Online load balancing on related machines

Abstract

In the load balancing (or job scheduling) problem, introduced by Graham in the 1960s (SIAM J. of Appl. Math. 1966, 1969, jobs arriving online have to be assigned to machines so to minimize an objective defined on machine loads. A long line of work has addressed this problem for both the makespan norm and arbitrary ℓ q -norms of machine loads. Recent literature (e.g., Azar et al., STOC 2013; Im et al., FOCS 2015) has further expanded the scope of this problem to vector loads, to capture jobs with multi-dimensional resource requirements in applications such as data centers. In this paper, we completely resolve the job scheduling problem for both scalar and vector jobs on related machines, i.e., where each machine has a given speed and the time taken to process a job is inversely proportional to the speed of the machine it is assigned on. We show the following results:• Scalar scheduling. We give a constant competitive algorithm for optimizing any ℓ q -norm for (scalar) scheduling on related machines. The only previously known result was for the makespan norm. *The load balancing (or job scheduling) problem, introduced in the seminal work of Graham in the 1960s [19,20], asks for an online assignment of jobs to machines so as to minimize some objective defined on machine loads. A long line of work has addressed this problem for both the makespan norm (maximum load) and for other ℓ q -norms of machine loads (e.g., [9, 26, 2, 16, 2, 15, 10, 18, 22, 3, 8, 11, ?, 4, 12]). In this paper, we study this problem in the related machines setting, where the processing time of a job on a machine is inversely proportional to the speed of the machine. The only previous result for this problem on related machines was a constant-competitive algorithm for the makespan (maximum load) objective [11]. However, in many situations, other ℓ q -norms of machine loads are more relevant: e.g., the 2-norm is suitable for disk storage [?, ?], whereas q between 2 and 3 is used for modeling energy consumption [?, ?, ?]. This led to constant-competitive algorithms for arbitrary ℓ q -norms of machine loads for the special case of identical machines (all machine speeds are equal) [?], and to O(q)-competitive algorithms for the more general unrelated machines setting (processing times are arbitrary) [4,12]. But, this problem has remained open for related machines. Moreover, recent literature has further expanded the scope of the job scheduling problem to vector jobs that have multiple dimensions, the resulting problem being called vector scheduling [13,7,28,25]. This problem is very relevant to scheduling on data centers where jobs with multiple resource requirements have to be allocated to machine clusters to make efficient use of limited resources such as CPU, memory, network bandwidth, and storage [17,29,27,14,24,25]. Recently, Im et al. [23] showed that for vector scheduling with the makespan norm, competitive ratios of O(log d/ log log d) and O(log d + log m) are tight for identical and unrelated machines respectively, where d is the number o...