Asymptotic Behavior of Random Discrete Event Systems

A large class of dynamic systems-such as the material flow in production or assembly lines, the message flow in communication networks, and jobs in computer systems-can be modeled by Discrete Event Dynamic Systems (DEDS). These are systems where the occurrence of events is determined by the system itself and not described by time. Examples of such events are the beginning or completion of a task in an assembly line or the arrival of a message in a communication network. Current research on DEDS uses a number of methods. Among these are the logical approach to automata (see, e.g., Wonham and Ramadge, 1988), the perturbation analysis of trajectories (see, e.g., Ho, 1987), simulation, and the temporal approach, which we shall follow in this article (see, e.g., Cohen et al., 1985). In these models, activity times at a node of, for instance, a production network are successively determined by combining the activity times at other nodes during previous activity cycles with delay and/or transport times. The aim is then to describe the dynamic temporal behavior of the network, given the knowledge of the nature of the delay and transport times and the initial state of the system. An important aspect of the temporal approach is that it permits a conceptual simplification by use of the so-called max-algebra to describe the models, yielding an analogy to conventional system theory. The elements of this max-algebra are the real numbers (together with --CO), and the only admissible operations are maximization and addition. In Cuninghame-Green (1979), a systematic theory parallel to linear algebra has been developed for the max-algebra, and in Cohen et al. (1984,198s) the use of the max-algebra in the temporal approach to DEDS has been discussed and illustrated.