This paper is concerned with the ordered statistic decoding with local constraints (LC-OSD) of binary linear block codes, which is a near maximum-likelihood decoding algorithm. Compared with the conventional OSD, the LC-OSD significantly reduces both the maximum and the average number of searches. The former is achieved by performing the serial list Viterbi algorithm (SLVA) or a two-way flipping pattern tree (FPT) algorithm with local constraints on the test error patterns, while the latter is achieved by incorporating tailored early termination criteria. The main objective of this paper is to explore the relationship between the performance of the LC-OSD and decoding parameters, such as the constraint degree and the maximum list size. To this end, we approximate the local parity-check matrix as a totally random matrix and then estimate the performance of the LC-OSD by analyzing with a saddlepoint approach the performance of random codes over the channels associated with the most reliable bits (MRBs). The random coding approach enables us to derive an upper bound on the performance and predict the average rank of the transmitted codeword in the list delivered by the LC-OSD. This allows us to balance the constraint degree and the maximum list size for the average (or maximum) time complexity reduction. Simulation results show that the approximation by random coding approach is numerically effective and powerful. Simulation results also show that the RS codes decoded by the LC-OSD can approach the random coding union (RCU) bounds, verifying the efficiency and universality of the LC-OSD.