Owen Dafydd Jones,
Fast, efficient on-line simulation of self-similar processes.

Thinking in Patterns: Fractals and Related Phenomena in Nature,
M.M. Novak Ed., pp 165-176. World Scientific 2004. 

Abstract: We describe a class of self-similar processes that can be used to fit 
self-similar data, and give a fast, efficient on-line algorithm for simulating them.

Introduction: Self-similar processes are of interest as models for internet packet
arrival data, high-frequency financial data, ECG and EEG traces, and various hydrological
and meteorological time series.
Simulation of self-similar processes has proven problematic, because they exhibit a
slowly decaying correlation structure (long-range dependence), which means that the 
individual elements of any sequence of observations X(1), ..., X(n) are strongly 
correlated.
In practice to date this generally means that it is not possible to simulate X(n) 
without simultaneously simulating X(1), ..., X(n-1), and this necessarily results in
an algorithm that requires O(n) storage to generate n steps of X.
More importantly, this means that if you have already generated n steps, then it is not
possible to generate step n+1 directly, instead it is necessary to generate all n+1 
steps from scratch.

One model which avoids these problems is the M/G/infty queue.
Unfortunately this model is not flexible enough to be useful in practice.
Here we present a new class of self-similar models called EBP processes (for Embedded 
Branching Process), which are flexible, readily fitted to data, and easily simulated.
Features of the simulation algorithm are
(i)   Scaleable: O(n log n) time to generate n steps.
(ii)  Efficient: O(log n) storage required to generate n steps.
(iii) On line: can generate a new step on demand.