  Online algorithms are usually analyzed using the notion of competitive ratio
which compares the solution obtained by the algorithm to that obtained by an
online adversary for the worst possible input sequence. Often this measure
turns out to be too pessimistic, and one popular approach especially for
scheduling problems has been that of "resource augmentation" which was first
proposed by Kalyanasundaram and Pruhs. Although resource augmentation has been
very successful in dealing with a variety of objective functions, there are
problems for which even a (arbitrary) constant speedup cannot lead to a
constant competitive algorithm. In this paper we propose a "rejection model"
which requires no resource augmentation but which permits the online algorithm
to not serve an epsilon-fraction of the requests.
  The problems considered in this paper are in the restricted assignment
setting where each job can be assigned only to a subset of machines. For the
load balancing problem where the objective is to minimize the maximum load on
any machine, we give $O(\log^2 1/\eps)$-competitive algorithm which rejects at
most an $\eps$-fraction of the jobs. For the problem of minimizing the maximum
weighted flow-time, we give an $O(1/\eps^4)$-competitive algorithm which can
reject at most an $\eps$-fraction of the jobs by weight. We also extend this
result to a more general setting where the weights of a job for measuring its
weighted flow-time and its contribution towards total allowed rejection weight
are different. This is useful, for instance, when we consider the objective of
minimizing the maximum stretch. We obtain an $O(1/\eps^6)$-competitive
algorithm in this case.
  Our algorithms are immediate dispatch, though they may not be immediate
reject. All these problems have very strong lower bounds in the speed
augmentation model.
