonymous referee for pointing out mistakes in ....

D. D. Botvich and N. G. Duffield, "Large Deviations, the Shape of the Loss Curve, and Economies of Scale in Large Multiplexers", Queueing Systems, Vol. 20, No. III-IV, 1995.

....bound gives the correct exponent. So far, only a bufferless link confronted with demand that is constant over all time has been considered. The notion of effective bandwidth can be extended to cover sources of data that vary in time, but that are statistically stationary and mutually independent [45], 46] 47] Let X ji [a; b] denote the amount of data generated by the ith connection of type j during an interval [a; b] We assume that the process X is stationary in time. Set ff j (s; t) 1 st log E[e sX ij [0;t] 6) For t fixed, the function ff j is the same as the oneparameter ....

D.D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, 293-320, 1995.

....t) Gamma s(b ct) 14) scaling resources to C = Nc, B = Nb, and N ae j sources of type j. This result is based on Large Deviations theory, which is also applied in several other approaches above and has been widely used in providing general results on the asymptotic behavior of log P (Q B) [4, 13, 21]. Here, we briefly review related Large Deviations techniques. In [21] for example, Glynn and Whitt show that for a large class of stochastic processes log P (Q B) GammaffiB: 15) However, for many important types of processes, such as self similar or other long range dependent processes [3, ....

.... (note that compared to Equation (3) which shows similarity, the large deviation results only show log similarity) Recent work has focused on the asymptotic behavior of P (Q B) when the number of sources, the queue size, and the service rate are all proportionally sent to infinity (e.g. [4]) This limit is quite a different limit from the one in Equation (16) However, such results have generated approximations such as the one in [4] that when applied to Gaussian processes produce the same expression as the MVA upper bound discussed above. This approach also allows for the use of ....

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D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers, " Queueing Systems, vol. 20, pp. 293--320, 1995.

No context found.

D. D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queuing Syst., vol. 20, pp. 293--320, 1995.

No context found.

D. Botvich and N. Du#eld, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

No context found.

D. D. Botvich and N. G. Du#eld, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems Theory Appl., vol. 20, no. 3-4, pp. 293--320, 1995.

No context found.

D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems, vol. 20, pp. 293--320, 1995.

No context found.

D. D. Botvich and N. G. Duffield, "Large Deviations, the Shape of the Loss Curve and Economies of Scale in Large Multiplexers", Technical Report DIAS-APG-94-12 , Dublin City Univ., May 1994.

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D. Botvich and N. Duffield, "Large deviations, the shape of the loss curve, and economies of scale in large multiplexers," Queueing Systems Theory Appl. Vol. 20, 1995.

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