Section: New Results

Applications

Performance evaluation of an emergency call center

Participants : Xavier Allamigeon, Stéphane Gaubert.

Since 2014, we have been collaborating with Préfecture de Police (Régis Reboul and LcL Stéphane Raclot), more specifically with Brigade de Sapeurs de Pompiers de Paris (BSPP) and Direction de Sécurité de Proximité de l'agglomération parisienne (DSPAP), on the performance evaluation of the new organization to handle emergency calls to firemen and policemen in the Paris area. We developed analytical models, based on Petri nets with priorities, and fluid limits, see   [55], [56], [65]. In 2018, we performed specific case studies, with several students of École polytechnique: Laetitia de Coudenhove, Julie Poulet, Céline Moucer and Julia Escribe.

Tropical models of fire propagation in urban areas

Participants : Stéphane Gaubert, Daniel Jones.

As part of the team work in the ANR project Democrite, we developed a model of fire propagation in urban areas, involving a deterministic analogue of first passage percolation. We showed that the fire tends to propagate according to a polyhedral shape, and derived metric limit theorems, exploiting discrete convexity results à la Shapley-Folkman. We validated this approach by simulations, on the fire following Kobe earthquake in 1995. The polyhedral shape is also apparent in historical fires, like the great fire of London (1666). These results are announced in [28].

Smart Data Pricing

Participants : Marianne Akian, Jean-Bernard Eytard, Stéphane Gaubert.

This work is in collaboration with Mustapha Bouhtou (Orange Labs) and with Gleb Koshevoy (Russian academy of Science).

The PhD work  [81] of Jean-Bernard Eytard concerns the optimal pricing of data trafic in mobile networks. We developed a bilevel programming approach, allowing to an operator to balance the load in the network through price incentives. We showed that a subclass of bilevel programs can be solved in polynomial time, by combining methods of tropical geometry and of discrete convexity. This work is presented in  [80] and also in [34]. In a followup work, presented in  [81], we managed to extend these results to wider classes of bilevel problems, and to relate them to competitive equilibria problems.

Game theory models of decentralized mechanisms of pricing of the smart grid

Participants : Stéphane Gaubert, Paulin Jacquot.

This work is in collaboration with Nadia Oudjane, Olivier Beaude and Cheng Wan (EDF Labs).

The PhD work of Paulin Jacquot concerns the application of game theory techniques to pricing of energy. We are developing a game theory framework for demand side management in the smart grid, in which users have movable demands (like charging an electric vehicle). We compared in particular the daily and hourly billing mechanisms. The latter, albeit more complex to analyse, has a merit as it incitates the user to move his or her consumption at off peak hours. We showed the Nash equilibrium is unique, under some assumptions, and gave theoretical bounds of the price of anarchy of the game with a hourly billing, showing this mechanism remains efficient while being more “fair” than the daily billing. We proposed and tested decentralized algorithms to compute the Nash equilibrium. These contributions are presented in  [102], [23].

Another work, by Paulin Jacquot and Cheng Wan, deals with limit theorems for atomic games with a large number of players [30], [42].