We study an online unit-job scheduling problem arising in buffer management. Each job is specified by its release time, deadline, and a nonnegative weight. Due to overloading conditions, some jobs have to be dropped. The goal is to maximize the total weight of scheduled jobs. We present several competitive online algorithms for various versions of unit-job scheduling, as well as some lower bounds on the competitive ratios.We first give a randomized algorithm RMIX with competitive ratio of e/(e − 1) ≈ 1.582. This is the first algorithm for this problem with competitive ratio smaller than 2.Then we consider s-bounded instances, where the span of each job (deadline minus release time) is at most s. We give a 1.25-competitive randomized algorithm for 2-bounded instances, matching the known lower bound. We also give a deterministic algorithm EDF α , whose competitive ratio on s-bounded instances is 2 − 2/s + o(1/s). For 3-bounded instances its ratio is φ ≈ 1.618, matching the known lower bound.In s-uniform instances, the span of each job is exactly s. We show that no randomized algorithm can be better than 1.25-competitive on s-uniform instances, if the span s is unbounded. For s = 2, our proof gives a lower bound of 4 − 2 √ 2 ≈ 1.172. Also, in the 2-uniform case, we prove a lower bound of √ 2 ≈ 1.414 for deterministic memoryless algorithms, matching a known upper bound. Finally, we investigate the multiprocessor case and give a 1/(1 − ( m m+1 ) m )-competitive algorithm for m processors. We also show improved lower bounds for the general and s-uniform cases.

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We consider the following buffer management problem arising in QoS networks: Packets with specified weights and deadlines arrive at a network switch and need to be forwarded so that the total weight of forwarded packets is maximized. Packets not forwarded before their deadlines are lost. The main result of the article is an online 64/33 ≈ 1.939-competitive algorithm, the first deterministic algorithm for this problem with competitive ratio below 2. For the 2-uniform case we give an algorithm with ratio ≈ 1.377 and a matching lower bound.

We study the problem of scheduling equal-length jobs with release times and deadlines, where the objective is to maximize the number of completed jobs. Preemptions are not allowed. In Graham's notation, the problem is described as 1|r j ; p j = p| U j . We give the following results:(1) We show that the often cited algorithm by Carlier from 1981 is not correct. (2) We give an algorithm for this problem with running time O(n 5 ).

Our main result is an optimal online algorithm for preemptive scheduling on uniformly related machines with the objective to minimize makespan. The algorithm is deterministic, yet it is optimal even among all randomized algorithms. In addition, it is optimal for any fixed combination of speeds of the machines, and thus our results subsume all the previous work on various special cases. Together with a new lower bound it follows that the overall competitive ratio of this optimal algorithm is between 2.054 and e ≈ 2.718. We also give a complete analysis of the competitive ratio for three machines.