D. Heath, S. Resnick and G. Samorodnitsky (1998), " Heavy tails and long range dependence in on/off processes and associated fluid models", Math. Oper. Res. 23, 145--165.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....spectrally positive L evy processes reflected in their infimum) In particular we derive the joint Laplace transform of the time to first exit and the overshoot. The aforementioned stopping time can be identified in the literature of fluid models as the time to buffer overflow (see for example [1, 13]) Together Universit e de Pau, e mail: Florin.Avram univ pau.fr y Utrecht University, e mail: kyprianou math.uu.nl z Utrecht University, e mail: pistorius math.uu.nl 1 with existing results on exit problems we apply our results to certain optimal stopping problems that are now ....

D. Heath, S. Resnick and G. Samorodnitsky (1998), " Heavy tails and long range dependence in on/off processes and associated fluid models", Math. Oper. Res. 23, 145--165.

.... almost perfectly matched by a Pareto distribution with parameter # ##### (dashed line) The analysis of queueing models with multiplexed heavytailed renewal arrival sequences, e.g. On Off processes, is difficult primarily due to the complex dependency structure in the aggregate arrival process [10]. This stems from the wellknown fact that the superposition of renewal processes, in general, is not a renewal process. An intermediate case of multiplexing a single heavy tailed process with exponential streams was investigated in [11] 4] 12] In [4] it was discovered that these hybrid ....

D. Heath, S. Resnick, and G. Samorodnitsky, "Heavy tails and long range dependence in on/off processes and associated fluid models," Mathematics of Operations Research, vol. 23, pp. 145--165, 1998.

....of the stationary buffer distribution of a statistical multiplexer fed by a class of long range dependent discrete time M G 1 input processes. More precisely, we obtain the exact asymptotics describing the tail of the buffer distribution for the Pareto case. Several other papers, e.g. 7] 8] [9], 10] 3] address this topic, but most of them rely on large deviation techniques or focus on the continuous time case. Based on the approach presented in [11] and [12] we obtain the asymptotics of the buffer occupancy using generating functions and the elementary form of the Tauberian theorem ....

D. Heath, S. Resnick, and G. Samorodnitsky, "Heavy tails and long range dependence in on/off processes and associated fluid models," Tech. Rep., Cornell University, 1997.

....P (D x) 1 Z 1 x P (X Y s) ds: De ne the renewal counting process t = 1 X n=0 1 [0;t] S n ) t 0 ; 1.3) and set t = E t . We denote the ON OFF process generated by the source by W t , where W t = 1 if t is in an ON period, 0 if t is in an OFF period. 2 Heath et al. [8], using an explicit construction of D and the stationarity of (S n ) have shown that the ON OFF process W is strictly stationary with mean EW t = on = The main result of [8] yields an asymptotic relation for the autocovariance function (k) of W if 2: as k 1 (k) const k ( 1) k = ....

....generated by the source by W t , where W t = 1 if t is in an ON period, 0 if t is in an OFF period. 2 Heath et al. 8] using an explicit construction of D and the stationarity of (S n ) have shown that the ON OFF process W is strictly stationary with mean EW t = on = The main result of [8] yields an asymptotic relation for the autocovariance function (k) of W if 2: as k 1 (k) const k ( 1) k = 0; 1; 2; 1.4) Hence, if 2, the autocorrelations are not absolutely summable. In this sense, the ON OFF process W exhibits LRD. Next, consider a network of M iid ....

Heath, D., Resnick, S.I. and Samorodnitsky, G. (1998) Heavy tails and long range dependence in ON/OFF processes and associated uid models. Mathematics of Operations Research 23, 145-165.

.... were explored in [67, 53, 68, 63] see also Chapter 5 of this dissertation) Parallel to the modeling approach through self similar long range dependent processes, a more analytically tractable approach using fluid renewal type models, in which inter renewal times are long tailed, was explored in [2, 52]. Queueing results in these two papers rely on the classical result by Pakes [88] on the subexponential (long tailed) asymptotics of the waiting time distribution in a GI GI 1 queue (or earlier work of Cohen [32] which considered a regularly varying GI GI 1 queue) Recently, the result of Pakes ....

