On the optimal stochastic scheduling of out-forests

This paper presents new results on the problem of scheduling jobs on K ³ 1 parallel processors so as to minimize stochastically the makespan. The jobs are subject to out-forest precedence constraints, i.e. each job has at most one immediate predecessor, and job running times are independent samples from a given exponential distribution. We define a class of uniform out-forests in which all subtrees are ordered by an embedding relation. We prove that an intuitive greedy policy is optimal for K = 2, and that if out-forests satisfy an additional, uniform root-embedding constraint, then the greddy policy is optimal for all K ³ 2