## Section: New Results

### Stochastic processes, queueing, control theory and game theory

Participants : Eitan Altman, Konstantin Avrachenkov, Alain Jean-Marie, Vijay Kamble, Danil Nemirovsky, Natalia Osipova, Alonso Silva.

#### Boostrap techniques for simulating the M/M/1 queue

Participant : Eitan Altman.

In [89] , E. Altman, J. Rojas-Mora and T. Jimenez (University of Avignon ) present several conclusions on how to simulate faster and with higher accuracy. Having chosen the M/M/1 queue, the authors are able to make use of a rich knowhow in the simulation theory as a reference, as well as of simple known formulas for the performance measures of the queue. The authors present both generic simulation aspects that would probably be useful for any implementation, as well as some aspects that are specific to the simulations in ns-2. They illustrate how bootstrap techniques can improve performance in the sense of reducing the number of simulations and of increasing the accuracy.

#### Tensor approach to mixed high-order moments of absorbing Markov chains

Participant : Danil Nemirovsky.

In absorbing Markov chains first moments and non-mixed second moments are determined in matrix form. Mixed moments of higher orders cannot be represented in a matrix form and have not been calculated so far. In [127] , D. Nemirovsky succeeds in computing them in closed-form by using a tensor approach.

#### Advances in game theory

Participants : Eitan Altman, Konstantin Avrachenkov, Vijay Kamble, Alonso Silva.

##### Foundations of evolutionary Games

In evolutionary games, various populations have an impact on each other through a large number of local interactions each involving a small number of players. Each such interaction defines a fitness that depends on the actions of each player involved. In [17] , E. Altman, Y. Hayel, H. Tembine and R. El-Azouzi (University of Avignon ) add a notion of individual state to these games and consider the situation in which an action of a player determines not only its immediate fitness in given stage but also the transition probabilities to the next state. The player no longer maximizes her/his immediate fitness but rather maximizes the expected average cummulated fitness over time. The authors define an appropriate notion of equilibrium and propose a way to compute it. Applications to energy management problems are described by [134] .

##### Foundations of population Games

Population games are similar to evolutionary games with the difference that interactions are not any more local but involve a continuum of players. In [97] , P. Wiecek (University of Wroclaw , Poland), E. Altman and Y. Hayel (University of Avignon ) study a dynamic version of such games where each player has some individual state, and the actions of the player determine the transitions between the states. They use this formalism to solve a power control problem in a CDMA system (where all mobiles use the same spectrum of frequencies at the same time). To do so, they adapt the theory of anonymous sequential games by Jovanovic & Rosenthal, J Math. Econ, 1988.

##### Stochastic games with delay sharing information pattern

Non-cooperative game theory has gained much interest as a paradigm for decentralized control in communication networks. It allows one to get rid of the need for a centralized controller. Decentralizing the decision making may result in situations where agents (decision makers) do not have the same view of the network. The global view of the network state cannot be available to an agent as fast as the information on its local state. Incorporating into the decentralized control paradigm this information asymmetry renders it applicable to a much wider class of situations. In [47] , E. Altman, V. Kamble and A. Silva model the above information asymmetry using the one-step delay sharing information pattern from team theory and generalize it to the context of non-cooperative games. They study its properties and apply it to a distributed power control problem.

#### Branching processes with queueing applications

Participant : Eitan Altman.

In the last several years, E. Altman has been developing the theory of
branching processes with non-Markovian immigration process, and
has been applying it to a large number of queueing problems:
polling with correlated vacations, the infinite server with correlated
arrivals and more. Branching processes can be written in the form
X_{n + 1} = A_{n}(X_{n}) + B_{n} where B_{n} is the immigration and
A_{n} is a subordinator (non-decreasing
Lévy process) with the following central properties:
(i) it is infinitely divisible and (ii) it has independent increments.
In [13] , E. Altman extends this class of processes by
dropping the requirement of independent increments in the process {A_{n}}_{n} .
The equations that define this new class are called
“semi-linear stochastic difference equations”.
Explicit expressions for the first order-moments of X_{n} in steady-state under
the weak assumption that the process {B_{n}} is stationary and ergodic
are derived.

#### Advances in queueing theory

Participant : Alain Jean-Marie.

In conjunction with A. Ben Tahar (University Hassan I , Settat, Morocco), A. Jean-Marie has completed the analysis of the fluid limits in the Multiclass Processor Sharing queue, extending it to the Discriminatory Processor Sharing service discipline. An analysis of the accuracy of fluid approximations has been added to the work [61] , [113] .

#### Stochastic scheduling

Participants : Konstantin Avrachenkov, Alain Jean-Marie, Natalia Osipova.

The problem of optimal service scheduling in queues with impatience has been considered by A. Jean-Marie and E. Hyon (University of Paris X ). The problem has been solved in [124] for a single-server, discrete-time queue with geometrically-distributed impatience. For this simple system, the optimal policy is “always serve” or “never serve”, depending on a simple criterion on costs, the impatience probability and the discount factor.

In [21] K. Avrachenkov, P. Brown and N. Osipova (France Telecom R&D ) analyze the Two Level Processor Sharing (TLPS) scheduling discipline with the hyper-exponential job size distribution and with the Poisson arrival process. TLPS is a convenient model to study the benefit of the file size based differentiation in TCP/IP networks. In the case of the hyper-exponential job size distribution with two phases, the authors find a closed form analytic expression for the expected sojourn time and an approximation for the optimal value of the threshold that minimizes the expected sojourn time. In the case of the hyper-exponential job size distribution with more than two phases, the authors derive a tight upper bound for the expected sojourn time conditioned on the job size. It is shown that when the variance of the job size distribution increases, the gain in system performance increases and the sensitivity to the choice of the threshold near its optimal value decreases.

In [87] N. Osipova (France Telecom R&D ), U. Ayesta (Laas-Cnrs ) and K. Avrachenkov apply the Gittins optimality result to characterize the optimal scheduling discipline in a multi-class M/G/1 queue. The authors apply the general result to several cases of practical interest where the service time distributions belong to the set of decreasing hazard rate distributions, like Pareto or hyper-exponential. When there is only one class it is known that the Least Attained Service (LAS) policy is optimal. The authors show that in the multi-class case the optimal policy is a priority discipline, where jobs of the various classes depending on their attained service are classified into several priority levels. The authors find that the Gittins policy can outperform by nearly 10% the LAS policy.

#### Singular perturbation theory

Participant : Konstantin Avrachenkov.

In [27] K. Avrachenkov, in cooperation with V. Ejov, P. Howlett (University of South Australia , Australia) and
C. Pearce (University of Adelaide , Australia), considers a bounded not necessarily compact
linear operator A(z) between Hilbert spaces which depends analytically on a perturbation parameter z .
If A(0) is singular the authors find conditions under which A^{-1}(z) is well defined on some region 0<|z|<b
by a convergent Laurent series with a finite order pole at the origin. Under certain conditions the
results can be extended to Banach spaces. The results can be applied to singularly perturbed Markov chains
with continuous state space.