D. Heath, S. Resnick, and G. Samorodnitsky, "How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails," Annals of Applied Probability, pp. 352--375, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....another instance of an analytically promising model. This is because M G # processes have both a renewal and Poisson structure. Samples of recent results and additional references on both fluid and discrete time queues with M G # arrival processes can be found in [11] 4] 14] 15] 16] [17], 18] 19] 20] However, the understanding of multiplexing a finite number of heavy tailed On Off arrival processes is quite limited. General bounds can be found in [21] 22] In this paper we derive explicit asymptotic results for approximating the stationary overflow probability and loss ....

....active are equal to (# # , # # , # #### ) respectively. Random variables # ###### # ## ###### # # # # ### are i.i.d. Because of the probabilistic sample path techniques that we use in the paper our proofs require the following minor technical assumption. Similar assumptions can be found in [17], 28] and, most recently, in [25] Assumption 1: The capacity of the queueing system satisfies the following # ## # # # ### ## # ## # # # # ### # # # ### # # # ### #### # # # # where # ### # ###### # # and # # ### # # # # ##. Remark: If this assumption is not satisfied, by choosing ....

D. Heath, S. Resnick, and G. Samorodnitsky, "How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails," Annals of Applied Probability, pp. 352--375, 1999.

....the so called M G 1 process, represents another instance of an analytically promising model. This is because the M G 1 processes have both a renewal and Poisson structure. Recent results and additional references on both fluid and discrete time queues with M G 1 arrival processes can be found in [5, 21, 28, 33, 11, 16, 30, 18]. However, the understanding of multiplexing a finite number of heavy tailed On Off arrival processes is quite limited. General bounds can be found in [7, 12] In this paper we derive 2 10 4 10 5 10 6 10 7 10 4 10 3 10 2 10 1 10 0 File lenght, bytes Figure 1: Log log plot of ....

....being On are equal to (r i ; ae i ; p on;i ) respectively. Random variables f on; i) on; i) j g n i j=1 are i.i.d. Because of the probabilistic sample path techniques that we use in the paper our proofs require the following minor technical assumption. Similar assumptions can be found in [16, 25] and, most recently, in [36] Assumption 5.1 The capacity of the queueing system satisfies the following c 62 ( M X i=1 [k i (r i Gamma ae i ) n i ae i ] k 1 ; kM ) 2 M O i=1 [0; n i ] and M X i=1 n i r i c: Remark: If this assumption is not satisfied, by choosing ....

D. Heath, S. Resnick, and G. Samorodnitsky. How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails. Annals of Applied Probability, pages 352--375, 1999.

....another instance of an analytically promising model. This is because M G 1 processes have both a renewal and Poisson structure. Samples of recent results and additional references on both fluid and discrete time queues with M G 1 arrival processes can be found in [11] 4] 14] 15] 16] [17], 18] 19] However, the understanding of multiplexing a finite number of heavy tailed On Off arrival processes is quite limited. General bounds can be found in [20] 21] In this paper we derive explicit asymptotic results for the stationary overflow probability and loss rate in a finite ....

....active are equal to (r i , ae i , p on;i ) respectively. Random variables f on; i) on; i) j g n i j=1 are i.i.d. Because of the probabilistic sample path techniques that we use in the paper our proofs require the following minor technical assumption. Similar assumptions can be found in [17], 27] and, most recently, in [24] Assumption 1: The capacity of the queueing system satisfies the following c 62 ( M X i=1 [k i (r i Gamma ae i ) n i ae i ] k 2 M O i=1 [0; n i ] where k = k 1 ; kM ) and P M i=1 n i r i c. Remark: If this assumption is not ....

D. Heath, S. Resnick, and G. Samorodnitsky, "How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails," Annals of Applied Probability, pp. 352--375, 1999.

....another instance of an analytically promising model. This is because M G 1 processes have both a renewal and Poisson structure. Samples of recent results and additional references on both fluid and discrete time queues with M G 1 arrival processes can be found in [11] 4] 14] 15] 16] [17], 18] 19] However, the understanding of multiplexing a finite number of heavy tailed On Off arrival processes is quite limited. General bounds can be found in [20] 21] In this paper we derive explicit asymptotic results for the stationary overflow probability and loss rate in a finite ....

