Vyacheslav M. Abramov

Non-cooperative routing in loss networks

The paper studies routing in loss networks in the framework of a non-cooperative game with selfish users. Two solution concepts are considered: the Nash equilibrium, corresponding to the case of a finite number of agents (such as service providers) that take routing decisions, and the Wardrop equilibrium, in which routing decisions are taken by a very large number of individual users. We show that these equilibria do not fall into the standard frameworks of non-cooperative routing games. As a result, we show that uniqueness of equilibria or even of utilizations at equilibria may fail even in the case of simple topology of parallel links. However, we show that some of the problems disappear in the case in which the bandwidth required by all connections is the same. For the special case of a parallel link topology, we obtain some surprisingly simple way of solving the equilibrium for both cases of Wardrop as well as Nash equilibrium.

Some Results for Large Closed Queueing Networks with and without Bottleneck: Up- and Down-Crossings Approach

The paper provides the up- and down-crossing method to study the asymptotic behavior of queue-length and waiting time in closed Jackson-type queueing networks. These queueing networks consist of central node (hub) and k single-server satellite stations. The case of infinite server hub with exponentially distributed service times is considered in the first section to demonstrate the up- and down-crossing approach to such kind of problems and help to understand the readers the main idea of the method. The main results of the paper are related to the case of single-server hub with generally distributed service times depending on queue-length. Assuming that the first k−1 satellite nodes operate in light usage regime, we consider three cases concerning the kth satellite node. They are the light usage regime and limiting cases for the moderate usage regime and heavy usage regime. The results related to light usage regime show that, as the number of customers in network increases to infinity, the network is decomposed to independent single-server queueing systems. In the limiting cases of moderate usage regime, the diffusion approximations of queue-length and waiting time processes are obtained. In the case of heavy usage regime it is shown that the joint limiting non-stationary queue-lengths distribution at the first k−1 satellite nodes is represented in the product form and coincides with the product of stationary GI/M/1 queue-length distributions with parameters depending on time.

A large closed queueing network with autonomous service and bottleneck

This paper studies the queue-length process in a closed Jackson-type queueing network with the large number N of homogeneous customers by methods of the theory of martingales and by the up- and down-crossing method. The network considered here consists of a central node (hub), being an infinite-server queueing system with exponentially distributed service times, and k single-server satellite stations (nodes) with generally distributed service times with rates depending on the value N. The service mechanism of these k satellite stations is autonomous, i.e., every satellite server j serves the customers only at random instants that form a strictly stationary and ergodic sequence of random variables. Assuming that the first k-1 satellite stations operate in light usage regime the paper considers the cases where the kth satellite station is a bottleneck node. The approach of the paper is based both on development of the method from the paper by Kogan and Liptser [16], where a Markovian version of this model has been studied, and on development of the up- and down-crossing method.

A Large Closed Queueing Network Containing Two Types of Node and Multiple Customer Classes: One Bottleneck Station

The paper studies a closed queueing network containing two types of node. The first type (server station) is an infinite server queueing system, and the second type (client station) is a single server queueing system with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. It is assumed that there are r server stations. At the initial time moment all units are distributed in the server stations, and the ith server station contains Ni units, i=1,2,...,r, where all the values Ni are large numbers of the same order. The total number of client stations is equal to k. The expected times between departures in the client stations are small values of the order O(N−1) (N=N1+N2+...+Nr). After service completion in the ith server station a unit is transmitted to the jth client station with probability pi,j (j=1,2,...,k), and being served in the jth client station the unit returns to the ith server station. Under the assumption that only one of the client stations is a bottleneck node, i.e. the expected number of arrivals per time unit to the node is greater than the expected number of departures from that node, the paper derives the representation for non-stationary queue-length distributions in non-bottleneck client stations.

Asymptotic Analysis of Loss Probabilities in GI/M/m/n Queueing Systems as n Increases to Infinity

Abstract The paper studies asymptotic behavior of the loss probability for the GI/M/m/n queueing system as n increases to infinity. The approach of the paper is based on applications of classic results of Takács [24] and the Tauberian theorem with remainder of Postnikov [17] associated with the recurrence relation of convolution type. The main result of the paper is associated with asymptotic behavior of the loss probability. Specifically it is shown that in some cases (precisely described in the paper) where the load of the system approaches 1 from the left and n increases to infinity, the loss probability of the GI/M/m/n queue becomes asymptotically independent of the parameter m.

On Existence of Limiting Distribution for Time-Nonhomogeneous Countable Markov Process

AbstractIn this paper, sufficient conditions are given for the existence of limiting distribution of a nonhomogeneous countable Markov chain with time-dependent transition intensity matrix. The method of proof exploits the fact that if the distribution of random process Q=(Qt)t≥0 is absolutely continuous with respect to the distribution of ergodic random process Q°=(Q°t)t≥0, then n

Statistical Analysis of Single-server Loss Queueing Systems

Q_t xrightarrow[{t to infty }]{{law}}pi

Conservative and Semiconservative Random Walks: Recurrence and Transience

n where π is the invariant measure of Q°. We apply this result for asymptotic analysis, as t→∞, of a nonhomogeneous countable Markov chain which shares limiting distribution with an ergodic birth-and-death process.