TY - JOUR
T1 - Simple Derivations of Properties of counting processes associated with Markov Renewal Processes
AU - Ball, F.
AU - Milne, Robin
PY - 2005
Y1 - 2005
N2 - A simple, widely applicable method is described for determining factorial moments of (N) over cap (t), the number of occurrences in (0, t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t -> infinity. The factorial moment formulae combine to yield an expression for the probability generating function of (N) over cap (t), and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.
AB - A simple, widely applicable method is described for determining factorial moments of (N) over cap (t), the number of occurrences in (0, t] of some event defined in terms of an underlying Markov renewal process, and asymptotic expressions for these moments as t -> infinity. The factorial moment formulae combine to yield an expression for the probability generating function of (N) over cap (t), and thereby further properties of such counts. The method is developed by considering counting processes associated with events that are determined by the states at two successive renewals of a Markov renewal process, for which it both simplifies and generalises existing results. More explicit results are given in the case of an underlying continuous-time Markov chain. The method is used to provide novel, probabilistically illuminating solutions to some problems arising in the stochastic modelling of ion channels.
U2 - 10.1239/jap/1134587814
DO - 10.1239/jap/1134587814
M3 - Article
SN - 0021-9002
VL - 42
SP - 1031
EP - 1043
JO - Journal of Applied Probability
JF - Journal of Applied Probability
IS - 4
ER -