Young Researchers Seminars, Maths Applications & Models (Sfragara, Vardanyan)

Matteo Sfragara (Stockholm University)
Queue-based random-access protocols for wireless networks

In this talk, we discuss mathematical models that address fundamental challenges in wireless networks. We first introduce Carrier-Sense Multiple-Access (CSMA) protocols, distributed algorithms that involve randomness to prevent the devices to transmit simultaneously and hence their signals to interfere with each other. These models can be viewed as interacting particle systems on graphs, where the interference is captured by a hard-core interaction model. We describe how they exhibit metastability and how understanding metastability properties is crucial to design mechanisms to counter starvation effects and improve the performance of the network. In particular, we focus on random-access models where the transmission rates depend on the queues at the nodes. We discuss three different network topologies: we start with complete bipartite networks, we then generalize our results to arbitrary bipartite networks, and finally we explore dynamic bipartite networks in which the interference graph changes over time, which allows us to capture some effects of user mobility. This talk is based on three joint works with Frank den Hollander (Leiden University), Sem Borst (Eindhoven University of Technology) and Francesca R. Nardi (University of Florence).

Viktorya Vardanyan (Università di Trento)
Mean field games with terminal state constraints

We investigate a mean field game (MFG) characterized by state dynamics modeled through stochastic differential equations influenced by both idiosyncratic and common noise. These dynamics are subject to the constraint that the terminal state variable resides within a nonempty convex closed set. Additionally, the mean field interaction affects both the state and control in the dynamics and the costs. Motivated by the work of Ji and Zhou [Comm. Inf. Syst. 6(4), 2006], we introduce an auxiliary MFG problem and establish the stochastic maximum principle (SMP) for an auxiliary optimization problem with fixed flows. Through the formulation of a suitable forward-backward stochastic differential equation (FBSDE) of conditional McKean-Vlasov type, we demonstrate the existence of its solution and confirm its role as a MFG equilibrium. Furthermore, we apply these findings within a financial context. The talk is based on a joint work with Luca Di Persio (University of Verona) and  Luciano Campi (University of Milan).

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