A storage model with self-similar 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: Self-similar, 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 self-similar with a surprising accuracy (see Leland et al. [9]). Traditional traffic models based on the Poisson process or, more gener...

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 short-term 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 self-similar model are discussed extensively, and the notion of ideal Free Traffic is introduced. Keywords: LAN traffic, long-range dependence, self-similar, 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...

In a Galton-Watson 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...

The volume of the n-dimensional 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.

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 inter-cell interval. 1

We propose an analytically tractable model of loading-dependent 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.

A (generalized) stochastic fluid system Q is defined as the one-dimensional 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...

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. Reed-Solomon code etc.).

: Fluctuations of aggregated connectionless traffic are modelled with the fractional Brownian motion. The applicability and interpretation of the self-similar model are studied both by comparison with Ethernet traffic data from Bellcore and conceptually, defining an ideal notion of Free Traffic. Insight into the three model parameters is obtained by finding their counterparts in a burst model with heavy-tailed burst length distribution. A bandwidth allocation formula based on the corresponding queueing system is discussed. Guidelines for short-term traffic prediction are found using an exact formula for the conditional expectation of a future value given the past. Directions for further study are indicated at the end of each section. 1 Introduction: fractional Brownian traffic The principle of a connectionless network is that all data is sent in relatively small datagrams that are labeled with the destination address and routed independently. No bandwidth needs to be reserved. The con...

The probability distribution of the number of lost packets within a block of consecutive packet arrivals into a finite buffer is an important quantity in various networking problems. In a recent paper, Cidon, Khamisy and Sidi introduced a recursive scheme to derive this distribution. In this paper we derive explicit expressions for this distribution using various versions of the powerful Ballot Theorem. The expressions are derived for a single source M/M/1/K queue.