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We evaluate the performance of Whittle index policy for restless Markovianbandits, when the number of bandits grows. It is proven in [30] that thisperformance is asymptotically optimal if the bandits are indexable and theassociated deterministic system has a global attractor fixed point. In thispaper we show that, under the same conditions, the convergence rate isexponential in the number of bandits, unless the fixed point is singular (to bedefined later). Our proof is based on the nature of the deterministic equationgoverning the stochastic system: We show that it is a piecewise affinecontinuous dynamical system inside the simplex of the empirical measure of thebandits. Using simulations and numerical solvers, we also investigate the caseswhere the conditions for the exponential rate theorem are violated, notablywhen attracting limit cycles appear, or when the fixed point is singular. Weillustrate our theorem on a Markovian fading channel model, which has been wellstudied in the literature. Finally, we extend our synchronous model results tothe asynchronous model.

Exponential Convergence Rate for the Asymptotic Optimality of Whittle Index Policy