Cyclic scheduling in a stochastic environment

Cyclic or periodic schedules can be implemented in a job shop where demands for various products have a stable rate and mix. Numerous results on cyclic scheduling in deterministic settings are available, but studies considering uncertainties such as machine failure are rare. This paper examines the behavior of cyclic schedules in a stochastic environment characterized by random machine failures that may delay execution of tasks and thus may cause the actual production to deviate from a specified cyclic schedule. The authors intend to understand the behavior of cyclic schedules under uncertainty and to find those cyclic schedules that are the least disturbed by occurrences of machine failure. The cyclic scheduling problem of one or multiple machines can be formulated into a convex program for which the objective is to minimize a weighted sum of expected ergodic delays experienced by tasks. For one-machine schedules, the ergodic distributions of delays, as well as their expected values, are displayed. For multiple-machine schedules, a necessary condition for ergodicity is presented and stochastic lower bounds on task delays are derived.