Discrete Event Systems (DES's) are those whose states have
logical or symbolic values rather than numerical ones [19],
[20]. In addition, state changes are driven
by internal and/or external events rather than by time on a
global clock. Such systems arise in practice in the areas of
communication networks, robotics, traffic systems, flexible
manufacturing systems, and many others. Typical characteristics of
DES's include: occurrence of events at discrete times;
concurrency or parallel activities; asynchronous behavior,
which refers to the aperiodic evolution of system behavior due to
variable completion times of some of its activities; choice,
which occurs when more than one activity is possible at a given state;
mutual exclusion, often seen when resources are shared; and
real time constraints. Real time constraints are commonly used in
describing safety properties in a real system and so are of critical
importance. Two examples of DES's are given below in Figures
1 and 2.
Several tools have been introduced to model and control DES's. Among
them, we have: formal languages models, pioneered by Ramadge and
Wonham [19], [20], [26]; models based on Petri
Nets [17]; models based on temporal logic [14]; and
algebraic models. In this paper we review some of the tools
currently available to model DES's, and we illustrate through our
working examples the modeling stages that they encompass, or the
process they follow in creating the control model. Our purpose for this
is to provide some measure of modeling power by identifying
some of the advantages and disadvantages of each of these models.
In the first part of this paper, we give a brief overview of four
specific models. We do this from the point of view of Ostroff's
modeling framework [15]; that is, we identify for each model,
the synthesis of the plant P, the description of specifications, 
, and the
design of the controller, C. We start with the Ramadge & Wonham
(R&W) framework [19] [20], which uses formal
languages and state machines. Next, we describe Timed Transition
Models, which are used by Ostroff [14] in synthesis of
controllers for real-time DES's. This model is also based on state
machines. Then, we discuss Petri Net models, and we conclude with
Net Condition Event Systems (NCES). NCES models are an
extension of Petri Nets and are used in the controller synthesis
method by Hanisch & Rausch [6], [22]. In the
second part of this paper, we illustrate these methods on our two
examples, the Two-Pusher Example, and the Three-Machine Example.
Finally, we provide some comparative analysis for each of these
examples, and we pay special attention to how the descriptiveness and
representation of the plant models affect the models' capabilities for
handling various specifications, and the implementation of the
controller synthesis algorithms.