.... structure of an aggregate arrival process may be very complex, even though the appearance of each individual source may be truly innocuous (like an On Off source) The complex autocorrelation structure of the aggregate source obtained by multiplexing long tailed On Off sources was examined in [52]. General bounds for multiplexing long tailed fluid processes have been obtained in [24] In [16] a limiting case of an infinite number of On Off sources with regularly varying On distribution was investigated. In the same chapter a case of two sources was solved, in which one source had regularly ....

D. Heath, S. Resnick, and G. Samorodnitsky. Heavy tails and long range dependence in on/off processes and associated fluid models. preprint.

....Remark 3.11 Expression (3. 10) may be rewritten as: Z t 0 (P[I (u) 1jI (0) 1] Gamma P[I (u) 1jI (0) 0] du; which shows that the roles of A 1 and S 1 are strictly equivalent in terms of long range dependence (a phenomenon already noticed by Brichet et al. 10] and Heath et al. [27]) This is the main difference with the problem of the long tailed stationary buffer content distribution, which depends only on the tail of A 1 (see Subsection 3.3) Lemma 3.12 Assume that the distribution of A 1 or S 1 is non lattice. Then for any measurable, non null function f : R R ....

....and 3.12 once again, thus obtaining Formula (3.9) which proves long range dependence. 3. A single fluid source 14 Formula (3. 9) suggests the stronger result: Cov(r (0) r (u) u 1 r 2 p(1 Gamma p) pP[A 1 u] 1 Gamma p)P[S 1 u] It has actually been proved by Heath et al. [27], Theorem 4.3) in the special case when P[A 1 t] is of the form l(t) t 1 ffl , where l(t) is a slowly varying function and ffl 2 (0; 1) and P[S 1 t] o (P[A 1 t] The classical results of Karamata on functions of regular variation (see Lemma 7.7 or [7] then yield: Cov(r (0) r (u) ....

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D. Heath, S. Resnick, and G. Samorodnitsky. Heavy tails and long range dependence in on/off processes and associated fluid models. Technical Report 1144, School of Operations Research and Industrial Engineering, Cornell University, January 1996. To appear in Mathematics of Operations Research. References 41

....This suggests that the corresponding ftp (file transfer protocol) traffic is heavy tailed. The analysis of queueing models with multiplexed heavy tailed renewal arrival sequences, e.g. On Off processes, is difficult primarily due to the complex dependency structure in the aggregate arrival process [15]. This stems from the fact that the superposition of renewal processes, in general, is not a renewal process. An intermediate case of multiplexing a single long tailed arrival sequence with exponential processes was investigated in [5, 21, 31] In [21] it was discovered that these hybrid queueing ....

D. Heath, S. Resnick, and G. Samorodnitsky. Heavy tails and long range dependence in on/off processes and associated fluid models. Mathematics of Operations Research, 23:145--165, 1998.

....suggests that the corresponding ftp (file transfer protocol) traffic is heavy tailed. The analysis of queueing models with multiplexed heavy tailed renewal arrival sequences, e.g. On Off processes, is difficult primarily due to the complex dependency structure in the aggregate arrival process [10]. This stems from the well known fact that the superposition of renewal processes, in general, is not a renewal process. An intermediate case of mul INFOCOM 2001 2 10 4 10 5 10 6 10 7 10 4 10 3 10 2 10 1 10 0 File lenght, bytes Fig. 1. Log log plot of the empirical ....

D. Heath, S. Resnick, and G. Samorodnitsky, "Heavy tails and long range dependence in on/off processes and associated fluid models," Mathematics of Operations Research, vol. 23, pp. 145--165, 1998.

.... almost perfectly matched by a Pareto distribution with parameter ff = 1:44 (dashed line) The analysis of queueing models with multiplexed heavy tailed renewal arrival sequences, e.g. On Off processes, is difficult primarily due to the complex dependency structure in the aggregate arrival process [10]. This stems from the well known fact that the superposition of renewal processes, in general, is not a renewal process. An intermediate case of multiplexing a single heavy tailed process with exponential streams was investigated in [11] 4] 12] In [4] it was discovered that these hybrid ....

D. Heath, S. Resnick, and G. Samorodnitsky, "Heavy tails and long range dependence in on/off processes and associated fluid models," Mathematics of Operations Research, vol. 23, pp. 145--165, 1998.

