The Stochastic Quickest Path Problem via Minimal Paths

The quickest path problem, a version of the shortest path problem, is to find a single quickest path which sends a given amount of data from the source to the sink with minimum transmission time. More specifically, the capacity of each arc in the network is assumed to be deterministic. However, in many real-life networks such as computer systems, telecommunication systems, etc., the capacity of each arc is stochastic due to failure, maintenance, etc. Such a network is named a stochastic-flow network. Therefore, the minimum transmission time is not a fixed number. The transmission time can be reduced if the data are transmitted through several minimal paths simultaneously. Focusing on a stochastic flow network with multistate arcs, this paper studies the stochastic quickest path problem. We evaluate the probability that d units of data can be sent through two minimal paths simultaneously under the time constraint T. Such a probability is named the system reliability. A simple algorithm is proposed to generate all (d,T)-MPs and the system reliability can then be computed in terms of (d,T)-MPs by applying inclusion-exclusion.