A semi-Markov process with an inverse Gaussian distribution as sojourn time

Let Xn denote the state of a device after n repairs. We assume that the time between two repairs is the time τ taken by a Wiener process {W(t), t ⩾ 0}, starting from w0 and with drift μ < 0, to reach c ∈ [0, w0). After the nth repair, the process takes on either the value Xn−1 + 1 or Xn−1 + 2. The probability that Xn = Xn−1 + j, for j = 1, 2, depends on whether τ ⩽ t0 (a fixed constant) or τ > t0. The device is considered to be worn out when Xn ⩾ k, where k ∈ {1, 2, …}. This model is based on the ones proposed by Rishel (1991) [1] and Tseng and Peng (2007) [2]. We obtain an explicit expression for the mean lifetime of the device. Numerical methods are used to illustrate the analytical findings.