Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process

Abstract

This paper is concerned with two problems in connection with exponential ergodicity for birth-death processes on a semi-infinite lattice of integers. The first is to determine from the birth and death rates whether exponential ergodicity prevails. We give some necessary and some sufficient conditions which suffice to settle the question for most processes encountered in practice. In particular, 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 second, more difficult, problem is to evaluate the decay parameter of an exponentially ergodic birth-death process. Our contribution to the solution of this problem consists of a number of upper and lower bounds.

Geomatric ergodicity and quasi-stationarity in discrete-time birth-death processes

“…This transformation is similar to the duality transformation used, for example, in van Doom [5], [7] in the context of continuous-time birth-death processes, and a proof of the above results may be given on the basis of the corresponding results for continuous-time birth-death processes and the transformation described in (2.20); we shall not give the details. It should be noted that the polynomials {QJ(x) } y of (3.9) need not be random walk polynomials, so that the measure ijr* need not be a random walk measure.…”

“…Further information on the random walk measure x\r can be obtained by transforming the sequence of random walk polynomials into a sequence of birth-death polynomials, as suggested by Karlin and McGregor [15]. In this way many results from the theory of birth-death processes in continuous time (see Karlin and McGregor [13], [14] and van Doom [5], [7]) can be translated in terms of random walks. Pursuing this approach we define 20) and…”

“…The present paper constitutes to some extent the discrete-time counterpart of van Doom [5] and van Doom [7], which discuss exponential ergodicity and quasistationarity, respectively, in continuous-time birth-death processes. However, a study of limiting conditional (or quasi-limiting) distributions as in [7], which turned out to be considerably more complicated in discrete time, has been relegated to a separate paper, see van Doom and Schrijner [9].…”

Extinction Probability in A Birth-Death Process with Killing

“…If α 0 = α then the problem of determining α 0 can be reduced to that of finding the decay parameter in a pure birth-death process, for which many results are available (see [5], [6], [12], [18], [19], [23], [25], and [27]- [29]). Indeed, defineX := {X(t), t ≥ 0} to be the birth-death process on C with birth and death rates…”

Spectral Gap of the Erlang A Model in the Halfin-Whitt Regime

“…To establish Theorem 12, Gamarnik and Goldberg used the framework of Karlin and McGregor [20] for birth-death processes, and the result of Van Doorn [9] on the spectral gap of the M/M/s system. In [14] the starting point is the discrete M/M/s model, and its spectral gap is then analyzed in the Halfin-Whitt regime (1).…”