Disregarding Duration Uncertainty in Partial Order Schedules? Yes, We Can!

Abstract

Abstract. In the context of Scheduling under uncertainty, Partial Order Schedules (POS) provide a convenient way to build flexible solutions. A POS is obtained from a Project Graph by adding precedence constraints so that no resource conflict can arise, for any possible assignment of the activity durations. In this paper, we use a simulation approach to evaluate the expected makespan of a number of POSs, obtained by solving scheduling benchmarks via multiple approaches. Our evaluation leads us to the discovery of a striking correlation between the expected makespan and the makespan obtained by simply fixing all durations to their average. The strength of the correlation is such that it is possible to disregard completely the uncertainty during the schedule construction and yet obtain a very accurate estimation of the expected makespan. We provide a thorough empirical and theoretical analysis of this result, showing the existence of solid ground for finding a similarly strong relation on a broad class of scheduling problems of practical importance.

Estimating the probability of meeting a deadline in schedules and plans

“…In [35] RCPSP/max (Resource Constrained Project Scheduling Problems with minimum and maximum time lags) is studied, where the durations of activities are not known with certainty. Similarly, in [7] a simulation approach is used to evaluate the expected makespan of a number of Partial Order Schedules (POS). The evaluation in the paper shows correlation between the expected makespan and the makespan obtained by simply fixing all durations to their average.…”

“…A few aspects distinguish our work from the presented scheduling papers. First, both [35] and [7] provide guarantees only in probability. These are good for application where such guarantees are sufficient while we provide stronger guarantees.…”

“…Second, our work is on approximating CDF, i.e., the probability of missing deadlines. Clearly, unlike expectations, this cannot always be directly derived from averages and variances as in [7] or by sampling as in [34]. For example, consider the following two tasks.…”

“…While most tools aim at good average performance of the plan, in which case one may ignore the full distribution and consider only the expected resource consumption [7], our paper focuses on providing guarantees for the probability of meeting deadlines. This type of analysis is needed, e.g., in Service-Level-Agreements (SLA) where guarantees of the form: "response time less than 1mSec in at least 95% of the cases" are common [8], Section 10 discusses additional related work.…”

Minimizing value-at-risk in single-machine scheduling

“…The deterministic counterpart of our problem is a conventional single-machine scheduling problem, in which all processing times take on their expected values. Bonfietti et al (2014) conduct a study on the performance of the deterministic model -obtained analogously by replacing the random activity durations by their means -for minimizing the expected makespan in resource-constrained project scheduling and job shop scheduling problems. Their empirical results depict a very high correlation between the expected makespan and the makespan of the deterministic problem.…”