Abstract
We consider schedules of unit-duration tasks with random dependencies. Formally the schedules are obtained by imposing a random partial order of vertices in an Erdős-Rényi graph. For such schedules, we provide asymptotic formulas for the parallel execution time and for the required number of processors in a greedy allocation scheme as a function of connection probability. We also derive asymptotic bounds for the number of required processors in an optimal allocation scheme. We test our results for small schedule sizes using simulation and conclude that the convergence to asymptotic results is achieved relatively fast in practice. In our simulation experiment, we define and compare the efficiency of Mixed Integer Programming and Constraint Programming approaches to find the exact solution of the optimal allocation scheme and conclude that Constraint Programming is more efficient in our setting. In the last part of the paper, we provide preliminary results of similar analysis for random schedules generated from arbitrary graphs.
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