The Distribution Function of Opportunity-awaiting Time for Random System with Objective Domain

The description of opportunity-awaiting time (OAT) is addressed for a class of random systems with objective domain. The faster transition probability matrix (FTPM) and slower transition probability matrix (STPM) are defined to estimate transition probability matrix which is used to compute the distribution of OAT. This estimation is employed to deduce the OAT distribution function whose error is less than the expected value. Then OAT is proved to be phase-type (PH) distribution, so are its upper and lower bound which arbitrarily trend to accord. The analysis and example demonstrate that the distribution of OAT lies on the variance and the correlation coefficient of system output with fixed objective domain, which provides the foundation for researching the opportunity-awaiting control (OATC).