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We establish stability criterion for a two-class retrial system with Poisson inputs, general class-dependent service times and class-dependent constant retrial rates. We also characterise an interesting phenomenon of partial stability when one orbit is tight but the other orbit goes to infinity in probability. All theoretical results are illustrated by numerical experiments.

We deal with zero-sum two-player stochastic games with perfect information. We propose two algorithms to find the uniform optimal strategies and one method to compute the optimality range of discount factors. We prove the convergence in finite time for one algorithm. The uniform optimal strategies are also optimal for the long run average criterion and, in transient games, for the undiscounted criterion as well.

A random walk can be used as a centrality measure of a directed graph. However, if the graph is reducible the random walk will be absorbed in some subset of nodes and will never visit the rest of the graph. In Google PageRank the problem was solved by the introduction of uniform random jumps with some probability. Up to the present, there is no final answer to the question about the choice of this probability. We propose to use a parameter-free centrality measure which is based on the notion of a quasi-stationary distribution. Specifically, we suggest four quasi-stationary based centrality measures, analyze them and conclude that they produce approximately the same ranking.

In this note we study multivariate perturbations of algebraic equations. In general, it is not possible to represent the perturbed solution as a Puiseux-type power series in a connected neighborhood. For the case of two perturbation parameters we provide a sufficient condition that guarantees such a representation. Then, we extend this result to the case of more than two perturbation parameters. We motivate our study by the perturbation analysis of a weighted random walk on the Web Graph. In an instance of the latter the stationary distribution of the weighted random walk, the so-called Weighted PageRank, may depend on two (or more) perturbation parameters in a manner that illustrates our theoretical development.

We consider systems of tandem blocking queues having a common retrial queue. The model represents dynamics of short TCP transfers in the Internet. Analytical results are available only for a specific example with two queues in tandem. We propose approximation procedures involving simple analytic expressions, based on mean value analysis (MVA) and on fixed point approach (FPA). The mean sojourn time of a job in the system and the mean number of visits to the orbit queue are estimated by the MVA which needs as an input the fractions of blocked jobs in the primary queues. The fractions of blocked jobs are estimated by FPA. Using a benchmark example of the system with two primary queues, we conclude that the approximation works well in the light traffic regime. We note that our approach becomes exact if the blocking probabilities are fixed. Finally, we consider two optimization problems regarding minimizing mean total sojourn time of a job in the system: (i) finding the best order of queues and (ii) allocating a given capacity among the primary queues.

We study the interaction between the AIMD (Additive Increase Multiplicative Decrease) multi-socket congestion control and a bottleneck router with Drop Tail buffer. We consider the problem in the framework of deterministic hybrid models. First, we show that trajectories always converge to limiting cycles. We characterize the cycles. Necessary and sufficient conditions for the absence of multiple jumps in the same cycle are obtained. Then, we propose an analytical framework for the optimal choice of the router buffer size. We formulate this problem as a multi-criteria optimization problem, in which the Lagrange function corresponds to a linear combination of the average goodput and the average delay in the queue. Our analytical results are confirmed by simulations performed with MATLAB Simulink.

We consider jamming in wireless networks in the framework of zero-sum games with α-utility functions. The base station has to distribute the power fairly among the users in the presence of a jammer. The jammer in turn tries to distribute its power among the channels to produce as much harm as possible. The Shannon capacity and SNIR optimization are particular cases of the proposed more general α-fairness SNIR based utility functions. Specifically, we consider two α-fairness utility functions, based on SNIR and Shifted SNIR. This game can also be viewed as a minimax problem against the nature. We show that the game has the unique equilibrium and investigate its properties. In particular, in several important cases we present the equilibrium strategies and the Jain's fairness index in closed form. It turns out that there is an important difference between SNIR and Shifted SNIR α-fairness utility functions. In the case of the SNIR based utility function all users obtain nonzero powers when α > 0. On contrary, when the Shifted SNIR based utility function is used, some users with bad channel conditions might not receive any power at all. We have also detected a surprising non-monotone behaviour of the Jain's fairness index in the case of the SNIR based utility function.

We study the interaction between the MIMD (Multiplicative Increase Multiplicative Decrease) congestion control and a bottleneck router with Drop Tail buffer. We consider the problem in the framework of deterministic hybrid models. We study conditions under which the system trajectories converge to limiting cycles with a single jump. Following that, we consider the problem of the optimal buffer sizing in the framework of multi-criteria optimization in which the Lagrange function corresponds to a linear combination of the average throughput and the average delay in the queue. As case studies, we consider the Slow Start phase of TCP New Reno and Scalable TCP for high speed networks. © 2009 Elsevier B.V. All rights reserved.

We analyze the performance of an Additive Increase Multiplicative Decrease (AIMD)-like flow control mechanism. The transmission rate is considered to increase linearly in time until the receipt of a congestion notification, when the transmission rate is multiplicatively decreased. AIMD captures the steady state behavior of TCP in the absence of timeouts and in the absence of maximum window size limitation. We introduce a general fluid model based on a multi-state Markov chain for the moments at which the congestion is detected. With this model, we are able to account for correlation and burstiness in congestion moments. Furthermore, we specify several simple versions of our general model and then we identify their parameters from real TCP traces.

There are a few procedures for computing the Laurent series expansions for the mean passage time matrix and for the deviation matrix of a singularly perturbed Markov chain. We suggest here a method for computing the first terms in these expansions in a way which highlights the system dynamics in various time scales.

We study singularly perturbed linear programs. These are linear programs whose constraints and objective coefficients depend on a small perturbation parameter, and furthermore the constraints become linearly dependent when the perturbation parameter goes to zero. Problems like that were studied by Jeroslow in 1970s. He proposed simplex-like method, which works over the field of rational functions. Here we develop an alternative asymptotic simplex method based on Laurent series expansions. This approach appears to be more computationally efficient. In addition, we point out several possible generalizations of our method and provide simple updating formulae for the perturbed solution.