Extinction probability in a birth-death process with killing

We study birth-death processes on the non-negative integers where $\{1,2,\ldots\}$ is an irreducible class and $0$ an absorbing state, with the additional feature that a transition to state $0$ may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence as $t\to\infty$ of the probability of absorption at time $t$, and relate it to the common rate of convergence of the transition probabilities which do not involve state $0$. Finally, we derive upper and lower bounds for the probability of absorption at time $t$ by applying a technique which involves the logarithmic norm of an appropriately defined operator. \u

Birth-death processes with killing

The purpose of this note is to point out that Karlin and McGregor's integral representation for the transition probabilities of a birth-death process on a semi-infinite lattice with an absorbing bottom state remains valid if one allows the possibility of absorption into the bottom state from any other state. Conditions for uniqueness of the minimal transition function are also given. \u

On the speed of convergence to stationarity of the Erlang loss system

We consider the Erlang loss system, characterized by $N$ servers, Poisson arrivals and exponential service times, and allow the arrival rate to be a function of $N.$ We discuss representations and bounds for the rate of convergence to stationarity of the number of customers in the system, and display some bounds for the total variation distance between the time-dependent and stationary distributions. We also pay attention to time-dependent rates

Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes

We consider nonhomogeneous birth and death processes and obtain upper and lower bounds on the rate of convergence. Homogeneous birth and death processes and birth and death processes on a finite state space are studied as special cases.Birth and death process Differential equations, Ergodicity, Bounds, The rate of convergence

Bounds and asymptotics for the rate of convergence of birth-death processes

We survey a method initiated by one of us in the 1990's for finding bounds and representations for the rate of convergence of a birth-death process. We also present new results obtained by this method for some specific birth-death processes related to mean-field models and to the $M/M/N/N+R$ service system. The new findings pertain to the asymptotic behaviour of the rate of convergence as the number of states tends to infinity.\u

Lower bounds for the rate of convergence for continuous-time inhomogeneous Markov chains with a finite state space

An approach is proposed to the construction of general lower bounds for the rate of convergence of probability characteristics of continuous-time inhomogeneous Markov chains with a finite state space in terms of special “weighted” norms related to total variation. We study the sharpness of these bounds for finite birth–death–catastrophes process and for a Markov chain with large output intensity from a state. © 2018 Elsevier B.V