Discrete-Event Simulation of Continuous Systems

Computer simulation of a system described by differential equations requires that some element of the system be approximated by discrete quantities. There are two system aspects that can be made discrete; time and state. When time is discrete, the differential equation is approximated by a difference equation (i.e., a discrete time system), and the solution is calculated at fixed points in time. When the state is discrete, the differential equation is approximated by a discrete event system. Events correspond to jumps through the discrete state space of the approximation. The essential feature of a discrete time approximation is that the resulting difference equations map a discrete time set to a continuous state set. The time discretization need not be regular. It may even be revised in the course of a calculation. None the less, the elementary features of a discrete time base and continuous state space remain. The basic feature of a discrete event approximation is opposite that of a discrete time approximation. The approximating discrete event system is a function from a continuous time set to a discrete state set. The state discretization need not be uniform, and it may even be revised as the computation progresses. These two different types of discretizations can be visualized by considering how the function x(t), shown in figure 1a, might be reduced to discrete points. In a discrete time approximation, the value of the function is observed at regular intervals in time. This kind of discretization is shown in figure 1b. In a discrete event approximation, the function is sampled when it takes on regularly spaced values. This type of discretization is shown in figure 1c.