Bounds on the Rate of Convergence for a class of Birth–death Process with Two Types of Catastrophes

This paper examines a special class of inhomogeneous birth-death process with two types of catastrophes. The first catastrophe, denoted as the α(t) catastrophe, results in the complete annihilation of all individuals in the system, while the second, β(t) catastrophe, leads to the partial demise of individuals, leaving only one survivor. In addition, we deal with general type of impatience. Namely, we assume that the incoming arrival rates decrease and the service rates increase as the queue length increases. By analyzing the forward Kolmogorov system and employing the logarithmic norm of the reduced infinitesimal characteristic matrix, we derived an estimate for the convergence rate to the limiting regime in both the l1 norm and the "weighted" 1D norm. Additionally, we provided an estimate for the expected number of individuals remaining in the system. A numerical example is presented, wherein the process X(t) is analyzed with specific intensity functions, illustrating the behavior of probabilities p<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf>(t), p<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf>(t), p<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf>(t) along with their mean values.