....suggests that the corresponding ftp (file transfer protocol) traffic is heavy tailed. The analysis of queueing models with multiplexed heavy tailed renewal arrival sequences, e.g. On Off processes, is difficult primarily due to the complex dependency structure in the aggregate arrival process [15]. This stems from the fact that the superposition of renewal processes, in general, is not a renewal process. An intermediate case of multiplexing a single long tailed arrival sequence with exponential processes was investigated in [5, 21, 31] In [21] it was discovered that these hybrid queueing ....

D. Heath, S. Resnick, and G. Samorodnitsky. Heavy tails and long range dependence in on/off processes and associated fluid models. Mathematics of Operations Research, 23:145--165, 1998.

....assumptions on the input streams. These range from queues with fractional Brownian motion inputs due to Norros [9] general Gaussian processes with negative 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 ....

D. Heath, S. Resnick and G. Samorodnitsky; Heavy tails and long-range dependence in on/off processes and associated fluid models, Mathematics of Operations Research, Vol. 23, 1998, pp. 145-165.

....This fact is further supported by empirical research in Leland et al. 12] and Crovella and Bestavros [7] The latter authors studied the tra c on the World Wide Web. They found evidence of Pareto like tails in le lengths, transfer times and idle times. See also Crovella et al. 6] Heath et al. [9] study the ON OFF model at the source level. They construct a stationary version of the ON OFF process of an individual source. Assuming heavy tailed (Pareto like) lengths of ON and OFF periods, they show that the ON OFF process of an individual source necessarily exhibits LRD in the sense that ....

....a large deviations result is used, which is presented in Section 4. In Section 6 we give the proof of the convergence to fractional Brownian motion. 2 The ON OFF model We commence by considering a single ON OFF source (such as a workstation) In our presentation we closely follow Heath et al. [9], and we also adapt their notation. During an ON period, the source generates tra c at a constant rate 1, e.g. 1 byte per time unit. During an OFF period, the source remains silent; we assign the value 0 to it. Let X on ; X 1 ; X 2 ; be iid non negative random variables representing the ....

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Heath, D., Resnick, S.I. and Samorodnitsky, G. (1998) Heavy tails and long range dependence in ON/OFF processes and associated uid models. Mathematics of Operations Research 23, 145-165.

.... on the basis of Donsker s theorem and its generalizations, fBM can be viewed as a natural limiting approximation to a broad class of more physically plausible models, that describe how traffic in a network is generated from its individual sources; see Heath, Resnick and Samorodnitsky (1997) Heath, Resnick and Samorodnitsky (1998), Konstantopoulos and Lin (1996) and Whitt (1998) Because of both theoretical and statistical evidence supporting fBM as a possible traffic model, there is significant interest in trying to reach an understanding of the implications of such long range dependent traffic for the performance of ....

Heath, D., Resnick, S., and Samorodnitsky, G. Heavy tails and long range dependence in on/off processes and associated fluid models. Math. Oper. Res., 23:145--165, 1998.

....( Y ) n ) n=1;2; g where (D Gamma ; D ) is a random vector independent of fX n ; Y n g with distribution P [D Gamma x; D y] Z 1 x y P [X on Y off s] ds = Z 1 x y 1 Gamma F on F off (s) ds: 2. 1) Here is an explicit construction of the stationary on off process (see Heath et al. 1998) for a one sided version) Define three independent random vectors B; X ( Gamma;0) on ; X ( 0) on ) Y ( Gamma;0) off ; Y ( 0) off ) which are independent of fX on ; Y off ; X n ; Y n ; n 1g as follows: B is a Bernoulli random variable with values f0; 1g and mass function P [B = 1] ....

....is defined in terms of fS n ; n = Gamma1; 0; 1; 2; g as follows: I(t) B1 [ GammaX ( Gamma;0) on ;X ( 0) on ) t) 1 Gamma B)1 [ GammaY ( Gamma;0) off GammaX on ; GammaY ( Gamma;0) off ) t) X n6= Gamma1 1 [Sn ;Sn Xn 1 ) t) 2.2) Remark 2.1. A useful fact (Heath et al. 1998, Corollary 2.2) is that for any t 0, conditional on I(t) 1, the subsequent sequence of on off periods is the same as seen from time 0 in the stationary process with B = 1. In particular, conditionally on I(0) 1, looking forward into the future produces an on period with distribution F (0) ....

