Markov-Modulated Processes & Semiregenerative Phenomena by Antonio Pacheco, Loon Ching Tang, Narahari U Prabhu

By Antonio Pacheco, Loon Ching Tang, Narahari U Prabhu

The e-book contains a suite of released papers which shape a coherent remedy of Markov random walks and Markov additive approaches including their purposes. half I presents the principles of those stochastic methods underpinned via a pretty good theoretical framework in keeping with Semiregenerative phenomena. half II provides a few purposes to queueing and garage platforms.

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If the observed range of a process contains a semiregenerative set as a subset, then the process is semiregenerative. We take the view that semiregenerative phenomena are important in themselves and therefore worthy of study. For the analysis we use techniques based on results from Markov renewal theory. U. 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.

119) where Hj (A; t) is the conditional distribution measure (given that J is in state j during t units of time) of a L´evy process and Bjk is the distribution of Markov-modulated jumps as in Sec. 12. Let f (x, j) be a bounded function on R × E, such that for each fixed j, f is continuous and has a bounded continuous derivative ∂f /∂t. The infinitesimal generator of the process is defined as the operator A where Af (x, j) = lim h−1 h→0+ k∈E R [f (x + y, k) − f (x, j)]Fjk (dy; h). 119). 17. 122) ∂x July 30, 2008 15:50 World Scientific Book - 9in x 6in Markov Renewal and Markov-Additive Processes where dj is a constant, τ a centering   −1  τ (x) = |x|   1 and µjj is a L´evy measure.

61). We may view our process as describing the occurrence of an event E such that the rate of occurrence λ is modulated by the underlying Markov process J, and hence a random variable denoted by λJ(t) at time t. The following properties of this process are then direct extension of those in the standard Poisson process. (a) The means. 85) where e is the column vector with unit elements. Now let ∞ nPjk (n; t). 78) we find that the ejk (t) satisfy the differential equations e′jk (t) = ejl (t)νlk + πjk (t)λk , l∈E Adding these over k ∈ E we obtain d E[X(t)|J(0) = j] = dt k ∈ E.

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