2023, Fall

Abstract: This talk is dedicated to the queueing systems in which each customer requires a random number of servers simultaneously for the same random time (multiserver job queues). Such systems possess the non-work-conserving property (idle resources may coincide with a non-empty queue). These models are tailored to study supercomputer-like systems as well as modern multicore/multiprocessor computers (as well as smartphones), where programs may use multiple computing cores. In this talk we address the stability conditions and basic performance of the model which are obtained by matrix-analytic method.

Video

Abstract: Wе investigate the asymptotic behavior of Green functions associated to partially homogeneous random walks in the quadrant $\Z_+^2$.

With mild conditions on the positive jumps of the random walk, which can be unbounded, a complete analysis of the asymptotic behavior of the Green function of the random walk killed at $(0,0)$ is achieved. The main result is that eight regions of the set of parameters determine completely the possible limiting behaviors of Green functions of these Markov chains. These regions are defined by a set of relations for several characteristics of the distributions of the jumps.

In the transient case, a description of the Martin boundary is obtained.

In the positive recurrent case, our results give the exact asymptotics of the invariant distribution.

These limit theorems extend results of the literature obtained, up to now, essentially for random walks whose jump sizes are either $0$ or $1$ on each coordinate.

Our approach relies on a combination of several methods: probabilistic representations of solutions of analytical equations, Lyapounov functions, convex analysis, methods of homogeneous random walks, and complex analysis arguments.

Video, Slides

Abstract: We consider a homogeneous continuous-time Markov process on the state space Z_+ = {0, 1, 2, ...}, which we interpret as the movement of a particle. The particle can only transition to neighboring points in Z_+, meaning that its coordinate changes by one unit with each change in position. The time spent by the particle at each point depends on its coordinate. The process is equipped with a branching mechanism. Branching sources can be located at each point in Z_+, and we do not assume that the intensities are uniformly bounded. At the moment of branching, new particles appear at the branching point and continue to evolve independently of each other (and the other particles) according to the same laws as the initial particle. This branching Markov process corresponds to a Jacobian matrix. Formulas for the average number of particles in an arbitrary fixed point Z_+ at time t > 0 are obtained in terms of orthogonal polynomials corresponding to this matrix. The results are applied to some specific models, obtaining exact values for the average number of particles and finding their asymptotic behavior for large times.

Video

Abstract: TBA

Video

Abstract: An approach based on "integral intensity" - without assuming existence of the hasard function of the service time distribution - is offered for estimation of recurrent characteristics in the queueing model under the investigation. The method is based on the natural markovization of the model, on some embedded Markov chain, and on an affine Lyapunov function.

Video