In this paper we introduce and analyze a new class of service policies called multi-phase gated service. This policy is a generalization of the classical single-phase and two-phase gated policies and works as follows. Each customer that arrives at queue i will have to wait K_i cycles before it receives service. The aim of this policy is to provide an interleaving scheme to avoid monopolization of the system by heavily loaded queues, by choosing the proper values of interleaving levels Ki. In this paper, we analyze the effectiveness of the interleaving scheme on the queueing behavior of the system, and consider the problem of identifying the proper combination of interleaving levels (K_1,...,K_N) that minimizes a weighted sum of the mean waiting times at each of the N queues. Obviously, the proper choice of the interleaving levels is most critical when the system is heavily loaded. For this reason, we to obtain closed-form expressions for the asymptotic waiting-time distributions in heavy trafficc, and use these expressions to derive simple heuristics for approximating the optimal interleaving scheme. Numerical results with simulations demonstrate that the accuracy of these approximations is extremely high.