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55
A Storage Model With SelfSimilar Input
, 1994
"... A storage model with selfsimilar input process is studied. A relation coupling together the storage requirement, the achievable utilization and the output rate is derived. A lower bound for the complementary distribution function of the storage level is given. Keywords: Selfsimilar, fractional Bro ..."
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Cited by 301 (13 self)
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A storage model with selfsimilar input process is studied. A relation coupling together the storage requirement, the achievable utilization and the output rate is derived. A lower bound for the complementary distribution function of the storage level is given. Keywords: Selfsimilar, fractional Brownian motion, Local Area Network traffic 1 Introduction In a series of papers (e.g. Leland [8], Leland and Wilson [7], Fowler and Leland [4], Leland et al. [9]), researchers from Bellcore have reported and analyzed remarkable Local Area Network (LAN) traffic measurements challenging traditional data traffic modelling. The Bellcore data are both very accurate and extensive in time, and their most striking feature is the tremendous burstiness of LAN traffic at, practically, any timescale. More than that, the statistical analysis has shown that the traffic is selfsimilar with a surprising accuracy (see Leland et al. [9]). Traditional traffic models based on the Poisson process or, more gener...
On the use of fractional Brownian motion in the theory of connectionless networks
 IEEE Journal on Selected Areas in Communications
, 1995
"... An abstract model for aggregated connectionless traffic, based on the fractional Brownian motion, is presented. Insight into the parameters is obtained by relating the model to an equivalent burst model. Results on a corresponding storage process are presented. The buffer occupancy distribution is a ..."
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Cited by 219 (6 self)
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An abstract model for aggregated connectionless traffic, based on the fractional Brownian motion, is presented. Insight into the parameters is obtained by relating the model to an equivalent burst model. Results on a corresponding storage process are presented. The buffer occupancy distribution is approximated by a Weibull distribution. The model is compared with publicly available samples of real Ethernet traffic. The degree of the shortterm predictability of the traffic model is clarified through an exact formula for the conditional variance of a future value given the past. The applicability and interpretation of the selfsimilar model are discussed extensively, and the notion of ideal Free Traffic is introduced. Keywords: LAN traffic, longrange dependence, selfsimilar, prediction 1 Introduction In this paper we are considering the modelling of traffical phenomena in a connectionless network. The principle of such a network is that all data is sent in relatively small independen...
A Polytope Related to Empirical Distributions, Plane Trees, Parking Functions, and the Associahedron
"... The volume of the ndimensional polytope for arbitrary x := (x 1 ; : : : ; x n ) with x i > 0 for all i de nes a polynomial in variables x i which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this po ..."
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Cited by 40 (2 self)
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The volume of the ndimensional polytope for arbitrary x := (x 1 ; : : : ; x n ) with x i > 0 for all i de nes a polynomial in variables x i which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this polynomial as the volumes of chambers in two dierent polytopal subdivisions of n (x). The rst of these subdivisions generalizes to a class of polytopes called sections of order cones. In the second subdivision, the chambers are indexed in a natural way by rooted binary trees with n + 1 vertices, and the con guration of these chambers provides a representation of another polytope with many applications, the associahedron.
Enumerations Of Trees And Forests Related To Branching Processes And Random Walks
 Microsurveys in Discrete Probability, number 41 in DIMACS Ser. Discrete Math. Theoret. Comp. Sci
, 1997
"... In a GaltonWatson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p ..."
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Cited by 38 (15 self)
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In a GaltonWatson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p j+1 for j \Gamma1. The formula for this distribution is a probabilistic expression of the Lagrange inversion formula for the coefficients in the power series expansion of f(z) k in terms of those of g(z) for f(z) defined implicitly by f(z) = zg(f(z)). The Lagrange inversion formula is the analytic counterpart of various enumerations of trees and forests which generalize Cayley's formula kn n\Gammak\Gamma1 for the number of rooted forests labeled by a set of size n whose set of roots is a particular subset of size k. These known results are derived by elementary combinatorial methods without appeal to the Lagrange formula, which is then obtained as a byproduct. This approach unifies an...
