The Integral of a Step Function Defined on a Semi-Markov Process

Consider an m-state, irreducible, recurrent semi-Markov process (S.M.P.) and a step function f( · ) which takes on the value vᵢ, i = 1, ⋯, m, when the S.M.P. is in state i. We study the integral of f( · ) between 0 and t. The Laplace transform of the characteristic function of the integral is obtained in a general form by use of matrix notation. In the case of a stationary semi-Markov process the transform of the expected value of the integral is inverted in closed form. Asymptotic properties of the expected value of the integral are derived by applying “Smith’s Key Renewal Theorem”.