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• We consider a new variant of the online interval scheduling problem. • This variant formulates best-effort requests and responses as an online problem. • We show some upper and lower bounds on the competitive ratios of this variant. 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 release times , deadlines and 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 ≥ 3 , the competitive ratios of a greedy algorithm are at most 4/3 and at most 3, respectively. Also, for each m ≥ 2 , we show a lower bound on the competitive ratio of any deterministic algorithm. [ABSTRACT FROM AUTHOR]

Online scheduling is one of the most basic online problems. We are given a sequence of jobs in an online fashion, and they must be assigned to one of m ( ≥ 1 ) equivalent processors. Each job j with the processing time p ( j ) is given at the release time r ( j ) . An algorithm decides which processor is used to process j and must schedule it for time p ( j ) after time r ( j ) on the chosen processor. Many criteria have been proposed to evaluate the performance of scheduling algorithms, and Muthukrishnan et al. (FOCS 1999 and SIAM Journal on Computing 34 (2), 2005) used the stretch of a job j in an objective function, which is defined as the ratio of the amount of time which j spends in the system to its processing time p ( j ) . The cost of an algorithm is the total stretch to schedule all the given jobs, and our goal is to minimize it. Muthukrishnan et al. considered the case in which preemption is allowed, that is, any job can be stopped during the process and after a while the process can resume. In this paper, we show how to construct instances to obtain various lower bounds on the competitive ratio for each m . In the instances, m jobs are given regularly for a sufficiently large time span after a maliciously chosen time. The processing times of the given jobs are taken depending on the remaining processing times of uncompleted jobs at the start time of the “burst” of jobs. We prove that for the instances, the stretch of a job completed before the burst hardly affects the evaluation of a competitive ratio. Further, we provide a job sequence given before the burst for each m . Then, we can improve the previous lower bounds for any deterministic online algorithm for each m using the instances. For example, we obtain lower bounds of 1.228, 1.257 and 21 / 17 ≈ 1.235 for m = 1 , 2 and ∞, respectively. Moreover, we obtain the first non-trivial lower bounds for any randomized online algorithm for all m . [ABSTRACT FROM AUTHOR]