Time version of the shortest path problem in a stochastic-flow network

Many studies on hardware framework and routing policy are devoted to reducing the transmission time for a flow network. A time version of the shortest path problem thus arises to find a quickest path, which sends a given amount of data from the unique source to the unique sink with minimum transmission time. More specifically, the capacity of each arc in the flow network is assumed to be deterministic. However, in many real-life networks, such as computer systems, telecommunication systems, etc., the capacity of each arc should be stochastic due to failure, maintenance, etc. Such a network is named a stochastic-flow network. Hence, the minimum transmission time is not a fixed number. We extend the quickest path problem to evaluating the probability that dd units of data can be sent under the time constraint TT. Such a probability is named the system reliability. In particular, the data are transmitted through two minimal paths simultaneously in order to reduce the transmission time. A simple algorithm is proposed to generate all (d,T)(d,T)-MPs and the system reliability can then be computed in terms of (d,T)(d,T)-MPs. Moreover, the optimal pair of minimal paths with highest system reliability could be obtained.