SOME DISTRIBUTION AND MOMENT FORMULAE FOR THE MARKOV RENEWAL PROCESS.
Abstract
A Markov Renewal Process (M.R.P.) is a Markov chain, in which the time taken to move from one state to another is not fixed, but is a random variable, whose distribution depends on the two states between which the transition is made. If f sub ij (t) denotes the number of transitions from i to j (i,j = 1, ... , m) of such an M.R.P. in the time interval (0,6), F(t) = (f sub ij (t)) is called the transition count matrix of the M.R.P. The distribution of this matrix is derived in this paper; its first and second order moments are obtained and asymptotic expressions for these moments, when t is large, are also derived. These expressions are in terms of the basic quantities p sub ij, Q sub ij of the M.R.P. and not in terms of recurrence times, as obtained by Pyke, and hence more suitable. Distributions, moments, and cross moments of cumulative processes associated with the M.R.P. are also derived. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jun 10, 1969
- Accession Number
- AD0691297
Entities
People
- A. M. Kshirsagar
- R. Wysocki
Organizations
- Southern Methodist University