quantum simulation, Poisson Pareto Burst Process Quantum Simulation - Rare Event Simulation by means of Cloning, Thinning and Distortion R. G. ADDIE University of Southern Queensland Toowoomba, Australia addie@usq.edu.au January 24, 2003 Abstract. A method of rare event simulation, termed here quantum simulation, and known also (with some variations) as population Monte Carlo, and Sequential Markov Chain simulation, is applied in this paper to rare event simulation of communication systems. The technique described in this paper generalizes importance sampling, importance splitting and the Monte Carlo method which uses a collection (population) of particles. The term quantum simulation is used for the rare-event simulation technique presented in this paper because the entire ensemble of simulations resembles the parallel universes model of quantum mechanics. By using cloning, thinning and distortion it is possible to design simulations with the speed of importance sampling and the flexibility of importance splitting. A particularly difficult system is investigated in this paper by means of quantum simulation, namely a buffer fed by a Poisson-Pareto Burst Process. The simulations are able to confirm an analytic approximation for this model to a degree not previously achievable. Furthermore, this approximation was originally developed as a consequence of the investigation of this model by means of quantum simulation. Keywords: Poisson-Pareto Burst Process, Long-range dependence, Importance Splitting, Importance Sampling, Quantum Simulation 1 I NTRODUCTION Quantum simulation, population Monte-Carlo, sequen- tial Markov chain simulation, . . . [13, 7, 11, 6, 1] makes use of spontaneous generation of clones (copies) of simu- lation processes which then proceed with an independent random number stream. Processes are also thinned (killed) to ensure that the total number of processes stays within reasonable bounds or remains constant. The cloning rate may be state dependent, and in particular cloning rates may be chosen in such a way that events of interest occur more frequently. Individual threads (processes, clones) do not necessar- ily progress with the same dynamics as the original system being modelled although as an aggregate, the entire collec- tion of threads can always be viewed, by using an appro- priate transformation of the statistics of the collection of threads, as an unbiased model of the original system. This method of simulation is a generalisation of im- portance splitting, also known as the Restart Method, as introduced in [18] and further developed in, for example, [5, 10, 12]. Quantum simulation is also a generalisation of importance sampling, eg [8, 14]. Quantum simulation provides a framework in which both importance splitting and importance sampling can be described. Importance sampling makes use of an analytic formula for a change of measure which transforms (“dis- torts”) the model under consideration into one which can be simulated more quickly. In the case of quantum simulation, an arbitrary change of measure can also be introduced, in addition to cloning and thinning, and a formula for the conversion of the statis- tics collected from the simulation back into those appro- priate for the original model, will always be available, al- though it is not necessarily a Radon-Nikodym derivative, as it would be in the case of importance sampling. In the case of quantum simulation the inverse transformation formula is calculated by the simulation program. The papers [5, 12] on a method known as Direct Prob- ability Redistribution, a method which computes a suc- Submission 1