In this paper, we find the optimal index coding scheme with transmission time allocation that minimizes overall outage probability in a single-input single-output broadcast channel. In our system model, a transmitter has a set of files (i.e., a file library), and each user requests a single file in the library. Meanwhile, each user has a memory so that may already know some files. When the transmitter can only use limited transmission time to serve the users, the overall outage probability varies according to the index code and transmission time allocation. To find the optimal index code and time allocation, we first derive the optimal time allocation for a given index code. Then, we propose the optimal index code searching algorithm. We also propose a pruning algorithm, which reduces the searching complexity while maintaining the optimality. Our simulation results show that our proposed scheme well minimizes the overall outage probability, and our proposed pruning algorithm significantly reduces the complexity of our proposed scheme.