Recovering cyclic schedules from delay

A closed single-server system is considered in which n items are scheduled to circulate at a fixed period. Each service is recognizable and is schedul ed for its individual point of time; it is non-preemptive and its length de pends only on which item is being served. Aberrating from this desired sche dule, some of the items start out with delays. While an item is delayed, th e time between a departure it makes from the server and its next arrival at the server is shortened by an item-specific parameter. The aim is to recov er the regular schedule as early as possible, or minimizing the sum of dela ys on services. All services must be executed even if delayed. A greedy algorithm for a tractable subproblem is given. The overall problem is proved Sigma(2)-hard; some subproblems are proved NP-hard. For one of t he latter, an approximation algorithm is given whose performance ratio appr oaches one if the maximum delay is large enough relative to other parameter s. It is proved that without this natural restriction, there can be no algo rithm with asymptotic performance ratio one.