Erik A. van Doorn

TL;DR: In this article, the authors studied the quasi-stationary distribution of a birth-death process on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1 · · ·} is an irreducible class.

TL;DR: This paper contains a survey of results related to quasi-stationary distributions, which arise in the setting of stochastic dynamical systems that eventually evanesce, and which may be useful in describing the long-term behaviour of such systems before evanescence.

TL;DR: In this paper, a complete solution is obtained for processes where, from some finite state n onwards, the birth and death rates are rational functions of n. The authors give necessary and sufficient conditions which suffice to settle the question for most processes encountered in practice.

TL;DR: The deviation matrix of an ergodic, continuous-time Markov chain with transition probability matrix P(·) and transition probability matrices P(t) and P(d) is the matrix D ≡ ∫ 0∞(P(t − P) dt as discussed by the authors.

TL;DR: In this article, the authors proved the convexity of the analytic continuation of the classical Erlang loss function as a function of x, x >= 0 and the uniqueness of the solution of the basic set of equations associated with the equivalent random method.