Transient Analysis of Immigration Birth–death Processes With Total Catastrophes

Abstract

Very few stochastic systems are known to have closed-form transient solutions. In this article we consider an immigration birth and death population process with total catastrophes and study its transient as well as equilibrium behavior. We obtain closed-form solutions for the equilibrium distribution as well as the closed-form transient probability distribution at any time t ≥ 0. Our approach involves solving ordinary and partial differential equations, and the method of characteristics is used in solving partial differential equations.

Level Crossing Ordering of Skip-Free-to-the-Right Continuous-Time Markov Chains

“…Later, the analysis was extended (see [5]) to semiMarkov processes (SMPs) and a set of sufficient conditions for the level crossing ordering of skip-free-to-the-right SMPs was derived (see [5,Theorem 2.1]), whose translation for skipfree-to-the-right continuous-time Markov chains (CTMCs) X (1) and X (2) is as follows:…”

“…(a) The transition probabilities of the embedded DTMCs (at transition epochs) associated to X (1) and X (2) are ordered, in distribution, for any common initial state.…”

“…In this paper, we essentially show that if X (1) and X (2) are two uniformizable (see [10]) skipfree-to-the-right CTMCs with (infinitesimal) generator matrices Q (1) and Q (2) , respectively, and if there exists a positive real constant α ∈ (0, 1] such that j ≥k q (1) ij ≤ α j ≥k q (2) ij for all i and k,…”

“…where q (k) ij , k = 1, 2, are the entries of Q (k) , then X (1) is slower in level crossing than X (2) . This result extends [5, Corollary 2.1], which established the same conclusion for α = 1.…”

“…for X (1) to be stochastically smaller than X (2) in the usual sense if their initial distributions are ordered in the same way, where the CTMCs do not need to be either uniformizable or skip-free to the right. The sufficient conditions (1) for the level crossing ordering of uniformizable skip-free-to-theright CTMCs may lead to positive conclusions in situations where some of Kirstein's conditions (2) and Irle's conditions ((a) and (b), above) do not hold.…”