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Rate control under heavy traffic with strategic servers. (English) Zbl 1409.60131
Summary: We consider a large queueing system that consists of many strategic servers that are weakly interacting. Each server processes jobs from its unique critically loaded buffer and controls the rate of arrivals and departures associated with its queue to minimize its expected cost. The rates and the cost functions in addition to depending on the control action, can depend, in a symmetric fashion, on the size of the individual queue and the empirical measure of the states of all queues in the system. In order to determine an approximate Nash equilibrium for this finite player game, we construct a Lasry-Lions-type mean-field game (MFG) for certain reflected diffusions that governs the limiting behavior. Under conditions, we establish the convergence of the Nash-equilibrium value for the finite size queuing system to the value of the MFG.

60K25 Queueing theory (aspects of probability theory)
91A13 Games with infinitely many players (MSC2010)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
93E20 Optimal stochastic control
60H30 Applications of stochastic analysis (to PDEs, etc.)
60F17 Functional limit theorems; invariance principles
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