On the equivalence of [mu]-invariant measures for the minimal process and its q-matrix

In this paper we obtain necessary and sufficient conditions for a measure or vector that is [mu]-invariant for a q-matrix, Q, to be [mu]-invariant for the family of transition matrices, {P(t)}, of the minimal process it generates. Sufficient conditions are provided in the case when Q is regular and these are shown not to be necessary. When [mu]-invariant measures and vectors can be identified, they may be used, in certain cases, to determine quasistationary distributions for the process.

[1]  Frank Kelly,et al.  Reversibility and Stochastic Networks , 1979 .

[2]  W. Veech.  The necessity of Harris’ condition for the existence of a stationary measure , 1963 .

[3]  C. K. Cheong.  Quasi-stationary distributions in semi-Markov processes , 1970, Journal of Applied Probability.

[4]  David G. Kendall,et al.  Unitary Dilations of One-Parameter Semigroups of Markov Transition Operators, and the Corresponding Integral Representations for Markov Processes with a Countable Infinity of States , 1959 .

[5]  David Vere-Jones.  SOME LIMIT THEOREMS FOR EVANESCENT PROCESSES , 1969 .

[6]  F. P. Kelly.  Probability, Statistics and Analysis: Invariant measures and the q-matrix , 1983 .

[7]  J. Walrand,et al.  Sojourn times and the overtaking condition in Jacksonian networks , 1980, Advances in Applied Probability.

[8]  D. Vere-Jones.  GEOMETRIC ERGODICITY IN DENUMERABLE MARKOV CHAINS , 1962 .

[9]  E. Seneta,et al.  On quasi-stationary distributions in absorbing continuous-time finite Markov chains , 1967, Journal of Applied Probability.

[10]  David C. Flaspohler.  Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains , 1974 .

[11]  David G. Kendall,et al.  The calculation of the ergodic projection for Markov chains and processes with a countable infinity of states , 1957 .

[12]  Richard L. Tweedie,et al.  SOME ERGODIC PROPERTIES OF THE FELLER MINIMAL PROCESS , 1974 .

[13]  F. Kelly,et al.  Sojourn times in closed queueing networks , 1983, Advances in Applied Probability.

[14]  E. Seneta.  QUASI‐STATIONARY BEHAVIOUR IN THE RANDOM WALK WITH CONTINUOUS TIME , 1966 .

[15]  J. A. Cavender,et al.  Quasi-stationary distributions of birth-and-death processes , 1978, Advances in Applied Probability.

[16]  J. F. C. Kingman,et al.  The Exponential Decay of Markov Transition Probabilities , 1963 .

[17]  Rupert G. Miller.  Stationary equations in continuous time Markov chains , 1963 .

[18]  G. Reuter,et al.  Denumerable Markov processes and the associated contraction semigroups onl , 1957 .

[19]  T. E. Harris.  Transient Markov chains with stationary measures , 1957 .