Poisson Simulation outperforms Markov Simulation

Markov Simulation and the more recent Poisson Simulation are two fully consistent ways of modelling, applicable to the same types of problems. A Markov model is based on a detailed description of every situation, called state, that a system can be in and every possible transition between these states. This approach allows powerful analysis and makes Markov models a fundamental part of the theory of stochastic processes. However, Markov models are also frequently used for simulation. Here the detailed description gives huge, inflexible models, slow execution and problems with model fitting and validation because the natural parameters are spread out as parts of the transition probabilities. These problems are effectively eliminated in Poisson Simulation, where the model is based on a small number of state variables (compartments) and where each parameter is explicitly represented only once. This paper first presents Markov Simulation and Poisson Simulation and demonstrates how these fully consistent methods are related. Then a systematic scrutiny of the merits and demerits of using Markov Simulation and Poisson Simulation in a modelling project reveals that Poisson Simulation is superior to Markov Simulation in every phase of the project and can handle considerably larger and more complex models. The power of Poisson Simulation also extends far outside the field of Markov models. A number of illustrative examples are included to demonstrate the differences and advantages of using Poisson Simulation instead of Markov Simulation.