Recovering cyclic schedules from delay

Abstract

A closed single‐server system is considered in which n items are scheduled to circulateat a fixed period. Each service is recognizable and is scheduled for its individual point of time;it is non‐preemptive and its length depends only on which item is being served.Aberratingfrom this desired schedule, some of the items start out with delays. While an item isdelayed,the time between a departure it makes from the server and its next arrival at the server isshortened by an item‐specific parameter. The aim is to recover the regular schedule asearlyas possible, or minimizing the sum of delays on services. All services must be executedeven if delayed.A greedy algorithm for a tractable subproblem is given. The overall problem is proved Σ2-hard; some subproblems are proved NP-hard. For one ofthe latter, an approximationalgorithm is given whose performance ratio approaches one if the maximum delay is largeenough relative to other parameters. It is proved that without this natural restriction,therecan be no algorithm with asymptotic performance ratio one.