In this paper, we investigate a delay-aware channel allocation problem where the number of channels is less than that of users. Due to the proliferation of delay sensitive applications, the objective of our problem is chosen to be the minimization of the total average queuing delay of the network in question. First, we show that our problem falls in the framework of Restless Bandit Problems (RBP), for which obtaining the optimal solution is known to be out of reach. To circumvent this difficulty, we tackle the problem by adopting a Whittle Index approach. To that extent, we employ a Lagrangian relaxation for the original problem and prove it to be decomposable into multiple one-dimensional independent subproblems. Afterwards, we provide structural results on the optimal policy of each of the subproblems. More specifically, we prove that a threshold policy is able to achieve the optimal operating point of the considered subproblem. Armed with that, we show the indexability of the subproblems and characterize the Whittle's indices which are the basis of our proposed heuristic. We then provide a rigorous mathematical proof that our policy is optimal in the infinitely many users regime. Finally, we provide numerical results that showcase the remarkable good performance of our proposed policy and that corroborate the theoretical findings.