The Theory of Timed I/O Automata presents a new framework for describing and analyzing the behavior of timed systems. This mathematical framework uses timed I/O automata (TIOA) for the representation of systems. The TIOA framework is a special case of the hybrid I/O automaton (HIOA) modeling framework. The authors have used knowledge gained from developing the HIOA framework to revise earlier work on timed I/O automaton models. Their purpose is to have a timed I/O automaton model that is compatible with the new HIOA model.Designers of real-time systems or timing-based algorithms can use the TIOA framework to describe complex systems and to decompose them into manageable pieces. This framework is a natural basis for computerized modeling and analysis.This lecture includes:* Mathematical preliminaries* Definitions of external behavior for timed automata and implementation and simulation relationships between them* Trajectories and timed sequences, which are useful for describing the behavior of timed systesm* Composition and hiding operations for timed automata, along with operations for adding bounds that relate TIOAs to other timed automaton models* Compositionality results for TIOAs as well as special classes of progressive and receptive TIOAsKaynar, Dilsun K. is the author of 'Theory of Timed I/O Automata ', published 2006 under ISBN 9781598290103 and ISBN 159829010X.