Section: New Results

Advances in methodological tools

Participants : Eitan Altman, Konstantin Avrachenkov, Alain Jean-Marie, Philippe Nain.

Perturbation analysis

In [17] K. Avrachenkov, together with R. Burachik, J. Filar V. Gaitsgory (Univ. of South Australia, Australia), study a linear programming problem with a linear perturbation introduced through a parameter $ϵ>0$. The authors identify and analyze an unusual asymptotic phenomenon in such a linear program. Namely, discontinuous limiting behavior of the optimal objective function value of such a linear program may occur even when the rank of the coefficient matrix of the constraints is unchanged by the perturbation. The authors show that, under mild conditions, this phenomenon is a result of the classical Slater constraint qualification being violated at the limit and propose an iterative, constraint augmentation approach for resolving this problem.

Zero-sum games

In [18] K. Avrachenkov, together with L. Cottatellucci and L. Maggi (both from Eurecom, France), study zero-sum two-player stochastic games with perfect information. The authors propose two algorithms to find the uniform optimal strategies and one method to compute the optimality range of discount factors. The convergence in finite time for one algorithm is proved. In particular, the uniform optimal strategies are also optimal for the long run average criterion and, in transient games, for the undiscounted criterion as well.

Approximations in semi-Markov zero-sum games

In conjunction with E. Della Vecchia and S. Di Marco (both from National Univ. Rosario, Argentina), A. Jean-Marie has pursued the studies on the Rolling Horizon procedure and other approximations in stochastic control problems. Their first study on convergence conditions for average-cost MDPs has been published in [23] .

They have then turned to the case of discounted semi-Markov zero-sum games. Generalizing previous contributions of the literature, they have established existence conditions and geometric convergence results when action spaces are compact and rewards possibly unbounded. The bounds they obtain hold for the Rolling Horizon procedure as well as for variants called Approximate Rolling Horizon [91] . In the same semi-Markovian context, they have also performed a sensitivity analysis of the model with respect to its parameters: cost function, discount factor, transition probabilities and state space [90] .

Retrial queues

In [84] K. Avrachenkov and P. Nain, in collaboration with U. Yechiali (Tel Aviv Univ.), consider a retrial system with two input streams and two orbit queues. More specifically, there are two independent Poisson streams of jobs feeding a single-server service system having a limited common buffer that can hold at most one job. If a type-i job (i=1,2) finds the server busy, it is blocked and routed to a separate type-i retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. The authors study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a boundary value problem. Various performance measures are calculated and numerical results are presented.

Branching processes

In collaboration with D. Fiems (Gent Univ., Belgium), E. Altman introduces in [41] non-standard new branching processes and applies them to evaluate queueing processes. The processes are characterized by replacing the standard Algebra involved in the definition of branching processes by the max-plus algebra. Among the applications introduced are (i) polling systems with infinite server, and (2) new Cruz type bounds for systems with feedback.

Standard branching have been used in the past to study polling systems. In [30] V. Kavitha (LIA/Univ. Avignon) and E. Altman have revisited this method and applied it to spatial sensors, that receive or send data via a mobile relay or base stations. They derive conservation laws for this continuous state space polling system which allows them to compute optimal polling strategies.

D. Fiems (Gent Univ., Belgium) and E. Altman have further used in [24] semi-linear processes, which extend branching processes, to compute expected waiting times in polling systems with generally distributed walking times (the standard i.i.d. assumption is replaced with the assumption that the walking times are stationary ergodic).

In [22] , the problem of parallel TCP connections is studied by O. Czerniak and U. Yechiali (Tel Aviv Univ., Israel), in collaboration with E. Altman, for a model in which, when the sum of throughputs reaches some value, there is a loss. It is assumed that the connection to suffer the loss is chosen according to a round robin policy. The expected throughputs of the connections are computed using an approach based on multitype branching processes.