This paper proposes a backup resource allocation model for virtual network functions (VNFs) to minimize the total allocated computing capacity for backup with considering the service delay. If failures occur to primary hosts, the VNFs in failed hosts are recovered by backup hosts whose allocation is pre-determined. We introduce probabilistic protection, where the probability that the protection by a backup host fails is limited within a given value; it allows backup resource sharing to reduce the total allocated computing capacity. The previous work does not consider the service delay constraint in the backup resource allocation problem. The proposed model considers that the probability that the service delay, which consists of networking delay between hosts and processing delay in each VNF, exceeds its threshold is constrained within a given value. We introduce a basic algorithm to solve our formulated delay-constraint optimization problem. In a problem with the size that cannot be solved within an acceptable computation time limit by the basic algorithm, we develop a simulated annealing algorithm incorporating Yen's algorithm to handle the delay constraint heuristically. We observe that both algorithms in the proposed model reduce the total allocated computing capacity by up to 56.3% compared to a baseline; the simulated annealing algorithm can get feasible solutions in problems where the basic algorithm cannot.