Yakov Satin

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.

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.

On a Queueing Model with Group Services

An analogue of M t /M t /S/S Erlang loss system for a queue with group services is introduced and considered. Weak ergodicity of the model is studied. We obtain the bounds on the rate of convergence to the limiting characteristics and consider two concrete queueing models with finding of their main limiting characteristics.

On M n (t)/M n (t)/S queues with catastrophes

We consider the Mn(t)/Mn(t)/S queue with catastrophes. The bounds on the rate of convergence to the limit regime and the estimates of the limit probabilities are obtained.

Some bounds for M(t)/M(t)/S queue with catastrophes

We consider the M(t)/M(t)/S queue with catastrophes. The bounds of the rate of convergence to the limit regime and the estimates of the limit probabilities are obtained. We also study the bounds for the mean of the queue and consider an example.

On Truncations For SZK Model.

We consider a class of inhomogeneous Markovian queueing models with batch arrivals and group services. Bounds on the truncation errors in weak ergodic case are obtained. Two concrete queueing models are studied as examples.

On Mt /Mt /S Type Queue With Group Services.

We consider Mt/Mt/S-type queueing model with group services. Bounds on the rate of convergence for the queue-length process are obtained. Ordinary Mt/Mt/S queue and Mt/Mt/S type queueing model with group services are studied as examples.

On stability for M t /M t /N/N queue

We obtain some stability bounds for nonstationary Erlang loss queueing model. Two specific examples of queues with close to periodic arrival and service rates are considered.