In the online shortest path problem, the arc costs are the online parameters in a network and their values are not known for decision-makers in advance. So, the online decisions are made by arriving any node and with respect to some statistical information of the leaving nodes and not by the realization of the arc costs; however, the decisions for the traversed paths are not changed or rejected, and they are established permanently. The online decision criteria are defined as expected path lengths related to the available statistical information in the online manner. The online expected stochastic shortest path lengths are computed by two novel online adapted formulas. Then, the competitive analyses are presented for the obtained online stochastic optimal solution against the offline optimal solution. The expected competitive ratios are revealed differences between the application of some local and global information, so that it shows e–2 improvement for the local statistical information against the global statistical information, relatively. Numerical results and the average case analyses compared the obtained formulas as the online optimality indices. The formulas are applied in both acyclic and cyclic networks, and the obtained competitive ratios are verified by some numerical examples.