....active are equal to (r i , ae i , p on;i ) respectively. Random variables f on; i) on; i) j g n i j=1 are i.i.d. Because of the probabilistic sample path techniques that we use in the paper our proofs require the following minor technical assumption. Similar assumptions can be found in [17], 27] and, most recently, in [24] Assumption 1: The capacity of the queueing system satisfies the following c 62 ( M X i=1 [k i (r i Gamma ae i ) n i ae i ] k 2 M O i=1 [0; n i ] where k = k 1 ; kM ) and P M i=1 n i r i c. Remark: If this assumption is not ....

D. Heath, S. Resnick, and G. Samorodnitsky, "How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails," Annals of Applied Probability, pp. 352--375, 1999.

....the so called M G 1 process, represents another instance of an analytically promising model. This is because the M G 1 processes have both a renewal and Poisson structure. Recent results and additional references on both fluid and discrete time queues with M G 1 arrival processes can be found in [5, 21, 28, 33, 11, 16, 30, 18]. However, the understanding of multiplexing a finite number of heavy tailed On Off arrival 2 10 4 10 5 10 6 10 7 10 4 10 3 10 2 10 1 10 0 File lenght, bytes Figure 1: Log log plot of the empirical distribution of file sizes on five file servers in COMET laboratory at ....

....being On are equal to (r i ; ae i ; p on;i ) respectively. Random variables f on; i) on; i) j g n i j=1 are i.i.d. Because of the probabilistic sample path techniques that we use in the paper our proofs require the following minor technical assumption. Similar assumptions can be found in [16, 25] and, most recently, in [36] Assumption 5.1 The capacity of the queueing system satisfies the following c 62 ( M X i=1 [k i (r i Gamma ae i ) n i ae i ] k 1 ; kM ) 2 M O i=1 [0; n i ] and M X i=1 n i r i c: Remark: If this assumption is not satisfied, by choosing ....

D. Heath, S. Resnick, and G. Samorodnitsky. How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails. Annals of Applied Probability, pages 352--375, 1999.

....sources) We can state the following result about expected time to buffer overflow. Note the complexity allows only analysis of expected times and that only the order is obtained. For functions f; g we write f g to mean 0 lim inf x 1 f(x) g(x) lim sup x 1 f(x) g(x) 1: Theorem 6 ([19]) Suppose the session length distribution F on is heavy tailed 1 Gamma F on (x) x Gammaff L(x) ff 1; and r Gamma on 0 is non integer. SUMMARY xvii Then E(fl) fl 1 Gammak 1 1 Gamma F on (fl) k : If 1 Gamma F on (x) x Gammaff ; ff 1; then E( fl) fl ....

D. Heath, S. Resnick and G. Samorodnitsky. How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails. Available as TR1204.ps.Z at http:// www.orie.cornell.edu/trlist/trlist.html; To appear: Annals Appl. Probab., 1998.

....with heavy tailed on and or off periods. Such models offer some mathematical tractability and an explanation of observed long range dependence in the packet count per unit time data. There is a large body of recent literature which uses such models for modeling network traffic. See, for example, [1, 6, 14, 28, 9, 17, 29, 34, 35, 15, 36] and the references therein. The basic fluid model, which we call the classical on off model, consists of a single idealized source feeding a server. The single channel of this model alternates between an on state, in 1991 Mathematics Subject Classification. Primary 60K25; secondary 90B15. Key ....

....and then make the assumption that when buffer content reaches this level, excess arriving work gets lost. The probability that this happens is the so called loss probability or loss fraction. Some results on time to buffer overflow in such models with heavy tailed on periods are given in [12] [14], 23] and [36] In [36] the influence of heavy tailed input on loss fraction and mean buffer content for particular fluid queueing models is investigated. In this paper, we present a single channel on off model with TCP like control mechanism which is designed to make buffer overflow extremely ....

D. Heath, S. Resnick, and G. Samorodnitsky. How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails. Ann. Appl. Probab., 9:352--375, 1999.

....subject of active research. This classical work on heavy traffic approximations has little relevance to recent work in communication networks, which explains self similarity of network traffic by means of on off models having infinite variance and heavy tailed transmission time distributions. See [2, 28, 20, 21, 31, 22, 32, 11, 10, 12, 24, 16, 15, 14]) A recent stimulating paper by Boxma and Cohen [4] studies the stationary waiting time for the GI G 1 queue under the assumptions that the system is under heavy traffic and the service distribution has infinite variance, while the interarrival distribution tail is of smaller order than the ....

