The formalism of probabilistic languages has been introduced for modeling the qualitative behavior of stochastic discrete-event systems. A probabilistic language is a unit interval-valued map over the set of traces of the system satisfying certain consistency constraints. Regular language operators such as choice, concatenation, and Kleene-closure have been defined in the setting of probabilistic languages to allow modeling of complex systems in terms of simpler ones. The set of probabilistic languages is closed under such operators, thus forming an algebra. It also is a complete partial order under a natural ordering in which the operators are continuous. Hence, recursive equations can be solved in this algebra. This is alternatively derived by using the contraction mapping theorem on the set of probabilistic languages which is shown to be a complete metric space. The notion of regularity, i.e., finiteness of automata representation, of probabilistic languages has been defined and it is shown that regularity is preserved under choice, concatenation, and Kleene-closure. We show that this formalism is also useful in describing system performance measures such as completion time, reliability, etc., and present properties to aide their computation.
['Vijay K. Garg', 'Ratnesh Kumar', 'Steven I. Marcus']
A probabilistic language formalism for stochastic discrete-event systems