In online networks, some parameters are not initially known by decision-makers, especially arc costs that are revealed over time, thereby online decisions are made without complete knowledge of the future events. Three kinds of statistical information are available in terms of online manner arrival of the last traversed nodes: the exact traversed length, the average shortest path length and the shortest path length. Three different stochastic models are considered and the related stochastic online decision criteria are obtained, such that the best competitive ratio is 2.3130. Under the assumption that the online decision-maker is informed about the intervals of the arc costs, and after some constant competitive ratios are produced, 2.3130 is determined as the best obtained lower bound competitive ratio against some previous works.