OLD AND NEW EXAMPLES OF SCALE FUNCTIONS FOR SPECTRALLY Negative Lévy Processes
, 2009
"... We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussi ..."
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Cited by 27 (11 self)
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We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as crossreferencing their analytical behaviour against known general considerations.
Smoothness of scale functions for spectrally negative Lévy processes
, 2006
"... Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that stan ..."
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Cited by 25 (8 self)
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Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.
A LoadingDependent Model Of Probabilistic Cascading Failure
 Probability in the Engineering and Informational Sciences
, 2004
"... We propose an analytically tractable model of loadingdependent cascading failure that captures some of the salient features of large blackouts of electric power transmission systems. This leads to a new application and derivation of the quasibinomial distribution and its generalization to a saturat ..."
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Cited by 22 (8 self)
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We propose an analytically tractable model of loadingdependent cascading failure that captures some of the salient features of large blackouts of electric power transmission systems. This leads to a new application and derivation of the quasibinomial distribution and its generalization to a saturating form with an extended parameter range. The saturating quasibinomial distribution of the number of failed components has a power law region at a critical loading and a significant probability of total failure at higher loadings.
On the Dynamics and Performance of Stochastic Fluid Systems
, 1998
"... A (generalized) stochastic fluid system Q is defined as the onedimensional Skorokhod reflection of a finite variation process X (with possibly discontinuous paths). We write X as the (not necessarily minimal) difference of two positive measures, A, B, and prove an alternative "integral representati ..."
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Cited by 19 (5 self)
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A (generalized) stochastic fluid system Q is defined as the onedimensional Skorokhod reflection of a finite variation process X (with possibly discontinuous paths). We write X as the (not necessarily minimal) difference of two positive measures, A, B, and prove an alternative "integral representation" for Q. This representation forms the basis for deriving a "Little's law" for an appropriately constructed stationary version of Q. For the special case where B is the Lebesgue measure, a distributional version of Little's law is derived. This is done both at the arrival and departure points of the system. The latter result necessitates the consideration of a "dual process" to Q. Examples of models for X , including finite variation Levy processes with countably many jumps on finite intervals, are given in order to illustrate the ideas and point out potential applications in performance evaluation. Keywords: RANDOM MEASURES, STATIONARY PROCESSES, PALM PROBABILITIES, QUEUEING THEORY, LIT...
On Loss Probabilities in Presence of Redundant Packets with Random Drop
"... The purpose of this paper is to study the loss probabilities of messages in an M/M/1/K queueing system where in addition to losses due to buffer overflow there are also random losses in the incoming and outgoing links. We focus on the influence of adding redundant packets to the messages (as in e ..."
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Cited by 18 (5 self)
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The purpose of this paper is to study the loss probabilities of messages in an M/M/1/K queueing system where in addition to losses due to buffer overflow there are also random losses in the incoming and outgoing links. We focus on the influence of adding redundant packets to the messages (as in error correction coding e.g. ReedSolomon code etc.).
Jitter in atm networks and its impact on peak rate enforcement
 Performance Evaluation
, 1992
"... Cells arriving to an ATM network experience random delays due to queueing in upstream multiplexing stages, notably in customer premises. This is the phenomenon of jitter and the aim of the present paper is to study its in uence on peak rate enforcement. We rst introduce some general characterization ..."
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Cited by 17 (0 self)
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Cells arriving to an ATM network experience random delays due to queueing in upstream multiplexing stages, notably in customer premises. This is the phenomenon of jitter and the aim of the present paper is to study its in uence on peak rate enforcement. We rst introduce some general characterizations of jitter and then, describe twomodels of jittered ows based on simple queueing systems. We discuss the objectives of peak rate enforcement and study the impact of jitter on the dimensioning of Jumping Window and Leaky Bucket mechanisms. A useful synthetic characterization of jitter appears to be a remote quantile of the cell delay distribution expressed in units of the initial intercell interval. 1