Shortest paths in stochastic networks with ARC lengths having discrete distributions

In this work, we compute the distribution of L', the length of a shortest ( s , t ) path, in a directed network G with a source nodes and a sink node t and whose arc lengths are independent, nonnegative, integer valued random variables having finite support. We construct a discrete time Markov chain with a single absorbing state and associate costs with each transition such that the total cost incurred by this chain until absorption has the same distribution as does L'. We show that the transition probability matrix of this chain has an upper triangular structure and exploit this property to develop numerically stable algorithms for computing the distribution of L' and its moments. All the algorithms are recursive in nature and are illustrated by several examples. 0 7993byJohn Wiley & Sons, Inc.