Belief Functions Theory

Updated May 24, 2020

A Time/Utility (née Time/Value) Function (TUF) specifies an action's (e.g., task's) application-specific utility depending on its completion time (C). By convention, a TUF is concave. It has a critical time (even if it is linear) after which its utility does not increase. A conventional deadline (d) is a simple special case, a downward step TUF having utility values {1,0}. More generally, a TUF permits downward (and upward) step functions to have any appropriate utilities {u1, u2}. Tardiness is a simple special case whose non-zero utility is the linear function C-d. More generally, a TUF allows non-zero earliness and tardiness to be non-linear. Thus, one useful interpretation of utility can be timeliness, providing a rich generalization of traditional action completion time constraints in real-time systems. TUF utility may include negative values. TUFs and their utility scales and values are derived from domain-specific subject matter knowledge. The optimality criteria for scheduling TUFs are maximal utility accrual (UA)-which can be interpreted as actions' collective timeliness-and predictability of that accrued utility (while respecting dependencies and resource constraints). The scheduler performs application-specific trade-offs between accrued utility and its predictability. The TUF/UA paradigm is intended for (but not limited to) open-world systems, so imperfections in the scheduling parameters are inevitable. Some of these imperfections may be amenable to stochastic scheduling. Others are too major and complex for orthodox (e.g., additive) probability theory. Imprecision may call for using fuzzy set theory. Epistemic uncertainties--e.g., ignorance of, or conflicts among, scheduling parameters--may be present and require an appropriate kind and degree of resilience in the UA algorithmic techniques. Thus, UA schedulers may base utility accrual and its predictability on more general uncertainty models, such as a (potentially "fuzzified" version of a) belief-based theory (e.g., the Transferable Belief Model, etc.). Online UA scheduling efficiency is often enhanced by implementing the scheduler in hardware (e.g., custom RISC-Vs, GPUs, FPGAs, ASICs). The TUF/UA paradigm has been particularly successful in military combat systems, because of the extreme uncertainties in those environments.