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D. Heath, S. Resnick and G. Samorodnitsky (1998): Heavy tails and long range dependence in on/off processes and associated fluid models. Mathematics of Operations Research 23:145-- 165.

....t g, which is 1 iff t is in an on period. Thus, for t D (0) Z t = 1; if Sn t Sn Xn 1 ; some n 0; if Sn Xn 1 t Sn 1 ; some n and if 0 t D (0) we define Z t = 1; if B = 1 and 0 t X (0) on ; 0; otherwise. A standard renewal argument gives the following result ([18]) Proposition 1 fZ t ; t 0g is strictly stationary and P [Z t = 1] on : Conditional on Z t = 1, the subsequent sequence of on off periods is the same as seen from time 0 in the stationary process with B = 1. It is easiest to express long range dependence in terms of slow decay of ....

....Z t = 1, the subsequent sequence of on off periods is the same as seen from time 0 in the stationary process with B = 1. It is easiest to express long range dependence in terms of slow decay of covariance functions so we consider the second order properties of the stationary process fZ t g. See [18]. A SINGLE CHANNEL ON OFF COMMUNICATION MODEL. v Theorem 1 The covariance function fl(s) Cov(Z t ; Z t s ) of the stationary process fZ(t) t 0g is fl(s) on off Gamma Z s 0 F off (s Gamma u) F (0) on U(du) on off Gamma F (0) on U (1 Gamma F off ) s) on ....

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D. Heath, S. Resnick, and G. Samorodnitsky. Heavy tails and long range dependence in on/off processes and associated fluid models. Math. of OR, 23:145--165, 1998.

....level. Moreover, as asserted in Theorem 3. 3, the tail of the stationary distribution of fX n g is always light tailed (even in the case when L is heavy tailed) This is a completely different result from the classical case where the stationary distribution becomes heavy tailed too (see [12] and [13]) A SINGLE CHANNEL ON OFF MODEL WITH TCP LIKE CONTROL 11 We postpone the proof of Theorem 3.3 until after the next two propositions. The consequences of Theorem 3.3 are not needed in their proofs. We discuss first the support of any stationary of fX n g under different conditions on L and Y . ....

D. Heath, S. Resnick, and G. Samorodnitsky. Heavy tails and long range dependence in on/off processes and associated fluid models. Math. Oper. Res., 23(1):145--165, 1998.

....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. Heavy tails and long range dependence in on/off processes and associated fluid models. Math. of OR, 23:145--165, 1998.

....and Samorodnitsky ( 67] constructed an example of a single exponential server fed by a long range dependent input which had queue lengths and waiting times which were heavy tailed. Mathematical studies of the connection between on off inputs with heavy tailed on periods appeared in [77] and [45, 46, 37, 61, 48, 47]. The infinite source Poisson model was studied in [70] and [61] Attempts to explain observed self similarity in network traffic have largely focused on heavy tailed transmission times of sources sending data to one or more servers. The common assumption is that transmission times have iid random ....

....in the interval (1; 2) The theoretical reason is that mathematical analysis of models has been based on renewal theory and without a finite mean, stationary versions of renewal processes do not exist and (uncontrolled) buffer content stochastic processes would not be stable. See for example [37]. iii) F has relatively thin tails so that the variance is finite. This includes classical models for telecommunication. Section 2 defines the infinite source Poisson model and defines the basic descriptor processes: number of active sources at time t, cumulative inputted traffic to the server ....

D. Heath, S. Resnick, and G. Samorodnitsky. Heavy tails and long range dependence in on/off processes and associated fluid models. Math. Oper. Res., 23, 145--165, 1998.

....As a more structural traffic model, a superposition of a large number of ON OFF type sources whose activity periods are heavy tailed, has received considerable attention. Such models are approximated by fluid models with M G 1 inputs, sometimes referred to as infinite source Poisson models. cf. [13, 14, 16, 15, 18, 33, 34, 35, 28, 19, 5]. The M G 1 input model described in [17] is of this type. Using a distributional limit theorem, Konstantopoulos and Lin ( 17] explain the suitability of an totally skewed stable L evy motion as a macroscopic traffic model for a high speed network switch. The limit process is self similar, but ....

