  This work introduces a natural variant of the online machine scheduling
problem on unrelated machines, which we refer to as the favorite machine model.
In this model, each job has a minimum processing time on a certain set of
machines, called favorite machines, and some longer processing times on other
machines. This type of costs (processing times) arise quite naturally in many
practical problems. In the online version, jobs arrive one by one and must be
allocated irrevocably upon each arrival without knowing the future jobs. We
consider online algorithms for allocating jobs in order to minimize the
makespan.
  We obtain tight bounds on the competitive ratio of the greedy algorithm and
characterize the optimal competitive ratio for the favorite machine model. Our
bounds generalize the previous results of the greedy algorithm and the optimal
algorithm for the unrelated machines and the identical machines. We also study
a further restriction of the model, called the symmetric favorite machine
model, where the machines are partitioned equally into two groups and each job
has one of the groups as favorite machines. We obtain a 2.675-competitive
algorithm for this case, and the best possible algorithm for the two machines
case.
