In this paper, we consider the service caching and the computing resource allocation in edge computing (EC) enabled networks. We introduce a random service caching design considering multiple types of latency sensitive services and the base stations (BSs)’ service caching storage. We then derive a successful service probability (SSP). We also formulate a SSP maximization problem subject to the service caching distribution and the computing resource allocation. Then, we show that the optimization problem is nonconvex and develop a novel algorithm to obtain the stationary point of the SSP maximization problem by adopting the parallel successive convex approximation (SCA). Moreover, to further reduce the computational complexity, we also provide a low complex algorithm that can obtain the near-optimal solution of the SSP maximization problem in high computing capability region. Finally, from numerical simulations, we show that proposed solutions achieve higher SSP than baseline schemes. Moreover, we show that the near-optimal solution achieves reliable performance in the high computing capability region. We also explore the impacts of target delays, a BSs’ service cache size, and an EC servers’ computing capability on the SSP.