In this paper, we consider an edge computing network supporting ultra-reliable low latency communications (URLLC) operating with finite blocklength codes. We derive the end-to-end (E2E) reliability, while both the delay violation probability and the decoding error probability are considered. In addition, we propose an optimal system design to minimize the E2E error probability by optimally setting the target decoding error probability for preserving blocklength/time for each transmission link. In particular, we solve the corresponding optimization problem by proving its convexity. Via simulations, we validate our analytical mode. In addition, we evaluate the considered network, and characterize the impact of delay constraint, target decoding error probability and packet size on the E2E reliability of the considered edge computing network.