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We use the Markov chain approximation method to construct approximations forthe solution of the mean field game (MFG) with reflecting barriers studied inBayraktar, Budhiraja, and Cohen (2017). The MFG is formulated in terms of acontrolled reflected diffusion with a cost function that depends on thereflection terms in addition to the standard variables: state, control, and themean field term. This MFG arises from the asymptotic analysis of an $N$-playergame for single server queues with strategic servers. By showing that ourscheme is an almost contraction, we establish the convergence of this numericalscheme over a small time interval.

A numerical scheme for a mean field game in some queueing systems based on Markov chain approximation method