#### Whittle Index Policy for Multichannel Scheduling in Queueing Systems

In this paper, we consider a queueing system with multiple channels (or servers) and multiple classes of users. We aim at allocating the available channels among the users in such a way to minimize the expected total average queue length of the system. This known scheduling problem falls in the framework of Restless Bandit Problems (RBP) for which an optimal solution is known to be out of reach for the general case. The contributions of this paper are as follows. We rely on the Lagrangian relaxation method to characterize the Whittle index values and to develop an index-based heuristic for the original scheduling problem. The main difficulty lies in the fact that, for some queue states, deriving the Whittle's index requires introducing a new approach which consists in introducing a new expected discounted cost function and deriving the Whittle's index values with respect to the discount parameter $\beta$. We then deduce the Whittle's indices for the original problem (i.e. with total average queue length minimization) by taking the limit $\beta \rightarrow 1$. The numerical results provided in this paper show that this policy performs very well and is very close to the optimal solution for high number of users.