ABSTRACT
This chapter deals with Markov processes which have parameter set and state space or subsets of it. The chapter considers homogeneous Markov chains. Hence no confusion can arise if only Markov chains are referred to. The classification concepts already introduced for discrete-time Markov chains can analogously be defined for continuous-time Markov chains. The intuitive interpretation of the Markov property is the same as for discrete-time Markov chains. Transitions between the states of a continuous-time homogeneous Markov chain are controlled by its transition probabilities. The transition probabilities are comprised in the matrix of transition probabilities. The chapter discusses some structural properties of continuous-time Markov chains, which are fundamental to mathematically modeling real systems. Transition Times and Transition Rates is only possible to exactly model real systems by continuous-time Markov chains if the lengths of the time periods between changes of states are exponentially distributed, since in this case the 'memoryless property' of the exponential distribution implies the Markov property.
