Valentina I. Klimenok

Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory

Multi-dimensional asymptotically quasi-Toeplitz Markov chains with discrete and continuous time are introduced. Ergodicity and non-ergodicity conditions are proven. Numerically stable algorithm to calculate the stationary distribution is presented. An application of such chains in retrial queueing models with Batch Markovian Arrival Process is briefly illustrated.

A Retrial BMAP / PH / N System

A multi-server retrial queueing model with Batch Markovian Arrival Process and phase-type service time distribution is analyzed. The continuous-time multi-dimensional Markov chain describing the behavior of the system is investigated by means of reducing it to the corresponding discrete-time multi-dimensional Markov chain. The latter belongs to the class of multi-dimensional quasi-Toeplitz Markov chains in the case of a constant retrial rate and to the class of multi-dimensional asymptotically quasi-Toeplitz Markov chains in the case of an infinitely increasing retrial rate. It allows to obtain the existence conditions for the stationary distribution and to elaborate the algorithms for calculating the stationary state probabilities.

A retrial BMAP/SM/1 system with linear repeated requests

A retrial queueing system with the batch Markovian arrival process and semi-Markovian service is investigated. We suppose that the intensity of retrials linearly depends on the number of repeated calls. The distribution of the number of calls in the system is the subject of research. Asymptotically quasi-Toeplitz 2-dimensional Markov chains are introduced into consideration and applied for solving the problem.

Lack of Invariant Property of the Erlang Loss Model in Case of MAP Input

The BMAP/PH/N/0 model with three different disciplines of admission (partial admission, complete rejection, complete admission) is investigated. Loss probability is calculated. Impact of the admission discipline, variation and correlation coefficients of inter-arrival times distribution, and variation of service times distribution on loss probability is analyzed numerically. As by-product, it is shown by means of numerical results that the invariant property of the famous Erlang M/G/N/0 system, which was proven by B. A. Sevastjanov, is absent in case of the MAP input.

The BMAP/PH/N retrial queueing system operating in Markovian random environment

We consider the BMAP/PH/N retrial queueing system operating in a finite state space Markovian random environment. The stationary distribution of the system states is computed. The main performance measures of the system are derived. Presented numerical examples illustrate a poor quality of the approximation of the main performance measures of the system by means of the simpler queueing models. An effect of smoothing the traffic and an impact of intensity of retrials are shown.

The BMAP/G/1-->·/PH/1/M tandem queue with feedback and losses

Tandem queues are good mathematical models of communication systems and networks, so their investigation is important for theory and applications. In this paper, exact analytical analysis of the tandem queue of the BMAP/G/1->@?/PH/1/M type with customers loss in case when the intermediate buffer is full and with possible feedback is implemented. Possible correlation in the input stream is taken into account by of assuming the Batch Markovian Arrival Process (BMAP). Stability condition for the tandem is derived. Stationary distribution of the system states is derived. Loss probability is calculated. Dependence of the system performance measures on correlation in the input flow, buffer capacity, and feedback probability is numerically illustrated.

On the Modification of Rouche's Theorem for the Queueing Theory Problems

In analytic queueing theory, Rouches theorem is frequently used to prove the existence of a certain number of zeros in the domain of regularity of a given function. If the theorem can be applied it leads in a simple way to results concerning the ergodicity condition and the construction of the solution of the functional equation for the generating function of the stationary distribution. Unfortunately, the verification of the conditions needed to apply Rouches theorem is frequently quite difficult. We prove the theorem which allows to avoid some difficulties arising in applying classical Rouches theorem to an analysis of queueing models.

Erlang loss queueing system with batch arrivals operating in a random environment

We consider the BMAP/PH/N/0 queueing system operating in a finite state space Markovian random environment. Disciplines of partial admission, complete rejection and complete admission are analyzed. The stationary distribution of the system states is calculated. The loss probability and other main performance measures of the system are derived. The Laplace-Stieltjes transform of the sojourn time distribution of accepted customers is obtained. Illustrative numerical examples are presented. They show effect of an admission strategy, a correlation in an arrival process, a variation of a service process. Poor quality of the loss probability approximation by means of more simple models utilization is illustrated.

A tandem retrial queueing system with two Markovian flows and reservation of channels

We consider a tandem queueing system with single-server first station and multi-server second station. The input flow at Station 1 is described by the BMAP (batch Markovian arrival process). Customers from this flow are considered as non-priority customers. Customers of an arriving group, which meet a busy server, go to the orbit of infinite size. From the orbit, they try their luck in exponentially distributed random time. Service times at Station 1 are independent identically distributed random variables having an arbitrary distribution. After service at Station 1 a non-priority customer proceeds to Station 2. The service time by a server of Station 2 is exponentially distributed. Besides customers proceeding from Station 1, an additional MAP flow of priority customers arrives at Station 2 directly, not entering Station 1. If a priority customer meets a free server upon arrival, it starts service immediately. Else, it leaves the system forever. It is assumed that a few servers of Station 2 are reserved to serve the priority customers only. We calculate the stationary distribution and the main performance measures of the system. The problem of optimal design is numerically investigated.

The BMAP/G/1

Tandem queues are good mathematical models of communication systems and networks, so their investigation is important for theory and applications. In this paper, exact analytical analysis of the tandem queue of the BMAP/G/1->@?/PH/1/M type with customers loss in case when the intermediate buffer is full and with possible feedback is implemented. Possible correlation in the input stream is taken into account by of assuming the Batch Markovian Arrival Process (BMAP). Stability condition for the tandem is derived. Stationary distribution of the system states is derived. Loss probability is calculated. Dependence of the system performance measures on correlation in the input flow, buffer capacity, and feedback probability is numerically illustrated.