Stochastic Scheduling on Parallel Processors and Minimization of Concave Functions of Completion Times

We consider a stochastic scheduling problem in which n jobs are to be scheduled on m identical processors which operate in parallel. The processing times of the jobs are not known in advance but they have known distributions with hazard rates ρ₁, (t), …, ρ _n (t). It is desired to minimize the expected value of к(C), where C _i is the time at which job i is completed C = (C₁, …, C _n ), and к(C) is increasing and concave in C. Suppose processor i first becomes available at time τ _i . We prove that if there is a single static list priority policy which is optimal for every τ = (τ₁, …, τ _m ), then the minimal expected cost must be increasing and concave in τ. Moreover, if к(C) is supermodular in C then this cost is also supermodular in τ. This result is used to prove that processing jobs according to the static list priority order (1,2,…,n) minimizes the expected value of ∑w _i h(C _i ), when h(·) is a nondecreasing, concave function, w ₁ ≥ … ≥ w _n , and ρ₁ (t₁)w₁ ≥ … ≥ ρ _n (t _n )w _n for all t₁, …, t _n .