Birth and death processes with random environments in continuous time

Abstract

"A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process."

Parallel mutation-reproduction processes in random environments

“…Proof. This theorem is a generalization of Theorem 2.2 in [8] to the multitype situation, and its proof closely follows the pattern of the proof in [8].…”

“…We want to show an instability property of {Z(t)} t≥0 as in [8] which we follow closely. For this purpose, we make a further assumption on the random environment (named Environmental Assumption): {η(t)} t≥0 is an irreducible, positive recurrent, Markov chain in continuous time on a countable state space Y with jump times τ n ↑ +∞, n ≥ 0.…”

“…Remark 3.1. (Cogburn and Torrez [8]) If a Markov chain in a random environment, {Z n } n∈N0 , is proper, and states of Y × (Z 2 + \{0}) communicate (that two states communicate means that the former is accessible from the latter and the latter is also accessible from the former) and lead to Y × {0} (that means any state of Y × (Z 2 + \{0}) can reach the absorbing states Y × {0} with a positive probability), then necessarily (y, z) is transient and P (y,z) {Z n = z In this section, our main result (Theorem 3.4) follows by an application of a result (Theorem 3.3) due to Cogburn. Before stating Theorems 3.3 and 3.4, recall the definition of uniform ϕ−recurrence given in [17].…”

“…Assume that the splitting intensities {Λ α , α ∈Z 2 + } are all controlled by the same environmental process {η(t)} t≥0 , which is an irreducible, recurrent Markov chain in continuous time on a countable state space Y. Extending results of Cogburn and Torrez [8], we show the instability property of {Z(t)} t≥0 , i.e., P (y,z) {lim t→∞ Z(t) 1 = 0 or ∞} = 1, for every (y, z) ∈ Y × Z 2 + , given some additional technical conditions. The paper is organised as follows.…”

Jackson networks in nonautonomous random environments

“…For birth-death processes (and M/M/1/∞ queues) in a random environment there is a long history of investigations; see, e.g. [4], [5], [10], [21], and [37]. Related research is on service systems under external influences which cause the service process to break down or decrease availability of servers; see, e.g.…”