D. Heath, S. Resnick, and G. Samorodnitsky. How system performance is affected by the interplay of = averages in a fluid queue with long range dependence induced by heavy tails. Available as TR1204.ps.Z at http://www.orie.cornell.edu/trlist/trlist.html; To appear: Annals = Appl. Probab., 1998.

....the process as soon as the behaviour (2.3) is observed, even if the process is not self similar . 6 C.A. GUERIN, H. NYBERG, O. PERRIN, S. RESNICK, H. ROOTZ EN, AND C. ST ARIC A 2.2. The infinite source Poisson model. We now review the elements of a data transmission model used in [46] 47] [38], 70] and [61] Let f Gamma k ; k 1g be the points of a rate homogeneous Poisson process on R = 0; 1) so that f Gamma k 1 Gamma Gamma k ; k 1g is a sequence of iid exponentially distributed random variables with parameter . In the stationary case the Poisson process instead should be ....

....1 Gamma 1 ff : Provided the constant output rate r satisfies r, the X( Delta) process of (2.7) and (2.8) has negative drift and is stable. Being regenerative, X(T ) will have a limit distribution. However, high levels will still be exceeded by X( Delta) in algebraic time since, according to ([38]) the expected hitting time of high level fl (see (2.9) satisfies E(fl) i fl 1 Gammak 1 1 Gamma F (fl) k where k : inffj 1 : E(L 1 ) j rg is the minimum number of sessions running simultaneously which are needed to flip the mean drift from negative to positive. Here we use the ....

D. Heath, S. Resnick, and G. Samorodnitsky. How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails. Annals of Applied Probability, 9, 352-375,1999.

....self similar. This approximation is in the spirit of [32] Sections 4 and 5 give Gaussian process approximations to the content process and time to buffer overflow. 2. An infinite node, Poisson based communication model. We first review the elements of a communication model used in [19] 20] and [16]. Let f Gamma k ; k 1g be the points of a rate homogeneous Poisson process on R = 0; 1) so that f Gamma k 1 Gamma Gamma k ; k 1g is a sequence of iid exponentially distributed random variables with parameter . We imagine that HEAVY TAILS 3 a communication system has an infinite number of ....

D. Heath, S. Resnick, and G. Samorodnitsky. How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails. Available as TR1204.ps.Z at http://www.orie.cornell.edu/trlist/trlist.html; To appear: Annals Appl. Probab., 1998.

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D. Heath, S. Resnick and G. Samorodnitsky (1997a): How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails. Annals of Applied Probability, to appear. Available as hit-bounds.ps at http://www.orie.cornell.edu/ gennady/techreports.

....tailed burst lengths, where a burst is a period where packet arrivals are not separated by more than some threshold value [28] Analysts are largely in agreement about the self similar nature of aggregate traffic, at least at time scales above a certain threshold. Empirical [45, 2] and theoretical [40, 17, 18, 19] evidence supports the heavy tailed explanation of the self similarity. However, measurement studies diverge in their conclusions about the marginal distributions of cumulative traffic. There exists empirical evidence supporting a heavy tailed assumption backed by theoretical work [15, 23, 32] and ....

....between S. Resnick and H. Rootz en was supported by the Gothenburg Stochastic Centre. Alwin Stegeman s research is supported by a Dutch Science Foundation (NWO) Grant. 2 T. MIKOSCH, S. RESNICK, H. ROOTZ EN, AND A. STEGEMAN Model (i) the superposition of M ON OFF sources (see for example [45, 40, 24, 19, 17, 18]) and Model (ii) the infinite source Poisson model, sometimes called the M G 1 input model (see [19, 31, 1, 26, 21, 20, 37] In model (i) traffic is generated by a large number of independent ON OFF sources such as workstations in a big computer lab. An ON OFF source transmits data at a ....

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D. Heath, S. Resnick, and G. Samorodnitsky. How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails. Ann. Appl. Probab., 9:352--375, 1999.

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D. Heath, S. Resnick, and G. Samorodnitsky, "How system performance is affected by the interplay of averages in a fluid queue with long range dependence induced by heavy tails," Annals of Applied Probability, pp. 352--375, 1999.

No context found.

Heath, D., Resnick, S., and Samorodnitsky, G. (1997). How System Performance is Affected by the Interplay of Averages in a Fluid Queue with Long Range Dependence Induced by Heavy Tails. Technical Report #1204, School of Operations Research and Industrial Engineering, Cornell University. 7

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