Before revision:  The main focus of the present work is  the expectation of the hitting time (to a given target set) of the process with restart , for which we obtain the explicit formula.Observing that the process with restart is positive Harris recurrent, we obtain the expression of its unique invariant probability. Then we show the equivalence of the following statements: (a) the boundedness (with respect to the initial state) of the expectation of the hitting time ; (b) the finiteness of the expectation of the hitting time for almost all the initial states with respect to the unique invariant probability;
After revision:  The main focus of the present work is the expectation of the hitting time (to a given target set) of the process with restart .The formula is convenient when considering the problem of optimizationof the expected hitting time with respect to the unique invariant probability;