  The interval scheduling problem is one variant of the scheduling problem. In
this paper, we propose a novel variant of the interval scheduling problem,
whose definition is as follows: given jobs are specified by their {\em release
times}, {\em deadlines} and {\em profits}. An algorithm must start a job at its
release time on one of $m$ identical machines, and continue processing until
its deadline on the machine to complete the job. All the jobs must be completed
and the algorithm can obtain the profit of a completed job as a user's
satisfaction. It is possible to process more than one job at a time on one
machine. The profit of a job is distributed uniformly between its release time
and deadline, that is its interval, and the profit gained from a subinterval of
a job decreases in reverse proportion to the number of jobs whose intervals
intersect with the subinterval on the same machine. The objective of our
variant is to maximize the total profit of completed jobs. This formulation is
naturally motivated by best-effort requests and responses to them, which appear
in many situations. In best-effort requests and responses, the total amount of
available resources for users is always invariant and the resources are equally
shared with every user. We study online algorithms for this problem.
Specifically, we show that for the case where the profits of jobs are
arbitrary, there does not exist an algorithm whose competitive ratio is
bounded. Then, we consider the case in which the profit of each job is equal to
its length, that is, the time interval between its release time and deadline.
For this case, we prove that for $m = 2$ and $m \geq 3$, the competitive ratios
of a greedy algorithm are at most $4/3$ and at most $3$, respectively. Also,
for each $m \geq 2$, we show a lower bound on the competitive ratio of any
deterministic algorithm.
