By H. Jerome Keisler

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Prabhu (1988) ‘Theory of semiregenerative phenomena’, J. Appl. Probab. 25A, pp. U. Prabhu (1994), ‘Further results for semiregenerative phenomena’, Acta Appl. Math. 34, 1-2, pp. 213–223, whose contents are reproduced with permission from Kluwer Academic Publishers. 39 BookTPP July 30, 2008 15:50 40 World Scientific Book - 9in x 6in Markov-Modulated Processes and Semiregenerative Phenomena We present some basic definitions in Sec. 2, starting from that of semiregenerative processes, which states that a process Z = {Ztl , (t, l) ∈ T × E}, with T = R+ or T = N+ and E being a countable set, is a semiregenerative phenomenon if, in particular, it takes values only the values 0 and 1 and has the following partial lack of memory property on the first index with respect to the observation of the value 1: r P {Zti li = 1 (1 ≤ i ≤ r)|Z0,l0 = 1} = i=1 P {Zti −ti−1 ,li = 1|Z0,li−1 = 1} for 0 = t0 ≤ t1 ≤ · · · ≤ tr and l0 , l1 , .

2. Suppose J(0) = j. Then J(t) = j + n for Sn ≤ t < Sn+1 (n ≥ 0) ∆ for t ≥ L. , as we have already seen. If the distribution Fj has the exponential density λj e−λj x (0 < λj < ∞), then J reduces to the pure birth process. 8 Markov-Additive Processes: Basic Definitions We are given a probability space (Ω, F, P ) and denote R = (−∞, ∞), E = a countable set and N+ = {0, 1, 2, . }. 2. A Markov-additive process (X, J) = {(X(t), J(t)), t ≥ 0} is a two-dimensional Markov process on the state space R × E such that, for s, t ≥ 0, the conditional distribution of (X(s + t) − X(s), J(s + t)) given (X(s), J(s)) depends only on J(s).

7 BookTPP Markov-Modulated Processes and Semiregenerative Phenomena The Semi-Markov Process We define a process J = {J(t), t ≥ 0} as follows. 31). The process J is called the minimal semi-Markov process associated with the MRP {(Sn , Jn ), n ≥ 0}. Denote J(t) = Pjk (t) = P {J(t) = k|J(0) = j}, j, k ∈ E, t ≥ 0. 9. We have t Pjk (t) = 0− Proof. Ujk {ds}Pk {S1 > t − s}. 50) An easy calculation shows that t Pjk (t) = Pj {S1 > t} δjk + 0− l∈E Qjl {ds}Plk (t − s). 35) with gj (t) = Pj {S1 > t} δjk . 51) is given by t t l∈E 0− Ujl {ds}gl (t − s) = 0− Ujk {ds}Pk {S1 > t − s}.