In this paper, we propose a modification of the reliable routing algorithm in a stochastic time-dependent network. We consider a stochastic ontime arrival problem. Reliability means maximization the probability of arrival at a destination within a given period of time. Modification of the shortest-path algorithm is aimed to decrease the computation time of the algorithm. The base idea of the proposed modification is to select a certain subset of nodes and links of the graph which can be used for calculating the shortest path. We propose two methods for selecting subset of nodes: based on a bounding box and based on the k-shortest path algorithm. Experimental studies of the base and modified algorithms are carried out on the transport network of Samara, Russia.