## Section: New Results

Keywords : Nash equilibrium, stability, service discipline, evolutionarily stable strategy.

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

Participants : Eitan Altman, Konstantin Avrachenkov, Dieter Fiems, Robin Groenevelt, Alain Jean-Marie, Philippe Nain.

#### Advances in game theory

Participant : Alain Jean-Marie.

Conjectures have been introduced in game theory in an attempt to model anticipations formed by players in the absence of a complete information. Equilibria reached in a game with conjectures are called "conjectural equilibria". In static games, the structure of a conjecture is rather simple and well understood. When the game is dynamic in discrete-time, several structures are possible In a joint paper with M. Tidball ( Inra , Montpellier), A. Jean-Marie establishes the relationships that exist between these dynamic conjectural equilibria and the classical Nash-feedback equilibria [24] .

#### Stochastic recursive sequences

Participants : Eitan Altman, Robin Groenevelt.

E. Altman has pursued his work on a class of discrete-time stochastic recursive sequences of the form
X_{n+ 1}=
A_{n}(
X_{n}) +
B_{n} , where
X_{n} and
B_{n} are column vectors taking nonnegative values, and where
A_{n} is some non-negative random field. It is assumed that
{
B_{n}} is a stationary and ergodic process (allowing, in particular, for arbitrary time-correlations) and that
{
A_{n}} is a renewal sequence. In
[30] stability conditions of the sequence
{
X_{n}} have been established, and
E[
X_{n}] and
E[
X_{n}X_{n+
k}] have been computed in steady-state for any
k0 . These results have been applied to the study of the infinite server queue in discrete-time in the presence of time-correlated arrivals.

In [23] , [43] , R. Groenevelt and E. Altman use this formalism to compute the expected waiting time in two-queue polling systems with correlated vacations.

#### Advances in queueing theory

Participants : Konstantin Avrachenkov, Urtzi Ayesta, Dieter Fiems, Philippe Nain.

##### Scheduling

In [20] , K. Avrachenkov, in collaboration with U. Ayesta ( Cwi , Amsterdam) and P. Brown ( France Telecom R&D , Sophia Antipolis), analyzes a Processor-Sharing (PS) queue with batch arrivals. The analysis is based on the integral equation derived by Kleinrock, Muntz and Rodemich. Using the contraction mapping principle, the authors demonstrate the existence and uniqueness of a solution to the integral equation. Then, the authors provide an asymptotic analysis as well as tight bounds for the expected response time conditioned on the service time. Finally, it is shown how these results can be applied to the Multi-Level Processor-Sharing scheduling.

As a natural multi-class generalization of the well-known (egalitarian) PS service discipline, Discriminatory Processor Sharing (DPS) is of great interest in many application areas, including telecommunications. Under DPS, the mean response time conditional on the service requirement is only available in closed-form when all classes have exponential service requirement distributions. For generally distributed service requirements, Fayolle et al. showed that the expected conditional response times satisfy a system of integrodifferential equations. In [33] , K. Avrachenkov, in collaboration with U. Ayesta ( Cwi , Amsterdam) and P. Brown ( France Telecom R&D , Sophia Antipolis) and S. NÅ©nez Queija ( Cwi , Amsterdam), uses that result to prove that, for each class, the expected unconditional response time is finite and that the expected conditional response time has an asymptote, when the system is stable.

The asymptotic bias of each class is found in closed-form. The latter involves the mean service requirements of all classes and the second moments of all classes but the one under consideration. The authors also study DPS as a tool to achieve size-based scheduling, and provide guidelines as to how the weights of DPS must be chosen such that DPS outperforms PS.

##### Preemptive priority queues

In collaboration with Ger Koole (Vrije University, Amsterdam, The Netherlands), Dieter Fiems and Philippe Nain investigate in [66] discrete-time preemptive repeat identical queueing systems. The high- and low-priority interarrival times are assumed to be geometrically distributed and deterministic, respectively whereas the service times are assumed to be deterministic for both classes. As an application, the queueing model is used to assess waiting times of scheduled patients in the presence of emergency arrivals.

#### Evolutionarily stable strategy (ESS)

Participant : Philippe Nain.

In [68] , P. Nain, in collaboration with P. Bernhard, F. Hammelin (both from I3S, University of Nice Sophia Antipolis, France) and E. Wajnberg ( Inra , Sophia Antipolis, France), finds the ESS driving the behavior of foreagers competing for a common patchily distributed resource. The innovation of this work lies on the fact that an unlimited number of foreagers reach the patch at random arrival times, wheras previous studies considered that a fixed number of foreagers reach the patch at the same time. In this setting, the authors find the optimal leaving rule, namely when foreagers should leave a path not yet exhausted in order to find a richer one, in spite of an uncertain travel time.