Before revision:  We consider the competitiveness of online algorithms in four different models. Namely, we consider the following stochastic reward problems: when the stochastic graph is not known and the order of online arrivals is determined uniformly at random (i.e., the ROM model) ;
After revision:  We study this problem in the generality of a patience (or budget) constraint which limits the number of probes that can be made to edges adjacent to an online node. The patience constraint results in modelling and computational efficiency issues that are not encountered in the special cases of unit patience and full (i.e., the ROM model) ;