D. Heath, S. Resnick, and G. Samorodnitsky. Heavy tails and long range dependence in on/off processes and associated fluid models. Math. Oper. Res., 23(1):145--165, 1998.

....delayed renewal sequence by (4.1) fS (j) n ; n 0g : fD (0) j ; D (0) j n X i=1 (X (j) i Y (j) i ) n 1g: Then a stationary version of fZ j (t) t 0g is defined by (4. 2) Z j (t) C (0) j 1 [0;X (0) j ) t) 1 X n=0 1 [S (j) n t S (j) n X (j) n 1 ] See Heath, Resnick and Samorodnitsky (1996) for details. In a similar way we can construct a stationary version fZ j (t) Gamma1 t 1g defined for all real t. We take, further, the k stationary on off processes to be independent. In this section we consider a single server queue as above, with service rate r, fed by the k on off ....

Heath, D., Resnick, S. and Samorodnitsky, G., Heavy tails and long range dependence in on/off processes and associated fluid models, Available as TR1144.ps.Z at http://www.orie.cornell.edu/trlist/trlist.html (1996).

....on the less dramatic case ff 1. Heavy tailed session length distributions cause both the buffer content process fX(t) t 0g and the number of running sessions process fN(t) t 0g to possess a form of long range dependence. See Leland et al. 1994) Beran et al. 1995) Agrawal et al. 1998) Heath et al. 1998). It is well understood that long range dependence usually translates into deterioration of performance of the server. This is the case if one studies the steady state distribution of the amount work in an infinite buffer (see e.g. Choudhury and Whitt (1995) Boxma (1996) Jelenkovi c and Lazar ....

D. Heath, S. Resnick and G. Samorodnitsky (1998): Heavy tails and long range dependence in on/off processes and associated fluid models. Mathematics of Operations Research 23:145-- 165.

....has been observed in Heath et al. 1997) It has been argued in the literature that a decay in system performance is caused by long range dependence in the input stream. This has been observed in different situations by Duffield and O Connell (1995) Ryu and Lowen (1995) Erramilli et al. 1996) Heath et al. 1996), Vamvakos and Anantharam (1996) Liu et al. 1997) and Resnick and Samorodnitsky (1997b) A survey is in Boxma and Dumas (1996) Since heavy tailed session length is known to cause long range dependence in our model, and in similar models (Jelenkovi c and Lazar (1995) Boxma and Dumas (1996) ....

.... Vamvakos and Anantharam (1996) Liu et al. 1997) and Resnick and Samorodnitsky (1997b) A survey is in Boxma and Dumas (1996) Since heavy tailed session length is known to cause long range dependence in our model, and in similar models (Jelenkovi c and Lazar (1995) Boxma and Dumas (1996) Heath et al. 1996), Willinger et al. 1996) Resnick and Samorodnitsky (1997b) the loss in performance of our fluid queue we mentioned above is not surprising. It is also not surprising that the performance loss tends to grow as the session length distribution tails grow heavier, because the length of memory ....

D. Heath, S. Resnick and G. Samorodnitsky (1996): Heavy tails and long range dependence in on/off processes and associated fluid models. To appear in Mathematics of Operations Research. Available as TR1144.ps.Z at http://www.orie.cornell.edu/trlist/trlist.html.

....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 ....

[Article contains additional citation context not shown here]

D. Heath, S. Resnick, and G. Samorodnitsky. Heavy tails and long range dependence in on/off processes and associated fluid models. Math. Oper. Res., 23(1):145--165, 1998.

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Heath, D., Resnick, S., and G. Samorodnitsky (1997): "Heavy Tails and Long Range Dependence in an/off Processes and Associated Fluid Models", Math. Oper. Research, forthcoming.

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D. Heath, S. Resnick and G. Samorodnitsky, Heavy tails and long range dependence in on/off processes and associated fluid models, Tech. Report # 1144, School of Operations Research and Industrial Engineering, Cornell University, 1996.

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