Match-up scheduling of mixed-criticality jobs: Maximizing the probability of jobs execution

Abstract

This paper deals with a mixed-criticality scheduling problem: each job has a criticality level depending on its importance. In addition, each job has a finite set of possible processing times, and a known probability for each of them. Every job must be processed between its release date and its deadline. Moreover, each job has a weight corresponding to its payoff. This problem has applications in single machine scheduling of real time embedded systems scheduling, production and operating theaters.We propose a model that takes all the possible processing times of a job into account. An offline multilevel schedule is computed such that safety rules are satisfied, in every situation. This is achieved by allowing the rejection of low criticality jobs when higher criticality jobs need longer processing time, at runtime. The runtime schedule is matched-up again with the offline schedule after such deviations from the offline schedule. The offline multilevel schedule optimizes a non-regular criterion aiming to maximize the average weighted probability of jobs execution (i.e., the total expected payoff).Such a problem is strongly NP-hard. We first study the problem where the sequence of jobs is fixed: we show its complexity and provide a MILP formulation. For the case with two levels of criticality, we provide a dynamic programming algorithm. Finally, we propose a Branch and Bound method for the general problem (i.e., without a fixed job sequence).