Online Batch Scheduling of Incompatible Job Families with Variable Lookahead Interval

We consider online parallel-batch scheduling of f incompatible unit-length job families with variable lookahead interval to minimize the maximum completion time, where f is the number of job families which is known in advance. Incompatible job family means that a batch only contains the jobs from the same job family. At any time t, an online algorithm can foresee the information of the jobs arriving in the time interval (t, t + Delta(t)], where Delta(t) = (lambda - 1)t + beta is variable. When the batch capacity b = infinity and f >= 1, we provide a best possible online algorithm with competitive ratio 1 + alpha(f) for 0 <= beta < 1 and 1 <= lambda <= 1 + beta, where alpha(f) is a positive root of lambda f alpha(2)(f) + (lambda f + beta + 1 - f)alpha(f) + beta - f = 0. When the batch capacity (f <) b < infinity and f >= 2, we give an online algorithm with competitive ratio 1 + max{f/f alpha(f) +f, f/lambda f alpha (f) + beta + 1} for 0 <= beta < 1 and lambda >= 1, and prove that it is the best possible for lambda = 1 and 0 <= beta <= f+1-root f(2)+1/2.