## Diffusion approximations for controlled weakly interacting large finite state systems with simultaneous jumps.(English)Zbl 1391.60232

Summary: We consider a rate control problem for an $$N$$-particle weakly interacting finite state Markov process. The process models the state evolution of a large collection of particles and allows for multiple particles to change state simultaneously. Such models have been proposed for large communication systems (e.g., ad hoc wireless networks) but are also suitable for other settings such as chemical-reaction networks. An associated diffusion control problem is presented and we show that the value function of the $$N$$-particle controlled system converges to the value function of the limit diffusion control problem as $$N\to\infty$$. The diffusion coefficient in the limit model is typically degenerate; however, under suitable conditions there is an equivalent formulation in terms of a controlled diffusion with a uniformly nondegenerate diffusion coefficient. Using this equivalence, we show that near optimal continuous feedback controls exist for the diffusion control problem. We then construct near asymptotically optimal control policies for the $$N$$-particle system based on such continuous feedback controls. Results from some numerical experiments are presented.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60H30 Applications of stochastic analysis (to PDEs, etc.) 93E20 Optimal stochastic control 60J28 Applications of continuous-time Markov processes on discrete state spaces 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60K25 Queueing theory (aspects of probability theory) 91B70 Stochastic models in economics
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