Online makespan scheduling of linear deteriorating jobs on parallel machines

Traditional scheduling assumes that the processing time of a job is fixed. Yet there are numerous situations that the processing time increases (deteriorates) as the start time increases. Examples include scheduling cleaning or maintenance, fire fighting, steel production and financial management. Scheduling of deteriorating jobs was first introduced on a single machine by Browne and Yechiali, and Gupta and Gupta independently. In particular, lots of work has been devoted to jobs with linear deterioration. The processing time p _j of job J _j is a linear function of its start time s _j, precisely, p _j = a _j + b _js _j, where a _j is the normal or basic processing time and b _j is the deteriorating rate. The objective is to minimize the makespan of the schedule. We first consider simple linear deterioration, i.e., p _j = b _js _j. It has been shown that on m parallel machines, in the online-list model, LS (List Scheduling) is (1 + b _max) ^1-1/m-competitive. We extend the study to the online-time model where each job is associated with a release time. We show that for two machines, no deterministic online algorithm is better than (1 + b _max)-competitive, implying that the problem is more difficult in the online-time model than in the online-list model. We also show that LS is (1 + b _max) ^2(1-1/m)-competitive, meaning that it is optimal when m=2.