Scheduling with unexpected machine breakdowns

We investigate an online version of the scheduling problem $P, NC|pmtn|C_{\max}$, where a set of jobs has to be scheduled on a number of identical machines so as to minimize the makespan. The job processing times are known in advance and preemption of jobs is allowed. Machines are {\it non-continuously\/} available, i.e., they can break down and recover at arbitrary time instances {\it not known in advance}. New machines may be added as well. Thus machine availabilities change online. We first show that no online algorithm can construct optimal schedules. We also show that no online algorithm can achieve a constant competitive ratio if there may be time intervals where no machine is available. Then we present an online algorithm that constructs schedules with an optimal makespan of $C_{\max}^{OPT}$ if a {\it lookahead\/} of one is given, i.e., the algorithm always knows the next point in time when the set of available machines changes. Finally we give an online algorithm without lookahead that constructs schedules with a nearly optimal makespan of $C_{\max}^{OPT} + \epsilon$, for any $\epsilon >0$, if at any time at least one machine is available. Our results demonstrate that not knowing machine availabilities in advance is of little harm