Alexander I. Zeifman

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.

SOME ESTIMATES OF THE RATE OF CONVERGENCE FOR BIRTH AND DEATH PROCESSES

The ergodic properties of birth and death processes are studied. We obtain some explicit estimates for the rate of convergence by the methods of theory of differential equations. EXPONENTIAL ERGODICITY; LINEAR DIFFERENTIAL EQUATIONS

Some universal limits for nonhomogeneous birth and death processes

In this paper we consider nonhomogeneous birth and death processes (BDP) with periodic rates. Two important parameters are studied, which are helpful to describe a nonhomogeneous BDP X = X(t), t≥ 0: the limiting mean value (namely, the mean length of the queue at a given time t) and the double mean (i.e. the mean length of the queue for the whole duration of the BDP). We find conditions of existence of the means and determine bounds for their values, involving also the truncated BDP XN. Finally we present some examples where these bounds are used in order to approximate the double mean.

The N-limit of spectral gap of a class of birth–death Markov chains

We extend Zeifmans method for bounding the spectral gap and obtain the asymptotical behaviour, as N→∞, of the spectral gap of a class of birth–death Markov chains known as random walks on a complete graph of size N. Copyright

On truncations for weakly ergodic inhomogeneous birth and death processes

Abstract We investigate a class of exponentially weakly ergodic inhomogeneous birth and death processes. We consider special transformations of the reduced intensity matrix of the process and obtain uniform (in time) error bounds of truncations. Our approach also guarantees that we can find limiting characteristics approximately with an arbitrarily fixed error. As an example, we obtain the respective bounds of the truncation error for an Mt/Mt/S queue for any number of servers S. Arbitrary intensity functions instead of periodic ones can be considered in the same manner.

Perturbation bounds and truncations for a class of Markovian queues

We consider time-inhomogeneous Markovian queueing models with batch arrivals and group services. We study the mathematical expectation of the respective queue-length process and obtain the bounds on the rate of convergence and error of truncation of the process. Specific queueing models are shown as examples.

Perturbation Bounds for M t /M t /N Queue with Catastrophes

This articles focuses on M t /M t /N queue with catastrophes and obtains stability bounds for the main characteristics of the respective queue-length process.

Nonstationary Markovian queues

For a class of Markov queues, we estimate the decay function in different types of exponential convergence.

Stability of birth-and-death processes

Explicit bounds for the deviation of the distribution of the state probabilities of a perturbed birth-and-death process from that of a nonperturbed one are obtained. These bounds appear to be exact in time.

Limiting characteristics for finite birth-death-catastrophe processes.

General nonstationary birth-death process with possible catastrophes on finite state space is studied. The approach for obtaining the bounds on the rates of convergence to the limiting characteristics is outlined. Method for construction of the limiting characteristics is proposed. We also show that, as a rule, the initial conditions quickly become irrelevant. An example of respective process is shown in details.