Minimum cost routing on stochastic networks

In this paper, the authors consider the minimum cost transshipment problem in a directed network, having a single source node s with supply S and multiple demand nodes. The arc lengths are independent and exponentially distributed random valuables. The cost of shipping is $1/unit flow/unit length, and T is the minimum cost of shipping the S units so as to satisfy all the demands. The authors construct a continuous time Markov chain with an upper triangular generator matrix such that T equals a particular first passage time in this chain. This fact is used to derive numerically stable algorithms for computing the exact distribution and moments of T.